ML20081J012
| ML20081J012 | |
| Person / Time | |
|---|---|
| Site: | Comanche Peak  |
| Issue date: | 11/01/1983 |
| From: | Doyle J Citizens Association for Sound Energy |
| To: | |
| Shared Package | |
| ML20081H953 | List: |
| References | |
| NUDOCS 8311080240 | |
| Download: ML20081J012 (72) | |
Text
{{#Wiki_filter:_ UNITED STATES OF AMERICA NUCLEAR REGULATORY COMMISSION BEFORE THE ATOMIC SAFETY AND LICENSING BOARD In the Matter of APPLICATION OF TEXAS UTILITIES I GENERATING COMPANY, ET AL. FOR I Docket Nos. 50-445
! and 50-446 AN OPERATING LICENSE FOR COMANCHE PEAK STEAM ELECTRIC .I STATION UNITS #1 AND #2 I (CPSES)
AFFIDAVIT OF JACK 00YLE 1 Q: Have you read the Staff questions and answers in reference to the open 2 items as relates to the Walsh/Doyle allegations? 3 4 A: Yes, I have. 5 6 Q: Do you agree with their conclusions? 7 8 A: No, I do not. 10 Q: Have you read Question and Answer Nos.1, 2, and 3 in Dr. Chen's Affidavit 11 in reference to the diaphram problem on Support No. CC-2-008-709-A43K? 12 13 A: Yes, I have. 14 13 Q: Would you like to comment on Dr. Chen's conclusions? 16 8311080240 831104 PDR ADOCK 05000 G
1 A: First, the answer by Dr. Chen does not answer the question of how this 2 large support was designed, ar.alyzed and constructed with the errors in funda-3 mentals which he agrees were present. That is, the 16-inch diameter pipe 4 was capped with a 1/2-inch thick plate that was excessively flexible and over-5 stressed to such a point that to rectify the problem required a plate 27 times 6 as stiff (1-1/2 inches) and a weld which apparent 1y'is ~167 per cent of the 7 original weld (3/16 to 5/16). Beyond this, I believe that it is presumptuous 8 to assume that this problem would be detected in the so-called " iterative 9 design process." Problems of this nature, as a general engineering' practice, 10 are caught at the preliminary check-and-approval phase and not after construction. 11 12 Q: Do you have any coments to make in reference to Dr. Chen's coments 13 regarding the thermal constraint problems associated with support No. 14 MS-1-003-009-C72K (Questions 4, 5, and 6)? 15 16 A: As has been the case on a number of occasions, the NRC Staff presents 17 us with the bottom line only. How the numbers evolved and what parameters 18 were involved are not set out for comment. For this reason, I can only 19 coment on the limited information provided. CASE should be provided with 20 specific details and calculations regarding this in order for the record 21 to be complete in this regard. 22 One point which requires coment is the fact that a condition existed 23 which, according to Applicants' procedures and numbers, indicates that a-24 stress level of 30.8 ksi from constraint of expansion only existed in the 25 joint of the 10 and 20 inch stanchion assembly and that this had been overlooked,
1 as was a 27.8 ksi stress in the 32-inch diameter main steam line. These 2 are hardly insignificant stress levels. 3 An additional point to be made involves the coment by Dr. Chen at page 4 4, lines 15 through 17, where he states that an unverified code evaluation 5 indicates that the margin of safety for the pipe support welds are higher than 6 those that exist for the pipe. This is an interesting coment on the part - 7 of Dr. Chen, when one considers that the maximum stress at the outer point 8 of the stanchion joint is 30.8 ksi and this is for pipe diameters that are 9 10 and 20 inches with wall thicknesses that are 1.000 inches and 1.031 inches, 10 respectively, while the weld effective throat is only .375 (.707) equals 11 .265 inches. This inability to rationally evaluate problems from the infor-12 mation supplied graphically illustrates why more detailed information is 13 required and why additional hearings are necessary, with an opportunity to 14 cross-examine the Staff witnesses regarding their conclusions and the bases 15 for them. For example, where does this 30.8 ksi occur and what is the source 16 of the stress (moments, shear, etc.)? 17 One thing this particularly points out is the importance of including all 18 contributors to the stress in a component and the ' additional necessity to 19 include the sum of the individual spring rates to determine the actual loads 20 with reasonable limits which could alter the safety factor. 21 Finally, on page.4,' lines 10 through 14, Dr. Chen compares the 30.8 ksi 22 stress levels in the stanchions with an allowable of 42.0 ksi and concludes 23 that the' support is therefore acceptable. I am less certain than is Dr. Chen, j 24 at least until I actually see what the stress ratio is for the 30.8 ksi due 25 to thermal constraint combined with the. Level' B, C and D loads. l l 1
1 Q: Do you have any comment in regard to the deflection due to torsion in 2 wide flanges, Question and Answer 7, 8 and 97 3 4 A: Yes. But my reply will be limited for two reasons: first and foremost 5 is because the Applicants have discarded.use of this procedure, as stated 6 by Mr. Finneran at Tr. 7136/4-17 which states, in part: - 7 "I think it was probably a case of an engineer got too smart for himself 8 in designing this program." g Second, this problem was well covered in Section XII of CASE's Findings 10 of Fact regarding the Walsh/Doyle Allegations. 11 However, there are several points in Dr. Chen's affidavit which indicate 12 a trend on the part of the NRC Staff to write off problems without careful 13 consideration. These I will address. 14 First, in reference to the statement made by Dr. Chen at Answer No. 8, 15 page 6, where he states the use of this high torsional constant somehow had 16 a beneficial effect; "However, for frictional forces, the underestimate 17 of support deflection tends to result in a conservative estimate of friction 18 forces. " 19 This statement is inaccurate for two reasons: First, the forces and j 20 moments resulting from friction ~are based on a force obta.ined by multiplying 21 the thermal / dead load by a coefficient. Therefore, this deflection of the l 22 structure has little effect on stress except for the torsional moment on the 23 weld and reacting " columns." This, however, is a minor effect. Second and l 24 of more consequence is the moment in the cross beams. The moment and consequently 25 the stress in these members is dependent on t'he end condition. If, for example, 1 i 1
1 the beam is assumed as being fixed at each end (as is the case when one assumes 2 that the " columns" at the beam ends are torsionally stiff), then the moment 3 is equal to "P" (friction force) times the length of the beam divided by 8. 4 On the other hand, if one assumes that the beam in question is a simple beam, 5 then the moment in the beam is equal to PL/4. As may be evident, the beam 6 in question would have a stress level two times as high across the minor axis 7 as had been calculated by the Applicants. This not only is not conservative, 8 it is over-optimistic by a factor of almost two (for equations used abcve, 9 see AISC, pages 2-200 and 2-203, 7th Edition). Beyond this, the assumption 10 of such fixity is in violation of Appendix XVII 2421, Section III of the ASME 11 Code (a) and (b). 12 Further, Dr. Chen states that a reanalysis of the vendor certified 13 designs is not required (see Note 1, page 7). I disagree with this assessment 14 for the following reason: 15 If, as was shown above, the stress levels in the cross beams are under-16 estimated for the friction load, then the integrity of the support is not 17 assured and the allowables established by the Code for this loading case are 18 in doubt. 19 20 Q: In reference to the Richmond insert torsional moment arm problem, as dis-21 cussed by Dr. Chen in Question and Answers 10,11 and 12, do you have any 22 comments? 23 24 A: Yes, I do, and this may be summarized in a single statement which will 25 be elaborated upon below. Dr. Chen is not orily incorrect in his assessment,
1 he is in total violation of fundamental principles for load paths. 2 This is a problen that requires a certain amount of consideration, 3 and the best way to accomplish this is to evalua'.e the statements in order. 4 For a starting point, Dr. Chen includes a sketch (Figure 1 attached). 5 The most noteworthy feature of this sketch is that it states and shows in effect 6 that the NRC Staff inadvertently agrees with the position I have taken, with 7 the exception of the shapes of the load distributions shown. For example, 9 I have stated that the moment ann for consideration would be d. as shown on 9 the upper sketch of Figure 1. The Applicants and the NRC (and Dr. Chen re-10 iterates his position in this regard) state that the moment arm is d2 as shown 11 in the lower sketch of Figure 1. However, Dr. Chen indicates on Figure 1 12 that di equals d2. Beyond this, Dr. Chen states F 2 equals F2 , which is obvious. 13 But if dj equals d2 and the torque is the same, then FB (force in the bolt) 14 would be the same (F 2 equals FB ) regardless of which side of the washer 15 the coupling arm is selected. (This point by Dr. Chen is repeated in words 16 on page 10 of his affidavit.) 17 Basically, one must keep a major feature of this problem in mind, and that 18 is the forces involved. Regardless of the pattern developed as a result of the 19 load path and flexibilities involved, the sum of forces and moments on the top 20 side of the washer must equal the sum of the forces and moments on the bottom 21 of the washer, although the configuration of the loading blocks may differ. 22 Also, the loading path and major increment of the load transfer will follow 23 the path of strain degradation (see Attachments A and B). 24 In viewing Figure 2 Sheet 2, it is obvious that any torque in the tube 25 will cause the tube to attempt to rotate. However, the tube is captured between
1 the nuts of the Richmond insert and the' lower washer. The tendency by the tube 2 to rotate is therefore restrained by a line along the top of the lower washer 3 at:d the nuts.. The NRC Staff, in Figure 1 of Dr. Chen's upper sketch, appear 4 to agree with this evaluation. 5 As stated above, I disagree with Dr. Chen's representation of the contact 6 pressure for the washer lower surface to concrete (see Figure 1) where Dr. 7 Chen indicates the distribution is triangular with a peak at the outer edge 8 of the washer. Since the initial contact due to the torsion is at the tangency 9 point of the tube, the contact pressure as shown by Dr. Chen is not possible. 10 Because the Applicants are in agreement with Dr. Chen's conclusions, they also 11 are not in conformity with fundamental engineering principles. For Applicants' 12 view of the problem, see Figure 2, sheet 1, attached. 13 In the case of the real world load block, it is required to review the 14 problem. For a rigid plate to produce Dr. Chen's load block on the lower 15 surface of the washer, the load contact point would have to be located on the l 16 top surface of the washer,. as shown in Figure 2, sheet 1, detail "A". 17 For corroboration that this assumption of pivoting about the edge can 18 only result from a rigid plate loaded at the pivot edge, see Attachment C 19 (" Advanced Strength of Materials" by Jacob P. Den Hartog, Chapter. 5, page 20 162, Figure 109(b)). 21 In reference to the above, it is interesting to note that Dr. Chen has 22 included a number of caveats in his affidavit. For example, on page 8, note 2, 23 Dr. Chen qualifies his statements with ". . . If, the dominant deformation 24 in the assembly is due to pivoting around the edge of the washer . . ." (emphasis 25 added). But one must ask the question: How can the dominant defonnations cause
. 1 pivoting about the edge of the washer when the causal force is introduced at 2 least 1 inch inboard of this edge? 3 Dr. Chen's reference to the assembly as being snug has no bearing on 4 the initia1' reactions restraining the torque in the tube; therefore, his con-5 clusions based on this parameter are invalid, as will be shown below. 6 . The first point to keep in mind is the nature of this joint. It is made 7 up of a tube with a washer top and bottom, fastened to the concrete wall with 8 a 1-1/2 (or 1) inch A307 bolt to an embedded Richmond insert. Regardless 9 of how snug the assembly is installed, the fact remains there is no continuity 10 between the tube / bolt and lower washer. For this reason, the transfer of the 11 moment (torque) into a couple can only occur as I have shown, and that is the 12 reaction occurs at the upper surface of the washer at the tangent on the tube 13 and the bolt. The lower washer has only one load and that is the reaction 14 due to coupling and the resulting load path to the concrete will result in a 15 first moment of area equal to the first moment of force (area) which occurs 16 due to the concentrated line (contact area insignificant) load at the upper 17 surface of the washer. 18 The Applicants' procedures are therefore non-conservative by a considerable 19 margin. (page9) 20 A final point on this matter: Dr. Chen, at line 6 states ". . . newly-21 identified issue . . . " Dr. Chen is incorrect and he would have known this if he had read CASE exhibit 669 (Deposition Testimony of Jack Doyle), page 22 23 125, line 10, through page 126, line 9, which was presented as my testimony 24 during the Septenber 1982 hearings. 25
/
1 Q: Do you have any comment on the conclusions of Dr. Chen in reference to 2 the upper lateral restraint, question and answers 13 and 14 of Dr. Chen's 3 affidavit? 4 5 A: Yes, I have several coments on this issue. The first statement made 6 by Dr. Chen that I must take exception to appears on line 1 of Answer 13, when-7 he states that for the " assumed" wall characteristics, they were within the 8 allowables. There are two factors: One is the remark " assumed." Gibbs & 9 Hill has the drawing of this wall; therefore, they must know what the thickness 10 is. What occurs above and below the horizontal strip of interest is of little 11 consequence for a rough (optimistic) approach. Second, as was pointed out 12 in the CASE Findings of Fact (Walsh/Doyle Allegations), Section XIX - 7, the 13 results and parameters utilized by the Applicants are not in accordance with 14 the provisions of sound engineering fundamentals or with Applicants' FSAR. 15 Therefore, the argument' (without precise proof) that the stresses in the walls 16 and beam are within allowables is without merit. 17 On page 12 of his affidavit, Dr. Chen, in Answer 14, concedes that the 18 stresses will exceed the allowables if the combined accident pressure spike 19 and LOCA are considered, but he then goes on to unilaterally dismiss the possi-20 bility of these two events occurring simultaneously. He was told by both 21 Mr. Walsh and me that the FSAR required this simultaneous consideration (see 22 FSAR 3.8.110(b) the equation used by Applicants; see also CASE Exhibit 758I , 23 Sheet 40, Applicants ' calculations for upper lateral restraint). As CASE 24 indicated in the Findings of Fact (Walsh/Doyle Allegations), page XIX - 7, - 25 the Applicants used the wrong allowable. for .9Y of 50 ksi steel at 300 degrees, I Admitted at Tr. 6024.
1 in addition to failure to include all of the loads required even if LOCA , i 2 is not considered. I 3 Dr. Chen, at page 13, does however agree with CASE that the relief char-4 acteristics (concrete crushing, for example) are non-existent. 5 The major fallacy comitted by Dr. Chen is his attempt to maintain a
- 6- soft wall between the reactor cavity and the steam generator compartment.
- 7 On line 4 of page 13, he does concede that the other wall is stiffer, although 8 he does not elaborate on how stiff. I have already stated 'that a wall backed 9 by a significant slab is extremely stiff (see CASE Findings of. Fact (Walsh/
10 Doyle Allegations), page XIX - 7, lines 1 through 6). 11 On page 13, last paragraph of Dr. Chen's affidavit, he states that if 12 the wall were only 3.5 feet thick, the stiffness and as a result the load on 13 the wall would be well within the allowable. The fact is that the wall is 14 not 3.5 feet thick, but is in fact considerably (on the order of 7 feet, as i 15 alluded to by Dr. Chen on page 14, line 3) thicker. This fact may be noted 2 16 on CASE Exhibits 799 and 800 . From this drawing, three indisputable facts 1 17 may be noted. Fint, the upper lateral restraint is located at elevation 18 858'6"(approx.). Second, base of rail for the missile shield is located at 19 elevation 860'0" (approx.). Third, the thick portion of the wall therefore !l 20 extends for about 18 inches above the center line of the upper lateral restraint 21 and extends for more, than several feet below the centerline of the upper 22 lateral restraint. From the foregoing, it is obvious that the stiffness l 23 of the wall in question (without considering the two-way effects of the wall) 1 24 must be based on the 7'0" thickness or a precise calculation must be perfomed l l ! 25 to detemine if there is any benefit to be realized from the fact that of the ! l 2 Admitted at Tr. 6030. )
. ~ . _ , . . _ . _ _ . _ . _ _ _ _ _ _ .._ _ _ _ _ _ , _ - ~ _ ... _ . _ - _ ~
I eight foot band of wall being considered 2.5 feet have a 3.5 thickness and 5.5 2 feet have a 7.0 foot thickness. (But in so doing, one must keep in mind that 3 this is not a standard beam, but is rather a strip from a two-way slab and 4 therefore the effects tend to create more stiffness than would exist if this 5 were in fact a beam without edge restraints.) See Figure 3. The stiffness 6 of this wall is at least 8 times what Dr. Chen would have us believe. - 7 Dr. Chen is again incorrect when'he states that the 7 foot thickness 8 covers only 1/7 of the wall length, when in fact the 7'0" thickness (minimum 9 -- the outer ends of the compartment wall is backed by a slab which forms (see Figure 3, sheets 1 and 2) 10 the platform above the reactor)Aactually extends for the full length (width) 11 (horizontal) etc. of the generator compartment. The 1/7 that Dr. Chen refers 12 to is the depth of the wall (vertical). Since the beam dimension vertically 13 is limited to the 8 feet assumed by the Applicants and basically is in the 14 7-foot thickness area, the fact that the 7 foot thickness represents only 15 1/7 of the vertical height of the wall is without relevance. 16 As shown above, the conclusions of Dr. Chen are' based on false parameters 17 and therefore are.wi.thout merit. If the NRC Staff or the Applicants wish 18 to show that the walls of concern and the upper lateral restraint beam are 19 designed to safety standanis and proper fundamentals of engineering, then 20 let the NRC Staff show this by the acceptable methods; that is, by the calcu-21 lations that were performed to design the upper lateral restraint in the 22 first place, or by providing new calculations to show that the design is 23 proper. The NRC Staff's procedure of arguing the beam into acceptance by 24 establishing justification based on "what if" and then extending this to 25 conclusions is not the acceptable procedure t'o prove critical structure
1 1 is within allowables. This is not in compliance with the adequate safety 2 margins required by General Design Criteria II of 10 CFR Part 50. If the 3 NRC Staff believes this problem is solved, let them tell CASF and the Board 4 exactly what the margin of safety is in the walls and the beam. 5'
~
6 7 Q: Do you agree with the conclusiens of Dr. Chen in reference to seismic d 8 flexibility of the Richmond insert A307 bolt assembly, Question and Answers 9 15,16 and 17 of Dr. Chen's affidavit? 10 11 A: Absolutely not, for several reasons which will be outlined below. 12 For starters, Dr. Chen again states that Mr. Walsh was in error when 13 he said that the peak of the acceleration curve must be used when analyzing 14 the pipe supports for seismic forces (see Dr. Chen's affidavit, page 15). 15 However, the Applicants' own FSAR at 3.78.3.5 states that if the actual , 16 frequency of the supports is not perfonned, then the peak of the response 17 curve must be used, times a 1.5 factor. i l 18 The NRC Staff states in the SIT Report (see CASE Findings of Fact (Walsh/ j 19 Doyle Allegations), page X - 3) that 23 worst-case supports were analyzed for i 20 seismic to determine the effect on the structure. The SIT states that the 21 effect was negligible. 22 The problem with the SIT and the Applicants' approach is that in detennining 23- the natural frequency of the support, only the primary structure is considered 24 (see CASE Finding of Fact (Walsh/Doyle Allegations), pages XXVI - 12 through 25 - 15; see also Dr. Chen's affidavit, page 16," note 8; page 22, note 11). The
1 concept of analyzing a system of spring's by utilizing the characteristics 2 of the stiffest component is in conflict with engineering fundamentals.
~
3 See Board Notification 82-105A, NRC Staff Evaluation Regarding Allegations 4 of Potential Design Deficiencies in Class I Piping, Section IV, page 4, 5 " Dynamic Effects where it states: 6 "The dynamic interaction between the pipe and pipe clamp is a complex 7 design problem. From a design standpoint, there are many uncertainties 8 that could affect the actual system response such as consideration of 9 total support system flexibility, mechanical non-linearities, construction 10 and installation tolerances, and uncertainties in the dynamic loading 11 i ts el f. . . " (Emphases added.) 12 The actual stiffness of the pipe support system by engineering fundamentals 13 involves the inverse of the sum of the reciprocals of the independent stiffnesses. 14 This method of combining spring in series-is conceded by Dr. Chen to be the 15 proper method of analysis (see Tr. 6051/14-18 and 6194/21-23). 16 Before proceeding, several coments made by Dr. Chen in his affidavit f 17 must be disposed of, since the conclusions have a profound effect on the 18 interpretation of joint stiffness. Dr. Chen states "The Richmond insert has 19 little flexibility in tension. Rather, it's (sic) flexibility is predominantly 20 in shear." (Page 16, Answer 16.) Dr. Chen is incorrect in his assessment. The stiffness of the A307 bolt (1- inch diameter) in tension may be determined 21 22 by referring to Applicants' Exhibit 142E, which contains the load / slip data 23 for the Richmond / stud assembly. The first. factor which is obvious is that the 24 spring factor is not constant by the test information. For the' maximum listed , 25 load (58 kips), the displacement was .929 inches (however, for the test, the
. ~ , - - - - - , ,- -
, - - , . , - . , . - - - - , - . - - -, 7
- 14 1 bolt was prehoaded to 2000 pounds befor~e displacements were recorded). There-2 fore, the spring factor at 56 kips would be 56/.929 equals 60 kips per inch.
3 At 4 kips of load, the spring factor was (4-2)/.017 equals 118 kips per inch. 4 A reasonable average (non-conservative) would be at 12 kips of bolt load 5 at which point the stiffness factor would be (12-2)/.068 equals 147 kips 6 per inch. Since many of the supports of this type at Comanche Peak are of 7 the two-bolt (four bolt with coupling of a moment between two bolts on top 8 -- tension, and two bolts on bottom -- compression) or four-bolt patterns 9 in straight tension, the joint will reflect the spring factor in tension 10 of these configurations. See below. 11 A second problem introduced by Dr. Chen involves his rationale for 12 dismissing the stiffness effects of the 1-1/2 inch diamter A307 anchor bolt 13 .in shear. Dr. Chen first agrees (see pages 16 and 17 of his affidavit) that 14 "A brief review of the designs reflected in several Gibbs~ and Hill stress problems 15 indicate that a fairly substantial, portion of the supports could have such 16 configuration. " (High shear loads on A307 bolts.) But he then goes on to 17 eliminate the problem with this statement: 18 "However, the Applicant's preliminary pipe support generic stiffness 19 study demonstrated that varying support stiffness over a wide range 20 did not significantly affect the fundamental frequency of the piping 21 system. The range of stiffness covered is substantial (sic) larger 22 than any effect on support stiffness that might result from considering 23 shear flexibility of the support anchorage." 24 What Dr. Chen does not state is more important than what he does state. 25 For example, Ref. Attachment II to Dr..Chen's affidavit indicates that a l . l
. 1 system was analyzed with 48 supports an'd 2 anchors. The supports for this 2 system contained actual stiffness values which varied from a minimum of 3 14 k/in to 3600 k/in. However, if one disregards the maximums which only occur 4 at one node point for the maximum stiffness and one node point for the minimum 5 stiffness, then we have actual stiffnesses which all are above 46 k/in but the 1
6 actual loads are found to increase in 67 per cent of the cases over the loads 7 obtained by using the generic stiffness. In view of the safety factor of le'ss 8 than 2.0/1.0, this is not a point that may be safely overlooked (and this was 9 a system selected by Applicants which did not include Richmond or hardware stiffness). 10 Dr. Chen (on page 17) states further: 11 "In addition, the Applicants' tests of ~ the Richmond insert assembly tested 12 deflection versus shear load using both high strength ASTM-490, and low 13 strength ASTM-307 bolts. While the low ~ strength bolts exhibited sub-l 14 stantial shear flexibility, the high strength bolts did not. Since 15 the anchorages were the same in both tests, the test results demonstrate 16 that shear flexibility of the Richmond insert assembly is due to the [ 17 characteristics of the bolt material, rather than a result of movement t 18 in the concrete anchorage. Thus, any effect on support stiffness due to 19 considering anchorage flexibility would not have a significant effect 20 on piping system fun'damental frequency." , 21 However, the'very flexible A307 bolt will still contribute to the support 22 overall flexibility. Dr. Chen goes beyond reason when he states the following 23 in reference to the oversize holes at pages 17 and 18 of his affidavit: 24 "With regard to the potential for slippage of the support in oversized 25 base plate holes, there are reasons to believe that this effect on support 26 stiffness, if any, will be negligible. First, although the concrete 27 insert joint are nominally considered to be bearing-type joints, it is
1 the usual construction practice to torque the bolts so that they are 2 ' snug'. Thus, there will be initial fric ion which provides a restraint 3 to slippage. Second, the base plate configurations involve multiple 4 holes (3-4 or more inserts in a given base plate). The likelihood 5 that all of the holes in any given base plate would all be oriented in 6 the same direction is small, thereby reducing the amount of potential 7 slippage. Third, if frictional effects are ignored, and slippage is 8 included in the analysis, the results will not be significantly affected. 9 Inclusion of slippage would require the use of non-linear analysis which 10 is not the industry practice." 11 In reference to the above statements by Dr. Chen, several areas deserve 12 comment. First is the conclusion by Dr. Chen that in this case we may consider 13 friction (of unknown value) as eliminating slip, etc.; we need only consider j 14 the Codes. For example, the AISC at page 5-129,1.5.2.1 " Shear", 7th Edition 15 states, in part: , 16 "The amount of clamping force developed by shrinkage of a rivit after 17 ccoling and by A-307 bolts is unpredictable and generally insufficient 18 to prevent complete slippage at the permissible working stress. Hence 19 rivited connections and connections made with A307 bolts for shear are 20 treated as bearing type. . ." (Emphasis added.) 21 There is no caveat in this code statement which allows for friction 22 considerations if the shear will create flexibility problems.3a 23 In my Surrebuttal Testimony (CASE Exhibit 763), page 12, lines 7 to 24 24, I also covered this point when I stated: 25 "One ' factor involves the use of SA307 bolt material for friction joints. 3 See Tr. 6877 and 7042. 3a Additionally, any friction due to " snugging" will cease to exist if any tension is also applied to the bolt. All of these bolts are subject to combined tension and shear.
. 1 This material is not allowed for friction joints. See CASE Exhibit 752 (ASME Appendix XVII, Table XVII-2461.1-1. Dage_3_88; this was attached 2 to CASE's 4/20/83 Brief RegardTng Consideration of LOCA in Design Criteria for Pipe Supports), where NOTE 1 states ' Friction tvoe connections loaded 3 in shear are not penaltted. The. amount of clanping force developed by SA-307 bolts is unpredictable and generally insufficient to prevent com-4 plete slippage. ' Beyond this, A307 is not recomended for dynamic appli-cations, as may be noted in Salmon and Johnson ' Steel Structures Design 5 and Behavior,' Chapter 4 Structural Fasteners, at 4.1 Types of Fasteners. 6 At
"(inthe bottom to reference of A307 page unfinished 85 and continuing)on bolts , thesepage bolts 6" are(sic - should be 86) recomended for ' loads (which) are piimarily small and static in nature' (by inference 7 not recomended for dynamic loads). See CASE Exhibit 763F."
