ML20211Q873

From kanterella
Jump to navigation Jump to search
Seismic Analysis of South Texas Plant High Head Safety Injection Pumps, Summary Rept
ML20211Q873
Person / Time
Site: South Texas  
Issue date: 02/19/1987
From: Dudiak J, Kamenic M
WESTINGHOUSE ELECTRIC COMPANY, DIV OF CBS CORP.
To:
Shared Package
ML20211Q857 List:
References
ST-HL-AE-1937, NUDOCS 8703030207
Download: ML20211Q873 (31)
v • d • e


Text

,

7_____._...

ATTACHMENT oLi e' f ST.HL AE /M 'l PAGE / OF JD

SUMMARY

REPORT Seismic Analysis of the South Texas Plant High Head Safety Injection Pumps J

Westinghouse Electric Corporation Generation Systems Technology Division i

I

.h.

f[lf[U Prepared by:

(

J. G. Dudiak Reviewed by:

ad.[pd g/f9/gp N. E. Kamenic i

f, 3030207 nyopp3 ADOCK 000004?0 4

Vim R-1

ATTACHMFS C ST HL-AE n e

PAGE A OF TABLE OF CONTENTS Section Description Page 1.0 Abstract 2

2.0 Design Criteria 2

3.0 Design Loads 2

3.1 Seismic Loads 3

3.2 Nozzle Loads 3

3.3 Deadweight 3

3.4 Operating Loads 3

4.0 Methods of Analysis 6

5.0 Modeling Techniques 6

6.0 Sumary of Results 10 6.1 Natural Frequencies 10 6.2 Foundation Loads 10 6.3 Stresses 11 6.4 Deflections 11 6.5 Bearing Loads 11 l

7.0 Conclusions 13 1

8.0 Appendix A: Description and Validation 14 of Computer Programs i

l r

l I

r L

HI A

=

PAGEw OF3 D 1.0 ABSTRACT The South Texas Plant high head safety injection pumps are Pacific Pumps model 6x10-WYRF 17 stage vertical pumps. The seismic stress analysis of these pumps was documented in Appendix I of Pacific Pump design report number K-441 4

Revision 2.

The design report also included ASME Code pressure boundary calculations, operability calculations, a rotor torsional analysis and a seismic analysis of the Westinghouse motor which drives the pump. This summary report will address only the seismic stress analysis of the pump which i

was prepared for Pacific Pumps by consulting mechanical engineer Douglas B.

Nickerson. This summary will include the design criteria, design loads, methods of analysis, modeling techniques, summary of results and conclusions.

2.0 DESIGN CRITERIA The seismic analysis of the South Texas high head safety injection pumps conforms to the design requirements and acceptance criteria of the documents listed below. These documents define the design loads and load combinations as well as the acceptance criteria for stresses and deflections. Compliance with these documents ensures the structural integrity and operability of the i

pump assembly for the South Texas Plant service conditions.

Design Documents:

1.

ASME Boiler and Pressure Vessel Code,Section III, 1974 Edition with Addenda through Summary 1974 2.

Westinghouse Equipment Specification 678815 Revision 2, dated 9/6/73, General Class 2 Pumps 3.

Westinghouse Equipment Specification 952455 Revision 1, dated 6/30/76, South Texas Project Code Class 2 Pumps and Motors 3.0 DESIGN LOADS The seismic analysis of the high head safety injection pumps combines loads due to seismic responses, externally applied nozzle loads, equipment 2

me,i.mim i

ATTACHMENT J' ST-HL-A - / 93 '7 PAGE OF LD deadweight and operating loads. The actual values used in the pump seismic analysis are identified in the following text.

3.1 Seismic Loads:

Generic seismic loads were used in the seismic analysis of the high head safety iniection pump assembly. Since the natural frequency of the pump assembly is below 33 Hertz, a dynamic analysis was performed using the generic design response spectra shown on page 4.

This generic spectra.was selected to have substantial margin above the actual South Texas Plant floor response spectra for the location and elevation of the pumps.

3.2 Nozzle Loads:

The analysis applied actual South Texas Plant nozzle loads provided by the architect engineer Brown & Root. These loads were envelope values for all of the high head safety injection pumps in plant units 1 and 2.

The. loa'd values used in the analysis are tabulated on page 5.

3.3 Deadweight

The pump deadweight used in the analysis was obtained from Pacific Pump outline drawing 300-VN49752 Revision 12.

The deadweight breakdown follows:

2850 lb.

l outer barrel l

pump (lessouterbarrel)- 5000 lb.

I 5820 lb.

motor 1330 lb.

motor mount total

- 15000 lb.

l 3.4 Operating Loads:

The analysis included operating loads due to the motor power output and the l

hydraulic thrust on the pump rotor assembly.

3 eminum i.

n sx a

a

_..2 2

a s=-.-

.u.-

w a

-.a

..m...*...

...t.......'---..

4

,...e...,.

..s

,m

..s...... t...

g s

g....

.s.*...*..

..'I

.*..........t.

i.....

g

-.......~... z --

.J..

1 g ~t~t.

l s.

....,,....,....r_

._..... s. -

a

.6 g

.e i

I.1

...a

-_r r

- 1..

.. =.... -..

9

...:::,...v...: s::: ::.-....... 3. "..

.,..1 y.... _

n

(,.. :...

. ~~ u..

g AHAU MNI#7-

. p.:. 3., :.c.,...:..

GENERIC SEISMIC

.. _ ' " ~ " ~ ~ ~ ~ ~

.st.H L. AE~ -/

, m.. '

_ '. ' ' ". _... ~ ~ '. " _ _ '

RESPONSE SPECTRA

.fA. QE. >5. 0 F

~

~

?

--. *e

....a........_._..

i........

...._..t.

.. f......

.. a.

. s....... -
i........

... ~ ~_ _.. s w..- t.. o,.,... -

.::g:..,........,.,.....

...l...,t..._.

p 1

. i

..I ~... f [ ~......:: :-

.r

.l s ::

...:..r. :e....: ~..s... yJ. )... t.

.2.....:

s-

.............s.

......s......... g..:::.......:.......

.........g.....

r...

,...s.

_..i...,..-..

.**.6...

_. _.,.. *....: ~.. s...,

......l...%.

.r.,

eg y

i

...t........

g.

.1...

~ *. e.: tn,...

.:--~

......, s.

.....v

.s.t.............s...........

......r

s. v.

.i....

.t...........r......i....

..s.

..s...........:.............g.

...........r......

,3

...s.......

..s.....,

l-e.,. :..,....; :,..: :.. o -

'... r.a.,.....

t 4......;

......)... - -

g

..a.

...s.....r......

... _.....i

... t. c,

.1.,i....,..f%..

I

........k.

..... ~. 1 r

@... L Vv.

. f. s

.I..:..,:.;

.. s.

.t.

3.

. s..

... ~..,.

.........s.......,.............s..,.t..s......

..r....

...s......--.._

.......w...

................tu

.....1....

.....g..z..

. La.

4s......

e : *.:

t.. ;........ - *.a...".... g".,....c i

-:;.:n. W in o

to

......:~..

... s.. --..

~.

.e

.....t.:.~..

....i..

r.

....s..

.....s.....

i

..t 2.

k s...... 9.."%..

s n_........,.......,

....g.

.. m m.

3

,...3.

.s.....

-...a s

.. ~......

.f....

...6..

s..

L.

-.....,,........t

....._..s......

s.

...t.......................8..........

........ e............

I.

m

..t......:.

.....r.....

s l

...r............

..i......................

r.......

L s..

  • * * * * *. *. -**)....

-.. a.==

.z.a

, =.

i

......'8

..f*.**.**.....

    • '. r......

p.

. ~....

.......... =f'......s.....c...........

.i.....

t.._..

.. In 1

J

.. s......

i

..:......t......

.......r.... ~.......

.m..... - t..

............t..:_.

..a.

.. _..........a

. s.

.....t..

\\

n.................

t F._r......

.......s.........,....-s.t L..~~.....

...._..t.....

.r..

m...,..

.....i.,..

s.........s...._.............s.......__.............

I g...

.. :. i......

.....e._............r..~...

.........i.....r.....t...-.

r.... :...........

t......

_e

..s.....................

..s...

s........,.:.-

.........>.,..........t.......

s........

.....a,......

...s.s...........~...

....... ~

1

..r.......__

t.

g.........,......

...t...... 7.

t u...

y

\\

....i....

.....t......... -.....

._...s._...

i.

............ eJ. :n.

t r

..... : e...

........ n,

.n

.s.

......".s.....

n

.......f...

.L....o.......

....f*."."g"*........r...

... ~......... 1..

s

.9

...-......l.....

4..

t...

....s......

i

-. -....s.

.i.............t

...i.....:..._._...

4..

l t........

...s.

s

.....s...t.....

.....r................4

.t....

.....i..... -

...t....

......s.......a

....t....

1. r.

...:........{............

..m-.... _.. _... _.........

..e.

... _..-..............s.

....t.

....i..

... r..... t. s -

i

.s i

..L....a....-.,.

..~.a.....

. -......~ s.

. 6

-s.

..4.._............ -....

.t

.6

.4,,......,

.t_........,.....,

.....t,..,.,

L n.

n..

..e.

..t

..s...s..

......f....:.........

..t. t.e........

g......

1 t.

.~

=..s.

....._..._....l.

......t.t.........

.. t......

I i

...s.........

t..

....s....

....r..

r..

1..

....r x....<.

_..r........,....

1

.. -i..

.....6......

a.....

....\\........

.s... t....s s

..a..

J.......

....u....

.........r.........

. 3.

.. w... g..... -..

t

. =.. =...

1.

n.

e.,_

.....a... %.

. *... *. *..... I.