8 Dr. Chen obviously does not have an open mind on this subject, as is 9 evident from his statements at pages 18 and 19 of his affidavit: 10 "During discussion with Messrs. Walsh and Doyle, Mr. Walsh stated that 11 he disagreed with my conclusion that the effect of oversize bolt holes 12 on seismic accelerations may be safely disregarded for the design of 13 pipe supports utilizing multiple Richmond insert anchorages. Mr. Walsh 14 asserted that unless the friction between base plate and concrete due 15 to the tighten.ing of the ccncrete insert bolting is quantified, no 16 credit could be taken for that friction. A' question regarding relaxa-17 tion effects in the bolt was raised by Mr. Doyle. They also cited three 18 references,1/ which they claimed showed that: (1) the shear forces on (sic) 19 the base plates with several bolts are not equally sheared4 by the bolts; 20 (2) ASTM-A307 bolts are not permitted in friction type joint; and (3) 21 ASTM-A307 bolts are not permitted for dynamic loads. I do not agree 22 with these assertions. Although the joints utilizing ASTM-A307 bolts 23 are designed as bearing type joints, some frictional effects will be 24 present due to the method of assembly. No credit is taken for these 25 friction effects in the bolt code evaluation, and qualification of these
1 effects is therefore not necessary. I discussed matters relating to 2 load sharing in multiple-bolt assemblies under shear forces at the May 3 1983 hearing, Tr. 6883-4. Accordingly, this matter is not an open 4 issue. Finally, I find nothing in the references cited by Messrs. 5 Walsh and Doyle relating to ASM-A307 bolts which contradicts my 6 conclusions in this regard. 7 9f Fisher, J., Journal of Construction, AISC, 3rd Quarter (1981), 8 p. 87; Salmon, C. G. and Johnson, J. E., ' Steel Structures, Design 9 and Behavior,' CASE Exhibit 706; AISC Handbook (1978)." 10 Board Notification 82-105A, however, does seem to show concern in reference 11 to gapping (and by inference slippage)when at V-2 it is stated: 12 "The staff also expressed a concern that the E-Systems pipe clamps are 13 designed by E-Systems to the rated load (service level A and B). However, 14 according to E-Systems, the pipe clamps will lift off the piping at higher 15 -loadings (service level C and D). Ihus, the dynamic impact effect of ! 16 the clamp on the local pipe surface has not been evaluated by Gilbert 17 partly due to the piping designer's unawareness of the E-Systems clamp 18 design assumptions." (Emphasis added.) 19 (In addition, there is the deadband of snubbers and 1/16" gap on rigid 20 frames, which also add to the effect.) 21 Beyond this, in an effort to show that the code concerns in reference 22 to A307 is still a valid point, consider the 1983 Code of ASME Section III, 23 which states as follows: 24 a xygg.2461.4 Slip Resistance-Friction Type Joints. The maximum slip resistance to which a friction 25 type joint may be designed shall not exceed the value of P,, calculated in the following equation: Po = mnT,k, t
I where P, = maximum slip resistance of thejoint,Ib 2 m = number of shear planes per bolt n = number of bolts in 'hejoint 3 T = imtial clamping force per oolt, Ib (not to exceed Il5,000A,, where A, is the tensile 4 stress area of the bolt, sq in.) k, = slip coefficient for the particular surface 5 condition taken from Table XVil-2461.4-1 Since the joint clamping force will be reduced by 6 any direct tension load on thejoint, the T, va!ue shall be re'duced by an equivalent amount before substi-7 tuting in the above equation. SA-307 and austenitic steel bolting shali not be used for friction typejoints. " (Emphasisadded.) 8 9 It is obvious from the above that if A307 bolts cannot be designed into 10 a friction joint for normal non-nuclear and nuclear applications, then any 11 relief assumed as a result of friction joints is not a valid engineering 12 approach (particularly when there is not even a cursory attempt to state what the friction value is alleged to be). 13 To appreciate the effects of considering the actual stiffness of a 14 support system as opposed to assuming an unreasonable generic stiffness which 15 exceeds by a wide margin the stiffness of the anchorage alone or the stiffness 16 of the hardware alone, consider the following: 17 Referring to CASE Exhibit 669B (Attachment to Doyle Deposition / Testimony), 18 Items 8T and 80, a Class 1 support with two (2) 1-1/2 inch diameter A307 bolts 19 loaded in two-way shear and tension with a size 1 snubber, the following spring 20 factors exist without considering the primary structure or the local effect 21 of the snubber bracket on the tube wall: 22 (1) (147 k/in x 20) / 7 (see Item 80) equals 420 k/in from coupling. 23 (2) 40 k/in (assume 2 bolts in shear- non-conservative) 80 k/in. 24 For 40 k/in factor, see CASE Exhibit 834 4, Appendix A, Test No. 7, 25 lines 3 and 4 extrapolated. 4 Admitted at Tr. 6471. 1 I
1 (3) Size 1 snubber x 3'15/16" C/C (see Item 80) 3 in, dia. pipe. 2 Stiffness factor equals .about 93.5 k/in. minus 32 x .368 k/in (rate 3 for. each inch of transition tube) equals 81.2 k/in (see Attachment 4 C, Sheet 3). 5 The stiffness factor for this class 1 support, not including the primary 6 structure or local deflection, is therefore the inverse of 1/80 + 1/420 7 + 1/81.2 equals 67.1 k/in. This is far less than the generic stiffness used (CASE Exhibit 669B) 8 by Applicants. See also Items 8V and 8Wg in which the bending stiffness (an 9 unknown quantity by testing) of the bolt is the minimum stiffness and is far 10 less than either the tensile or shear stiffness. 11 For another example of the effects of tensile stiffness of the A307 anchor 12 bolt, consider items 13CCC and 13DDD (CASE Exhibit 6698) for which the stiff-13 ness for the snubbers and A307 bolts only would be as follows: 14 (1) 2 snubbers size 1 x 15-13/16 C/C (with std. clamp and not more 15 flexible U-bolts) equals (2) 93500 minus 11.81 x 368 equals 178 k/in. 16 (2) (4) 1-1/2 inch A307 bolts x 147 k/in equals 588 k/in. 17 For this condition which does not include the extra flexibility for extra 18 long U-bolts, flange bending stiffness (see Section C-C, Item 13DDD)', diaphram 19 stiffness of item 3, bending stiffness of item 17 or the stiffness of the primary 20 structure as shown in item 13 CCC would be the inverse of 1/178 + 1/588 equals 21 137 k/in. Obviously, this structural conglomerate will have an actual stiffness 22 far less than is shown above. See also items 13JJJ,13D0 (with 1 x 4 plate 23 24 inches long in bending); etc. There are many such examples, but the above 24 will suffice to 11'lustrate the point. 25 I must add the stiffness problem as described above for A307 bolts does
- 2i -
1 not exist in the friction joint. The r'eason for this is as you load the 2 plate with a tension force, it relieves the preloaded compression in the 3 plate at the bolt joint. This compression is the result of torquing the bolt. 4 In other words, the bolt does not stretch (in a friction joint) due to applied 5 loads until the value at the friction joint equals zero (0) (with minor variations); in other words , until joint separation commences. (SeeFigure 6 7 5.) 8 (One other point which should be mentioned is that " snug tight" is a 9 very nebulo;us value. This was emphasized during the October 1983 hearings, 10 during the discussion by NRC Staff witnesses Johnson, Westerman, and Taylor, 11 at Tr. 8813-8817. See also AISC,1.5.2.1, discussed at page 16 of this 12 affidavit. In addition, Mr. Johnson referred to the fac t that there were 13 no visual gaps between the bolts and the place they are supposed to be (Tr. 14 8814); this is irrelevant and immaterial as far as preload pressure is concerned.) 15 . 16 Q: Are you in agreement with the conclusions as stated by Dr. Chen in reference 17 to generic stiffness values, as outlined in his answers to Questions and Answers 18 18, 19 and 207 19
- 20 A: No, I am not, and will coment on Dr. Chen's conclusions on this point.
I 21 On 'page 20, Answer 18, Dr. Chen makes the following statement: 22 "As indicated in the SIT Report, in computing the response of a piping 23 system to complex loading combinations, it is important to assure that 24 piping supports are sufficiently stiff so that they do not adversely i 25 affect the response of the piping. system. The Applicants use generic l l i e w -- , , - - . - . - .
1 stiffness values in its (sic) calculations of p. ping system response. 2 The use of generic stiffness values is comon practice and is acceptable 3 provided that the generic stiffness adequately represent the stiffness 4 of the installed support.E/ 5 10/ In discussions with Mr. Walsh and Mr. Doyle they asserted that the ASME Code and an unidentified Reg. Guide require piping stress 6 calculations for Class 1 systems to use actual stiffness values of the pipe supports rather than a generic value. Although this 7 is somewhat different from the issue remaining open in the SIT Report, we have reviewed the Code and Reg. Guide 1.124 ' Service 8 Limits and Loading Combinations for Class (sic) Linear-Type Com-ponent-Supports' and do not find that they specify the calculational 9 techniques to be used for dyanmic system analysis. Nor do they require or prohibit the use of either actual calculated sti:ffness 10 values or generic values. Both techniques have been accepted." 11 (Emphasis added.) 12 Although I do not argue with Dr. Chen's statement that it is important 13 to assure that piping supports are sufficiently stiff so that they do not 14 adversely affect the response of the piping system, I disagree with the approach 15 used by the Applicants. . 16 As noted in the statement (emphasized) above, the generic stiffness must 17 be adequately representative of the actual stiffness. When the Applicants 18 use a generic stiffness of 1,000,000 pounds per inch but the A307 bolts which 19 form the final attachment to a hard point (the concrete wall) have a stiffness 20 of only 147,000 pounds per inch tension and 40,000 pounds per inch shear, 21 1,000,000 pounds per inch stiffness for the total support would hardly be 22 considered as representative. Also, the same can be said of another compo-23 nent which makes up the chain of elements which form the support system; that 24 component is the hardware which may have a stiffness value of as little as 25 24,000 pounds per inch for a snubber (SMS 1/4), or a maximum of 240,000 pounds
1 per inch (SMS 10), the most common sizes. Also, the struts have a similar 2 problem, although not as severe. For example, for NPSI size 06 strut, the 3 minimum stiffness value is 180,000 pounds per inch to a maximum value for NPSI 4 Size 12 of 893,000 pounds per inch (again the most common size range). See 5 Attachment D. 6 The results of variations in stiffness (actual vs. generic) may best be 7 noted from Dr. Chen's evaluation of Applicants' analysis of the 6" component 8 cooling system which was analyzed using 'hetual"stiffnesses (Applicants' inter-9 pretation of actual). This evaluation by Dr. Chen may be found on page 24 10 of his affidavit and is as follows:* 11 "In the case of these two lines, approximately 97% of the supports 12 were within the range of 10 times generic to 1/30th of generic. The 13 results for the 6" lines showed that the natural frequency computed using 14 actual support stiffness was less than 1.3% different.from the frequency 15 computed using generic stiffnesses. 16 out of 48, calculated loads were 16 lower using actual values and 28 of the remaining 32 loads were less 17 than 25% larger. For the 9 load values that increased by greater than 18 25%, the greatest increase in load was about 500 lbs (824 lbs to 1371 19 lbs), and the largest percentage change was 200 percent (41 lbs. to 124lbs.). For one anchor (the surge tank nozzle) two components of 20 force and two components of moments increased. One component of force 21 and component of moment decreased. The maximum increase in force and 22 ' moment components were both 25.7 percent. For the second anchor (an 23 intersection of the 6" line with a 24" header) all components of force 24 and moment except one component of force increased." 25
- The fact is that because of the extremely high generic stiffness used by Applicants, no one knows what the loads are at the supports without a re-run of the pipe stress analysis with either the actual (using all contributors) stiffness or at least a reasonable generic stiffness.
l
- i 1 From the statement of Dr. Chen, the actual generic si;iffness of 1/30 2 is equal to 33,333 (assuming 1,000,000 lbs./in. generic stiffness) pounds 3 per inch, which is slightly lower than the shear stiffness of a 1-1/2 inch a diameter A307 anchor and is slightly more than the manufacturer's listed stiff-5 ness for a size 1/4 snubber 45 inches C/C.
s But when using these " actual" (Applicants' term which did not include 7 all contributors to the actual stiffness) stiffnesses, the loads on 75 per 8 cent of the 48 listed supports increased. And this system, it must be re-9 membered, was selected by the Applicants. 10 A final point made by Dr. Chen was in reference to Support CC-1-107-11 008-E23R, which is the support mounted to a 1/2 inch plate torsionally loaded. 12 The Applicants and NRC Staff went to great effort to prove that this support 13 was 0.K. as designed. They pr.ovided a Stardyne finite analysis which showed 14 that the deflection problem was a non-problem. See Staff Exhibit 2075, SIT 15 Report, page 40 and 41, where they state: 16 "In computing the response of a piping system to complex loading com-binations such as those which include a seismic event, it is important
~
17 to assure that piping supports are sufficiently stiff so that they do not adversely affect the response of the piping system. The Appli-18 cant uses generic stiffness values in its calculations of piping jg system response. The use of generic stiffness values is common practice and is acceptable provided that the generic stiffnesses adequately represent the stiffness of the installed supports. The 20 Applicant and its piping analyst, Gibbs & Hill, indicated that they believe that the use of their overall deflection guideline of 1/16 21 inch maximum deflection under service B cond.ition loads will result 22 in supports whose stiffness is adequately conformed to the generic values used in the piping stress analysis. In discussions with the 23 Applicant, the Special' Inspection Team noted that in the absence of review of the particular supports, it was unclear that the 1/16 inch 24 deflection guideline in fact results in support stiffness comparable to the generic stiffness used in the piping stress analysis. The 25 Applicant agreed to provide a study demonstrating that supports designed in accordance with Applicants criteria and guidelines have 26 sufficient stiffness to assure that they do not adversely affect the ! response of the piping system. This matter remains unresolved." 5 Admitted at Tr. 6289
1 My response to this may be found in my Surrebuttal Testimony (CASE Exhibit 2 763), page 19, and is as follows: 3 "The Applicants have submitted calculations (finite element) which resulted
. in .0608 inch deflection (see CASE Exhibit 763H). The calculations indi-4 cate-that tests show deflections of .055 inch (by exactly which methods and procedures we are not sure). In any event, the load used was 153 5 lbs.; the load on the support at the time of the last hearings was 209 lbs. (see CASE Exhibit 669B, item llTT). Even using 73% of the original 6 load, not including the accelerated hardware and structural steel weights and using the most sophisticated method (and least conservative methods),
7 the support barely makes it. Including the acceleration of hardware and steel as required may prove the support inadequate even at 153# loads. 8 Additionally, the stiffness factor for this support is about 250#/in. -- hardly the generic stiffness used in pipe stress." 9 (This would be equivalent to 1,000,000 pounds per inch for a 6-inch 10 diameter.) 11 Since the apparently optimistic report (although open on generic grounds), 12 the NRC Staff have discovered that the efforts to show that the support was 13 acceptable were wasted. See Dr. Chen's affidavit, pages 25 and 26, where 14 he states: 15 "A recalculation of support loads was also done by Applicants in connection , 16 with support No. CC-1-107-008-E23R. These calculations were performed 17 in response to the open item identified on p. 43 of the SIT Report. 18 This' support was significantly softer than the generic stiffness value 19 (approximately 1/360th, Tr. 6398), and much sof ter than adjacent supports. 20 At the site, I reviewed preliminary support load calculations utilizing 21 the actual stiffness for support No. CC-1-107-008-E23R and the generic 22 stiffness for other supports in this 6"' piping system. The computed 23 load went up substantially at the affected nodes (over 6 times). This 24 piping system model was revised to include the effects of adding a 3" 25 line to the affected 6" line. The support design has also been modified f, _ . _ _ , . , , - - _ . ~ .
1 by CMC 94130 (July 29,1983) to in~ crease the thickness of the plate, 2 which will stiffen it." (Emphasis added.) 3 I have no further coment on this support because this was, as Applicants 4 stated, the only vendor certified drawing which I cited in my August 1982 5 deposition. Andlyet it almost got through the Applicants' supposedly in-l 6 fallible system which is supposed to catch everything. This fact speaks
- 7 for itself.
8
~
9 Q: Do you concur with the findings of Dr. Chen in reference to bolt bending 10 (A307 bolts) as outlined in his affidavit, Question and Answers No. 21 and 22? 11 12 A: No, but I shall hold my arguments to a minimum since I believe we covered 13 this subject extensively in our Findings of Fact (beginning on page VII - 12). 14 It is obvious that all questions relating to the A307 anchor bolt are 15 academic until it is determined why A307 bolts were used to begin with, 1 16 when all authorities warn against their use. Obviously, then, the first 17 order of procedure in this matter is an evaluation of the A307 material 18 for thermal and seismic environments. 19 20 Q: Are there any other coments you have on the material presented by 21 Dr. Chen? 22 23 A: Yes. Within the documents presented by Dr. Chenwere the " calculations" N . 24 perfomed by Dr. Rajan which he claims show that support No. CC-1-028-034-25 S33R is acceptable. The comments by Dr. Rajan in this regard may be found j
- 1 at Tr. 6651/11 to 6655/7 and include,in part, the following statements:
2 " WITNESS RAJAN: The Staff looked at the support in question, and not only do we -- did we find an error in the computation of the stress; we also 3 disagreed with the simplistic approach that was taken to detemine that stress. 4 "Our evaluation indicated that the support woul.d ' result in stresses that 5 are close to allowable. 6 " JUDGE BLOCH: Mr. Rajan, you were using a more complex model than the one that was used by Mr. Doyle? 7 .
" WITNESS RAJAN: Sir, the support was not vendor certified. It was still 8 in a design stage, a creative design cycle. So it'would have needed a computer program to detemine the exact stresses, but we went beyond the 9 simplistic approach that Mr. Doyle did, and our evaluation was that it would be close to allowable.
10
" JUDGE McCOLLOM: What do you mean, close to allowable?
11
" WITNESS RAJAN: Sir?
12 13 " JUDGE McCOLLOM: Under or over? 14 " WITNESS RAJAN: By going to a more sophisticated model, we thought it could be shown to be less than the allowable. However, we were told 15 that it is still in the design, the iterative design cycle, so we did not go to th'e extent of exactly determining the stresses. 16
" JUDGE BLOCH: When you say more sophisticated analysis, what are you 17 referring to?
18 " WITNESS RAJAN: To determine the stresses exactly I believe it would take L a computer program. 19
" JUDGE BLOCH: In what sense was your model more sophisticated than the j 20 nodel that you described as simplistic?
21
" WITNESS RAJAN: Well, the support consists of two wide flange beams that A
are attached at a corner, and on the other end they are attached to the j walls with a plate and inserts. 22 I 23 "On the corner there is a downward component of a load which Mr. Doyle i assumed as simply would be acting on the web of one of the flanges. 24 q
"Actually, it would be shared by both the wide changes, and by making 1 25 some bounding assumptions, one can show that the stresses would be within o allowable.
i h
. I " JUDGE BLOCH: And your analysis showed with those bounding assumptions that the stresses would be within allowables?
" WITNESS RAJAN: Yes, sir.
" JUDGE BLOCH: I think just a moment ago it wounded like you were hedging 4 as.to whether it really was within the allowables.
5 "Which is the case? . 6 "Maybe I misunderstood what you said before. Could you clarify it for me?
" WITNESS RAJAN: We came close to allowable. That is all I can say.
8 4 " JUDGE McCOLLOM: Why do you say close to allowable? Is that because 9 of the coment that you made that it is in a different stage of the de-sign sequence, and at some time later that there is another' design stage
- 10 which will change something?
11 " WITNESS RAJAN: Yes, sir. I was told that it was being reanalyzed and perhaps might be even restiffened. 12
' " JUDGE McCOLLOM: Is this the ::ame kind of thing that happens to all of 13 these kinds of flanges , or webs, or what do you call it? These supports?
14 "This is not being handled separately as a.special case just because it has been brought up, has it?
" WITNESS RAJAN: No, it was not being handled as a separate case. '
16
" JUDGE McCOLLOM: Other such supports are typically handled in the same 17 manner that this one is? '
18 " WITNESS RAJAN: That is exactly right. 19 " JUDGE McCOLLOM: So that whatever corrections have been made as a result of the coments of Mr. Doyle are being made el.sewhere, also? 20
" WITNESS RAJAN: Yes, sir.
21
" JUDGE BLOCH: In fact, in terms of keeping score, because there seem 22 to be comments being made in this hearing about the score, this report says that the calculations were wrong, and that it was unrealistic, and 23 therefore it is resolved. But in fact you told us it is resolved because they changed the support.
24
" WITNESS RAJAN: No, sir. Our interpretation of Mr. Doyle's concern 25 was that he came up with an extremely large stress.
- - - wa-- g-, g, -nn-,-----,,,,,,,,,--r., -
, - , , , _ , . , _ - + - - - - . . - - . , , - . , , - , - - , - ,
.,,,, _ y, _ -,g.-.- , , - . _ . ,-,.w. ,-_.,,._,,.n-- ,,,,n__,, ,, ,n,r.,,,,
i . 1 "5c, when we looked at the support and we made a detemination, we found 2 that it is either with'in allowable, or close to allowable, and it is still in the iterative design cycle. So our interpreation was that it will be taken care of if there is a need to modify the support. 3
" JUDGE BLOCH: But just in terms of understanding what you have said.
4 the calculations were wrong, burdensome modifications are being made in-dependently as part of the iterative process by the Applicant? That is 5 the full explanation of the resolution of this particular concern? 6 "MR. RAJAN: That is correct. " (Emphases added.) 7 8 Thus, the entire conclusions of Dr. Rajan are without merit. Dr. Rajan 9 based his conclusion on erroneous parameters (see Figure 4); therefore, any 10 conclusion would be erroneous. 11 12 Q: Do you have any additional comments? 13 A: I have included as Attachments E, F, G, and H additional supportive 14 documentation for my affidavit. 15 I believe that it is obvious from the pregeding that there should be 16 further evidentiary hearings regarding these matters. The matters in contro-17 versy have not been resolved and cannot be adequately dealt with and explained 18 properly to the Board through affidavits. It could be better demonstrated 19 with models than with words and drawings, as I did in the September 1982 20 hearings. I could (and would) provide such models. 21 22 l 23 i 24 25
-,,,..,e- . - - . - - . . - _ _ , , . - - , _. , , , , , -.,.,. - -
l I have read the foregoing 29-page affidavit, which was prepared under i.. , personal direction, and it is true and correct to the best of my knowledge and belief.
@ Al Jack Doyle A DAT : fsypr,,l w } /988 STATE OF MASSACHUSETTS COUNTY OF 1)
On this, the / day of November,1983, personally appeared Jack Doyle, known to 'me to be the person whose name is subscribed to the foregoing instrument, and acknowledged to me that he executed the same for the purposes therein expressed. Subscribed and sworn before me on the [ day of November,1983. Oz d Notary Public in and to Massachusetts, County of e' tate of
~
ll Y MtS5iOri tMikts AFiUL IU,1987 i l l t -- _ _ _ _ - . . _ - _
1 1 Figure 1
\-
- n. 1 1
vv ~, i - v - - 3 . j ;_, , .,
.: ., -t _. m
' q'f -
, D)'
. 9 -
s - t Eg= E1 5. E
~
_ ,j
.h ,, 7 -
3 ,. . . yj S d :. T 1 L. Q' Q Ot -[
. ' q& -
Y
, i
\ "i p y1 Lt'I} _
1* A E/4 hf // .6 0, = $1
. 1 c.
3 t .\ g t d2. -- . _ _ _ p . fs - F, Ex: Es (- E ; V
- F- 4 2 =T
~
l . A l b ,:. -
~f////(Vti////{ !
.l
- 9'*'es-ease .. e ,
L
1 i=I G u rs E 2 3 SHEET 5 I 1
. =
'i '
) T54X4 .
l!/a JN. FICHMoHD lt4X4Xl _1 ;_ MT' h. A y.' .s o ' . . c' .e c(
, .*e oM. .:.- s:.s
- c. .
wo erect nscio amt ,e a .- u
<3 hew s 'X' 1.' e '_ . '; - ] f 3 b y 7 f y .
HoM.nat _' 2" ' ~ DETA)L 8 SHhDED AREA DEPICTS APPI_ICANT5 ~ METHOD ~~ CONThCY PRESSultE /. nuo .t.tfT / SEE CASE 649.8 ITEM 8G f 84 DETERMI115 'X" FRoM 'I TEM 8 K CASE 469B FOR mom. of /0" KIPS W/m No TENSlotl EGuit4 P = 6 77 h'IPS MOM. ARM EQUR/ 5 /0/6 77 = fo477 x :. EQUAL.S (2-I 177)3 = l. 569
Figure 2 2 i T3 4 X4X 3/8 (7"E.) l'/a 14. R/CHHOND f Ek4X4Xl T Q , J} d ~nCTUR L M OH RRM
, DlFF. NOT SIGNIr.
2
, 'g.'
' X
' .e+p t .
c. 10
,'f 1
[R i (Rz
= .RSSUMED Horf. art 1 MotAENT ARM hRH EQURLS rs 4 D1VIDED BY 2 MINUS CoANER RRDIUS
; '. A RM = (+/ 2.)-1 = 1 /d.
ASSUME RIGID PLATi~ $ LOAD D13TRIBUT/0H 3? f 1 i N 1 i b ',. .4' ,
.y 4 -
j W SHADED AREA REPRESGHT5 PRESSURE DISTR)BvTlan BE6 SHEET 3
Figure 2 3 P FORCE = R, v
)R, EOR HT IN TUSE
. i; l
ll - ii I .O
*R
' . h ', ,
d' ,
~P= ONLY FORCE OH PLATE ns RESULT of 70RS/0M D/5TR)BU170/l OF"W
- NORHRL ~BbDy' TOW NORMAL FQUNDATION STANDARD METHOD
~FOR ANALYSIS p
_ /" ./" y- 's b 69" Af- x'N y w = 4" ,
, CoFJTACT PRESSURf (ave) P/4 LINE OF FORCE AT CONCRET DUE TO R,(P)
~SAME A 5 A T' TOP of /" PLATE SEE /4TTACHH6/1T '6' )
HECHANICS of MhTEtelALS
~ BY IG0K PAUL POPCV l SE6 ALSo INTR.70 7HETHE0eET7 CAL ANb EX1612) MENTAL. ANAld31S OF STRESS $ STRAld BY DURELLI ETAL :
nTT80HMENT'B' l
- - . - _ _ _ - _ _ _ _ _ . ._ _ . _ _ _ a
rJki 3 ..'1 iutPER LAT. RES ELEV. AfWtoW 858'- 6" ,
. #'o fog
-1 /
/
l/
. /
o I y - o.v b 4 l HORO*EONTRL STRIP or WALL l y, TRERTdD R5 A MEntt l Ro sents , i l SECTION of wnli 'A' l (I) 57/FFHESS of WALL W Af LEAST* 6/808 ^'//II (2) 871FFMESS of WALL 'B' FAR 140RE 7)lAAl W/1LL 'A' (9) THEREFOR. ST/EESS /!! UPPER L9T. RESTRA)Mr O) HILL BE OVER eoI<GI. FROM L,0.';I;. OllC/ \ 37RESS DUE TO LAG (STEEL /GokleRETE)/N DirF.
^)L9th(hK91ENT/HaeMAL ePER) ABodT 3 /<.SI.