. 8 *.N. *. e. -.....

.. 1

, s.t i..'"..ka.I**I..**....

+=.....l...

  • =*

e ".

3

.........,..1....

.t.

.f..

,4 _

...4.............

. 81

,.g..

o............-..n.t....-

r_

...).

.g

.... ~.....

-r l

=... - - _

- *... 4 1. I..

4-.....,........ e..

3

..... f. f... l.

8 ;.; ;.

.._......t.......4.

e o.

.1.**...n".o..........

g..

.t.........

l'

...,t-= w.... O g......l

.=.

.f...

..nI.

,.?.......

.==....t*.....4..

.....f..

t......-..1...-*""3."..

..........g...

.g

.......= k. a t

...g....

...I...f..

.........t.

.u,......

t.

...i..

................L... n

...........l......................

...t......

..~.:....s.....

..q...

.......... g. g.... a. -.........g t

3 i..:..

c y.

W~. -..

.....g..,...

i

-..,.. t s.... a.

....1.......L........+......f...

a.......

.. -.....e

  • J g,.g

. ~.....

. ~....

........,......, ~.... 8.....

H-'

g.

.. s. y.

.1_

.s..,

4

.f p--

n

............ t*g.

....a..I=.._.~.....,e-

8..g. 4 *. =........

I......,

t e e..t....e....e.

...1...e

.3. 8 8

........5..

-8..

t ?..=.......... *.

.a.

..e e g.

O ee e.

e.

.....g. w.

i e

..l...

..e. g...

8 i..

...........,,..........n.o.,.......

.. n.

.e.

o.g.n.r..

  • .t........

..I_........ p. g

.*g.

g,,,.

.u.

..o.

I

..3..

n.

g 1

$*. J...

R

.. :..a...,(

-}. L. e...,

M 4

--TN'.W.'..t... 'm ".'1?hTf2;.. s% p)T%.m..f. :.."... i$'****.". **. l*Ct'%.-i>-...,. _.,..... _... _. _..

&.e****--

^ * * * * * * * *.

7

-,, ~. -

ATTACHMENT A ST HL AE /93 7 e

o PAGE 4 OF ${}

N0ZZLE LOADS Load Load Nozzle Load Case Component Magnitude (plus/minus)

Discharge Faulted FX 2311 lb FY 2416 lb FZ 720 lb MX 5133 ft-lb.

MY 2635 ft-lb.

MZ 2285 ft-lb.

Upset FX 1849 lb.

FY 1933 lb.

FZ 576 lb.

MX 4106 ft-lb.

MY 2108 ft-lb..

MZ 1828 ft-lb.

Suction Faulted FX 1907 lb.

FY 1096 lb.

FZ 1677 lb.

MX 2256 ft-lb.

MY 5348 ft-lb.

MZ 3963 ft-lb.

Upset FX 1526 lb.

FY 877 lb.

FZ 1342 lb.

MX 1805 ft-lb.

l MY 4278 ft-lb.

MZ 3170 ft-lb.

l l

X direction is axial to nozzles Y direction is vertical Z direction is perpendicular to nozzles 5

0175v:1b/0113BF

ATTACHMENT k ST.HL AE /937 PAGE rf OF LD The following loads were used 995 HP maximum motor output maximum total hydraulic thrust - 10173 lb.

4.0 METHODS OF ANALYSIS The South Texas high head safety injection was analyzed using a combination of computer code results and hand calculations. The detailed description and validation of the computer codes is included in Appendix A of this summary report. The pump assembly was divided into four subassemblies for ease of analysis. The four subassemblies consist of the outer case,.the inner case including the first three pump stages, the upper fourteen pump stages and the driver mount-motor assembly. The dynamic models of the four subassemblies are shown on pages 7 and 8.

i The computer codes calculated modal frequencies for each subassembly using the lumped mass, multi-degree of freedom models. Based on the natural frequencies of each subassembly, accelerations defined by the generic seismic response spectra were used by the computer code to calculate modal deflections at each mass point and also modal forces and moments in the structures.

The modal forces and cioments were added vectorily by the computer code and total seismic i

induced stresses were calculated. Hand calculations were used to calculate stresses in the subassemblies due to nozzle loads and operating loads. These stresses due to seismic loads, nozzle loads and operating loads were absolutely summed at each critical location of the four subassemblies. These stresses were compared to the appropriate allowable stresses defined in the pump design criteria.

5.0 MODELING TECHNIQUES The_high head safety inject' ion pump was split into four independent subissemblias for modeling. These subasser.bly models represent the pump outer case, the inner case including the first three puup stages, the upper fourteen pump stages and the driver mount with motor. The division of the pump 6

v omianw

L i

i l

ATTACHMENT 1 ST-Hl. AE- /f37 l

PAGE g OF 30 l

DYNAMIC MODELS i

I t

O.

y 1

i

-1 l-si G'

-1 l-h I

8

.i F

-t I-1 e

.x i,

.._h d'h i

1 i

l

.Jr---

4.-

y 1

h Ol F

_ __' s a

l t

~

A 4

L(

y 4 e Cl e

-f t

t

,q p

..A

.kE i

N

'/b

-l I-C l

..W t

9 I

'4]S f.-'5

+ f.