FIGURE 3 j ,
.' ; / j; . ! ;
/ / i ,/ / /
i
,! ~
S G EL 86 ,! f
/ , ,/
'/ / ///// / //. / // '
'!,'ll!,I!
j \ l//
/ / Ni, ,
/ lj/ /
,/ /** g6 _ k' ,
~
\ '! / ,' / /.
s,/ ,/ j'
///// /////
/ ,/ / / / ///./ l ll
~
" - '- y E , /
* ~ ! /
,/ p - .s
/ - gi t
/ loPEN
-// ~~
?o
.X pr c. '
*^
/ ! ///f. ._ Y //./ / / / / l <
/ /,y '
A- N/ / / / // / / '/ / / /v' / / / // p
'U
^X-7 xfj e
i
/ i
' / ,'
/
- /
,f ,/ / l/ /
,i // ,
/ ,
u.N ,g
~
' l dpol -
U.'A R. / i i ,i j L. R9 \ lI -
, fy j'
/
I opol / '
'xl N 'x ~\ 'N x x x x s. ,
- s. 'x x
s . s N N vs4Ae Ar czcv sca + ' N 'N\\N- 'i s ,
' ' '~ N
- s. -
" x ..
N _/ - _ x -
,x - ~
p UPPEK~~t tiTERAC RESTRAINTRERCTS ~MA}HST-.
~QTREHEL'/ .571Fr WAU BACKED BY SLB6 AC~~
~OPOS/15 WALL (7'-0). APPLIcAMTs~Assi>HE'~
REncT10M,5 FOR OALC 21FrNE55 RT A$8 HotAIEVEK Ar PTd' '?' o' wnu. plct<6 UPJ_ SLAB AL EL 860', FROH Pr d' 77+EREP),EE .
'WnLE VnkIES FROM PT 'C' TO PT *B'-
PT'C' WALL 9'o" PT B 7'O PLUS FULL. SL88
~OV c'il R6))C70ft- STlFFHESS HORE T1thadt)S T DUE TO '7 F007' WALc. PTW 70 PT 8'.
PLnH VIEW RTELEV860' UM-72 LATERAL.RESTRRulr i
i l G u l! G 'l
+ I
[ - UR RAJAMS FillTY
.!RSSUMPTION nl.LF%,
p DET 'S' d N .' Tla Ei WHicH REACTS no- schlE TLOR D 'P' I5~5UPPORTED DET 'R,- . Br) FLANGE A "THus RE-LIEVING BEND *45 ON wt'.8 A. As DR RAJA M MNALYEED DICR ATHER PERCIkVEC THE PROB LEM.
- TilcoRRECT ~
- 0#
. Gh 9, 4 A~ciuhL FIXlTY osr'a' h)Q No senLE ~
DET C
~WCT1)hL AS BUILT SEE case EXH161T 6 to9 8
- ) TEHS A F f' 46.
. H OTE:
- 1. FOR THE herurrt. ns rsvoi T-o OH bt110H FLnHGE 2 13 SUPPORTED RT BOTH ENDS. PLAN 6EJ t$ Hor 5 UPPORTED, LORD 'P'is TRANsFSSD To suPl%tTS (wet.Ds (Do 4 u]t) BY OfRECT BelDIHs t 7Dxstod of edEB 'A'
- 2. STIFFHESS R,5SUNING FULL LENGTH Cr !))EB n' EFTECnv6 69uBI.3 SE1/ts GQUBL3 10Kh4
FIGu?E 5 t v. x3
' f N h i ' orMGIM A L
~COMPRESSEb ' /
' DEPTH or t+55t+T OF N l PLATE PL&TE '
N " 1 / _ l P EbY poty unogg. PRE-LOAD HA S ELONGATED w 'No FURTHER ELONGRTICN ChW occvR. UNTIL40hD
.RPPLIED EQUI =LS PRE LohD MHD SEPERAT10H occurs RTFRICTION PLANE EPICTIOil JoI NT FOJR D)!NRHIC. LOADS TEE ALSO CASEGXMIBIT' 669 6 ITEMS toe,7 f &
MOTE: 1M BERRIMG TVPE sotHT 80A.T STRRTS TO ST1tEhCH h5 3004 h5 nH'l LohD IS APPLIED.
1 i
~
- 1
. .. ATTACHMENT A 'j -
St VERSION h.: ..
;.v
+~
.i. .;_.
d.:
,.- Mechan.....ics I % . *** .- * ;.* .
r.
; .N ; . ,' *
.- *. <e.",.*.t; . '.
1 *. s( t E. h '- of.32MateriElsh4 &ll 6w(?f.n,;.
. u .,- fr +,.m%. *@.
.s t;& M , % .,... ' ,-f " k A'H > "'t 3 ,'~*n....
A., '? .,;^ p ., G.; x e-
./'
u, m-E.
.g.
e - pues,
- a.
.x<
g. _:.::.. g'l e .i .y . ., s
.y.
H q ~,.g. . j. r rs. .. x g ,, . - ..- i,* r, : * . :: y. p ,t,.,' . - ', 3;
- . .A .
- s w@.. . . . s.'
M~;' '. #s ma w 1 , n. ,. ...
.# , .:;*jf. .gaf- .-t.fi9tgid.e...mck' rFf'" 2
?, .L. .
- 7. .
.. y e
g,MW.. s , 4 ,, v,.
.t .x
- f. . . j,*f: ., ..
. ";.3 2;-a-
~ -
gg .
- 8. g.i i gg 3, :..,
b $ r... ER[wy~.POPOM..g h h '[.' *" Professor of Civil Engine,ering -y$ _, gg5[ . gg - a" ;;,>: * ': University of CahfornG, Berkeley
-:,,3 g y y .y.
- D.. , .Mcw.
.ww.t. . .%,
sdyu.p
. n. c s:.
' ..: t.t.c%
wwr
- -:Q: k: -
e
.. - a: . hh: -
v usw. . R
- lD;. 3 . ,1"Nyr-c A :.:=!;@, n WW' N I ^* Mht-
.s. .. .i,g .Nef..g: . ,. .y PM W
. .4. Y.F
~ e.~= - . L .. y .- #%ry? - -.
. ';ip.. ,py N9tW: .
.*ph... : . .V:fWRiife
. .w_ v.
.m .
.z r...o. .
. .. . jn.,g. f
- 2. h_ .
,, ge, . ,
.. t.%y,~
cf.'"f, L
.e.. . .gm
.m.
*D!g- w . .,@p:.
Text in Collaboration with:m% ,f H a cj a .
.4'.19
..Y 21.W4' p'*
c.',.Na
'U4d araJ'ari..n'sh7%.*d M W,Ph,1 h(t u,h 7.
..<-<- G. ..w a.i> e W;. .
a= w fI~. :
.t.m '-*www _
.,~ -
k, .-
. 2,', ' . Associate }tesearch Scientist A 'm, g.Ps;w.I.,, .
3 r,, m *
./>
Q -
[. LockheedMissiles &Sg sa, ym...ie, caritorn.i2fy~:. , .- ..
, g '"q+p, , , t. .,a.. ~
_~m aun lw:.FR ' -y, g %,. -Ms%yVL.'1 : ,
.#'LM(
+ 5.2255. .u.ws:a . GW.+ = f :. a . ,?-
M:..
.<of.e .pt.. J 1[4.. .. ;; - q- . .r.
W.,.
.:.M. . . E. e.
Problems
- m**,
4* w..ith. n., Assistance l .
'. lf *
. J.p [
-'(.*.. h ' , . i g,
.[2" :.n. , . .*'fG'.'.
'W
~.' . ' . ~- .4 '. .U.h...?e'U. .. :u g C
. l - . . ;. . . .A. .. - : ..' . ; * *
^'
j$l$th'T..k. ..$$.t.ern,ati&,fne:' . ,f' ?.f.P c:(.4f.;,3, p y
' ?_.$.- .. '
. ',.,, v ,, n :
; -' $. g.$f
- .. $0,Q . . ..,...
ONC$$$0*
. marti k.'y l '.'. ;
e _.b[:, t ,h,, .
*e:&W.r,N
" y::.; .-
;c .
. ., , s..p4 :$ 8 w>'.; . - t .;}.....,. q.,ttt.C. '.P 1,,4O A,. . '. 3.:*.Y_11.
g .. . . e'?' * ' ., . .
.-re %& 's&tm r xr:#itiv L.9;.d'-p; ..
j.' w; A
.:M -
3: . ?p:.p M '. i ..i.*w.p:.'- :% .s % t;: GW *
' M . " E '%i,.* T-hf :f.LW. , , , ' ,. : $ y'l'
, ,e/ fY ;
Wr. . &Ws.i: 3 :n,f5tD6 2 ' i' - !M V ~
'T*?H' !
PRENTICE-HALL, INC., EngleWood Cliffs /,Nevd Jersey tj.7632 l:;;g l . g. . - y v.
^- - :~ r- ,
.'.fi.
.' 7. Y.:
. f.Y . . . . , -
6 v - x:3 lw .'
~-
l l l a-.w. -,-e-. - - - - , - . - - - - - - - -, w -, . , - - - - -- --. - - y----,r-v.
&,=
Thus;-for example, in the tension experiment described in Example 2-4, i Nu i![N.'Y M t@EoWRminw M : 'se i. v ;- O diyidjiG a=l(70)l pii5w d bh.d.w/, @nY.Ohush5,0*G(IT-fLO.333)f.26.2.k.10*'kN/m*. 8' % dQ$kih;i;ik$ h .
~ ' YM NSMM[NMW!?YMiNWM mX ki.+'2-17 if6 ; - '
xN.;M i.0.wt%,,, !%5MNNMNM d@rE fundamental s formulas'of strcss an
*~'TidMMCQIMIA %the 'proceding- articleirof this chapteritisi
,,bideformationElfiuch deformations, -
h)46-sg:d %,dhan talu M,-[gthegsamfg-,orsp*
.;7 ,, jabogequatio,nsfexist in isotropic ~(
rate" 'agoigsgie F Fel- m** .g *
-crqsection'al' area of ad' axially ~
i F'"Niisii.s.u. inichiu
.- ,if ., ~lik rc.E~i$atitiEllyapplied.overa.
1 stes,' a4otm m 2-ts r.n.
.s. ..- ..
i
= . .ta, ke.s,plac6)T. hisi.s caused by the..
r must physically ~ con. us in a deforme. - .dT iM#inust itNEId%shintr7egiial ~
* ~ smo' untfaith'e"adjoi5in'giides off "aHimiiikules.'!hese ~
ns resdit from extensi6nAl and' shearing def6Em@ o' fving
* ~_ offmaterials E,'G/;asdiv and the appliedi' WddsI istdbed "siddist'r'b'ution- i are beyond d Itd ol' tills 4NieIms"arsheate'd'iWihTEarhematical theorfk ^
of " ' Ev'en'MWaced'aihthodsTulE.'th'e?sinipler cases have E bein~s'ENedMiniiiGiiFicaidiHiculties becoinit6o~ seas for many prac - ~ Ptically$5tl6dsht .'"Tthe'gro6p 6f pr' blems o that are not tractable
~
t mathe'distica ly[ **LJerical proceduies forinulated i on the basis
##ofinitEelim{entshce equations aren'oE Eidely used for the
^ alutidnit'c$nplen?Nf*DigitAcomputers' ire"in' dispensable in such ~
workf'AitehiaiivelyEMrimiidial tschniq6Fs'(dainly photoelasticity, fibriefij4is'6iisOd haienn 'de'veloped'to determine S the actual
.bstro'siilst'ibdllotp- r - et.Gwihinidun:r.ss wi.e4 M Iis"thi 'lext..
.m nw~~mnv . .
~ * ~ ' gnificant to'examinWualita'ively t the results o' f 6 more ' dild i t dEexiEple'~1n Fig2'-16(a) a short block isN
. ddii NfSFde'MThii IN ~ BGUIEtliis aiske'r$toblein 'could beind solved by E usin' W~ 1]
ve; M y,- .
'really'iohect? Reasoning
_ pparelit3h'af thMtrairiEriiustle maxitaum in thei
'vicinityMf the a6peidTeiceilisoce thetorresponding' stresses must also be '?
.inaxititutii4THat Tedirea.is"th&ansiirergiven by the" theory ofclasticity.1Theh end resu g aFstress'distributionj[at varioussections I6(b), (c),'and (d).'are For 4shown in/
~ ~
..,theY.:d[oii . a. [i.f.r .. disdiMEi8ti'iiiagrams ..+... Figs (,2 presen purposes ntu ton.ts su enttoj.usti these.results. Note- .
fhip'articiil619thdigitiiefMfihWohiialjtrsiat a sec'ti5n close to the applied .
*f cj&! force.t2Also'nold 5.%%> b #9 WILW4iG h u+ JJ4 d id- M fisiiiOystlyitis hak-smoothes out to a nearly uniformJ M J *S.Timoshenko, f. PCGoodier, TIwory ofEasticity (3rd ed.), New York: McGraw-
-1
. . " Hill.1970...p. . . , 60. Fi . 3M,isadopted
- w. t .
from this source. ,, . o l [ tin a purely e tic mesesimidusseress is infinite right under a " concentrated" force.
^ j
, , . h.
w .+m AIN-HOOKE'S LAW- XfAL IDAtr ?HOBLEMS',[ : M
... :., g :.
;A- 1,f.
- l, , , e, 'p: _
- yl, ' l
, 32,Yl h' abi.: o"- ' ;
I _ - - - - - . _ , - --my ,,-_.._7,..
,. %_,,._,_,-.,,-,_y_, --
, , , , , , , ,,,,g., _...__._-,,_,,r_.,._,, __,_...,,o,....,_,--,.x ,v,,,.,,.m.,., ,
1 1 l . \ l r ,., i *;.,3 i -
- A I 1P
\(P ' 1P EP' < 3
.a s
e t ..
...} c. % }.
, }
p
~.
t* g w----
*3 g ._ .
g- ,e j i a s - e.- f .s .... I
-- ' Y M 44ttilk X
- ca vs. = ,. p, .M t w s .e.. m
. . . . . g u..
,._l . ..r,..
- a asg . ~
2 575 a**
L< l
'f..*
--. .g.Js-i \ .gCass . < '. ' u.l.387 a.,, % _
.--b---.- e '- ~~"~~' ~
a- T --
#'"l'C l .027 # 7 .e .', b q%",<.s .a if --=~ -
-O't,*E . _ . _
. . t',7
.S* i M).
c n .~ i G h4I -
<* ' _ . p.M, b, , ". T.u.:,[!cg $. . , *,...W,
,z - , .n . .
., M. *:.
m&.
.,};.
Nf.,{;.I
,w
,'?.
--f;.g?,.
. $,' .-.W
;.v.-g; yL, -. - . :'; m . . r - .- - . . ,"
. -c Cg.W
.~~.:; ' . .r". . . ,9.. -- . ;Q. .._y , p.f-s Q 9l,?e ' '
v .e+4 4 ;. .tc
.M =7. s :.Yt_ 'fD 'a r's 6 *
&;*-%{(a) .
~W W #
W (d)' Y*' y, C*L,E 6 *~ T~ Q 41 Q (b[. ; -~~ ~~ TU$.E$ %+k? lh.A i - 'EM ~&~- i_ - -Q' MFig. 216.- Stress distributio, .cTM _ & E d k;.E n near a concentrated D"N force ;;5- .
.<. g?
h(.E. kme 'n.v&E q J.p .2. % F khWr stress:d.i.strib m.- .~
. : ~_ . Th.is:t ust. rates th.e fam,e.d.; Saint, V..e
~
khhYS Ation at..a...se.ction
..a.l t..o. the widtli:o..:3.h,e,.'b.
.t
_b.sl,o.w#..,N,
'i
. apid dissipa.ti.o Fof.M,s t ..
n 4 ,ar. f*m., q . .
,.p .. p;.
gg
+
. locab..i,l,d..s.t.ressesc.*Ihisprm.
- r. e.
w.
.1 may, p ..
erts-
.-.t. hat th.e..e..f.fect
. ~ . .
i iva
~or .ostressesN . n of.forc.e,. ..;..
N m m .e .- ^ whichs ~s..at a distance approxim, c ately,e tm,.ated,.
~
,.c.
k.. g .w... g" .
."%c;a.he .l to t .wid,th'or.; +..g r> hickne m.w.s o.f.s.b.o,dy,f
. - t s
&, w u w .- . . . . . . . . . - - ~ ..
r .r p,. r.m;,=, causes . . . w. .stress distribtit.e.. ion-
- t. hat - . .. e
.,r : g. 'true at a, distance equal member to themf.~.fo.llo.w...s.a.simpl6. of apph. . .-.q law'
~ - .
- 7. .- widdicdthe rom
. cr the pom,t r,
,.i ,,e so . < - - ~- -- -
tress .< n e .s.. cati.on o...f a.. concent.ra.ted. ,fo. -rce.M. *..,..hII.at.a.t ev.e.ry
..r ~.-
, Vp. _ s . . , _ . .
is investigated accur.a.tely t.he. . .. .. m -n.. ms is still.. ..,cor ectly given by Eq.1-1.
,"g",
L, 7
.,p-.w f-. Th. .is follows, sm.ce the equat.-
ions daamassinust alwa,., ys. rbe satisfed. .. matter N.o.. D ~ F
'3 7 .ge.".
' how ir.reguiar the na.tur.e Ef. _thW":Mution.T...t.T...given a member; an integral (or sum) derddever the whole area must b. u . .e equal to -
sectioniiir'ough o .- krnen@v wmtw ,$,099ggggfo,ce.aw
~
.m;g._ am. . mq . m, m: y.,,g . nn ,
. -Because of the mco soikm_g, foYthe above g..z ,Mr:;. p- .
e e-gx.w* . :: w.vq& ment.w ioned " peak,, o.. great rl*.ocalstrm==
. ~ p =conven.ient Wh%. -vsc.-w mrnv.
n.untered.- --.~ce--- heme ve has
-, be.e
;.;tM.k, -- %5Gj.2 mg the stress'by the..
.p.rp. en. :e. -- ... m . ..practic.e..
. w . .: ~Ih.is
..n sc he.m.r .e. c..onsists s np of- -~. . m .ele-mentary,d by a 'numbE qr~1-i csu.%. ly.e 2 and then;mul'tiplym.g the stress so .
A.,y ,.;z . equations (as' E N Nh.M. W..~ e com'p' tits .. called.disEe}s,s.concen,tni.si.o.n..fac.
- - c.o - . e . e.
torc
$" .::m. . . - ._ , . . . . .
tth.. .s n..um. ber_w.ll be d.es.ignate- d by i
.i . . . . - - , , . ._ .. . 7 r. . values. a the s. tress-co.nc.e..n.t. ra -
s- _fac_torad.epe.n.d on.ly .o.n t,h.e' geo.. p . . .mem,b,'e.r_. ~Ilies.e TA c_J * . . w il ble in technical'.i.kca.m.,. variam.satab . ropo. rtion,s af, th M" J . u.u c in. b.!'d.ramsiW - afa.c.to.rs func..w.e, t ar .
- a. va eometr.ical a
param. eters sc ~
- of tr_y- ,,- les an.d. graph.s'. a U,. sing th. cis ..u scheme, ~-
p...,c.g#".!#rq .-
, r Eq.1c.i.on..
4
. 1 may now'be of
.. .mthe rewritten
. ..g.q .... : . p ej..
as c ...
,.y Mfg:,4. u_,n,>fC kiW';qk4 *.
Em=. % r.ine:t w ." m m e w + WN ~-. ca_r: i.w .i.:
- ,m.,.. ..a - V.~ x,.. . w.v". m~.m:u.. .-:::
.__ . . ._wy p.o.pr~
.s. r' ,c .
3r
*W.
~
s.?
$5 N$.*: ?.5.R. W~~ W..,$.,m. . . .. . . . ..$N4 a,:ta nW.w$ N .&%
m ~,.r.1 n.w.n_. :-
. n.w.
. n .c. m: . .. .
"= n v,.=~
wher.e.-K is the stres.swidth . . . .
. . .fa,cto. r concentratio.nc . .below of tlie.. member, b=. 2.575.
W. Y? FNMM " ' HEnES = 2.575W,F '99... ** W.;8. E n . . - . .
> u. u ,e w . x.a c. -e
- .c
~
,w w.,
t - m.
- ' 93.,y.I.E
~
m. o -c . . . me n c:,;-- . ..,;... .
'R. JJ.Roark and W.C. Young, formulas for Stress and StrdinT 5th ed.).'New' York : .l
?- .- . ., .
Q.r-s C *~g 4 .- U - ' -
-4 y; 1- McGr.aw-Hill.1975.' ".:":.: :. c t ^ -/ Ot.m. > . " ',','d.'.-
. , ,. . p.i.: .J '. u .
4, .
*:, .i; c.,y: ..*..e.
, 2% x 7 . -. . H.
th .
'l
- .A " .','JC 7~ 5
~
40 [ ART.211 STRESS CONCENTRATIONS . i
a.T
.v: r 4 o-J " ', . 'l .( 4 .', , ..i , . ,
. ~ . . .g.- -. .,
. . . w. :&,. hb O :
~
.o :r . :) . . .o *
,gp y.-g g- '"d.,*n~'.~,-
.. .(
l
'~
Q -} z - e y*
* "g~ .. i 1 1 - 1 f'. l. ), , ' {. .
? g; mpm:
.y.
> .;. t. . x.. ,.
- M ..,t.
. ;. . ~ *J u,g - y- .
. mto w ... :-
->.c- r t w .. > xi. 3 . ,
( N v.,
~
(.).w,,, I
.f" d
- d'r l', #, h c'
[ ,.,- o ,; 4 M - g ~.: ~ ' ... &
- 1. .r c:
. w b:
e 1.Q $~a.; <* ..
'.c. ...
, s. *-
*;p.. - ..
, w; y: r. '. - n. -
n,
,m-y. MW
, f,6, ).1
'.y " v. :x.gp*l ' ~ ; - JW* ~.%.:~h. @?' e:'
- ~
. pi .t .
=0 ::~. - :
M
~ .'.- & M u ,* * . ;e ,.. .;..s. ~:#
.. :s.'?.
m.-
- u s s- b, ' -
. 3.x,,.s . m'p,&
z a9 ,, , n v.1.d.n
.._w,
.., r; cm.
p f a eM. ..= <
-:. . e r
.. n .;? Y . ' 8h .. > . ym
- .y(.h * *
- ~.
k;UI.}_..- '
~, #Q
..)}.f.[.*l, G.[
;~;
ca
*M .Ij:Q ~ gi QQ..', ,
C: l,.j;ui-* ::
,,9 Sf- '
", . . - &3g &y%- s. $.g >,< , ;.
. /3.C ~r-
- Z ' d.,-3_,. _ r-N.M-y g @g;p p.t.N E
-M *
...M.' e . - 68 8., d: *.Od *4' iN7~
g 'U- -yPfJe ; CA6% w Q 2: t"% 1.e!d J. ?? ?gd.b c .v4x..a.
- h p 8" .%- c'j D "'f.N- G :~[ ,.E 8.E/ l @/'*7.t,) * .9 , 8 :
~
'@* ~ - rc' '"JPJ - ~ ~!i~!~ 1 - - 2 F'W j.*"cP:"t-F l 8'$ Yd..- Nh.
i O$E** - ~,' '.. . e v.,. - Q , o? $4+ cM
'p. h"f*
k- ,1 c?c- -b
,g% e >1 Mg'. ., .; 4+-u - 5:; .-
,W.W:-&?
- t -
( _-
.l&fl. Q . . ~, ^+g$$n .$& [M .Y.4 ;;ay >.F m ',8,&~. R. ,' '" ' 6 % M'f"d @.
QM?$W--E4$M$% Et$Qlg{"$WNP&:"$fM"Ty.g :je# ? 84W 3 e w,hew{ memM. m gj k h g s.,g MhhTNf'fdW**.).NiuQ k d* k ' Y.'.d, N N
- bY hadh ND Ewew w' N E R -
E cr hh-R m i m e q- [{- E "7J
=E .{ j',(.E g' u
yFggt.p& - - ~ ~ _.- ... ~ ,,, M._ I'.". Q @~ M"M W % ;. g ., p r . . . . ,._: .~. ; ..n _ w m 3 - " , y_ y .-?c,..:,y g g; N m e m A,
. p y.
1-p - - C- 3.. . v. p.
,,M..' U; .
- ,% . s ~
* ((g$9E^ t
^
?
- <- s > QQ&&Qg{'
5 W , S ' g, 3
? &,,f .
~ '
~
%,{ N ' - e .
e 'l"
~'#
.-- g
~
' ~' -v r
.4e
~
r[ ; . t@ ., .,g. . %g/...
, ;M.
- 'JMygg F .' ,
M e.%p@.m ; l 7* 1 1 -
-- <=, .x -
?i -
. . m
. . ^-*'*' ' g ""-" 4 :
.. , 9:=pp- ,
]
I 4. 4 2 [ ,' !'
~ M.u.' ,> .4n g,') s. ' .-
l Qv%s %sr.,e :s. , ., a 4
. -m
- w_.
+-'"~~~ . - -~Sl""
-r s
a-
.Q jg,p ;{.-
,,,, ,w
[*, . .M[."ih .' l g
#*tl.?',%
' ~
~
!F; dhd- '
**"'." + +
,g ?- . 4 4;.g .
- -.--~ <
' p. *--+
'Q
.%-w
,.*'*Pg;,;
1 ~ ' _ w,,,, g .,v..;,.. (e. . .a g gv ..- , m - -
% m. y t.g, , - .
.a.. ,7,- . .t.
e-e nu
, . , *e * - -
- ' d U M ,/ #
',dU i' t J.
l " g .^ p% 'V;, f
' a s - , ; , ge.';f-y I*
- '^ -
- d * -. r. o 5 Y834@ 'T i :. . -
1 A,ur-.n* rce.e :e --- : -- _ _g_ _ _ __,. sy ~a . ~
-tg ~
'[ I - .sn .; i :- .-- : ;
vamru
$r.- 2 :*/ p a;.
-.: .i m yr%<z .- . nQ%ygy :x. *1t
.* , y,. w:, . . . . , --_
,, . y;;&; *y_. 7 _. ..~~'
- M W ._ . x&
n Qgg g.7%a%v - _, so -g w: g. % . % h'},,p 74 4_ v ;iG m. ,f mer i
.~ - e ; a .t-
.... .. r -- -
y:=c:=:- e m. c.~'-2:*@ sA,, --- =*s--2,.m o 9 >c ;. :.et,
' .e . -. ::. ::,. m-?: ;
am um.ve yykMsM - T-e-t-W W ( s vW. e 4y 5 f.Ar -5ia;,,?.,Q6m=F~~w M Dm.T J .:n,C.Qb -fr r+ M 1 i~On.*.f; r %~~ l - f 'J i.. .Mv C.~ J &WJ '6Cd 7.. c..,,.si-
, '. 1 .. ~.a .(b>Ef.-
-1.2 't/l.C..m 4 ,s ,.eO
. 'l- ,y . ~,g, g.g .
h'd. et ~, % ,y. ;fg
- M .=(~-
../.~ r.f r. * - .m a 9.' -
u:..:z.~w
.;+.b w' M-' e~-
= p m- 4,4. 1*-*aj.-
g -
.A. J 9 um .wQ-k.r.
n n
' ' W ~--r%.b.5W w.w&o cr .
w
- r m: Nd.- .,.:
m? T ;,Jf%.
. g ' *. ;
'. %y." A % :.w w L 77;n9?-ry .