~

's E

E f.

i i

g' 7

,-g

,m.

O 14 l

~

l 0-i.

ATTACHMENT A r

6{

ii fj Li-rn AE /939 g

PAGE4 OF W T 1-U

.G.,I 1:--d T

e-

+W i

usi e u

l C

9-4 8

I 3

O 1 e, e-

~

I e

~

te g

s 1

l 2"

l T

4 l

s l

og 9

e i

..r S,

i Il fl m.

e g

. n e

s

~~

9 m.

i um 1

a.

q)

_ ; y e

enum o

e-

ATTACHMENT A ST.HL AE-/937 PAGE /c OF Sc assembly into four separate vibrating systems was justified by the experience of the author in the analysis of vertical pump assemblies. The vibration of a vertical pump motor mount-discharge head-motor assembly does not usually transmit to the pump column assembly (inner and outer cases). This is due to the fact the motor mount and discharge head are rigidly fastened to the foundation by a heavy base flange with anchor bolts. On the other hand, the pump column connects to the foundation through a flat base plate which allows considerable flexibility under trunion loadings. This flexibility was accounted for in the outer case model by applying an angular spring constant calculated using the methods of " Formulas for Stress and Strain" by Roarke and Young.

This modeling methodology justif'ies the decoupling of the driver mount-discharge head-motor assembly from the pump column.

l

[

The models of the inner case (including the first 3 pump stages) and the upper i

14 pump stages account for the effect of the water surrounding the respective subassemblies. Hydrodynamic masses representing the water were calculated l

using the method of ASME paper 70-FE-30, "The Effects of an Annular F'luid on the Vibrations of a Long Rotor" by Fritz.

The hydrodynamic masses were added to the mass points representing the structures, thus the water contained in the pump case was considered to vibrate with the metal elements of the pump.

l l

l The pump shaft / rotor assembly was not modeled as a separate vibrating element i

in the seismic analysis since Pacific Pumps performed an operability study and rotor torsional analysis which are included in design report K-441 Revision 2.

The effect of the pump casing deflection on bearing loads was evaluated, however, by assuming that the shaft deflection exactly follows the pump inner casing deflection. The assumption is justified by the close spacing of l

bearings along the length of the pump shaft. The bending moment in the shaft was calculated by multiplying the casing bending moment by the ratio of the moments of inertia of the shaft and casing. The shaft moment and bearing spacing was used to calculate bearing loads, which were converted to equivalent bearing pressures.

1 9

01&lt/011307

ATT.ACHMENT ol ST.HL.AE /937 PAGE // OF 3D 6.0

SUMMARY

OF RESULTS This section summarizes the calculated results of the seismic analysis for the South Texas Plant high head safety injection pumps.

Included are natural frequencies for each of the four pump subassemblies, foundation loads, calculated stresses, deflections and bearing loads.

6.1 Natural Frequencies:

The predominant natural frequencies for each subassembly were calculated using the modal frequency computer code. The frequencies are summarized below:

Operating Speed - 3570 RPM (59.5 Hertz)

Outer Case - Horizontal - 11.5Hz, 69.3 Hz, 188 Hz, 339 Hz

- Vertical

- 138 Hz Inner Case (plus 1st 3 stages) - Horizontal - 7.9 Hz, 31.2 Hz, 81.4 Hz, 203 Hz

- Vertical

- 180 Hz, 561 Hz Upper 14 stages - Horizontal - 34.7 Hz, 123 Hz, 255 Hz, 389 Hz Driver Mount / Motor - Horizontal - 16.3 Hz, 71.0 Hz, 232 Hz None of the calculated frequencies falls close to the pump operating speed so no operational resonance difficulties are expected.

The 2nd mode frequencies of the outer case and the driver mount / motor are closely spaced. 'This was accounted for in the analysis by applying an amplification factor to forces and moments for the 2nd mode.

6.2 Foundation Loads:

The calculated maximum loads on the foundation at the outer case flange mount are summarized below. These loads are due to the combinations of seismic responses, nozzle loads and operating loads.

Fx = 10403 lb. (shear, total on all anchor bolts)

Fy = 16454 lb. (tension on highest loaded anchor bolt includes effect of overturning moments)

. F2 = 8414 lb. (shear, total on all anchor bolts) 10 onunw 1

-r;

.__a.

~

' ~

ATTACHMENT A 7

ST HL AE-/9.3'fD PAGE 61 OF Mx = 891,859 in-lb (overturning moment about foundation neutral axis)

My = 111,406 in-lb(torsionalmomentaboutfoundationcentroid)

Mz = 841,708 in-lb (overturning moment about foundation neutral axis)

6.3 Stresses

The analysis calculated stresses in the critical areas of each of the four pump subassemblies. These critical areas were chosen such that the highest stress levels in the pump assembly were evaluated. The areas of the pump assembly which were not analyzed will have lower stress levels and are considered to be acceptable. The table on the next page summarizes stress levels in the critical areas of each pump subassembly due to the most limiting combination of. seismic, nozzle and operating loads. The stresses are compared to the normal case allowable stress levels as defined in the pump design criteria. All calculated stresses meet the appropriate allowable stresses.

6.4 Deflections

The four pump subassemblies were modeled as beams with lumped masses in the i

modal analysis.

The maximum seismic induced deflections were calculated for each mass point. The greatest calculated deflection in each subassembly is given below.

l Outer Case

.075 in.

Inner Casa Plus 1st 3 Stages

.210 in.

Upper 14 Stages

.003 in.

Driver Mount / Motor

.008 in.-

The pump Inner Case (plus 1st 3 Stages) and the Upper 14 Stages deflect in parallel with the pump shaft and rotor elements. The relative deflection between the stationary and rotating elements is negligible.

6.5 Gearing Loads:

.The loads in the water lubricated pump shaft bearings were calculated based on the deflections of the inner pump casing. The maximum bearing pressure was only 5.65 PSI, which is negligible.

The loads in the pump shaft bearings will have no detrimental effect on the bearing function.

11

>=wenst l - --

~.

?

ATTACHMENTol ST-HL AE- /93 7 PAGE /30F $b

SUMMARY

OF STRESSES Normal Case Calculated Allowable Subassembly Location Material Stress (PSI) Stress (PSI) l Outer Case Suction Nozzle SA-182F304 11214 24900 i

Suction Flange SA-182F304 13112 24900 Suction Flange Bolts SA-193 Gr.87*

11262 37500 Lower Shell SA-182F304 5423 24900 Anchor Bolts **

A-36*

19227 29000 Inner Case &

MountingFlange(circ.) SA-182F304 10894 24900 1st three Mounting Flange (long.) SA-182F304 7306 24900 stages 3rd Stage Flange Bolts SA-564 Gr.630 31481 42000 3rd Stage Inlet Flange SA-564 Gr.630 22699 42000 Bolts 3rd Stage Bowl SA-296CA6NM 2146 17900 Upper 14 Maximum Bowl Stress SA-296CA6NM 1223 17900 Stages Driver Mount Motor Mounting Bolts A-193 Gr. B7 2596 37500 Driver Mount Bolts SA-193 Gr. B7 7521 37500 Driver Mount A-36 1052 28950 l

Discharge Head Flange SA-193 Gr. B7 18625 37500 Bolts Discharge Head Base SA-193 Gr. B7 34234 37500 Bolt:

Discharge Head Base SA-182 F304 3600 24900 l

  • Actual material is supplied by the plant site. Materials identified are l

those typically used in these applications.

    • Calculations based on eight 1-1/4 inch diameter anchor bolts.

12 eim wenaw

... ~. _ - - - _ _. _ _ _ _ _ _,,... _. _ - - -.... -. - _ _. _.. _.. _ _ _

ATTACHMENT A-hfdIh'0 b

7.0 CONCLUSION

S The analysis of the South Texas Plant high head safety injection pumps calculated natural frequencies, critical stresses, deflections and bearing loads. The natural frequencies of the pump assembly do not coincide with the pump operating speed, therefore, no unusual resonances are expected during pump operation. Stresses in all critical areas of the pump assembly were calculated for combined seismic, nozzle and operating loads. These stresses were well within the normal case allowable stress levels identified by the pump design criteria. The deflections resulting from the worst combination of operating and seismic loads are minimal and will have no detrimental effect on pump operability. The bearing loads are negligible and will not affect the bearing function.

It is concluded that the Pacific Pumps model 6x10 WYRF vertical high head safety injection pumps are adequately designed so that no structural damage or loss of function will occur when the pump is subjected to the South Texas Plant seismic and nozzle loads.

i l

i I

s 13 oin-wount a

m

  • ST-HL-AE-R37 APPENDIX A PAGEis OF3D DESCRIPIION AND VALIDATION OF COMPUTER PROGRAMS

. =.5=$-

"BMDAT", "CA!BE", " CON 3M", "KDIDF" AND " STRESS" Introduction In order to solve the mariy structural dynamic problems of modern engineering, a set of computer programs have been prepared. These prograns have been focused on the problem of determining modal frequencies and modal displacements. There exist a number of finite element stress programs.

2 These can be used to obtain dynamic information but at considerable cost.

By using lumped mass models of simplified systems closed form solution of modal frequency values can be obtained at far less cost. For many vibration problems these solutions ar's more than adequate. The programs herein described carry this approach beyond obtaining vibration data to the determination of nodal displacements. This serves to provide the using engineer with data ff?

so that he can evaluate interference problems within the r.achine. These l

7 displacements are then used to determine internal dynamic stresses within the structural members.