%C', W,::Qst' t4 ; -- ..c.; d:gr
-,- q ;a:; -
- 1. -' . ~
- Q'
-- - \
\
^: : w;y
\
,S
; :-W.D
- d. 5, **. . -
~ f :.j:; v ,
W =;C.5; ~. ,.
-$lu.Q.,bk@.,.
bl l
, ; ; #'~,*.-
' 2,My2 wyp
,,'.d--,,
M i +- .h 'e, mm. p.._ % _ _ _. c. _ ,,--
;+.y- y..- .q a ,-..y--
,- ,. g,,,,g-,,,.7,,,- g g g --
m.. ***'.*.a---*'
- . ,, n - - - -
L9.g.. = ,- -- _j .%
-_ 7~~..,,,f, o 7 o.
. ,, w.ry - 1:g: A,
.s . .:.w _%ese.
+ c.;., ,.;.
wcl%.e A - fs.. e.,
- , .,-....~,--r,,.,=- cqw . _r
, qu.- :: -_ p, 2 &o ;M,-, -.4,-
.. - %__, ,y; o
;g. .--- . ..___- _ _ _= . ;c_1 seo
! os a v : . e 'y.,,y, m, , , 4 .-R ( E-y"- -.n.. --_,.*s;-- u.m m, y= - - _-,.,~,--.'_,,,, s-_ - , WgQ
., . % n.g .
~,
sgg re .a, ..
- y. b .g h 7g;,g / .,;- ,
- 2. ., g 3 7
- - . y ,- p
. . .O c y
w;.: 9 q- , g n g gj w coqw_ gen i
-~ . eg. y . g i
1 s. -
; ;.,..; ,..-. , . , ., ;. ,m ~. -n - .
^ C. , ,l *.?' -.
~
~ -
i i " 1, * , ' . , ,[h , ' 9 5
- r n .
.. ...~. ,
# , 6 e (
l l
*M,_
.k..s .T .,
~"-
( {.: ' t h.'T
- N; ,-y. ? Q 3_.- __
T.. .: 7.;;- _ _ _ _ __ _ _ - .. , 3
l
, ohhadeM e a .d'.d ape ? - -
,h8 h lO 5IM.T.I.eAb.eI W.) 1_ l_ # - ---- C * % %" , a.& gap. W E . . . . . ,;,.q;,;,.,,,.',',
'l j' ',,
;.'r.*.'
' * ' 1 , .*/I - 7 ";
. ;,g-Q. y~ t.?
- W . ; , Ch k ': m l. - ,
s ,-. ~
% t*J
> 10s -. C',.a
~s*%
h . . .
, .f3' c;. :
, J.
.-) ,.' q g
] y.>4 I. (*- [ D -
. p,[ ,._
.d ~.t m'.q:h . "
,* 'g; '
t .d/ , , ,
!b
.s . fa. s-
.e.-
u . . . .
%~
,-, ?
lb p ,
' v&.
eab a g ..
.~..;.+
*:F Y d jj.c ,1 c3
.,,.s
&[? -*. *fkr&.?p " .1 " ' : ) f,) . ? , i ' L. h $ h $sE,;.1*
5 8 d p ' h*[T . y b, [s h . s.i
.'-. . . ,1 .t ',
- I , . . -
A
. * . - . - .y
- m. . . . . , .
'*..i'3 .. . ..
7Oin g g ', g F,. . n .I.b;t.
.-* .D * . , . s / 4..'.
-a . .
.: r; " k,D[#t' Y, 6
['. s ')b ; ,; M
. , e -
+ ,N . M f.@..c... g ".3's A..M c
a X -)-9 , 1 ~ [l , . g.. .. p f ' t.!
, M Q, .s Q.-fW. ;.
. 3
_j . j.d M. o
- 7. $~9 f k.d u ~ g1
. - 4> - -
A [.n73 e3j- 7;p/f
.. %,a
, p: . 4 L% ~ - P 3, g ,,n.
p ,
.*a.,n, r. 6 9. , 3*:.: Q. _ W. %:..s. . - .} .' ;.g . ::: - ..
.-i.c- ;3. , .. < 4..8:%r ~-
- . Q
. a, U ;y
* - . y ~. E w
. m :v ,h---~'. ,..<l r .. . *
- p. %'-* % n
~:=:1 t
- .gLp9 j4, - .o r 4 x' .4'.j7 -
e
..-c.
-,,.3~..
- sqg: .
- o. < E ga.,., mMi t o .. , . . - -
q w . /. 4 f 9 , -o n e .g.s. r:.ir M -l y*e . .1
. a+
T %m ', 7-F% g-w ax ; h.e t' ' ,
- { S-4
& "D ' ' *', 4., h . 53 A O: of .P b V .*
- 3 4 -4 m *Q*T,.f W; ~G ' ^~f M
-{ . > .}.n -c5'o. * $ l'kY N. Lf:
- l' - c. ' &4.%,
. M% ",V 1 :
C k-~* ' 1 ' 3 -- s - '- ~.
. ' . ' NM
~
- Vk
', ... S t+
"J.
H,g s . . . ' [.3 EU7 - > + ?
' ~
- ,c r ., . .
sf fd.+ gh R,h.RO rJl oh P {4 r p#. . m ,'pm,r :j- m .m e.'g3N . . ~ E gqgh&.n
,, .- M . . ,
.g 3 . .
&G l gh. Y
,e# f> j*
,.,Nf,sf.
. ,gt...
- f. <. O.* a e h -
w .& : . . . .\ ' . . ' -<c - w.i -
. i .
- h befS W W5 seg g-
^
T O ha 4 = 4kg'l -
,A . .u ,: *. ;, _
T.J '. H W . ..i '% "- = %, .E".3 " ~ M ?. M :'T !.MEM
), h,,,N{'
. L. A ir,4, ,4,$.
,w [*
.g.u..;3 .g. v. .~
g*.*3 g. . ej g. l. ge - l ,
' f ,' 'T @- M4 .* W e w'd.t whe ,3 g y w &g &' w. w ::y . u 31 'E.- -
l ji *,,- g.. .. , ,..
' i ts 7 s ,
' ' ,- y ' . . . . . ' . %*.
'*.- y d.
J ;c *."., '@%,a, g 0 a-- m m gg La-m '-e y w, %y7-:@.h -:; ~ g4 q,. . e' fw .. , L.7 ,4 }:mh - Qg, .g o 4.o.4v g.y.w - -
**"% .j g A-M dsU '" <T
- "' 5 9N W w J k D ' I f. )' M;;. ..,
4 '. & { . - q.%%h%
>' p ,s,:..D... -
. & ,? p5 A W ' f,.#
5 jh pcg g
.1 *
.g ,. y _m ~~~~_.g, m%
- ~ - -
E . -q 7' _ w.. ,, . b b
. k _
- n.w g :w
*9* *'
# wig
, q.Wv.n .7 mn v6 :c,,
T8 3[M h ~o5 , w* be$..- ,,. E.. , P- . .W ..a,. N. h" . n . '. *. t.&,M..g*.'" r.A;fM .g W..D;N'- .r.. " h j[m *WM.i. ff12 - . _ , o,acht.G
.. ,ok; .. 4.
s 9: M:~4.h w.; - a w a+M .m. 7,, yn'V%,ste ,ae.nm w y "
@3.m
.a
*;r
-.c?"" *, ~*; 7 E*fb*t % .m une-m N N yemy
'UN,fMdi i N,yw.w$MF,*g
,y*, wg g y no 4, 3.u.
{m'*are g p c..Ac2.
- v. w m g *j
, 6' y li4 _ _ -"O. 1*.W E2 G k 4,. % WW2 *E i ~'
' g - y- M. * '.3. e o ',g "@M , 46p.
tm. .%c, r%gd8Q,W 5-cv .-- iwAM C .. (gP a
- . . : :: -, s
.c o,eiv .e v: h%::ru-=" . -A.. .-e k ... . _ .. - ,;f_. ,M. @ ,2 pg .$ n.g. E E Mn. gthM 'g % . MEDJU,;MQ7. h,".
e -- - < M. -{. [. l V. j l . Vy - x x x. f,n.@,.fM*}A16h, a e9 aw - w%m .$ 'U[W v. S.x?g Po,yh.. 2 x l t .I ...i- Mx-
'1 :a v 4WN- c ,,
l h\ x
~
hh, bh.i m .w w AnsiWS. a~ -- i '-?$' fh ? 'y'6 I,@bb I.f*H ed a g o.gw ? .; A.k
~
- p.g.(g - -ayh.-
.b-- ,,
n .. . - 'x w '*h 4.N P ** . -eth - M *'~rt s'
.x
, t- s - x M 4.-(- ,T d TL W L.. . $ids. 4@'Y&_ k.N - A---N-- - - m'x .,. s _ . c. 3"3'ff,g; ,_. 'Mr ' '
).; ,ag '.74.. T %.- No", ^*R WW 4,MM%
c74 [%eiyQ" 2QipE{ . g.*3.*d!f *** t <@~ x? - Qc_,"~*;*. ;.g. MqT.
.L . "n
( :c.. y :.n y . ~ 4vW 3 .w v.~gf..r. ;}. aU. dw m gpg.* 44w.c. c1
~~
v..~.~ c . .
.y .. , .-
o
, ,, . . g s . .
g eJ
,..~.m...3..
4 h ::: c ic 1 s.
. v 6 g o ,g ' .. .o g g
"-~
a d g . c a.. w m
. ,g
- 7. - ~,,gg.
4 v , me, / F;, t;. n.yw,.. a";2.g.;c,, f_g,,, c 2 m .ayP 5.$ ,e . . . . 4=- n-. .; - v5:4
*E . .= @C':
a cz:.::. . s M inu.th;- . gte.; ,cs~., . O '"
"..' ' L*2" a6~.2G.F p e w &".N.L $ #8 .. l" . '".W [ . g , ; g@g -. vg . ng).cd5
*7*~ %. -y e } f , w =.,.n gg ag 3.o a. gleg gW g:. Mb - '
- rn 4 m
- i a a.w,- e= c -
, .u .. . o .g o- C %* M*****'
^. . .a s . .
D.'$, ln.k 4,9 ^ 3 y .m Q y.O 2d *mb h6,
' ' *'*Q l '; * ~.~- ' f.'=D.70 .** 3 ' t *[ lsl'
.g . m =c.my
+,,g.;: . g-.a. dap. .a -a m., .&,,a.
l y( . N . '.%y.t. s ww. . .. ,. g - l y.. eqa q
}e ..;..w
, . . . .mw c .- o y e
.u .m.% m.
e i n. . . .c .w ., o , . g u , *; h.s *g.. y y= 93 7...a. . m :. = r . .,. .: .m ~. q .y.;.c;.= ;; -e .c o w. g-e7 ~e ym.x. .ce:1.,7... .e-- ..' e p . o t'- ga n .,o p u. Y;_-_7 - Agr;w
- 7'f2 e .- g.f.o.+$S e p d ec ,., d.:d.,e$ Eo w 'l
, ^ t .m ,
W*2;T % .c < ? - - *g s E ,'_.m%,; on g,e.E,._;a
. f~r.u
- c g .- g y s.
m
~@
.m- y'c t.;:c;;Eg,ge,,.e.82 u .v.=J. u
. c 0;.yg-;
q:. G 73 t g g ,i E 72, - .,r ,su . . ;eg % . > .u c- k o, ec . . - -
+ -..w... .
, r 4
.c.; - . e: .- w c m~
y'
, . vz.tv.
ci .e c oa 2 y -v pm - o . W.:.' e' ,
" ' c
,N
' ~ h h b h .E<5 W - C .- -* C ,' . Y$A >g $
E~'E 91 .c ;;; p,m. .- m % r a ,,y O
,;, l
.r ,
. .ll \
i i*" - - ' - -eem- - - - - _ . _ , , . . , . - _ _ o _
~
,.4.'$- - - g
. :pyc3 O*. .a,.;* ,' ?,' .*['1'l;. M ~' * -
..,.,'%s ,
.*. *~' ,..,
,... ., A. .U,.a. i..i .' ,' . . , . .. . , . ., .
a' [., .[. ,,2_ . " 4,,j* . 7 ?.
-; , 3* ,,. i, , , ,J.j-[.N Vf,'; *.
"l t #*Wid;M..,p Adift.. re.
?
.:t.i ,"i h4
,'t%
. .ON.a.;I ,nS: ,.f ,
- ?~.e.m%,.
9.9.au . .. v... es - osf, s e4.. M %.e4 m *k %df'a.'.r..f UL' :'. A-w+.*;JL,'.v
..a . .w .~ v
.r**j.,*., .ir eans
. a v. ,j . . ~. s.a
.' +
u . e. sjv b %* w 4. c . r.r,-
* . sa. :. - ...m.- .. . .n.; <
< p, *. -.;, %
h -o; & er yes e,.q., m-u
- a. - t.h-ave.r.w.s.a
.> vd v.,. w n . >*.rrn::
t
,t g.e v: w W,y. m;,e,t.
.z r *a-;pS.xc .- ; . ;p y,: mj, , --
.; .ym y _ s .-
, ~; ; m. t _
I:$$hs: ~'" 5.5* *r'.bi*f* $ 5$ ha. $kS?*s*a"a $YY5$ YS
, i.24]$. ". w* ' hY5b? ..
m t%2.n w.i. Ms.:M~.3 :s ; .
=<w
..%:aur a:: . v-w.::.v.,m r e a*:4.:g. m
- k&5$ ]$5
.w ; &w$$5 wmpx--- - w$'5$
- - ?' Yb e.asse -
Wah&55 Y.N!$Nhe:iU$ek.."TEbifs??eh NYhYO$ my.x:ssa'www wth ugp4-em u m m. q ~- # - ---
-- ~'s.e.cq,w:x../.%*,Rs,7M % t 9 ,E:G0g L% h ..E;l'?4
%E M5 dbD m.E M'**EW*5W 2$iAA*ei:$ E'5mSNh'T.net d '" M T_ f=,5mE E2I$IE 4~787.MT4 8h$1 e
TO : Dr?g .m.4. E +t{M. yg r h__. - _ _ _ it ___ _- ,.~. - - r- - a.;- m.7
--~
,.9 - ---
_--~ ggy; , w,,, :q j,;, m... g, e-v 'w m.,pv - 4.- o g_ g, g,7,, p 3.= ** *..44L , ,,.;,cg_-e=W Q*'a , M 7_ _^_ ,f - OMWo r' 3.. ,
- f' "-*" "" . g"".T "'-"__'dmep _m, e[,;.[****.f.*QQI*$hwiimP 87 ?.
^
.----[ earnes . Q i,,p M ter,4+
r r .- e T:,-
.- ..aee-- m . T - - a :.gA
- 9runer=e 5.L -+ =Dibn f.
o 'e 3 .# r-n* * '<*J1 _ ,-: %_ Eh;_ - T G. ;?5,'-
- p. y $ . I
- - *mumme_ggs an.
-n .
_ a.a >4. g- m l - n . ir2-q ,. w m.-
.+_.
u . . ... m- s u
.. .. o . . . ~ . ,
dt.
- L vo w
>r 6 94[v , .a.O.1% een
;h.. .wC/.d(
.M Q -- Lvu 9# %
- m. . . 3
# $'A, _$ /M 1,. -
. _ . .m h1 32 , el ".. n.__ .,.__.__- c.,[z z '-
um.
. =
y ; ,R s m.a Igmy4;i- e ' M arg.;d'1l'if
- v .$ pa E 4'.'*d SC
.4.N;C27L
,gl. e' E A Whe. ~
- 7. - ~ .355$'u.
,.,,:., n : v V. r. ti m ,w w..;.".*
/:w. m d. ,- = y ,l
'x D M bM N.--I.O8
*JEY M t_veng Z S'I 9
i ' Y * - ,. - ,- . -
. 4 .b.bf .
Y A $# I ' l140 % ' w m J ... y :iYf . My or, written
.. somewhat gdifferentl@y,'rememberin' v .y yw:.:H .rp r?? .
V %.h!Fff D;I.p$h k +.7l'g
.Eq.l(61)
I p p; v$,nW' . (page " *'1' 107),i. BEN
; . ep R',xg MUkh}k1YY ~ hik r g @$ism'$g?Y Yh NY
? e p? ^h g@~ %c. .
n
+..g.g< N m a m, i,
;L.62
. ,.NEOp l3 7,3 w_ y a ,,,
m,. b b W . f ] WT#s%&MQ9
- II '# "
- $; d${.3j$
}%(7,
.;.; w
' N!M f ' "
We recognize from tho' deb {p [cNajidth ,e , t${ diQ1E BEAMS' ON AST C UNDATIOg'lW-MMt.i
~'
l. brackets repres'e ntsIthd plate membrane'solutionW.We'ses'jbynil' .thdt*l odi ition a"i$tliVseedd)l. g.}dQ%effTr$r*;Irikl1I,thaiii,,%y'
@oNd y 18Q ,.,, ..M9 4
rig 7ai "N*f,isam a be.g$q f,the pra f small'bendmg the plate th'cory alone is 65 per 82n MFo'!!argedeflectionaldeflectioEs r of pmMern.7 , , n ,,
- i:
*%'*"d n , rder't sustam tlie large~ wheel losas placed on it, the
" the order of the thickness) the plate geta'much7 stiffer than-thelendmgb Amy of this chapterindicates.d.5-7: mil must be s.upported almost along its entire. length, by closely spaced Themnllt Q.(8q illustmths I0fk] M-mep'b(/dbee^ gy g; Q{ [ iesr.Kiriv,esti ion ~ 'this probidm led (ab66t'.1880) i o rat b'endi~ n ddiffbeFind an to a theory , l i I o1150 Q dE' [tliodt : 9 1 VM7
- i. '
0 11 h0M W . M k- , k . g ProMens 70 to 00. . . : .0] M,f,gg3 g'. p groaf, mpo3dpceE ci{ilingi' ec'rdiinly, fdudamelntal S b'ut later'it was fouml that the theo font to railroad track but to many other
, applied'not -l
- h>M .f ' f $thg.iMW MgdM I gjgg , 'sitSintibnsIni.'filgWAn'iidi6igh[h r n'It g JN9*. y ;,br'idg6' deck' oil floor structure con-Y g .-n.> j ,: , ; 4 7. g y. .
gg p }}'f'gh han M \' a,' r.lW$; jJI.rt : % M . : ;t%.g ,Eachtihilind al'b'eain'o'filiis%ictwo'ik'is's'upportEd b[tho'many beams
' ' 'I ~~
py? D ' cross'ing it at right angles, and,these crossbeams assert reactions on t
> I. 4".s .b,- . , Q fLrst beam proportional to thofo' cal deficction. Each individual beam in 2 ,-
g, . , " ' i 5 'the"notwork thus.is placed; ny,ngelastic foundation consisting of all tho y to'id,very useful in the
.. l 23 - N .,M . 3, 7' ' S iij
@$ ll 51lnf nkirigh, roved
/C3 D" r
' t-M..". G g u 4 hssb'ealds.%'dhtt)ms hsi[of sliip '
au II .N'i h . . YrrIilar*sjitict'urc essures y %s ddd examplo,is"a}tliin'-ydlif cylindrical'she l
.n , ,
5 hs 4 v. p: . ,; w
. 48 q.g whichypry with th'o longitu'dmal coordmate z only and which are constant
' I with 0, circumferentiallygIf we cut out of this shell a longitudinal strip j' q}tM [N;'1.v; 'bDg J Mi. i ., . . ' 1,n , y j. :of width rdd'then'this strigi is is Pbeam,'.' subjected to some radial loading
(.
.,h jid hg31}e gtgTije M@theginjlspageacti,on 5
ho tli shell m the~foim of hoop s'tr'forces esses onfrom the the re h,;Q >. # . .h h g.g ph o # i
, p. c % . maininig gr (2 dj a ,ty' sides, h(aym, g.the em [Silgle*,*g,' lie'tw'6cn jhd
. ~
yJ2p #M[. b; " ' d h:.-L(~- %g ! y;p % g .:0m( y This k JN "% DTed I' dN p'jgdf y~ $illlie resultant'in dishis" sed 'ori the radial'dirEcti6nfi. yhd'1'Gi.%'N ' 4 '/ '.i.L ' . " gin tho' direction tio'Er' ade#regarding tho "" rail 2a'd ' de tN'oY n
. N. h i..
g* ,g@%j[,bg if g [D I a lkuihir the clast i lbeha #'f[d'!to}th(i7f soYis M
*# ',^' ..
- ! nf uhg, %g y'
yj, Jtd gif 4 \Q Q , J pa.eef: b k y V)M.o.'"hiQ'lpw(84 . E . i I A! ljil, g.g , g pqqgg.yj ;
-v ere pis the local downward leflecti6n of the foundation under the rail;
. 'O* S tT-Pi'i N M&[,@ )M
. .h iAMM W .Z p.d Y Qis the downward {and -g the upw'ard) force from the foundation on tho
- 3. ' .
HdMI M ". O g l. tth - 4 M2 'fodadation modulus," measured -
? V.ph&Q {J p - h 4 g E' unit'lcrfgth] Q yi aAwd r, }%, way.ofNilTand k j my m.a m n h Q & @
- m m -,,s . . t em a. .
N, [f. k$ ' {. ' i
. 8'l $ ' '"4 in'unita of g/r/~s= lb/l /ir'i$$lb 1 o usu ailrosatrac j!} ,
F ; constant has a value'of the order of k = 1,500 lb/sij in., which means that " .yhy ., b, F. ~ t. [liy F,1*Q .j (85)
- dphi;Mgi i gj' - .s .
if the then thelongwholerail rail isispushed uniformly uniformly.1;j loadedqvith g =h,1,500 Ib per, gmamde running h' in y ' M j g} *fy sg!u'hg g . gon,' interpretatio
,g iil t[tli'o'fotindatiori r[bh ; assumption '(84)',has the'greatNdva'ntigeMepiyn,@h'eiMde*Isimple pg as , ncan Mbe; itfitiTa ppmbegF%e}s'olvo'tho'"rb nYarchuation.
u g also is in fairly g'ood' it run be criticized on two points.1.The first;pnd most importatit is' ilia uc%,, e. quation L p, g );.,by[{,
=.,, o usu,al substip y; f nn actual soil behaves non-linearly,' becoming gradually stiffer for grhats titN 80D g g e ; g'4 i g .g h'f n .,y g . .. y{
b j e
~
/*fI p ilpilections. gher foro;the rglatio fjj regnt f M (;
4 rather ,than a straight.hne,,.g an
=. h'$
f, d$fi-1e slopehg' tics l Q,y, p W. ! ibe' omin~g largcr'with'increasirii;' fst$ eg ig n ggy p' ,h,%yp;~ , , , ie5mathema I# c
.nomena'is~ extremely complicat 9 p%n'dUi@i's;kso"t afo f.5h; 4"! [qgg,9%n/ hf 4 ' we
%,ll na in otlior cases wo work out 'a Imcarl theory g {, use[it3 as far as '
it gog N ll id @entiss jloyintions inim it in a ~ ~alitativfrpa'riEO*dj gThT pj In y "g v 6kd .' W . ' . p jql N! j!:
~
W j O f gn$intlipi
. p pl.#is nto}ol[o$o Etl,48t} yjd } liou[ lliustin n p@ efl j y( l d
WI/ ).9 % h9m.p *. . N d"
- i utwM compie : ,;,
t by'th6'lottd att thla}t poin g ; El. 1 2 n is comp ..nc. 3 This,'oflcotOfo,j n{tirf*aheemeripj he ap u - jl1 -
'"[t, or .$ Id tEo =l I
other behavior of most soi loads'nearbp.}ls, at first sight. Wo'do' not 'cohsider 'cfs' fjl$iids%1fe[direcd% but the L. f object!on k"h 1N M a o ,pg yg 39{not'as M ot,in seriougaQ}
' (our fourth roots of -l as the fate %The point -I can .
e tangu u an i l i t,h rem , in ,3 ap g soil; ther'o always is a rail,in I t y 1 g ,
, 3 ,
q f
- v. 1r*lllll g(eprhqpearT$ un gj6-M 4 Kri
- n. [ %ourth" roots are at' angular '
y , d c.o, .Ji; 3 W. g Mi ! q M,00
.. h .
hl
- S, .
M y[
.[ - %
y
?
.t ril tu,,
/that Point n., etc.,trom Do Afoivre's u" , ..-
r v. % . , r : Jtheorem.p[,by,/ - - p i, idi6 i %yil i t M. o . .. h,d N'g 5, L i % 24 p y y ,,iM.4. W nr dd M;i'iin 3 .niv.%G I er . WrV e-
-+ G G.e rit .d as can e..veytfi yg i T1"
;. a M,ptor i 0W i-Me 'wbsorbed'w.th 3; .
- can be a
t.%Ipt,jncSqlir,Qijigtpirq{. L t we , : et % I j?i MEdf"'g ' y* 3' ' A loading p, of the rectanguin diagrammatil:' shape'shTEn,$lacM d ly k' ' W. r
~'
Fio. 92. the e dl (without a rail in twtween) will cause a soil deflection somewhat, like that hU @YM I 7';;k h.p 4 '. < "+ I.* ! 2 (SG) shown in (a), while the nmthematical assumption (8 t) demands a, deflection diagram (( %% gg,% jytj'ggd:jippi,; j .j;,n. - , p- . g; continuous loading p (F,g. 92a) on the,r. ,l,,th,e?.N,.,:< i ai m.im,o -. . p sT.gsrgtlie defigtson ", tlitQ,q fdyc solutions.of.thgroduc ofgthe.raj g g grsg ,g gg . diffeiential g,g y equation'become.
;,g .g y =n,..e
, y
}'
will oc quite smooth and tito reaction from the;grognd,gs also sngi thly g =,Q ,,gg distributed over a comparat,ively great length.. ,y ; , _ , ,,
- hmv we are ready to set up the dijferential equalson of the , grail.' SIhe Qf yyy gener'al g - solutiosof n.. i
- this eqitati[on*containing four integr the downward loading per unit length on th'e' rail'a'ridj is'tse'dowpward' MJ ,l.ydr'dNY'Nid d[h L F j
' reaction forco from thoidunda't' ion,T beam equation (which the reader t hen thGil o]l6 iM' d 5 obkt'lio"ciassilal g rita}r"
( *
'#*[
,r Be- #* eGCe' -}.!De .T'],4 t
b 'W 7 EI dx. p =. M t" p +.g $ N. y%PNM h t ; egressible in' terms oI.'trigonometrid y functions; . th'usMfM
- .A W "
'N J
...,,n y n. M. 4tiiQ. .p. 4o %;A
. e., : y c. 'tm , c. y ,ic*[Cl cos #r + Cs. sin.1]q (87)
) . .' e., [Cico
; .: - .r -
r.. .. r. ;amum t p'
,y where El is the bending stiffnesa of the rai1c4Sjbl ujin .v.e Chhth p'the previous A,' B, C, 7 (89 9 g . gt m: 8 t:hidhhnstaSts a some mannerghich is h'o,f no interest to user.The solut
}
- Emphasis added by Jack Doyle,,, (i g .
g gy g , i,d M M . % If' [ j k h h y.h[ . fi [,.c
t [p .WN..}g.144 bp1 Mn @h:..e W ON\ Mkd*_ ELASTIC::FOUNDATION . %'h, e . dq
. $MGMECAMpWE ff[IpyINf'INi poQ b M5
,;y G ~ ' BEAMS y .% . A y. m n t . , t..