Although the approach taken does not allow for the analysis of highly complex structures it provides a solution to a lar6e i

number of dynanic problems where the use of the very large finite element program is not justified.

1 Approach The approach,taken is to break the' entire computer pro 6 ram into elements which can be assembled in various ways to solve as many as possible vibration problems.

A data storage routine is loaded with a model of the device to be analysed for use with the programs. Data are taken from this file and used to calculate a static deflection matrix by one

==.:

of the alternate programs. In all cases the static deflection matrix is 31 made up of real deflections caused by actual weights. Thus the. engineer's 14

ATTACHMENT A ST.Hl.. AE- /937 PAGE noOF.30 2

s feeling for proper magnitudes can be applied to evaluate the reasonable-I ness of results. The static deflection matrix'provides' the spring rates for each resonant mass which is used to calculate the resonant frequencies for each mode.

A second advantage to the use of deficctions rather than the use of characteristic spring rates is that using this method modal frequencies are calculated in ascending order. Thus the fundamental fro-quency'is the most accurate rather than the highest modal frequency. Re-sponse values and real, dynamic deflections are calculated for each lunped mass for each mode. Based on the dynamic beam shape modal moments can be computed.

The moments are then added vectorily and caximum stress values calculated. By provid5 ng separate programs for each operation, different combinations can be used to solve a broad range of problems. The built in flexibility helps reduce the cost of the design process since all calcula-CI tions need not be made for each configuration.

Rationale The data storage program, "BXDAT" performs no calculations. It merely accepts the characteristics of the beam to be analysed and stores them in a data file for future use. The characteristics filed for each section or j

lumped mass are:

Modulus of Elasticity Moment of Inertia Radius of Cyration Distance from Inft End Static Weights Each of these items may be different for each section of the bean. There may be up to 21 lunped masses spaced at any distance from the left or top 15

~

r--

~., - -

m

- +--,,

m, - - - -,, - -

_=

ATTACHMENT d--

ST.HL-AE- /937 PAGE /7 CC SD 3

end of the beam. The beam section appropriate to e'ach weight is the one immediately to its left. Input is fro:n the keyboard. The program provides for checking and correcting the data before they are stored. It is also possible to modify an existing data table.

There are three different beam deflection programs designed to calcu-i late a static deflection matrix. The reason for this matrix form will become apparent when the method of calculating resonant frequencies is discussed. There is a program "CAK3M" which is used to calculate the static deflection' matrix for a cantilever beam. This program applies each weight, in turn, to the beam.

For each weight the deflection at each mass point is. calculated. The resultant matrix has rows of static deflections of each station with an actual weight applied at only one point. This program also includes the effect on all these deflections of a spring rate

.5fi for the " built in" end of the beam. Thus the flexibility of the supporting

= g base plate to bending of the pump column can be modeled.

In addition, the location of a support anywhere along the column can be included. When this option is elected a deflection matrix with no support is calculated and, using the deflection at the support, the load l

necessary to return the beam to zero deflection t10 inches is determined.

Having these loads a matrix of the deflection caused by them is calculated and the difference between the two matrices taken. This difference matrix l

is the deflection matrix of the supported beam.

(

The program for a simply supported beam provides for up to 10 supports and up to 21 lumped masses.

In order to solve the redundancies present for more than two supports the three moment equation is used. As with the pro-gram for a cantilever beam a deflection matrix is calculated. This matrix Lk L[

has a line for the static deflection of each mass for the wei ht of one of 5

i 16

ATTACHMENT A ST-Hl. AE-N31 f,

PAGE /F OF 30 4

them applied at its location. The program is named CONSM".

Supports may

(

be applied at any mass point and values of "EI" must be constant between adjacent supports.

The thirti program capable of calculating a static deflection matrix is " BEAM". This program uses the concept of transfer matrices for the vari-ous beam elements. The field elements use a vector representation of the classical strength of materials equations. These elements represent the characteristics of a wei htless beam. Point elements, available at the 6

Present time, a're a weight which is applici between any two field elements; a support wnich must have a spring rate; and a torsional spring. This pro-gram requires the beam elements be entered, in order, starting from the left end. Having entered the beam elements the boundary conditions at each end of the beam are supplied. The program calculates a transfer matrix for the beam for the first weight from the left end. This matrix is the product of the element matrices for the entire beam excluding all but the first weight matrix. Usin6 the boundary conditions, end vectors are calculated. The end vectors are the deflection, angle, moment and shear at each end of the beam. Using the left end vector and a transfer matrix formed for each weight point, point vectors are calculated for each weight point. The deflection at each weight is the element of the static deflection table sought. This same procedure is repeated for each weight applied to the beam, in turn.

All of the above programs use classical strength of materials theory.

In other words, inelastic deflection of the structure is not permitted. The basic deflection equation used is:

2 M=EIh (1) which may be integrated twice to get:

l l

i e

17

ST HL.AE-12p ATTACHMENT 5

PAGE 19 0F 3D (2) y=yo + 0,x +

y l,

This equation for the deflection y can be generalized for a section of a beam with portions extending from both ends by letting:

WR = M (3) y and W=S (4) g where M is the moment at point i due to weight W, and,S is the shear y

g y,y be the deflection at the left end of at the same point. If we let g

9,3 betheangleatthispoint, equation (2)becomes:

the beam portion and 1

MX S (X )2 i1 g y 01"01-1

  • E I EI gg gg M(X)

S(X) i i i i yi " yi_1 + 8.1 2E I MI X+

1 gg gg where the variables are taken at point i, except x.

E and I which are the length, Modulus of Elasticity and Moment of Inertia respectively, of i

the beam from point i.-1 to i.

The program "MDIDF" utilizes the static deflection matrix computed by either "CAKBM", "CONBM" or "3EAM" to calculate the resonant frequencies and modal d'flections of the beam. The rationale upon which the program is based e

is derived as follows. This derivation is taken from Den Hartog, Chapter 4, i

j withsomemodifications(1). Consider a weightless bar with a number of masses distributed upon it. If the deflection from equilibrium is given I

!. (

(1) Den Hartog, J. P., Mechanical Vibrations, 4th Bl., McGraw-Hill, New k

York, 1956.

1 I

18

,,,.,,,,...-------...-,,----e_,

n,

- - -. ~ - -. -

ATTACHMENT A ST-HL-AE 1939 PAGE aDOF3 D g

as x and the corresponding mass as m then g

y i:[

Nmy+kx31-0 (6)

But for this case the spring constant, k,

is difficult to evaluate since it depends on the relative deflection of all other masses. In order to avoid, this problem Den Hartog introduces the concept of " influence numbers" which are defined as, "fhe deflection of mass 1 caused by a force of 1 pound at the location of mass 2."

Prom this he writes the simultaneous equations:

y1 y y - QM2*******-"4"jj MW E

xy

- oc 2"-*21"11-"22"[2*******-*2j"jj x

N I

(?)

(I MM

-a E

i2"22*******-*ij"[Ij x1 = - ec11 y y Using the usual method for solving the equation of a harnonic os-cillator, we let sin ut (8) y =, A1 x

Son 2

M=-Au sin wt (9) y y

For these programs, in order to keep the calculations on a real, physi-cal basis, where the engineer's experience can guide him, static deflections are used instead of " influence numbers". The relation between the two is simply:

"ij " b W

(10) 3 where 8 is the static deflection of the beam at section i due to the 33 19

^ ATTACHMENT A gitgek

~

weight W) at position j. Thus a static deflection curve for the beam I

with a single load, W), is calculated to get each of the S 's for each g

value of J.

It is now apparent why the beam programs were set up to cal-culate deflection matrices as they do. Substituting in equation (7) and remembering that Ot where i and j are not equal, we get g3 = m31 2

2 2

A

=C A

+8 A

y yy y 21 2 31 3 2

A

+S dd A2"C 2y 22 2 (11) 2 2

A

-+S A

A1=Syy y 2i 2 6 A) 31 2

Dividing by and rearranging the equations:

f<

(S

)Ay+S jl ) = 0 A

A 21 2 * *

  • S yy

( }

-[z)^2 a

^1 + (822 8^3-0 32 12 l

(

2i 5 * * *

(Sji A) = 0 yyy+8 d

A A

In order that there be a solution for equations (12) the determinant of their coefficients must equal 0.

If we let

=x (13) 2 W

where A is the eigenvalue of the determinant we have an eigenvalue problem which may be solved for all possible values of the eigenvalue X.

In the 3U computer program the Hessenberg method is used to find the eigenvalues and 20

ATTACHMENT A, ST HLAC- /437 PAGEgtA0F 30 g

andeigenvectorsofthedeterminant(2). There will be as many eigenvalues

,'{

as there are weights on the beam. So there are as many modes of vibration as there are weights used in the model of the pump.

i The eigenvectors represent the deflection of the beam for each mode however, they are not actual values since the program normalizes them to unit vectors.

In order to derive real values from them, partition factors i

are calculated. These partition factors distribute the deflection betweea modes for each weight point. '!he real maximum deflection at any weight point is the product of the partition factor, the response acceleration 1

at the mode frequency, the eigenvalue of the mode and the eigenvector 4

l

)

corresporiding to the mode frequency and the weight point. The mode fre-quency is calculated from the eigenvalue by equation (13).

A more complete description of the method is described in Chapter 12 of Earthquake Encineerinz (3)andChapter3andtheAppendixofStructuralDynamics(4).