2 g'e-. -... ,. 3 ,q .p p. m [. e[-20. cos Bx e d !gm. 5 4 ','. its definition equation (SG) of the' symbol # is'the most general solution 4 . b' of the beam in those sections where it'carnes no load p. Most of ou SN.'y"2 ' bO ' 2r1i 'M ~. N
'[ ' (C ;cos'#r.rip2 (-C M.)
examples will be cf beams with concentrated force loadings P; then Eq.$ s.ui#x]
.[ $I6 holds in the stretches between the, fo.rdi.s.4 WM
~s" 'ilfe
,k. ,that,r .' co Mb
'M,# c ] A h .".N:, #
*further and to verify m.-.4 imensionless
. s (we hg.kyet'see,n.c.fi.r.st. rc. m..v erA.will'do:well. - ' 'IY - series one hne o b. mat. ion px must be d. . . vo ney. the c.os;mb a * .
- y ,e ~ s t then checks'.Eq,#(85,"e Substituting z = 0 into the above exp 5 m..), so that # .is an mverso length;.. f.whic.h ,w,r. th,e. phy-n. )1-g ,- o e shall.,see r di. a -
. O. significance later. Next we notico that Eq.,(87)' describes'" damped si sons; the boundary conditions,become , _
, h waves," the C., C, terms being' damped % hen going to'the left (in N/ 7,pd.l[S.Q , k -x direction), the C., Citerms ,being'dahipediwhes' going 5 to,thoji' W y@7 W ' Thus if our beam is very lonsthen tha'C,y Citerhis'showNe'r[16ge'det flections to the right and the C', C. te'rnis"verp.large deflections to theleft' dd.[2-
' k $I#e'2Q F *
(C.i-QglMcA'$ WP ish6g hg ; , $ t 22. The infinite Beam. Nowweapply Eq.(87) toour firstspecific case;I ioniwhich NE - E'N NF *' a beam or rail of infinite !cngth both toleft and rightiloaded with a si T ;S: ~-- AMgV . jg pg" pj - e,r'a . 4~- 3 l S j 1 - I .;=. 8EI#*l : 2hbb I popputn whdo, ount Ipq nt tioforop P in tho middloMO.MTi g %ca';l ]g , .J) g g,@;g ;pp$ci,p . , , F y [, differunt ynhtos ul load'itself,Witt.thojepstant hos jygk,)) y dot
; q o'rtthe solution of the infinite' beam, loaded gc, $r qid80)..iflius we'fiwhich is not true f.tg.(87 and right, then (bo entire. eft beam and yould at right.P[If be expressiblo theyin termsissl th o damq'y on account d'forco c ~
P at the center,'(FigSO4), of tho~ kF. i- presence V.{f' i.
.~
"; I,' of the loa jO -
.jhqW ppj"'m , ph g'r)N d * '
the half beam to the right,(z }l}0),,jivo,see that the C., C. teems _le[to .
@P,_g .h 'hg infinite deMiii5iis y at'infinits'x,"whichils7bviously rcontraryQo boundary conditions. If we make C .= C. p.0, then onlyfthopecondl2ff ... Wa M 't ,
2{g
. ,p MP.Rr-
,g, f
- 2 f %p d ,. .
i '
,Mr1 remains, which gives zero deflection at z -emda accordance with ' fact.# .F 'it
! Se +hg + n Pi l N.;,, Thus we say that C, = C. = 0 is necessitated by the boundary conditions? Dj; , ). . , ' , '
. 4. - at x = m, and we shall always see that .there will be.two conditions fori f' M [:)"34' , ,
Y# , W G N D } idil $ du6id " " y cach at x = 0."end" TVe kiio~# of thenothingbeam, totaling fourjonditions5,There about .tho*defl&tioii'opabout tho', bending remains thn$]l y 3 '. - {
- 3. t' .' M. # " '
i d TJ moment at the load P, but wis'canSay"sMic'tlIing"ab'ouI,th'o'slopoiddj . et/ection. h '.W, g hty . t [ about the shear force there. Unless the beam cracks,' the' slope must b'el ,,
' " ' " ~ "' M -
N~ gj i horizontal, from symmetry. Also, making two cuts, immediately to! the l ' M;,, s/ ops . V ye f !cft and to the right of the load P, equilibrium of the short center fecp ,j,I' %. p iti.a y*
. .' s ,
', h; % .. .
$y 7 requires that the two shear forces together. equal P, and. symmetry- q, .
< { '.e.f quires that each is P/2. Thus" th'econditions iinmediately~ to' the ri '
;3,,3,%g rj: ,,,,, N M ,:N t ~V ' ' . .
O'* W l'.[. s I- ' -l-cf the load are C v%y:r k;
.,.d,. '>M:.As+ M d??
y& {~ y .3 9 ,. 4.,g , , pl@ g. mhu m.s [N . OpN<.
,g.
3 . e , ., . . - a, o .
...c Np:? z=0 y'=0 S=El p"'.4 E P 6bg. .g ' WJ "
- TQ 2 b .SDM N
@## NT# f' d
M - . y. . A J r
.r ; . dDd.M..m.hh.%,M.$dyP1 b
C 6 !3 - G.4, 4 ,q' - . - s
%vr:
wie
. L . '1 t %,4.p .g.1$ , * *'f.W h-
'g.S,.3
,.,9 m gm 4. *%C R^..
10 r To evaluate these, we necd',tho .Ae va.ri.n,g g rjyativ/,.he 4 E4 5 .
.jr E4 c es o which*; aft i e N- .
N - w pw 9 Lifinite beamti central force lo P.9'Ihe shapes of the curves are given mak.mg C, = C = 0 m. Eq. (87), ,.' m ' are. miig
;,, g ..Q g 4's,g
' g;, ;~1 _
..84., i americally by the table of page 146, and the magnitudes are determined by I:qs. (8'O
, , . "*, Li , , '
taJ the table on 't . , . K:
%.5 t ; Vf/ - 9 i W Y e y .* page 146..i..d.'g,,y o' ' ~
- b y = e [C, cos - mcossx .ed-F C. sin'sx]&#dehq.g% Mmettoniiih ikia'rious derivativ'es vil' I occur tikio and sigain in this
, , f.O . v' = #e-' I(-C. 4- C.)3 'p,jha't i{,be'conWEdn{enieQtii give them shorter notations:
- Emphasis added by Jack)oyfey;#ggpCgins4.gg 3jj .g
-Q.
+.- w. .. o. . , + _..
~
a . vaHh+ , b.w,;; m% , m x wc. w ,
h ,.. ,
& L (146 .q ,3nEAMf 03, LA[sric[FQUNDAT ONQQ Q Q , g. .QTQTHEllNFINITE' BEAM Q W 147 ' ~
hli$(&lQ* 3' h 'j. h*j.(;f F,(px) = Wfk'co?px' WsYp'z)?h&.
& & & fy & p3..,
Deflecth r 0.3. = 3;F i N_, hWvpJg.gcb [p % %y
' LF,(pz) = eT #
' ain fx =l r ;
' yI ">
MWI d4 p - -
..D, h;I , [g S F,f. .k' k
's F.(pz) = e-#'(cos #x - m71[, g = F;, , , , .9 ,, Q h "gy.ghp.$t !/ .3; ,'(89)
. F.(pz) = e.,, coa px d ,1hfM%{ , F;W) ~ / ~f Fm p'.M~% MidL' VAenf5
* . Mf. NFi.
i 9 01:'. 4 J;' . i20 A i[ N jd'UN
-ll z, t
U-W I
yJ Q;; + i .
- c. .
F,(hr) 20}F: a l ,
*' f,
" . 01,'i 'that the rail is pushed .
+ -
L M di NUMERICAL Y 1.U rkE F F6xertons s sp above the cr. .ipna.1 ground loyc1 at"some distance from the load P. 13y. :, [this of c 150 fiiciaisithatyi *gr6drid is" supposed to pull down
## @#) . .'.FS83)['( hk)Mh M[.Mfh a hhj .Ajotg of tal tensitypabout 4. per, cent.of that of the. 4}
' I. 6000;
' , . ..' 0,000 .
$m@ l dod)
L) .
- kl'hQ i g g '
a 13'% kO ((tIfY ea;k(n((; a'lthougidhey ' are Q0 M,!$h, .,
' [.%
th' oA i'NI$'.d. gol433 N. .f U fIIlllh 0 @Il emlistidn'om't ny Equa%u!,(q9)' apply only to'the right half of P *
'h o.7a3 .. o sio g i'3A
{.g',..o!oo9'
}d[
143 tr 0.8 0.635 ' ". o . 322 o
$ (ip'01;.the left;Imitjf th'at j figdrq,is'syminetrical with the right half in O
f ..l0.313 3 h oi
't is 4 J, y" and Nntisy;nmetrical in' ~ .and ][p ',dN I' , .
Mr+M'l i jp)199N! "'
- M M " VM " N. ' ' / VD ! M '
1.0 0.508 0.310 4 FGY%2 L o kN Nbh o.lil o
- l 1.2 0.300 .o.281 f%o/172 l ..o'109 I (*f2 1
4.- ' " '
~ '
.*.L- l7
{$ d " I* # " ""
$ i 1.4 0.285 '0.243 // G y o.20t,.,' ',QoM42[,s %Qy ;i.( 4 @.,Q .,
- j u w g a L%
i g- L... %* ' u - - S w^# t ..
"'N"& 'T ~ ' ' T~*
. 2.0 .. o.007-oiO5' t z oh5.i 2 2.2 ' V,0.o24[ 2"'o.123g o.000 g {vd,179 f }' o 065 b iU 2,2 l R.k '. l [;,'d '! ^, ~ lM
-0.006',
- 0.025
' f- 0.06 f,g 0.038 g7,128g g.-o.102cg 0
.go.0641 o',oof Wy2d h 2,g t
q *.. y . ya y$
. m--~
m lid [_ .-m d gp% 2.8 -0.037 . .o.o20 % ._ g o,o78 2 2.f.o,o37 4 {J2.s' - v ,e f 30 - 0.0 12 \ o.o [ . ' hfh).7;[*:jdOh
.hDefleNtion.and bending-momenUdiagrams for an inGnite beam under two
. N 2 e;
3.2 . - 0.043 - 0.002 7 .. po.oss - 90;o0 - 2 .
;g
. fa.- . '3:g+ tie.rloaq.;
3.4 -0.oit - 0.009 2,y* do.02i o,032- . bjg'* e g ggy ydedgt gqp@g3.gf m' V ' .A. ." 3.0 - 0.037 -0.012 ef go,ona.. ! o.024 : 3,e Eh t Jhf gendra s~olutiong,69) yi .. nun}qrical values of the table of 3.8 -0.031 -0.o:4 {. h 0.004j'y o,ois: , .,3j mfinito. beams. loaded w,tt. y #4h .T'. MKl44 panjo!used N9 E$updosing[theyariou
,ppo fogsol)ymg,th blerg.p[Iindiv i
For * .% 4 E 20 l ' [ ft h '*n.cos$ ih o[7w Q D.MSIlh)Sshg)$p'a geam,,y@,tgegi4al distance I apart.f;, ,,.., q 4.4' 4.0 -
--o.010
-0.011 3 -0.on2'k
.o.cong o
,o,oo4g
, cote 4,o tEtniple,Qig.',j
,defle9twns @ause cpgrptel y4tlie!individualloa '
.. , .:.o.oto A
'o.coli 9d4 p,s,. khes, a'nd the tota'. deflectionWeach point is the sum of the~ two separate g
3 4.8 -0.oos o.cos O o.ooguri 4 ~ contributions. Numerical values for this como out of the table. We might d pp 9 m 5.0 - 0.005 -0.007., :o.008 i
,* m.a m, p/o.co2 @. 5. i lak for what distance i between the loads the def!cetion midway between
~
?
.n . w. . - ..% m .gq bads'hecomes equal 6 or less thEn that under each load. 4 The answer is ef }
With tlu.s new F notat.ion the solution of th;e y b'othwa%,in' finite' beam m[ #FMl/2) a i V F(0)gF(' l),5 dd.py trying several . v '
.'d'.
central force P is: ..: . gr},36,g. ,g .g ggg g g,jg;onye see that for'p 2.00 and fl/241.00'the two def!cetions - [V
.}
' [',k,% 7Dej@pl.5 .]k7 .* Emphasis" add'edihy'2ack Doyieg ,
idMi.N'db.8f b b h ' M M .D N .h h M N h d [ ' g_,_g ,
.i
' y .. e 140
- p. l n
Gf.148 : 1EAMS.ON L .c 33.q p 9 m . v .. . c ;eg p(>@, ELASTl0lFOUNDATIONi {j.%. ; 3. . L h.g,y.T[EBjlNFl?(ITE.
,, g ,f ,weg .q g ; q. pSAM .c;,. y '
f are almost the sarne. Thus if #iik slightly; larger 3han 2.00, we have tM . sg'thatM.pt F aq: $ h ,! . hi , s . , . . desired relation and I = 2.00/#., This length' dept.nds on the relati$ dO@f WgM'.24h;h%h3.fe yp[g,)gU c .. 3f.M S g" ;. - stiffness of the rail and the ground,[Eq.i,(86}];jor'a' given ground..k, M f] ' 'M ' h 'El ' .d il3M P
.de 4
s'
,u M M 4.4 k g , ,
i liotie1 Fo' erentiations using 19 results obta. in' 'd' e pr:viously m. length I = 2/# is larger fdi a .. r - usual, roadbed k = 1,500 lb/sq.' stiff a - m,o, and fo:r a heavy;130 lb sT .rail t.han,foNflexib f , suceces.ivo!
.' ;r,/yd
,! t c, ran ,o.f p[,id 4 iS8Tand Eq.
. s ..,.e..~ ,,
88 m. we calculate 1/# = 51, .y m The' table shows that for a smgle 1 W.~.
$ "i 'id[
~
M',(M)h'[r - the deflection extends.to 'E distance x = 2.3G/d = 2.36N51 in(= 10 ft on either sido of the list If then adjacent wheels irnaltrain'are/ppaced; closer than 20
'#x = 2.36, whicll' $' ^ pbp'fW men'ns .f.
W. . 7'J> .that it extends will nowhere lift from the grodnd . . . . : .sStWeen. M m .y k.h.:NMk.Lo h.,e'els.MAM_:.D.,ft, 2 e.s..y, E
).
thi .. gay;g,y.f..d.M. s. a.
< a.~D I @+ ^ ' "
.q g.. = . h- ; . ==
,1 y y y y :cm acW M w- g. g . -- ..- Q yc , '-u . . ,
g (00)
- p :n . w- ~ r . . . .
$;.qi h.eM hip mbgNQ'M - 55 * @R? ^.53f'l Elf,y=k F.(#x) . '
Mfb4 .l$.Pf T f,..'
's,
\M. . & upn EN } T g -' i .h
- . .. . .h Yf
^. ^- h%
V
,$;dgdygtip ) .
1 E.WW w '.J - slope . ,t.)G.% , .%QQ
-- W j,'f[di. four.fmictions:of pago I'l0 reappear, b'ut they aro brought out c!carly by l
gyh : c 1 sceltliatTthej sam (h < p".'h ,4 Ethifted j Keecond' down method o{ong,Inotc' ilenyatioY,MVo'g,Tlioiryasofi for,this ig" rail (Ing.
. ^ -
[ ,,gh.nlE? p t h dub' ject the 3 h -@gd, 'dMRhM,3 pull pair of forceiP.pt'distMici4Mari, a positivo one, P,'at the origin
'[ (8 hend/op mome.91 / 9 :.e g " :I.y ; Iadd alegative one,$-P, at f.[-y4.i .Th'en the deflection, by Eqs.
p p P :k ks; e N r lt f a ! ... y$,[1y!qlg. fig #x.h'&L:liwQt. di j.g. i { S .c jpMT%%. % %c i +F(#x)., i ( -#3) f l "- 3$ Pgggf. .d.j @$4,didhaddMk]bg/do-~ ng momen't Mdn the middle? d ., waticianswho areabout to giyo birth $
- r. nq ..,+d
'i-i r Fro. 06. Infinite beam with a So'n'c'entrated' ben itten as.lU. $ Fro. 97 A push-ruti load system P4 scribed by Eqs. (90). 1 i'.3::::q 4 *
'! M ,h..:. 'Q.y.;;A d'E .j f'~$p,%. if...h { ; 19 a diRerentiation, is rewT' (P3)s,F {S(x + 3)ll' ~ i i becomesaconcentratedmomnetMe = P4 ,
. .s t m.p - i! ' ' ., t 7 - .
A $ 4 . Another case of loading of fundamentallinportance is that of Fig 06f e' :g 6 .. i . infinite beam, loaded by a concentrated bending moment 31, in the cer.I$ [, ." ~ 2h ', * . ( . 41 . . ... ;.jsimultaneously grow x = 0. Wo shall solve this prob!cm in three different ways: by boundah $helet be ta e 3 n the forces'P grow, so that the product the limitf3 -, O the product P3 is a moment
; conditions; by superposition of the two solutions in Fig. 01; and finallyif $PJ remains constant.'.!Iri #
Maxwell's reciprocal theorem. wt o 7% p ;': 6 m " ch ihich we call'3f., and wo havez i, , +1 $' For the first method we return toanEq;(87)K:d remark that for -- x =511 Nc . a '..~ A. M. . . . , 0.WThen at'zN8 ; 3fod jj F l8(2 -N 3)I ' '- il821.
's i
')
j!P.$ }s'o j- ,-
#j the the boundary deflection M. .M'!4 conditions y
' ft o'm4 must remainare,f' finite
%,%eT [
f thiy*
~ I ~ r@!
g'.$2kjskg)h!M(.Mpd xQ2k@ rz M(ex = Cit 0 =h%i; dxWMEqs;(88)D]p MM l- t . t . ' , ' b k
'# 6 J,41, y =;.0 aDtf @3f@.7 Ely'J;= 6 2 'Ngaf NL i .
r 3m
. L. -: r n # # Ab e nthrdicchy,w. . L.#4 9 ).,n
~
~
hich,e is'the S result, m.m ;.'Eqs. (00).jTliu_ the senes of s.Jfunct.t ..ions in F.ig.OS are *4 i :J Ribstituting this into the four gencral~ expressions of page.r144 to 145g) g'thidgiivatives of those of Figq01.Q find m 6dmtsli.9di6- ' Mk!dMt [.Eqs.,,(90); from Fig. 04 0
~
,J . a 7 'J. M P g G** pA JI%)aJp ' @ The thiid'inanner of deriving'thisame result j g
ch m.n . a$fb s i n. w VI
; wh n u. m.me.
ifi -
. s %e, hN.mnge mg hh.Nf f M. b,,.Mw .
g ay n. q..hicciprocalitheorem w nM . .
!' . . b:.' it.N3M . . bid:' k d .d En N i.d;. i N S ..
,s._.k .m.u ;_ ,V
- _-_= - - - - -
~
T4M@4h!S' @51 $lM MEN' W;&q';M 3gg~ DR fM'ELfSTIClygghb;pr.tM' wP yf4 M M M
'151 F0llNDATION T, 9 a $T1152 FINI(R'BEAMM QM
# y6 g 6 , M ifT y ~% t, b. '
4-99 y ir g via e kfila(fsed be' free ofloading. For the r,ght i half fb ' l (work-ubsorbing cchiponhnti,efftho)"doflection at"liichtlon 2'y ? .:ca6 byfif dmt loadinipofatid lith. .W.
~
i timist' complote'theioneratio!Utio((87)hvith a particular integra! of unit load at locationW.oquhls'theypleff'octin' Pa'til' dtio,to a' unit " load
' ' at '2.'~;Wo~ apply.this t' o thofit6'ati$ipo I
' ssfload 1N~ hM 1 hi ' T 'idtan y pr Buch particular integralis simplo;
~
4[ h h, h hk*
- '. :S$jhkh,M 4' UIi
. f.
.hlIh[h, MMM 0
h [ ,
*"A -
' d j ~. d g b"~~ D~; gr$ till 'ex ! l .higl , no terms with e
- can ,.,.
. m[x.
.s.. 4 e u ,
Bos.rha. gy v.j%g L .. . , I'lu.08.' Tho' deflection at f causednomp only by , at 2aequalg un g%; the slope wwfw g mur, i so j ' 3 k 2 raused only by a unit load at 1." This s,tatonie,n of. Map,t-e 'slecipioc41'thWorent'lles t
- $): i 'd QN.f[ kj '
7' * ,'
* ' 1together the cases of Fig. Di and Fig. 00
' 1poment 3f. are applied simultaneously;bt7t'af different locationil and- 2.t Mu A N"" W+'
.f((8 ),i. holds w.th"out i clu.ngo. If wo
? , ' he lefphe, goner ; sol, t Tpon the work-abauihling compon'e'nt;of yefis-tidn' "loEa' tion'2 is a- oel pen itp ,; . , y by,nt
@ lIhun lop'p! Nhtijon3f %'Ivih!1 Q I
{q "PAlF,i)' 08c'at' j ,f g,. M, p(3hj)N T*@ "'h,,$$$*6- d. bill, }.Mnjht@o hid g f i l C hy@my11tlthit tilb $1lf, h to. pop , t g p g h. 4 c d ardi6w:somewhat different?from the previous j elopo the slope at 2 i(forIdrlbtilloin akotched in la, Ghon,4..,,~FiS P = 1 lb) equ'als the deflectio ott, k 0111 OS, d%it$),diat it 1,ihno,< .Bf.b no ,$ l' f abendini) macilwo do pbt kodw anything about y,"y', y", or y'"; all we know is that r lga h left asjust to the right of the ,5 moment 3f. ; = .1 in.-lb.vdue The:deflectig, unknown to us),is"sketchedjingg.fSja 7 i[Jhe heigh 9 r en dadticfagbejesa%gg
'SiEp[{hjab dis'tiibutio [cause l blljlinggg@,toi y9rses , y' h'a' o tiiero (tising tho derivan " '
.;y
' ~
y, exp fssiks ofja'gfD4 in5 p . I G$ tF cf the is todotted lino any'abovo Aff.is elative equal;to;tlie.depthTo'flthe full:liiiojt'P$twoi
. ' ^' '
TCM A -@ 0d g,7 (df,t P if this bo' truo 'for location',of'3fe[r' curves must bo of the same shape, merely; horizontally displayed with: f, l@P,Ttli'en'tho . gyMgesg$!TQ g.4.y, Wg)'q or
's respect to each other. By pulling Aff farther 3way from1P.it carrios.its' ' pe^ n. g')b,ly gv "rM: j'^ 4.* ..a k,. -I full-lino curvo.with it, while tho'. dotted lin jat'pt'fP.jTo[fini'sh>thD a q y ' 0} 3 g $;D y M[A:ccf L R' argument we'say then ilta't'thd ddfldhiohl dtYofojidriome'ni,'3fe'itl2/jh yk*h!J a %
t' h; 'only t-Jfos'F.(#x)/k,'which in that'P has been replaced'.bS$ e s diffe[,hich$alc5[nidiffer'ence LiMik;@royfqu#"dQrEesioi5fPgfj(f i
#i)M M k h $tYrfv h 3rwy .
Fiq1lg ff %p itor g,D bothI : M = '4t - B : i.MMP# . isderived by differentiatioa? WW$'@ > .M' w.yD,tp.1 p Hfi!e * - ! o and C t= ,-A = po/2k. Then with . P and AI.are unity. ' '.'o P ' t U 6f !IIIIIIII1II Thus the first'of '5 (the EqsT(90)lMQ jolving 'theso,'wo find @ or{Dg't " solution.tb notation of Eq ^ (88)7" lk; j " r u . .
% - 7 J$ gfgQg.[g/g g (,' % W-h h i :
t.{ FTi2h h)h l N/8d [ b h IkE t F * '~ jbWg M g W"j ,4 4 1 @9d ' :2 y : @N MK4Q venection 2 J r. i (M 3 G Naf. h s m, ! e d
.Yh
,p .g @j "I l 6 t,i p f=E -qx) M. 1 .: '
*M . .g a,,
' eM.4c JMr ... ..,,.. .
,jr's ,n I ch puslie'd o q h ben n- .'
""#h $p p
^ ME' pf ~(pV[W ,momentiand:p icar- orce, diagrams,tlien aro two, and three steps down
~F h' . . hh[.T; gd Fijs.388)[rcepe'ctNely, o ex(example is',an inf{igitq beam aded sinusoi.
aish'ohin th'o figure.' ,{ , dally,.. , Fro. 09. Infinite rail, loaded with[ coins' taint loa" din 5 Pe.on itsjght $1f ong _ Now we proceed to a foyexaNI ibh N thNe$ beck 15a[d kah [f gh Mi
]y g.gt2yjQ 1 Q rte.w.- h'M 4-_ e.
the rnii is not zero. virsti,,,3.rfga,3orfofti,rof1.ggro9)corry:aconc gyya p3 . dM w s @a k.$$wa s .a u. $ Wh 4MMy W M %W mm W @:q,41,y# ~ . ~ . ~ M. L ,. 4 - .f.4
. x. w .
- iir , t dO" ja".
i . ,i l l
.kYI* j2'g4g:'i '
.V ',i'
' U" V. .
s e, o. .e asee r r i
. esh ac y, h 3 .,yJ m nl j dsut l i it '
5 nml e
.' en o pg.h' na uaf l i u r ,
s a w' eae d r ~. -. i ms s t. ce aco n g o 'W 1 ,
~ __
aFu t wb a > c ea; ' ,
) gb s c ' es n h' :1 t ee h $.A A['3b d
e e ah x c f '- 6 ( mllio .d o ._
%:- g _d a n t
e y' l d f c ei b ht a e et o d w t t h o gd o rpa t" noam s mr a c _ nbmMW' n ge a _ idl rD e I , i l ial o eP I 1"
- .kl ly'
.la!'
heF t h hg.- u 'T h'a
,~ .
l asisah oihs a A,M. dnr# l_W 9 ,, , lulf l c e a d
,~
..t a t icre ahtW4, f
Ni?eng d v o t -ojh ; ' T 'r r ah t c l' t ul od hih'0 ytF< t a1
* ;/
h.i o s up n-i t i-g a no anb> t
'ot,- -; 'TT $r ,
.Err r r
t n f on w a s mml n eah otht r re o o m tg B' a e n
- q 2_l, bin'so h ed a
b o[y 5 n. , l pt o g* l - Ie ul a r c el a u Oia t wrs sib 1 kho e e nh
- T gf aneo t rf c n n
$ a 'g l ff Ti T. = -
t t a ' ' eh ngw i s yet 2,F' n a c u]- Mhrrr ; y:..o c r s T =
'io va yy$ i tJ, a
e .I5,'h I Ml c a o u'imN t r i Y 4
- h. f n ' f i
d2 ga f oi e m ph f mea dd t t u SMr k h E e vh' l - soh e t
'n e
D. .d adu i r g d b'df@.~.,.oa/ms. e, y . s *c i sl o g.,d p r e c t r'y nly df i
,a r
r h r 4h eb c . o d i, od%k.I4E% o a H' t# i t 1 t&
'Epdop'oO rl t 1
o M- dr o T " 4uende. 3 ,ch a T I. N-
'nl h i
t e t7e o h ) ,s /
/
- , hi2mUWai t a!
0 eyp Qigkjt%DT r e a r 1 d) osn. 2i Id: it nli 3 3,~5 t h ,a 1 f e4 J* , 6ug4 t r t a aI f I Fggf rt a l7 '- f i F 4gk t r y(h gc 'n 'e. cI o g"teM " ig'd 25 N..'e i j1 9 af 7 n g% . Nh' hwaF n a o) u e [Mr f T g'h ie. g c oj5a t f n'd e v i iS dv d
. t I
nfl,h@F, y E t i 1'e g e iM p pwi % c 0 E.n t
,~isf ' . c
- /
l fe a drfai [t d >)l e1 e n gd g;(,h,t gwh(gdegn.
, s .
f H ,. o l yt a P= i
,e h Te b,Ag;a,[,)2 o . ! I cf b r 2f{h g6
. y.o 10 m 4T?V Font' 1
l n t i T l p x. t r c if a gp v a me(i
- t Y . d a t,emhD J: c st a y s#
p e
' n'ne pl e;a h nan ml #TT %Mv @[P Ntpda h T'tl r of c
e d Eo [fa u8
#mgIm nhx& ri n }
o$)3der,'qd. g-
%Wm r r o 3. h e f g
.]r Snt(gm0rh 8,l al i pr j'e rv e'uls "'i
- Y Y )$,i1 t ai- F' 1 a M.l u g)1 ee sd gn gnini7htc 5 ,
hYt Mhe 2,E'ih = h- $0 n s (s at Uidii r y3 wlis t,d l {'t [; b(! ur} .j 4
- e rl y j ' ift a a .s e ffamP e .
e ao y
.gdp)( g ', 4 pw.. M ,
p, fpn,f i na rg,soa c,r eg s u gk 1 h 3A 1 1nnt y erEtn l e qh n yngm i s g 'd f.~ t t i 3WTjoi 5l abel te t r i
. o i r, e( % 'i g +
- fa cabv.geVh%
yh
.l
. t w ee v i.
o'
* . NU .Ifi..hI- a -o4U tl qg
.h er hd "hcf g g,p. r
- </. {D** 2 u i v dn 3
0gcnewg 1 rn a o ee t a n thit t
]shWP a y2,M'
'c s fi nI o i ggg,;g- r ns.W~'
f 4 alfeid ri e - g'I fh, ne T. fe -e pmfo. M eD """ t i er j cd e hao e t t
, f L Pi _
- a_
r 'I , t . I fft,.[ ff r
;'s t l '
;l. _
d g e 0 }1' ed fME4 " : 4 1 5 kp' )(,ahiu d
$ ( t.'e 'e sT' ir. h i t
( Mh:si pt 4 a t
'n' _
cf %dtie#e h' . - 9 s r t c s e a a c n dWW'a.Z9W e eC es e nii ee n c. i y^d e h{q.l awo ina r' a,'& . N t l ol t
,. '~ , %.
h pdi6b n)
.lt. ist l f c
. I po da s n iWEh i
m 6 d e p f. h 7 rt Q+,hes nh c e T. i. r. ' g ;x.lsa ( 1r :ni 9 l o1 o t t. aor t M hh s 'o t eh d oen PV.:.4..'1 . nwei t (. ( t. , ei s I x 2Thi m i { g :#am i uh i Mhb ff*M s te h T.,IN;h " h'e os g#e ei h.V
' t e s e t n N pi n) ui cgpa T,,h ,y +<,,., r' nn l u2 t n's h'2 o s(h hsslsbtN'Mho@*')jaf,l Xggi i
f I s t Q *M n n. /e be trh hM1:d Nih lwIi t c e d,
,e p inc
.ait l I _
N O 9.ai0 dg0 . . 'c ei sdu u J e$p's f u t e g a d. v ) q e os b i no uf or m n wdp.M a<ml h' }d T@r i ld f
- l I 3 os1 -
e Tp l s up t
*, i at s cnu z- ei 2 'o'lyd N
mle '. l A,iag^ i i e' h['l e os$dclz a m (6 a v9 'e l i f a e ye v p f.2. I d$,r t b, i nmd d'Ef c i g
- k.g. 'c 6 ,
Duh iF - - r s e f fTt c i, V t aoa e ' e 'e l 1 t . in; i at t l fN' hA'.ih n
,s.
l -
,e f
8 o l , U 1 al' r n s *L /r % h beow
- 3. rPp't f s thlE'e Sr/alb b 'e @
. i -
gg I e g a Q i l p(ei n a~.M Mbmm nin odn@nt s e u "t, m ,d D *h. .U J'
')
l g c ,1 ii i 's(/. o 'o i
'e r_ (
&m.n;y.f(g;Q F h' oue , e ' l t n a d ji .t4 i nY t h # i'
, m. E a gEp l
I tg t ns r u n p !s ' Gh Ml aei o .~n i f nt a i C t.e bil h s inh . (' et J, /2kff$sbitaig r y i i l
's o Qp'h n h gn on l I /
wv,. n)ng4.,i tWh%d'o h ; e a T. esMa se s ol i yih x mo hya s ( s
' l S l i :
at 1 "o t tf t ne e - )da o
$ os Mpm na y, hi ev,i;.E st (r d nfn 6f l O u a'v t da 'o i v 4.h' ,-
'( i o s s t s n.
- e. -
l [tdl c ii 'c ol c o ei t e E w y: j=.i g
- ki , a i r s. t il l' t . } s il i
e d oi nh' u sMMd a,MCMC e w h6 hl$vl t. eiG 1b mE1
$e " e bm's. M'i(8 5 ,i
/.
l ea s 1 ) . m
~N {4 Nl g sTq a e ~
hnl t if t i e mw e s r ph e4 oai et 'tc 'h w.d.f fl. fef
. ma d!7 h p ;I 4 J 0dtn a cp 4 of s e r'E 2
k n w wic'i n a
- 1. ) at a,nrech ns { '. yo1
- p. nci a pa s 9+x W betdhi t hsihl gd mh
.S, ai I q
'.. J 1 I r al s0p n' = e ou t
h pt l p t E hM - y P !d v a (. u
;qt 'm tdipJtaiat s hf nt e aio;y e(i aI c rI l
e ei m l NEAB df. . al ein "y l h)i o l yeEi c l a s r oc0tl pet e e pu m nh u . vh aiwt o 'aB a tx a t n I M.$.3 l a n yb u eu i ti n r 's s a 58 " i 4,y'
. M d e ih b e b e)m!C. - s nst r elr o oit s st eeAo cl,n/w r n mh '
DN )g' s o eec r( . ' . g( hl feyn' 2 d ahe h e hv :, l
'h ae su C n. b h7 st e
% nt u thi r q n n vn i
t 8 e lr asuoh et t fdt o =lowt
..b eamnt t
i knoEd g ;M
- k. - is a,i gca e n
,( > o emtn's i"
A e e. ha )<
,t h N. (h n A6
)
r s* e . q
=
npo od a oVf'o a e sd ,a vuas cra n t i . i aE h fy QW. amqu egort ,t t r l i. g ep ph t u
.$h- h. ('n's ae~i V r 0oo n s e r u ,. " uiar
- af l .
i t
^( .
0 iir ,o wrn ne v 0' wix 'e o ,fo es s wg t s r }N 1 tcci t e e s e e r e oah p2 r 1 gu t l t e b
..- S .ee ru c m nre wt t w= n m 't io r ass L:,., . whuut i
l in u ofl eeo l ot t i s r n l l N paao f pb S
> r i eyt l
- t. f- >
FddF St p h C' aoowb rI at s e {c . <
'7
$. ~
m
,_- U
. (f f, h'k ((b,U i' ,
}p , ' $ , , ' ;i' ~
a,l;g i
!I* ffj' j'!' i' j} .. . ;jhy
.,i o
. ik , .
$QShhfidNFINITEtBEAh!S&M I? 155
.:h DflAMQ ELASTlq4Q(JNDATION th H >
7@ i m 0 %j54*k
- A..i YW,Whjw;.?; SQamchiixprmdin.m&
t
. :& k kNext.3')f.
%we The y My1Q e s s.a: t solve mali the caso of 6itm bf all semi-infinito Fig.f.104:~the th'eso terms beam withoc'e an en f amplittides accoidmg16 Eq/.
& . ' ' lIn the middle of a field; hence the maxirsuin' deflectio'n isi i . . 4 soment M.D IIero the end cond;tions are .. .
g k . g.g4) g. . h yp EIy",', = 0 ; ,, n_
' , W. "
2h 2h r 21M(RW2/#l)*; (3@2/#l)' ., h tobtlN+ C.= kkf h E* y-li,Q 7 Q.',% ,y%3fl ,,i'n.Ui .h{ .O j fliich leads M
}} .
k h.2@ 0 A h I'- g,. p 7 ;13: This series is very rapidly convergent; 6 . ' '{ {. N . 1 ;Q.'h.1,4:(%,g- Q, q y b ,
. . . ' . Qexamplo, if #l y ir' the deflection is V 5 ..
',; y
~ '; I <
, ;? .
- !@i f ,
Tit..Eq.1(86), lank.carrylng 6tititheidifferentiations with Eqs. (88),'the [ #### #
, !!O,, h.. i; {.qQ%v....%.. .&d1. k' ganl .. k. r.O@rk result for thofsemi-infinite beam.icith end moment M. is
+-:
. ;y" . o' p mxt +..~ a 0.0001) [p;" > , "' .^ orn.
0.00314. 0.0004;-t,f . s W~h. - eX.> yk M y.Q. Jps'{Py 1(#z) 1 . s
. ce - 41 , . 1: m,c1 ' i.
y' [ f/@f .F'..?. M @;i W 11b te Wtm". B h h M. ..L !
$ $ 1 ;i '/dk
- g[$"f keh %hor
' De(lectio
,'! p.
b0ic en-f,, A
< 4 iM23. Se F 103rphich'eityh .d l de9p t@r4rihM S' >>/
*%):in.. 6
' M~ ~ iiht/ grallifwEich*lidli'o'nb
,bu,ig3 eE ath $y ., .,p Q.,
i'%g5load'P I ]fS.. at tlut ed - QQ, *. ' ."T.. r. EIS (03)
,m e'
),
momen/L h.scarriefa' don' ciiGathd M y" . Y. sheor; ,JCgge T 0 k 5 apphes { here p (8'l),fwith' dip eral.jsolition,7E as m all previous cass Qh 49eQqSheaniq.N.y%=M
?"h, EI
- .% fp* ' 'r )Oh,M f D,hn ;,t; ..% ( . ,
')::2#M.F (#x)'.1! a. , .. i :L k .-
e 1 f~ W Fro. 103. Semi-infinite beam WTho n h d.%%4 N : & . ~:" ' , with an end load P. The solution f.y i ,'conditionsfa't;tlio'left-hand ag'hese two'casf Figsl.103'and:1W,.orei:i e d at.34.4 gyppgp 4* pp;4 c'.g" ' ~ I ere illu trated as described by x l f . 4f , se theM M be dcrided from N ,,
~ !*
n
. , g EI - g<7 g hf.Ndi:g!.,, $
i, f4s.' (80) and (90) for the bothway infi-" h' a. / e t .de//ection # i
.h... d M $of p T ersai a's'will n'ow' be 4 Gwn Conside'rp' tliTvafinfinitdida'nijk [9 O P ' ' S With this substituted.Ninto the (te[expression.Wk.h.fb $ gges1 beam,7and vicej h(orce,'Q.
j,
]
['# g C.'=. ,
.P. simultaneously. The.bendmg 8/@e ,
f,if ,
~ ; ~e .t . Jg y y ; s, so that the general solution [ remembering Eq. (86)] is . g). i noment in the beani just to' the right of 7- ..ILN deloads, by Ega. (80) and (00),is i 'i~ Y J
ti g, ; , : I.'N. ( 2sP _, . .ho rMij i
;< V " T e ,{ cos #x =F7;,F.(#x)'<
u-2EPP , f,ffj pve i.-yy. gg moment 7' 44.'.2eiUp#gd%g F @sk, !f@@pdsgM.% .- Differentiating this three times'bye theleads rules of Eqs (88) TMlWL;4 f,M ;JN g.h g,q; # -
,g /J
- l. to the.res , _ ,
; for the semi-infinite beam tvilh end load P: , .,i.. s . . H f i ad th car force that placo, ,p)j: Fro.104. Illustrates Eqs. (03) for S ,U - -, ^
. j 3, .p . . .. the semi-infinite beam with an Deflectiongiy =,f
.s Mxk *((y h' ([-.M .+ 9# 8,
,,)M. j ; h* " **",5 3I*
,j m . l' . i . . J filgi k O
~'
Ed h. nyoyalue$ fWW.C.'M W t~ H-
'R y{t sp .and M4 so that the bending moment [
- t. *. W& .. oppa justthe '
SI PO
.m h' ") w!2 k @.II0 .N.k t.iF ' kin,thd.beanI'jtist'.'id tho9glit of the loads is zero, or P. -2#M..
If wo now
'i -
- p. ~ ?, g. .'M , [ Q ,.<
. Then'the sh' car forco {here is' S k (P./2) - (P./4) = P./4.
Moment - F #2)cp .. . at cif from the bothway, infinite beam the whole left half including the
.hMEM.Ely" = ke,(h,$k ' %';h7, . % cpnta.inin[t,h ,loadsf. pnd,M.L.yo havo,left a semi-infinite be, a h;y.g; g a shear
~
't:
hadelwith orce P.fj and;wn hout end moment;:.But that semi - Shear.3 ..[E7 ppm (' M T,M (#.yp' II Ui M NA
- f. )gA.m.#p/'y$.ay[hl$3-[p.w$g$js:n%..D.,M.$,..M.l.c',
a 3,;,4 .g . , ,7
. $ hM.,
m m' h, KO,'.y .;'y(r -C ~%4331;u
- g. .. . ..
' . ,;; fhghQ'.'.d.(Z.l.3.tdr.;,q.! {'
LL. : dN. '.u . t ..
;. . .L.
- ;...c21.h,h v . 2 - A i;n Mit. ..
m
l g ,
' BEAMS ON ELASTIC l FOUNDATION; f l * ('g p,.y> ! 157
. y.MdT)Syyl.INFl$1TE* BEAMS l
t
'.16G &
. . , p s : .. . .a '
infinite beam deflects just as tho\bothway infinito ono under the lona Metida p./k and an'upwaid deflection' caused by X of the shape shown [ P. and M. = -P./24 Multiply by 4, and sthe semi.-infinite beamjd pig.103.fThe resulting deflection'at the hinge.must be zero, so that Fig.103 under load P. has the same characteristics as,the: sum of31i %hs " fronhEqs.' )ptP beam of Fig. 94 under load 4Phand,the'be,amdofJFig'."06funder load $ pc M ri M i
%4f. SMii#1 +!.- w. ': .n; r 2
M. = -4P./2#, or .
;$,Y%<t%ip,ivl @;;[$ f . g gikg % yj Pi. o)T Sr . . ,-
y=4P.hPk4 F. I. f ,
' i' M '
fc a
"Shh because of the definition equations (88) for'the-F function.; These"sh the results (02), which were derived differently before.. _To find the results h i s _. h.- WS"F y = Pf [1 - F.(gx)]'
~-
[ ~ ~ T .- _ k. t03), we adjust P. and dl. so as to make the~ shear force S zem ' leg ~ P~ '" . , , . c M"4 . .N'""g_1 only a bendinr inument. Titis y Jeg to r Jar pp*an,e/eijis{d i ig hN, yMhdg[*~ % c .y'. 7-
. Nnw wo allti .
Im) tunt 10lr '}flio honm of Fig /91rdo flio opposite:dergFig frojn carries a .by llpli 1 ; TTTT TTT'TTTY""b Y.' V ' two ahear fotoos P/2 each,"one'oforf 'theeach 'sido .,iondy' load?_Therefore. w vi EY " - hg IidN; ' take a semi-infinite beam and load it;with"P./2.and'with a" moment'3/, simultaneously. Then we adjust the moment M." to such*n' value as tb h E #MN lskNEMM NNb.9+# ' #5-E' Mt ohm h make the end slopo zero, which gives us half th% bothway infinite beam:i 2a(P./2) P.[Eqs. 9 @* : 7 W E pd (02)] 4@,. t ka 73 % N .M
.F.[Eqs!(03)]"MD - @i%" @*
ga . y. . D ga ;,+ .-
~
y= l rwp ' k .. 7.g; p . 9. . i . . J :,. p W. q Ch j @f@ iyy ab.M g . ##. : y%y.i<!fg% , ' ..
~
. y' = - g p'k F' '~ T44: #'#'ig[by Eqs.,(88))),6
- g;'d ' j hac105.@ Deficetion diagram of a semi-infinite beam under uniform loading p. sup-d b a verticaHy immovable hinge at the lef t end.
Y At the origin both"MNdT l f"'*d h "" d**d' I ""d"U "I ""NgN F, and gnd F. niebe 1.00/.NfAlddMSd%)}Inksj '7
~
i y' must zero%.1IIencep .p'N N Nexample M Mwe as for t o end deflect.ion' t(Fig.106) of a semi-. r; . .: ., s another 0P.,_4f'j'A. k k
~.
,. .g g'[t c P
.y i
\..q} $ injinite beam;on'an' elastic foundation, loaded only by a lynding p. oncr a hretch I adjacent to the free end.: A solution of this probiere can be found
.,e 4#
' Then .y 1Qj [. '[J JQ..'MS[ '
The general. attack is as follows: hy using many of the p[revious infinhy.in bothjesults.4 y= k F. 2 .
. h 1Yi fNby 1 F. c s..tk'L 2 g w. w [z..; s (8.).,J. hh U Q' pponsi bea tgcKingt9 'chE;We find the deflection of directions.(Fig.
gerg~stret .
!I This is the result, Eqs. (80), kvo t oti to e **T g-loaded',byf[lIEload N.arbithr[ point [ oft beamjoutside the loaded stretch l) by using ,
.t As an example of the application of the'genciN. I$cYults7Els".kO2[and i % result Fig. 04gintegrat'edper.the stretch I m the manner of Fig. 05, .
? ', (03), to other cases of loading we take Fig.:105 'a' semi-i ' nite beam M IT
** W N"hin,."*"Yd P
g soluyongg,ca, a,d,[,p.& ,tead "dz ins of for tho two loads P of Fig. 95
'?
jected to a uniform loadirig'p.'allhioiig'itill$hgth{:f."
" '%sisphfe'dffree 3"h.bothweg mfini,te l$am_pendmg at, pomt,B, the edge of the loadmg moment M. a' its end. The " free" support *is#cNdude$to a .hing'e Uhich res'ulv ,
in a vertical reaction force"X only,Nith' zero'p'dYomen,dOFarTft!/ then cut'off the belim a41ie. edge of th'o loading and imposo on it F en a . right of this hinge the beam under tho'influbnce'of tlio steady load p wi!! 4that pomt a force and a moment equal and opposite to the two quantit,es i hat palculated. This leaves the. semi-infinite beam, loaded by p. over the deflect to an amount p /k, and if.the hingel'orceXwere absent, tho'ently . beam would go down parallel to itself byihis' distance pi/k.j The totf dretch I, by..-M. and;-- S., with zero moment and shear force at its
~
deflection then can be looked'iipon as the'ef position of4his cone I % Thus,aH ljoundary,congong saMod, ad & solubon M h - q
- p. . . . # pm g.,q pobleid'consas'talf m i - .' : &,
. ci..mt . .:m.qhd k &'- f k], ; Y::Q.P.?w; &.,.ff. MN,the Ilf'QQ superpos,ition of three case a - .
a :- :
- . * . .a .. . 2h .. ., s. , c.;) .;; h . _ s ..
NNk' :W h{ b pth0 , ,.^' 1810
- r. ,. j .~. g c: 7 ,p q f .& SEMI-!N,TINITS DEAMS m asAuss olpstss '
g..o. .nc.jr0UNDANOm g:;;.ylygyp' V:vM..h.us . 'u.,.. A.es,7, y9Ito semi-infinito boayn.uturer, loading -P.,Tak
- . .+
16 distance a, which >vas hehl coristant so far, can now bo mado variabh.,
' beam under loading p,1,
- the semi-infinito ono.under loading e-3I.g'r @hg. 4; i 4 'ed then the above expression la the deflection curvo of the portion of the
~ ..The mathematical steps of this, process nogfollow:);E.% ts.#g . 7Mbonding'-moment and shear-forco curves l
' tThe deflection at A duo.{o a~ load eleine'nf, sdfisbyity(80) . g!! Nib
; M to tholright i pf,the load.)h;csyect differpntia{g(gtr,h , J. to'of with the help o A
f: f~ .e q: p. tQf . y g;QQ 3 . , dy, = ;Fi[s(. z y)]S [ QP
- M 3 J .r. (F.(pa) -- F,[s(a + I)]l 3 w.W J q@4Mff sfEIy'/
w 4 = - -i j . 4, i .
. ji, g, w ' .,g ,.p,r,.,"g ..
- a. ,.
_ g t . . i . IIITTIT w t,..,hh..bh. ! .a q . f yh S hy - (sa) v- F.[s(a + !)]) ti e if t itic lieco o { l ' 'Mdr ~ --q@ %@ MV " J&:$dP!@N.Mk'f'OMU',{1 @{ /@U.% T M i.$ W PN F,(pl) e < b) ' - fTTTTWI1BW'A,,&.i&G .. g li'N?!!
~ l CWr ip f.qh ;4 ,, y 'e
( , y !; f
!ig, ; [ ' k'd '
{W l ,, : 4f p$4!$tcts;.e@.
,t Wikip d -q j .. ,
' y , J.
a 4
- tQd .g , j if r;i . g %y,aro 6hown I'd,Ti jl06c withi heir actual directions for small values
-g e
I d- ". ;f aplyTha solution to ourjroblem is the ' superposition of the casa, Fig.
'('
ggIIII M wp ' ..g$li@A
.106c, d,tand'e?Wd have nd analytical expression as 'yet for the curve.
i
- c. .,.l a y j ', '
fin 100c.6 the left'of point $ (the*above formula for yoonly holds to
. @.Qd*8N
- . ~
dl
"'" W h )g %a
~ -
*#t L;[.f.g A % g Aflection'curvo'as a^
nih'ol@{,The,only.fo Ault'and 'those of 'Eqs.302 anE(03)',
! 3
.g .
- m. M a %b. M $.e.. ~' im9; .9 G .'k. [K ? f,yx n p,{frh g n# 1
. .a m aS x s_
, y 4
- 5. .1 N .,p d
. m ,. pC...bp p g' 2h ai .23 d Vl .T n. i, if Q% iirl.'Y' 7 Q ,44M A * ,*
- Fro.106. Semi-infinite beam with constant' loading p. oyey a tinite. portion Lon! -
Wh inWIA :lSM;E4 &!ul A " m
?, The solution is a superposition ( ,w n .- qppg of the cases 6,'d, and i GN E,. 77 g
, . . 4g;,'r{(
6. (2j:.k.h; y F 6 % Q (-lF. r-QF.) = Pg [1 - F,(s!)] because point A is at distance a'f z'from th'e' loa'd.$The deflectioii@f . b.; y We see that this answer for the end deflection checks the two special cases of the bothway infinite beam due 'to the" total stretch I of loading is' $ ,
)
s g.9 : y " g i = 0 and p m +~ for which!the caso is simple.
)- y,= 24 PS . f.,.3 F'[#(a gjgi,V z)']
D Q i
]kh.JFinitelBeamsyrhe'. N N M.,% calculation ofcbeams of finite length on a ,
- 3, ,7 f,
-!:64;g;p 9e g j $ elastic is foundatioii:ia nFinoro'diflicult'.in'. principle than
- that of i g.
#:,g2Ej.).bONid)p
- h dr9nt'egration-constantsscantboldetermined from.th~6~four-boundary .' ,
ionditions,ttwo at'eacli endJJThis, IIowever, is simploj'in prifi'cip~ le"'i nly;" M 8 k//
?.Al
= 2k N f@68S!.hh
,W.W
. .. y '/fi. ?by
,f ,h. M N in actual practice the labor in'v'oly'ed'in cahying it out is very large. hfany have been' worked'out by latient calculators,'and their results have
,,W94 J i
By Eqs. (88) we have F = -(1/#)F$o'thit
. .N;%gh3&fQ. 4N# JM.'M !I titled." th assembled
'"f.k bvw ~ " and mb ttabulated iniaivery:useful boo Beams'on Elastic:Foun'dati8n.",'yWe shall discuss only a singlo y -
l pygsgdd < MUniversity
~
of Michigan Ppesl An'df.A,erbor,1946.if".y d f* dri.M'5.2k w : , F' " p2 dF.(#ril.c',7 rpaMv @w%~,pIq 2k J. - m > ts e 7 < .~ yf f], .-
. x'
*. id ed by...lhick Doyre.
.jEmphasi J.}4.n.hd'i ;.2.1:a c .hb li: 4II d i.b.is d d . h [ S hi i i. . . . . . . . . . . 2 ..'t
- - - - - - - - - - - - n. _ , , _
y " u.g.. w 3 ,- .g ' f ,:,:: p; h A ~ g.n.( . ,.jee-[. g q
~
[ 4,s.. - - NDa J[fN$w$[!h,. ff Q
.J q
M hl p' - NM- ~hh h hM I i . i I, gyAMSy
, 1Gl?
'BEABfS'ON3 E EUNNhTIQ . l Of w Lt.,o ._g Ofkg.
n . z.
- '1160.... @.;{ g.v.< . yv
' i x - , .
e[STif.x
- r. 9 .
;a-- Mw . .. n v i . .. .
We see that for a beam i 3( example here: that force P. in its center (Fig.107).q yt 7ltgg ' N.g.g of a beam e "of tohd length',l,' [sh6 load' d by:a conce'ntsY li Ta' relation'.which;is .' r flip
~
"tioj{0.9 hanti that foh beam longer *
.,[;I
- l. The deformation is symmetric 'a:Id foh th ki' on'dit1[idghis ar bea hl,igi% NThi's .g'onfirpp..hr* physical ?
! ; ey.;Vt1-o r M tti%lidr"' i bbd$ila e drt it'is"relatively'so stiff U . . ;.
5 ~"n P,, 9, k m oory l' [---y2 c V2 "g > ,4,ge z 40 N PEIy!'!; U . Oils n'ii'gid:podylan
.b*nd'if the beam'is sorocponding lon(that th6'cor$
infinito,bcam cf Fig.extremely 01 simp dwwhg .ik y= fE' 7 MO WO uld extend t of Small floctio then by cu 4 g Mhea forces'or'be}tmg nding moments the; mfinite ..ad ' i
.-..g,wos.rs.g.
- A. ;
. Mh. , ; n - -
"
- g'of'IEyr2% conse qu %p9 foFth'Efiiillo licam'as'well.lW ' T d ]
~
fer tiateN;( I) ti an{-f cations, {0 y f and r th61nterinMiME'adariY then we should refer to IIctenyi s.obliged book. _ llowever,to i;o-into in manyserious compli- '
. y - ggte.;substitutMhese%n. it.d ionsitimi Emiot.ical cases a good approximation is obtained by using either the stifT- '
[ N'"! PI.hy% i, loa
% j M g'I( . Q'Or;1h6; ' ng'pegm' cory'(#1 > 3)for by interpo-ial caso of loading 'py i -
hh%, %H@f % YH i( t.liEii {o the a l l
' h b{p 4 %,M' 'i" ,*
'^
b, 10 paper l llons, Ltd h1 [h6 hroboss many pages:o 2go . i ' g0 'g,..., @[h.h,h '8 M
' forever, and the answer for.th9 ,defectidnlys/gf(x)@ccupidsjtlircet lines of print in Ihtenyi's book:.too,large;and, cumbersome [even forkey . ;
4 L. 4
;3. .. h g[M ..?"1,Ii.I,N
,, d .. y. , . 5 4 '
h printing it here. We do reprint.My.th ?miic~li " " [ans, li ~M
, deflection at the center aini,attihggd!
-- .L h ) ;
Q% j!hbg (Qth d(L@ M y,ja;r ,t. d . % ;;d
~~
%!g,
, . .fM.; ',' Md Ww%g
~
.,p lje, , , ~
b,iG nn :_: iw d I a. .. < U x
?
- c
..a D f
[
- y. ..,"% $:'#!?@. #'d .
^ B 2P.B cos'(Bl/2)?cosNu,(#l/2)
' 41
'r?.N' ' k% ~ Q' %
\ W G.' g "No ;i& h L : '! Af ' g l
Mb f, .yddhn Q*f , . f _
.. . }4 j
. , e on pag g u i
pg gi ,"..n g,,. - 4%r 4 1 These results are well worth exammmg.Wm Eq.j86) , ing of #: itsoi, 146, and Figs. 94 to 100 we' remember:the physical mean AE. ng.y,re,94 p.s .I, loaded with P. in th;e center (Fig.107). l j m.108Z. Thifinite beanto.f totallengt h the reciprocal of a length, so that'whed'#l beBomes~as'large as'5.thd YM,Ng @;$'N#*de 7((e a "HNINiat' Na
,I'
) flection of the infinite beam hasep' e ter' d'dogio nothinkgTho'characli ow o wnd deflection to the central deflection. Q
. of the defleef. ion curve,' Fig.\197,Mdepenhpftli@ialuotofr#1gl fisu i
# . . values of #1 the beam is stiff andfgoe dgwrpilNh' foEla )i. forts ' 6] nqFli.9107,.,were I P6 O Mofis'et i ..l. to L~
apply p I. ' #1 it goes down like' Fig. '04hfo'r l ints(mediate
.off#Ethei ion i
yaInes;rigillillody; g eyW2 8 {mppose t C g, oa defic'c[de ,
'100). 'The reader should verify . ,
curve is between 04p.fdF# those two:cxtremes@Ir$Eh' idstheid 'k *"h'"@,! ,, Iht cosh'#Q=%'Thj IIN h"$nder..is.%1ar'gg read 'that #l < 1, so ;4 that nominator sinh #l = =, and in theyumerator. i
" should refresh his memory.'od.hf[plioli yh*ci ortfaliy' 'ee t'iopt e am,, seo ljegen I defjegion is 4P/ki,.while
, the in case 'ths bearn is so.; flexible that'#l :
other end yM
$ a
.O. di6es the end deflectiori becomes iEr'o and ee q 8, dt,u nately. carried'b'y the ground, sop.that th l ' . ' I' ~as they should.t Alsd, for 'tli6'kh{
deflections both becom'e'etiual'to cP.7ki,1 t e'ErT a'$ cen
- th'ey;s du!MF6i a';lMam , anyjcaseit , .
un{f[10 it amas,of the si ade'd, isgures, Fig.:1005 and c, must be equal. of f'mite length, between these extr'emEsf.ws hafe r I.dfMMUf 1 .In.l case l the load in! Fig.100jwere applied at the one-quarter-!cngth u .se .. -4Q,{!%! y..e .14 cos (Bl/2);eos$ (# QL' . e might resolve',that Idad P statically into a component P/2 in . d
~
'. point,jw'tsE
'cen and.sTompEnen i 2MhE'end4For the stiff beam the do-
# y..... $- l ' cosh #f "
J h
%jg Q.)$jh4 .w f,f r 7,; .@y$%. t Qp b-b casn.mv #( 4 3
.g .n $u am, w.e..awww.,. _ __
2fh.p
.u a u
- k. .. .x mmu~ ~ t. ~4s , .
- d as 1 jf,, ,,'
.I - .
dhp h$O TWORK "E't3fS%IY 8: fl 163
. [ khhhh
$, [floction curve would tlien be a suberp'ositiori;ofTif.100Livith'a stend/ h.gwhIcl isihi tInifo' ht .
f g deflection. It is not porrmasiblo,:to apply .thisjproceduro to'the floxible' br} tho'.(middle portion:of the long) longitudinal in order to deflect tho g ; beam, i.e., the deflecti0n curva an ~of sfloxib,5bcam;under.the loaq"
%id . (h q lyiin's.1 ' land henco'i 'tho.f' fou
]tf
;h l/4 is not the sum 'of the, curves forf/2 a,t' 'fo? 3a Eero. .Wh gy %' %fnbqmth. kk t b %'g .kgiftu'diiialito;be Snd!,(l flong,.'qwhich is. s i
- .+ m mp t.. , tilm Miig. 9 4 ,'. JFir's%v"o'calcula'tiip,.f,1f the EI of thelongitudinalis stiffness the same {$Ne%al[d as
, e .A
, in J - > M$ j that of the crosaboams, Wo obtain on Substitution into Eq. 8G (page 143) 1 yggggfgggggggggg,ggggJfgg, g a.b. p;;ii.g ,'
d .n .it :i n,:..!,;Q m ,; _- .
'h'"'
d
. i ( , .
( a ., f g g t hf b (, .*] A l'** ..(. W A Y . h *o.'.*'
.7.c .r m . ftp't -
$, ne.,h' MyN i I . h'%
$ {I p 4 ,t
?l ',k. .i ,/ b 4 q 5 ,
.fr9 f ,, -
r. 1 u & se , e m v %rrrw -- i iw ; jh M>I
,7 7 l y p g f $ k hj2[qb y, b.p hie . ,
]gg s t
, [ '
-r. I ,.
/
Fro.109. Dcam with end load (a); When'the beant to compara y stiff,i o t
'f
^ D a t.!
E [- ,j I.-
)
and deflects rigidly (6); when it la relatively. flexible,.it is.ylong' .and deflects like 'F kM U
- a " ' 5
'i.4;!
O. , A. "i)MEM khN '" g [' k.jsM,d jfg [@Mk.
- 25. Applications; Cylindrica,1] Shells QJisgri,clall happ g .
d.M' Q.Qg. t of tho theory of this chapter":wasI. radroad. trac cttial raths, e ,
/ y,gy - e ~ 1. ( .
,h(,f, ,. ,M ,[. , &:# c'#
63.1.beams S
' not continuously supported;itiests'o'n#cr6dstics sym'o(15 i[5.[apan't.%d l 10.
['fhh,[f Factory floor o 16 . by'h;,th~,h~aving 0)-f t area spac 6 the tio spacing is small with respecs to tho' general' aspect:of the curps Figs. 01 to 106, the series of finitd.t,io rea.ctior. ' load ( can ,e repla$o,,d (.the)h d.1 le thertsame spanforwith a single the long and fo 64-fhort, t,'contralbeams.bacKbono. 9A ,For this,' example wo ac i the continuou3 load.mg md.icated .m ,th:ese,
- 4. - 4 4
j figures.;; thout chan.. ..gmgiany. g .g h , ur longittid nil Itas a lengtli b' w. 4a, so
.,.~
;f ,ties thing is suchmuch.' that #c $ 0.2 From the theory the table wdl apply meeof'pa'go
. 140 we sc[t[a]ly'*yo'r(larger
'or,ohcasol'ller
.g w2Qso bd Mfy. that the bearit really is "long." Tho spacmgiMho^epacingjbot
- the theory is still applicable, although some errors must tilen be expected.' j spacing #of the cross' girders is #c =. 5.25c/a = 5.25/16 = 0.33. This is Another application for the general th'co'ry,7vhich was discovered soon k h h theory will still be roughly after its origmal denvation for radroad trackqs t'o. grid yorks l of beand, ) 'not as closely spaced as'ono might wis , but t espilicablo.:
which wo shall now illustrato by the exampld thow:nsinsFig?110.] A singIp concentrated load P is placed'in the.middlejof ?a"recEnfulfr" fact'ory' floor gW, J y.g. gbpf_g rb
.g -
, , . : .,g p}P5.25[a,. h'c # .2.62Pa'c
~
flWl 3.%~ $2k3 . d6*t2a ni 48EII 4 SRI
~
We want to know the central' deflection [and"alsYthVdi$tfib'utibri1of th[
' "bNA' -
load over the various crossbeams supporting the'cdnfr'al longitudinal.T'llA . "" " * * " " " "" set of crossbeams, spaced at distanco e apart forms im"" clastic foundationi *I** 146:
*f""**#
""U ni i ?"hh.t'ablo for the longitudinal. The central deflhtio'nkf a'singfo*Erofbearh b 'itselfy 3
. uader a load pic, by simplo eaml1 ry g g 4p, 5 g ., 36.9 j,g,7g , ,
0.00 't0.72 CO.51- 0.31 ~ 0.18 -0.07 m:f h M tt .
%.,W[ m M-
' . ^. p,ca* >3" 48sI% n,-EMcM, u%M '
, b "E O ""&yO100. '
'" d ** C'
. . 1 Then we can writo . .g 4 . .. i 0I k i,dh1<.d 4
- .npn48sI:y.et;πgnW:,.mi4D .
g :aa .; 2.62ea c , w'Mu; g&T"%=. . ,f{48EIyh:48 QQQ i '- '
.1 :s p 6 Wa*c 3: 1
.Mb h}f ,1fqsrQ[%gtX,yy g, * /' y }
- j. h.
- I $; .
g.a
. ..m.,
Q,.. .~...%.;
. k.~ m y.a p.._. :ggp ,..m. ,
os .2., - WG , ,Y,yg?N W N S_ _5$ N ,u:u:n:v n qr -. .
*(N"c ;' - ~'m r '* mme
- i. . ? ,. . ; : '
~
kT!IN SEP 141979 AT.TACHMEN,T D . Y; t
.3 rr r.t- .
Ath.6 v .iv. 7 . l . y e a e W geh . T b Na d . a-" \
';.4) %
. 4999 in'dustn.es. inc. '> & H STATUS , :. . h - 1~
? gge%menploto
.i f
1 O newjerseyO7094 l
. . . . :. W j
3l
~
,, ;; ),' -6Y6550 telex 144435 Septenher 6, 1979 ; <;. Pic_,,,,___', g gy,
$9 .A. k%)f.p ; '!PS I-12-0 3 8 4 ; n.t r y ,>.
p .n g t .. ' m--- - 1 g- --- J ,
, f3 [h'em f
b
. .cf
" s".Utilitics ser rices, Inc. .
,t j IBox 1002 ,"".
i '4* g Nose', Texas 76043 : ww.-- - -- l ;y . b ion: J.T. Merritt, Jr./R.T. MolantejtrN3
.f - ,' iir!.e;w 1 " ~~
a tot;: Texas Utilities Gen. Co. ,,,
' gI' I
[ J. I R.' Conanche Peak Stm. Elec. Sta. g-V . Units.1 & 2 P.O. No. CP0046 A.1 g. W f}-
,} ;
y .
~
.,, N f' '
Subject:
Sct f fness values for Snubbe2 Landf ?ctrutsl
%u..en
.; 3
, e*i 9
,e,
['? M:. lGtintlenen:
+t,
') . . . NPSI is transnr ti:..; v nues of stiffness for snubber and
(_s) k. ^ *Atrdt assemblie.s
'I,
1 f or pipe sizes up to 12 inches in diameter. the stiffness values for the ninimum and
) f Y.; .[7@";.'gedave .
i f ,nil,$adimur, leng ths 'of the assenblies . provide: Please note that these f f,.'i' ahtes do not incorporate the stif fness of th's structural i M ME ddhgel nembers which ray be involved in the complete pipe W 4 % pport.
, j'f . g& ..,,
M% i.'QY Wp' 'kere are any enections please do not hesitate to call f $f ?ti f ijQ**i $';,h,f.i'h,. fpH % b; d Very truly yours, ,
\
.J. , tf. e 2;
.Jk.
J, ., %g . 19 y 3. MPS INDUSTRIES, INC.
- c. sy .L iN -
/ ,
#b . 7-Verlon S,tiocre
[h fM rf [h, Project Manager FF.. G. COX ' -
- i :- m '
., C -
h (' s. J. tatfo. RAD -
^
$ ) ihb o.H.WAc h _ 1 k[a'- k . .y f"1 nents':
-(l) 2 paaes .
hold!N ilu_ [ g e4 L ' int ,p% 7 % - . - .. 7.6d.l _.,+mg ,; , i ,q inaqpog ' csn, 2L,2A .
,9 2_ :d:-491,. ,
y p U.Db W las, B&R/ 11,lA ' ' . .,'~ ': {,y'Q g1 h, . Q~ P t p.t Triard.tU Jr/,j TusI,oL,lL,2A lD# . . mq W x E.m 4 umyl6.b. ,.M b'
;;g 4d1611okayr.!.cshu'it,,lAr.
i y .-
- v
. w .
....y-d.lg[.
1 f b'mkMW
~
eg .gm$ AMug 7e4+ %fM hdaghg4M 9y+QAg$$h4 Q.
.,rg 4W a .jh@i_Wig!!$yyg%
Ei- 3
L
;%cem e4 ..' C O t
4.gvug .u,6.,,, .- .. , -
' *M ,
p.: , I
~
br10fMNF3//d>Jrmwss.r '/$/u.cs : , iti$RM&Mi ' ' dj p]$ki$NtEt4Y$s.M$?r~'~Jrmwea it.;
.e% _ _ ,,, _
f Q'"y'
.sn m.as h,. jSffs , , 1 jQQf ~^ MQ.}.}'
.. % ' 4,37 xm e gg g ,,g,,,,1Qhy. n
- %n y eN>n:.v.g.ro s. ;y
- c}aQwd b ', l*'ll NIO " $2 7 /*llX/O d{jme m a .
g + &c n ~a.s , 2.cs m ' gg.c i. u x,o r My
+p@y g. . .a . a i g .,a l Q;Q y.p
,f , ' /s. 7
+ 1.-
cn xioC gg. 7 i,3g xio e el.4
'} Qj
,. a
- k. s.o e x n' i u.us" 4 w in?
srW , sa.c gr b f S . . r:.w S'l ~ d /5*. 7 450 XiG g2 7 /. 72 gro
)'
k4 ,m .+pe- 3 s.
' #.o p
h.i - l
- 7. c xiorl, n.1 z.01x,o'1 ; i ..i5 %.
y
~t . m. A
? j !* h !* b N/d *
[eg f au L .. .
.r. . i . .
og .y a 6.h en m+. .h. /. 2 5 X/O' l /2]. </ i hf
, . ! t s W)gre .n &qpy?
5.$. N{ h . 15 S v ,. I-2'59xt0'.I SS S u um m .- ,
/biNtof.A.k
.m m . .'L fyp/biof 2
.'Nif a , n,.
n m, . ~ ;.l,ta , 3 33 y)p f, 22 7 ,
/. $'h. ., to
~
h$gg$a v w d
,,, ,},9} 's$.:.v. g
- - - *. 4' . . . ,, ,t h. : . 4 '., p -- .,it a --
r.s ,, v A...W
'lpNh,%.m~.
., f; ~ , i ? 4 03 hos' ;
)QU&,s,i gg,y , ,og4yjgy;
.o ;w mi.3
.g .
q
..*2 ,o %. . x g..~, .m.,,
gg.
~ .},'jh), m. , .
- 5' p' Ek . W /7 ']', j.93kg5 ll7o4 gfI.
m., h5yyff
.. n
,.. n )
4ps
. . t. . _
~
~224. ; .
- 9. c/ l0
#l l22. + h. y f; py' w u.z :.- v o l -
m a.Wyygg
. . , - 1
- i. n xio ' .
.%:.y 9 g . 9 an :
M$hedW. syyag ,
- 2. c. m., r :
ss.r !
. ,. m , M,w@wm E MM
/57 3 " X'E B2 7 /a S2 X/o I.
a% mm., v .g j 4h wJ. H ;!'i /8,7 3.39 xic i ,i y2. 9 s /. 4 2 xto Q.~
.=
QlL
.o.i.pp7 ag m, ,.
- 4
+ - _l r
.. . , w
....c hf % , ! ll*h .? J* ' / x /D D ' !!a9.
~
Ea 7.'f.<'O I '. ,
~
Ul
$M" G.[i Wp -
l, e - ey A.: .< b. ; d
.h;h.u'y,Lyf, .,l,mEZ*h
$. Tc' $10 7 !:2 Z- J'Z X/0 b '.m
. . ,?.'f M. L
[9 Yf fe ., f.ff ffh a . H.n mlw,w, .c 2 . ,, ,,, , ,.u. , ,.,, ,,, , .g y w.w
- w. ,
4u w v.u , bb.. .g i., 1
;, ) ,m... , . w. w
- 2. ;.
?
^
~. .
6/ /- w; z~~ ,r; m j. *. iNESEA es:sEua[v 'F- ng q ;_ _ _. i /v.f 5. B; :- % :-it ww w~; ' Zns. as:.: 1 :w ~' Sn.c.~usu
~
'rcomir. c4 cf 7m:rica & **
G 9=:. ; uae, . >> A:sc"?h' . .M 7;: Au. EMS /)! .,
& z.g.o gro Y N *( ':
- 2. (,9 xic 4 h!f.. w .i
.s. f,'.I. 43 .
m
>em.4
~
W.; p 45~ - g ;
;.n) 2
.3 ? {'.;.b -. jyi.
~
n 1 ,1 2 49 X/O 48 2. Se tro y 4
, g.g7xm V 45 p,3g gg : .
$ gt,*
. ph.4 4 n&adh g f.
w
.a./{5] . ..
3 . 2.G 7 X/Y 48 236 xto f
- f.*w'*f
' Q ' n fte f F Q' Y,t ' .4 Tyf@.f,s 9 33 xto 67 7 03xn wl
'. x , .'
.p,Qrmy %..Q
.:.( y
^'
l [
. . , . + -
f,3 Xfo g ,13.</o Y '
~ ,.4 >
k Y
?lh '{$.$,. /. 2/ xto 78 8. C9 XtoN $. ..,h.e e m.
f .
.. w..,,
w . .
# gp c ,
' ?p.u 240N0' W
,4 ?e
, J!!@@Q{b h$U b $ /. S7 X/o #
$ 4J-t ,N[g. .
.t m . ,3?,fh. L,.
4 5 8 3 Kio *f G7 S.45 X/o y
$ &[9M,- ,,.-
J g.,;< m.
,y' ,. .
.==* d'""
~ 'St+, .-M.
, jj 6 $,i4h(a)I':b f.. N'l' 7f : 5.+3 Kio JN . ; i
~
! -l l pg u ? l _;. u xi. = 9 ; i. o x.v' ~q ry4e ; m
'. . ,kh)Nf/1
,. + .+ . l j' 551' Y
- 47 '
4 5 7 X to l .
. . .['?..
e,g. A ,6 ,, l,* 2 3 e4 5 > :t...,f ( i - 78 l $l -
' '35 i,f. 4kS ;' '. *f ;* 5 .
/. ' S YIO l 75 7 8?X/O : &))W'*[8 g
,, n n.y . s ~ wake
- + .
. . yw . y p~.p , p g 2. =s xis n i,c.ox,o?g,,A a
-1 . .. ,
%,- w n, .
$ /, /$ kl0 l$ $ *
>j.',. .
m am
<..a.>.. %. .. y#
3 m.. q,l -
. ..n9 'th.~ n
.[
. , . . ~,
a
~ .c.. ,
. isa Oy p' [.M i .J gF t. - .b.
;i i 4 J.4 :' i:.g:. . .-
! 3:
7,
'< '? *
* ' .'l ?>i!,'j';/'i pf 'i O,h'2,QN*p
- t[rj
$y. )b
{
*.. ' .r' 49 y J. -
' 79p M Mg;ca-1cq 3^
9 Ypi. .i w,a(t,l4[' . laqc ;,-h@1
^
- p, : t..
. , f.f .Q:;g-q w.geg!
t 3,.- oe e .. N. a q b
.pw. m&u;..y
,h, E
s & w* G wgM f w
- m. 8 e k. h q We
> s., m >;(. i vw ~... M.fW%pnh(y%
W. S ,,f 4 p .s !.6 k *
'. p:t
;) dyw m.p ..: . . . ' . .c%mq 3,,pN ..., ,, .
, .w,g t4f.- h ... ' . '
I$
,[$$
%e.
.h ', ',I,rym[ M., !' . , tg } + ' ' "
[Td', h2 o$J'~
.r .% ; A T' ,.y, ..N.g4 y .
j t . . 4j r . e
' r,6 'a ,
. i ,4 I .'
.Ir -.a Jr. a *a i t.it. .r y!*. s' .9,.
. %c Q) ;g! { . i s ; 1'". . # 3t y rt$%+4
, & y m ,. . : Y J l $ .,' bi), . 3
. v.= .
T -
..:p w'4.',
n.h p, [ . !#?. o,, ,. UNITED STATES
* /d[/J~[fy .
2 o NUCLEAR REGULATORY COMMISSION i
,E> ,E WASHINGTON. D. C. 20555 A /P) , f
/ '
p h
-fy 1 -.
' October 14, 1983 ' -
Mrs. Juanita Ellis - b President, CASE - 1426 South Polk Street Dallas, TX 75224 In the Matter of Texas Utilities Generating Company, et al. - (Comanche Peak Steam Electric Station, UnTts T and 2) - Docket Nos. 50-445 and 50-446
Dear Mrs. Ellis:
k In response to your informal requests during the May 1983 hearing session, enclosed 9 1ease find a copy of: (1) calculations for support No. CC-1-028-039-S33R, performed by Dr. J.R.N. Rajan; and (2) the SIT checklist utilized in its independent design verification of 100 pipe , d 1; supports (pp. 54-58 of the SIT Report). ' Sincerel'y,; 7
, ( , 0
Ge ry S7Mizuno f Counsel for NRC Staff
Enclosure:
As stated a
~
9
.1 b
,g
.A av 41
(.Y t k[. r-1
': C C-/-O 28-039 .sgs x '
}m;a
^
.s 4<~/C .
'^ .W -
.?
m~d+2 imiy G (/S(r / . ). Q
'3 +
- =-.y%'= (. 3 .
3 Eh.2xdy.375,,2,( s} J : y = l' lJ, ,wp -
- s L .> N l ) . -,
5 ; * ~h l.l - PL jf E.T 2 r 2,)' 4 7:T PL -
= - '
M= 1.
'P I_ - - ,
) g (/j. , -L . 2 5"~ 3 7 (s / + f_' 4- - . - #-
/j,I r ,
f l. 4
- /.2r t'
] ; /, [ * ,N,, .c s t, j g .. .
/* I Ti-w{'~a .ln3 naifCy0y/j.a,~,s~.b ;
- - 000i*u M4 - Lp; r
, p[] gY %
y y. 2 25 x l2 2 )
- ' j 75- )
g#
~c. gn
'~~1
; ' 'A yf,,,
,!; ."m' . ' l 0D. ) '
- e ., f(e, ,, [D ,
o -
' c, . ,
gn
- 7,
. , g. ,, *
~ f' y' ?. Z s . - .
N, A CCEP7%Q CDA/ .T/ CERED
,/S, /
.~
h . l:.} ') ?4
r
. From ASME CODE ATTACHMENT F 1983 Edition APPENDIX XVII XV112461.2-XVII-2461.5 parts haded in direct shear, expressed in ksi of TABLE XVi!.2461.4-1 (J
3 actual shear stress area available (appl (cable to the total nominal bolt area in the shear planes in this EFFECTIVE SLIP COEFFICIENT VS SgRFACE CONDITION case), shall not exceed: .
.- l (a) For ferritic steels! , , swface condition '. Y, ,
j t
.c g - - .
' ' ; clean mill scale 0.25 I'* " O.625,
*. ~
Grit-blasted carbon , 0.41 3- - . and low alloy high . ,- stanoth stal (b) For austenitic steels: Grit 4 tasted heat o.25
- . s 3; 3 x .
,5 . ..:,;_ . wire bnahed scond;
- or blasted
,q ;'Ig.,- . . _ .;
* *._+
031 O . (2) Threadng Noi Ekluded from Shear Planes. The allowable shear s_ tress F,. in bolts and . ,,heaned, c' e,"yzinc8'"*
' o.45 threaded parts loaded in direct shear, expressed in silicate paint ksi of actual shear stress area available (applicable
^
W
- to the total bolt root arca in the shear planes in this %"
case), shall not exceed:
- (a) For ferritic steels: it experiences tension only. Friction typejoints shall be designed as given in XVII 2461.4.
d F,, = 0.62S, J' -3 . XVII-2461.4 Slip Resistance-Friction Type Joints. The maximum slip resistance to which a friction (b) For austenitic steels: type joint may be designed shall not exceed the value of P , calculated in the following equation: g F,, = 0.625, 3 P, = mnTsk, where P, = maximum slip resistance of thejoint,Ib m = number of shear planes per bolt XVII-2461.3 Combined Tensile and Shearing t' = number of bolts m 8hejoint Stec.:s.> Ts = initial clamping force per oolt, Ib (not to (a) Bearing Type Joints. Bolts subjected to com- exceed ll5,000A,, where A, is the tensile bined shear and tension shall be so proportioned stress area of the bolt, sq m.) that either the shearing or the tensile stress, kips / sq k, = sh,p coefficient for the particular surface in. of actual cross-sectional area, shall not exceed
, C ndition taken from Table X.Vil-2461.4-1
" the value derived from the cIlipse equation below Since the j.omt clamping force will be reduced by (O when the corresponding computed tensile or shear. any direct tension load on thejo,mt, the 7, value shall ing stress is substituted: be reduced by an equivalent amount before substi-tuting in the above equation. SA-307 and austenitic f,8 f,8 steel bolting shall not be used for friction typejoints.
,- ; + ,- ; = 1 The allowable tensile and shearing stress values XVII 2461.5 Bearing Stress. Allowable bearing shall be those' derived from the equations given in stress on projected area of bolts in bearing type XVII-2461.1 and XVII-2461.2. connnections shall be (b) Friction Type Joints. A bolt in a connection designed as a friction type joint is not subjected to ,* , E s 1.5S" 2d gl~h shear (provided the joint does not slip into bearing);
LJ 421
ighgg gy v ___ D L O ,:U R N AW hv...k.,,!
. .: ~ ; : eed > >9p.4r .
i' Oh E.4, fr ,%,q#' ~ ~ - e ; AMERICAN
;; ; :> ;... INSTITUTE s'4 ..%
OF STEEL.'1'Ul(SIRVCI.;:y ..y p m:jy,;n,
, Q . (
) -
p L 4
.. A -
i u. t g
'[
'( 33 1 y WL '
"8$1 y r ATTACHMENT G pdUA E !
i~H .- G 7 k.
~
I
- N s b N ~'
dl,f0!5$frD S {'hf i p1e r==,#e-a$$}4lgy$dO u ps~s a 1
. t
~: -"
. ' " :s]:, Q k.:. u . a 4.;;f. -)
5 e u
, ,a+ % . ,v :
~$ "
f - d..j . q~@m.+91 . Y
+
g- ' / ' R ;p iv. ,
- .: $Ch~g.pf;j rr s 4 p),d et. '
l
.., i ^
s Il ? E
- . e .a.a Oh'
~
w ,e%n f s, q498 1
.wew /e.,m
, ww f'3 ;nhl' .
, u up t S. id b @i7?2 ;r!$v.N'.Wi
-@s(% wfp W
.r M. ,y t. 4 O: .n.n; e~ .- . .ai:aJ:.,1
- - :, : > v - i 5 Y
- I f. -
'l' 'Y' N
- s '.k',
h, ,.iMW ;
.e . %g~ .
ts 2 M % ,i%3A'[#XiaMV : f u: fy;. g , w - *) w<1 J k $',$ c3, ..'+\ y$* f;;.p. 4:4..p
* {)p;;. G s.*kh G vg '.% >>
, i j s; *.cg[b ,
- c. .nushhM..h(C. ..%[da
-n.:. ~ u . o
.laG,';, . % & . :y:,
l
Structura Jetaisin ndustria Buidings JAMES M. FISilER The recerd AISC lecture series on " Light and lleavy In- h??J.f C * %'& f-M&'.' dustrial Buildings" 8 generated considerable discussion ] / concerning details and design assumptions relative to (1) steeljoist and joist girder systems and (2) column anchor I' ,C - - 4T- I bolts. These two topics, ahhough unrelated, were of major i (tj " concern to many engineers and fabricators in attendance. < This concern centered around the apparent lack of appli-l cation of structural engineering principles to designs and l details. The purpose of this paper is to [wint out design and , , detailing problem areas associated with these topics, to help ; V j ' N designers avoid structural problems in future designs. 1sgm.,e m. STEELJOIST ANDJOIST GIRDER SYSTEMS t-Fig.1. Typscal derast ofjoist andjoist garder at column Bottom Chord Extensions-Open-web steel joists are designed by the manufacturer as 1. terally supported simple beams under uniform loading. Using a joist in any other If a 40.m. deepj. .oist girder was used, the resulting force way or loading requires special consideration by both the . in the top and bottom chords of thejoist girder would be design engineer and joist supplier. One common example approximately 50 kips. The detail commonly used for this of this is t'o provide a bottom chord extension in order to Ic nstructi n is shown m Fig.1. If not designed and achieve rigid frame action for lateral stability. Although IFP'. detai led properly this connection may result (if the syst it is usually more economical to use the roof diaphragm ts loaded) m the followmg: system or X-bracing to carry the lateral loads to rigid walls, this cannot always be done. The designer then may resort 1. Buckling of the bottom chord of thejoist girder. to bottom chord joist extensions. 2. Shear failure of the bolts connecting thejoist-girder As an illustration of the magnitude of the forces which to the column, which in turn can result in a secondary are developed through the use of bottom chord extensions, failure of the joist seat resting on top of the joist examme the following situation. Assume that a pist girder girder. has been designed to support a total roof load of 45 psf. This loading consists of a 15-psf dead load and a 30-psf live load. It should be noted that 13.5 in. of %-in. weld would be If a 40-ft x 40-ft bay system was used and assuming the required to uansfer the top chord reaction into the column bottom chords welded to the columns after the applica. cap. In addition,13.5 in. of weld would be required to tion of all dead load, the resuhing lise load end moment in the transfer the bottom chord force into the stabilizer bar, plus joist girder would be .U = b wl: = % (30 X 40)(40): an additional 13.5 in. to adequately attach the stabilizer bar
= 16,000 lb-ft = 160 kip-ft. to the column.
A related problem relative to bottom thord extensions occurs with tilt up or precast wall systems. Wall crat ks w ill occur due to the continuity created by the detail shown in James M. Fssher is Vice Pressdent, Computers:ed Structural Yig. 2. The designer is encouraged not to extend the hotwm Dcaign , Inc., Milu, cuter, IPsscon sin. chords in these situations, if it is necessary to do so. then the This paper uus presented at rAc AISC National Engmccrsng resulting moments and forces must he supplied to both the Con /crence in Dallas, Texas, on May 1, Im l. joist manufacturer and the wall designer. THRD OUARTER / 1981
4 I.& facturer of the attual loading conditions. The designer must also < heck both the roof deck capacity and thejoists for the now drifted snow condition. Leads in excess of 120 psf have been V. P "'" known to occur.
,$ Standing Seam Roofs-Tha deselopment of the standing l
_. .Q.y.l seam roof was a major breakthrough in the design of metal
&*g,gd i
1 roof systems. The system was first introduced in the late w ots '
' l '60s and today most metal building manufacturers either l % offer it or plan to provide it in the near future. The differ-
- I ence between the standing seam roof and lap seam rooflies Y
g,g*4 -
-pr- in the mannerin which two panels arejoined to each other, m.~ a s w m .4 -
The seam between two panels is made in the field with a N W suws. < tool that makes a cold. formed weathertight joint. (Note: 4 "' some panels can be seamed without special tools.) Thejoint M.W EW uwens is made at the top of the panel. The standing seam roofis als unique in the manner in which it is attached to the Ag. 2. Detailar precast or tilt up wall , secondary structurals. The attachment is made with a clip concealed inside the seam. This clip secures the panel to the purlin orjoist, but allows the panel to move under thermal The designer should not create continuity by arbitrarily expansion or contraction. using bottom chord extensions. If this is done, the connec- The special characteristics of the standinz seam roof tions must be designed for the imposed loads, and the re- produce a roof that is superior to other exposed metal roof sulting forces given to the joist manufacturer and other systems. A continuous single skin membrane results after design professionals for proper joist and connection de- the seam is made, since through the roof fasteners base sign. been eliminated. The elevated seam and single skin member Stepped Elevations-The situation shown in Fig. 3 occurs Provides a watertight system. Due to the superiority of the standing seam roof, most manufacturers are wdhng to offer commonly in areas where stepped roof elevation conditions exist. It is insufficient to select ajoist based on an equivalent mnsiderably longer guarantees than those of fered on lap uniform load (uniform load producing the same maximum seam afs. bending moment as the actual loading) and a maximum end Several Potential des,gn i errors can occur when using shear condition. This procedure will not guarantee that the standing seam roof panels with joists. It should be recog-top chord of the joist is adequate for the higher localized zed by the designer that standing seam roofs has e very uniform load, or that the diagonals in thejoist are adequate. little mherent diaphragm strength or suffness and, there-Since the dIsigner does not have access to thejoist member f re,cannot be relied upon to resist lateral m plane forces sizes at the time of design, he must inform the joist manu. r i pr vide lateral stabihty to thejoist top chord. Joists are typically designed assuming full lateral support to the top chord but,if a standing seam roofis used, thejoist must
, be designed considering this lack oflateral suppert. If the (y' ~[lN N inadequate lateral support to the joist is called to the at.
tention of the joist manufacturer, he can provide the re-i N quired support by designir:g thejoist top chord based on the I N discrete bracing points provided by bridging spaced closer l -'~~l than for standard designs. I
' Because of the very light dead load associated with the 5 ? standing seam roof,it should also he noted that deflection criteria (I./240) usually controls the joist size. In addition, because of the light de.nlload, roof uplift uiteria must be carefully con <idered.
Crane leadrjoists have been used to support un-derhung crane systems. Iloweser, th- joist supplier cannot simply be given the loading due to the crane with reac tions assumed to be at panel points In practice, the underhung crane beam readion will not he resisted at panel points, but i will in all likelih(u! he resisted in a manner similar to that Mg. I Snowdrrff conditionfer roo/Inccioadt shown in Fig. 4. The bottom c hord of the joist must be n4 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCT;ON h- _ __ __
l 1 r ~ t 1 t f m f t T t r-- t _ .. 7 A* d _ __. ._
\ gPo j ,
- f. Mil { f t
.D w N ,
, As
' w h i ._.
Fig.4. llangingcraneload ,
~
a,s checked for cornbined bending and axial stress. In addition, the welds in the joist must be designed based on fatigue F,g.1 Concrete shear cone developmentfor anchor bolt considerations. A superior method would be to design a math head harness over the joist so the load is applied to the top chord.
. Inadequate development of the anchor bolts for ten.
FloorJoists-One of the most frequent problems associ- sion ated with Door joist construction is Door vibrations due to . Inadequate development of concrete reinforcing human impact. This problem is likely to occur on open Door steel systems when a 2%-in. thick slab of lightweight concrete . Improper selection of anchor bolt material is used on spans from 26 to 30 ft. Damping resulting from . Inadequate base plate thickness partitions, file cabinets, heavy furniture, etc., will signifi- . Poor placement of anchor bolts cantly reduce the problem. If open floor areas must be used, . Shear in anchor bolts increasing the mass by increasing the slab thickness is in . Fatigue general the most economical solution. A full treatment of Guidelines and suggestions for each of the above problems vibrations of steeljoist concrete slab Goors has been pub. lished by the Steel Joist Institute.2 are provided below. In addition to the comments below, valuable design information relative to anchor bolts is contained in the ACI Journal, August,1978.This infor-BO'ITOM CHORD BRIDGING mation will be published in Appendix B of th ACI Stan. Bottom shord bridging is extremely important to the dard Code Requirementsfor Nuclear Safety Related Con-structural performance of a steeljoist Door or roof system. crete Structures (ACIJ/9) in the near future. Bottom chord bridging serves to: Development of Anchor Bolts for Tension-Anchor
- 1. IIelp align thejoist during erection. bolts that are not quenched and tempered and are I in. or less in diameter may be hooked to increase their pull out
- 2. Brace the bottom chord for wind uplift require-resistance. Quenched and tempered anchor bolts greater ments.
than I in. can be threaded and embedded in the concrete
- 3. Laterally brace the joist diagonals (in combination with a nut and washer.
with the bottom chord). PCI research has shown that hooked anchor bolts fail Item 3 is often an unrecognized function. Since the di. t y straightening and pulling out of the concrete. This agonals of ajoist, joist girder,or truss are in effect individual failure is precipitated by a localized bearing failure on the columns, they must be laterally supported to prevent their hook. licaded anchors or threaded rods with nuts and buckling out-of-plane. Bottom chord bridging in combi. washers fail by a concrete cone mode. See Fig. 5. nation with the horizontal 0exural capacity of the bottom The pullout capacity of a hooked amhor hoh or a har thord must provide the required lateral strength and embedded in the concrete with a nut and washer can be uit fness. calculated as follows:
- 1. Obtain the anchor bolt tensile capacity from AISC COLUMN ANCllOR BOLTS allowable stresses. See Table 1.5.2.1 of the AISC Improper design and detailing of anchor bolts and column .N!anual.'
base plates have caused numerous structural problems in 2. Obtain the concrete pullout value from Sect. 5.17 of ie.dustrial buildings. Problems relative to design and de- the PCI design handbook 4 for headed an hors, or check bond and bearing for booked anchor bolts. tailing include: W. THIRD OUARTER / 1981 m -
o Example-Determine the allowable pu!!out value of a For the har with washer and nut:
% in. dia. A307 anchor boli embedded 12 in. in 3000 psi A., = d (12) x (12 + 3) = 799,72 in.2 concrete. Assume (a) that the anchor bolt has a 4-m. hook; Ultimate concrete capacity then (b) that in lieu of the hook a threaded rod with a nut = 0.85 (799.72)(4 x 1.0 x y 3000) = 148.9 kips and washer is used.
W rking capacity = 148.9/1.7* = 87.6 kips Solution (a)-ilook: Use bolt tensile capacity of 8.8 kips. From the AISC Specification, Table 1.5.2.1: lt should be noted that the calculation shown abose was F, = 20 ksi based on an isolated anchor boh for which the failur: cone Tens.le i capacity T = F,A = 20 X 0.44 = 8.8 k.ips shown in Fig. 5 does not overlap with adjacent failure cones. The PCI handbook also contains equations and criteria for From the PCI Design llandbook: cluster situatiot. Bond strength = rd/1(250) Development of Reinforcing Bars-In addition to where d = bar diameter making sure that the anchor bolt is sufficiently anchored L = embedment length in the concrete, the steel reinforcing in the foundation 250 = ultimate bond strength in psi (non oily system must be positioned and detailed to provide a suitable - steel) development length. See Fig. 6. The reinforcing must be Bond strength = r(3/4)(12)(250) = 7,070 lbs developed in accordance with ACI (318-77) requirements. Since anchor bolts are often oily due to thread cutting, These requirements may dictate that the bars be hooked the designer may wish to neglect the plain bond capacity. or that the anchor bolts be provided in lengths longer than Further, pretensioned high strength anchor bolts should calculated above, so that the rebars can indeed be developed. not be designed on the assumption of transfer of pretension Tabulated in the PCI design handbook on pages 8-19 and by bond. Progressive loss of bond will result in transfer of 8-20 are development lengths for # 3 to # 11 bars in 3000, the tensile force to the head and a consequent reduction of 4000, and 5000 psi concrete. If the reinforcing ba'r is not pretension. positioned against the anchor bolt, then the development length la should be measured from the intersection of the Assuming uniform bearing on the hook, hook strength rebar and the assumed conical failure surface,
= cf,'dL' where d = 0.7 Selection of Anchor Bolt Material-Consult local fab.
f,' = concrete strength ricators for availability of materials. As a guide, use Table d = bar or hook diameter 1.C," Material for Anchor Bolts and Tie Rods." pg. 4 4 U = hook length of the Eighth Edition AISC Manual. Ilook strygth = (0.7) (3000) (3/4) (4) Base Plate Thickness 'lle design procedures illustrated
= 6,300 lbs in the section" Column Base Plates"in the Eighth Edition Total pullout capacity based on embedment = 13.37 kips
("I'5" b
- A multiplier of 1.3 times the loadfactor shown tmuld be con-Assuming a load factor of 1.7, allowable pullout capacity sistent with PCI recommendationsfor "sennrae"connec-tions.
= 7.86 kips Use allowable load = 7.86 kips 1 -gfyf=
Solution (b)-Nut and Washer Combination: Check pullout in plain concrete. From Sect. 5.13.2, PCI Ilandbook: y-e Ultimate concrete capacity = 4A,, (4 A g) g g f m , m, .
- - ~ - - c.
- o . '
where e = 0.85 l A, = area of an assumed failure surface T*#.I.""' )'- l l l' - N ! _ _~ , m For the case shown in Fig. 5: A, = diar (I,+ ds) lY *b l M' * " I, = embedment length (Fig. 5) 8-
! A d4= diameter of washer or head (Fig. 5)
A = 1.0 for normal weight concrete (PCI Section 5.6) F1e 6 Pier dcrad w, ENG2EERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION
r AISC hlanual may be followed. For small bee plates, the new methul ilbNrated in the hlanual can be used to obtain rcquired plaic thic kness, however, thinner base plates can - , be obtained using yield line theories, hietal building ' manufanuirrs have used yield line theories to establish base plate thicknesses with success for niany years. Placement of Anchor Bolts-There seems to be no guaranteed solution to the anchor bolt location problem. Since it can be assumed that anchor bolts will not be placed exactly as indicated on the drawings, overside holes in the hase plate are a must. The larger the anchor bolt, the larger
@ ' Q the mersize must be. The author's office has established a rule-of-thumb that the size of the hole in the base plate i should be approximately 1% times the anchor bolt diam- f. ,7 g , ,g,_,
cier. Shear in Anchor Bolts-The AISC Commentary states 2. Floor Slab:
" Shear at the base of a column resisted by bearing of the In many cases the condition shown in Fig. 8 exists. In column base details against the anchor bolts is seldom,if these cases calculations will show that many times the evrr, critical. Faen considering the lowest conceivable slip shear can easily be transferred from the column simply coefficient, the vertical load on a column is generally more by the bearing of the column against the floor slab. In than sufficient to result in the transfer of any likely amount some cases the shear must be transferred using hairpin of shear from column base to foundation by frictional re- bars or tie rc,ds. h!any prr.b! cms have cecurred when sistance, so that the anchor bolts usually experience only the hairpin bars are placed too low on the anchor bolts, tensile stress. as shown in Fig. 9a. The problem can be avoided as The above statement is true for most multistory build- shown in Fig. 9b.
ings; however, in industrial buildings uplift forces in con-ju'nction with shear loads may exist simultaneously, and
- 3. Thrust Bars ,
Thrust bars such as the one shown ,m Fig.10 are used the desic ncr must take proper measures to transfer these ,
~
m mdustrial buildings when shear forces become sig-shear forces. Several mechanisms exist for shear transfer; these wil! be discussed below: ,
- 1. Anchc,r Bolts:
The author does not recommend that more than two ' anchof bolts in a cluster be used to transfer the base. , shear unless all anchor bolts are " leaded in." The ra-j_ tionale behind this statement is that in alllikelihood only \t a two anchor bolts will ever be in bearing in a base plate connection. Shown in Fig. 7 is a base plate consisting C'1[(' _ l am~n of four 1-in. anchor bolts. Under normal conditions, only one of the anchor bolts will be in bearing as initially installed. Under the application of a shear load, the column will slip and rotate so that perhaps another .,
- ' ' * - ' * ' - * ~
anchor bolt could go into bearing. Due to the oversize holes, the anchor bolts may not be able to deform suf- _ esw ficiently so that all four bolts could be counted upon to carry the load. Anchor bolt strength in combined shear and tension ._cm
" 'f ' k v
h L' * ~ " "; ' "" will be wntrolled either by the bolt material in combined M shear and tension or by the concrete under combined .. .
;-d '
shear and tension. To check combined stresses in the bolt
~
N '-
^~
material, it is suggested that the AISC interaction equations be used. The PCI handbook contains proce-s 'N dures for determining the concrete strength. The steel '*M,0
. 4 '
'y / 'E CI a m os .,oe d h ere a m w designer should be extremely careful when working ""
with concrete strength equations, since they are always written in taltimate strength terms. Fug. L Transfer of shear throuch poor slab 87 l TK M OUARTER I 1981
e e
-- e- ,,.. .. _ a nificant. This method of shear transfer is positive and direct. The thrust bar should be fillet welded to the bottom of the base plate to develop its fr hending A " ' " o * * * strength. A design example is shown below c.uw, ~ ....
$'n'c
- N - P "' ' Civen:
NZ4"*"' ;e g ""yP"j . . Base plate detail in Fig.11, where: I .o g c== .
- i. . . 'h - \_~,,,,,, G=1 in.
. I; a -
- 1: -
- r. ** i =
- o = < ia
- V = 50 k,ps i
', 9" . c ** ' f,' = 3500 psi
. . . , b = 12 in. (length of thrust bar) m) Solution:
, Check bearing on plain concrete:
" * * ' '*""" From PCI handbook, p. 5-7:
6in e V. = (1.7V) = dC,(70A8)(s/d)1'3bd [*ssin ,i oa acis a where
,IANh D (1.7V) = factored shear = 1.7 X 50,000 lbs 7 ,7 f c = 0.70 II - ' - 8: C, r 1.0 (zero tension)
X = 1.0 (normal weight concrete) [ $ ' ' .] ".
' .LI.,, , ;)) ,' s=d/2
#* V. = 1.7 X 50,000
', "f . ',
I = 0.70 (1.0) (70) (V3,500) (1, 2) /3 (12)d Cb) d = 3.08 in. (say 3 in.) Fig. 9. Placement ofhairpin bars Compute thickness, assuming cantilever model: Mp(bar) = (1.7 V)(G + d/2)
= (1.7 X 50) (1 + 3/2) = 212.5 kip-in.
,,,,,,.7, g.l 2 Io'o's .'. <. ~ F3 = 36 = (212.5 X 4)/12 f..
a
~
{ ! M .'f E ,s t = 1.40 in.
# 4 Use 11/2 in. thick plate.
ll l S'ush Co6uMed / P L . A. N
~ V
. s.. i 1 >l q 3
_ _ .,..u .. _ ~ g[] ,
=-t
' ...'s~....
ra son,
, ~,==~- ~
'., 'L
'( * " "4 gsj n 5 - == . u x, s ,
Aacwon -- t U s%'$,,'s, a E%gg, w
~
.b . ,J,_ = s , d[ .._
.-- S,o,~r,~, a.',* - - -
Fig. ~10. Detad of thrtat bar' Fug.11. Dengn examp.'e M ENG.NEERiNG JOURNAL I WERICAN INSTITUTE OF STEEL CONSTRUCTION i
Compute fillet weld im size, D: the designer should specify a prrwedure for tensioning and
' 9 ' '"5P'C" "-
12'5 D= The designer should take into account prying action for 3 15(l.7)(21.0)(0.707)(12)d ',
tensile fatigue situations. A factor which must be considered 50 is the possibility of overload. A tensile overload can cause
, '21.0(0.707)(2)(12), '
1 yielding of the bolt and thus a partial or complete loss of the initial chmping force. Base plates for anchor bolts subject
= 0.608 in'. to cyclic fatigue iciading in tension should be conservatively Use bin fillet weld. designed to minimize or preclude prying action. See Guide to Design Criteriafor Boltedand Rivetedjoints,6 pp.266,
- 4. Friction:
. 267 and 279, and AISC Specification Section B3.
A method of providing shear resistance in the absence REFERENCES of gravity dead or live loads is to pretighten the anchor bolts and transfer the load by friction. Based on an initial 1. Fisher, James Af.and DonaldR. Burttner Light and liean preload load in the anchor bolts and a coefficient of Industria! Buildings Amencan /nstitutrofStecI Construction, friction of 0.4 to 0.6 between concrete and steel, an al- Chicago,Ill.,1979. Iowable shear load can be calculated. 2. Vibration of Steel Joist Concrete Slab Floors Technical A rough guide to estimate the torque required to Digest No. 5, St<<lfoist /nstitute, Arlmgton, l'a.,1974,
' tighten anchor bo'Its is as follows: 3. hianual of Steel Construction Eighth Edition, American institute of Stect Construction. Chtcago, Bl.,1980.
Torque = NPD 4. Design IIandbook-Precas: Prestressed Concrete Second where K es 0.2 for oily threads Editson, Precast Concrete Institute, Chicago, Ill.,1974. P = desired pretension in bolt 5. Uniform Building Code InternationalConferenceo/ Building Officials, It'hittier, Cal.,1979. D = diameter of bolt
- 6. Fisher.f. It'. andf. II. A. Struik Guide to Design Criteria Shown below is the calculation to tighten a 2-in. dia. for Bolted and Riveted joints John ll'dey & Sons,New 1'or4, A36 anchor bolt to F,/2 or 18,000 psi. 1974.
K er 0.2 BIBLIOGRAPHY P = 0.5 x 36000 x 3.14 = 56,520 lbs Anchor Bolts D = 2 in. Ardahardjo, R. and I.. Soltis Combined Shear and Tension on Grouted Base Details AISC Engmeering fournal, l'ol.16,
- N .1,1979, pp. 23-26.
Tor 9ue = er 1900lb ft Conard, Richard F. Tests of Grouted Anchor Bolts in Tension 12 and Shear ACIfournal, Sept.1969,pp. 725-728. Dehnding upon the steel erector, the engineer may Hawkins, N. Af., D. Afitchell, and C. It'. Roeder hioment find that, rather thar. specifying a torque for the in- Resisting Connections for hiixed Construction AISC Engi. stallaiion of large anchor bolts, the erector may only neering fournal, Vol.17, No.1,1960, pp.1-10.
. require the desired bolt load. hiany steel crectors prefer Eharod, U.f. Anchor Bolt Design for Shear and Tension A/SC to tension heavy anchor bolts by using a hydraulicjack. Engmetring feurnal, l'ol.17, No.1,1980, pp. 22-23.
In this way the preloar can be directly applied to the 8 AlcAfackin, P.J., R. G. Slutter andf. It'. Fisher IIcaded Steel bolt. Anchors Under Combined Loading A/SC Engmeering fournal, Vol.10, No. 2,1973, pp.13-52. Fatigue-in situations where the anchor bolts are Ollgaard,J. G., R. G. Slutter and/ ll'. Fisher Shear Strength subjected to fatigue loading in tension, special precautions of Stud Connectors in Lightweight and Normal Weight must be taken. Assured pretension in the bolts is important; Concrete A/SC Engineermg fournal, Vol. S, No. 2,1971, - however, the usual procedures for tensioning bolts in pp. 55-64. ! steel-to-steel joints are inapplicable or highly unreliable Shoup, 7'. E. and R. C. Singleton lleaded Concrete Anchors in anchor bolt applications. This is especially true of the Cll "' 8'f;' 9ylatP (n hor Bolts with Simple
, f turn-of nut procedure. The author suggests if net tensile Assumptions Cial Engmeenne, Apnt,1966,p. 03.
stresses are kept to low levels (6-8 ksi), fatigue problems should not occur, flowever, if the anchor bolts are not B j tightened uniformly, then the assumed equality of loading DI ] I'. and E. F.Sans/q Column Base Plates with Axial l among the bolts may not be true and fatigue problems can leads and hioments ASCEfournalof the Strwtural Diosron, result. In fatigue situations, the designer should specify that l'ol.106, ST 11, Nov.1930, rp. 2167-2181. all of the anchor bolts be pretensioned to at least a magni- Afaitra,N. Graphical Aid for Design of Base Plate Subjected tude which exceeds the applied design loading, and use of to hioment A/SC Engmeenng fournal, l'al.15, No. 2,197S, a detail which precludes reliance on natural bond. Further, pp.50-53. m TrtRD OVARTER / 1981 L !
. _ . . . _ . . _ 3, ,3g g Guide to Design Criteria for Bolted and Riveted Joints
.~ .
.f John W.'f Fisher
. John H. A. Struik Fritz Engineering 1. abo.ratory
' 1 Lehigh Uniwrsity Bethlehem, Pennsylunta I
e i 1 l l 1 A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY & SONS, New York e Lont' -- 5 y J:r, .
- Toronto t
4 I ! l
s I'
% I
~ ~ ' ' ' - - -
~
s e
- . l 4
Introduction l I . , n I g e , j ' .JT . 7..-f;,g* f 4 150 .
, . .b -
'q. = '"
1 100 F - -- A502 grade 2 A400 Wts _ I rivets g
- , A325 tWts 50
[
~ASC2 srese 1-'
rivets f 1 I t f e
.00 .16 .24 .37 strain (inlin.)
T'iZ7'.l7.yg,*, Fig. l.3. Coupon stress-strain relationships for different fastener materials. The ASTM A307 low carbon steel fastener is crimarily used in light structures. subiected to static loads. The high-s:rength A325 and A490 bolts are heavy hex structural bolts, used with heavy hex nuts (see Fig.1.1). A307, bolts .are .made of low carbon steel with mechanical properties as designated.,byj ASTM LA307. A325 bolts are made by heat-treating, quenching, and tempering medium carbon steel. Two different strength levels are specified depending on the size of the bolts (see Fig.1.2).'* The quenched and tempered alloy steel bolt, designated as A490 bolt, has higher mechanical properties as compared to the A325 high-strength bolt. It was (EHPHIStS ADDED)*
. . .. , , , . i
.0 _
j in. A490 bolt
.5 4m _- -
i
. _ j in. A325 bolt
.M -
,i
] F n. A307 bott 20 Grip length 3 j (jin. thread length included) 0
' ' ' ' ' ' ' ' I
! 0.05 0.10 0.15 0.20
! ' Elongation (in.)
i Fig.I.4. Comparison of bolt, types (direct tension). ', I g l T
- Emphasis added by Jack Doyle.
. f 1
i ! I
.}}