(

The matrix of deflections is entered into proEran "KDLDF" which calcu-i lates the eigenvalues and from them the mode frequencies. If he desires, the engineer inputs a response acceleration for each frequency and the

(

l program computes partition factors. A cautions since the eisenvectors from this program are normalized as unit vectors the " partition factors" l

calculated are different from the " participation factors" of Structural Dynamics since Biggs' eigenvectors are normalized differently. The program closes by printing out a deflection matrix giving the real deflection of each weight point for each mode.

l (2) "HSNBQR Eigensolution of a Matrix Using QR Hethod". CSCX Basic Library Prog. 01-1998 CSC Infonet,1970.

(3) Weigel, Robert L'. (Ed.) Earthouake Ennineering Prentice-Hall, Englewood Cliffs, 1970.

b (4) Biggs, John M., Structural Dynanies, McGraw-Hill,19%.

1 21 l

.. _,. _. -.. _ _.. -. - - _... _ _ _ _ _ _ - - -. ~..

_ _ _.. _._~ -

ATT ACHMENT A y'

ST Hl..AE. /937 PAGE 4bOF3D The program " STRESS" is designed to process the results

,(

calculated by "MDLDF" to obtain the dynamic conditions at each lumped mass.

This program uses the method of transfer matrices to evaluate conditions along the beam.

It is not

. suitable for the situation in which there are multiple sup-ports on the beam.

The methodology is the same as for pro-gram "EEAM" except the weights are replaced by dynamic forces and all loads are applied at one time.

Dynamic forces are calculated using the def1setions from "MDLDF" for each mode together with its eigenvalue.

The mass is taken from the table set up by "EMDAT".

The equation used is:

7j (15)

  1. j"-

1 1

4 I The vector sum of all modal moments and shears is cal-culated.

The printout presents shear, moment, slope and deflection at each mass point for each mode.

The vector sum of modal moments and shears together with the dyna =le stress in the beam is also presented.

The stress is es1-culated using the equation:

S=h (16)

Where r and I are the radius of gyration and moment of Inertia taken from the table set up by "EMDAT".

22

i ATTACHMENT d-ST.Hl.- AE A3')

PAGEa4 0F S&

10 Validation

(

In oztler to validate the series of computer programs several sample problems were solved by hand methods and by the computer programs. Since classical dynamic. methods were used no effort was made to validate the method. This has been done by many years of usage. The methods are de-scribed in a number of texts such as Den Hartog (1), Biggs (4), and TimoshenkoandYoung(5). It has been considered sufficient to check that the computer programs do, in fact, solve the problems presented to them 1n agreement with the intended methods.

The first problem presented was a cantilever beam having three weights:

5 Pounds 5 inches from the built in end 2 pounds 12 inches; and 6 pounds 1

20 inches from the left end. The beam was considered to be a steel rod 0

-3 0 50" diameter having E = 27.9 x 10 psis I = 3 06746 x 10 in. For

~((

this beam.R is 0.25".

Table I shows the hand calculated results of a static deflection matrix. Table II is the computer results.

TABIE I t

-3

-3

-2 2.43391 x 10 7 54513 x 10 1 33865 x 10

-3

-2

-2 f

3 01805 x 10 1 34586 x 10 2.69171 x 10

-2

-2

-1 1.60638 x 10 8.07514 x 10 1.86924 x 10 TABIE II 2.43391E-3 7 54513E-3 1 33865E-2 3 01805E-3 l'.34586E-2 2.69171E-2 1.60638E-2 8.07514E-2 0.186924 The hand calculations were made using an electronic desk calculator having 10 significant figures, rounded off to six significant figures to

!(l(

(5) Timoshenko, S. and Youn6, Ij. H., Vibration Problems in End neerine, Van Nostrand,1955 1

e 23

ATT ACHMENT A HM4433 u

correspond to the computer printout.

~

[

The eigenvalues of the hand calculated deflection matrix were obtained by solving the cubic equation. The values are compared with the computer solution in Table III.

  • TABIE III Hand Calculations Computer Results Eigenvakue Precuency Eigenvalue Preauenev 0.19986 6.99803 Herts o.199859 6.99805 Hertz

-3 2 55121 x 10 61 9392-2 551212-3 61 9392 4

4.05935 x 10 155 278 4.05915E-4 155 282 The correspondence is considered good enough to prove the computer programs '3MDAT", "CAN3M" and "MDIDF".

However, as a further check a sample problem appearinE in Dynam5cs of Vibrations (6) on page 197, Example 3 5 3 was solved by the computer programs. Calculating the participation

(

factors for the sample problem and then using the equation A

T'"

a,, =

$rn

(

to convert the normalized deflections to real deflections we can compare the frequencies and deflections from the sample problem and the computer run. Table IV gives these results.

The comparison between the results of the sample problem and the com-puter run correspond sufficiently closely so we can consider 'TMDAT",

"CAMBM" and "MDIDF" validated.

We will use this same problem to validate the program "FTRESS". Using the deflections from the sample problem we computed the second derivative (6) Voltera, E. and Zachmanoglov, E. C., Dynanics of Vibrations, Merrill, 1

1%5 24

ATTACHMENT 8-ST Hl.-AE /939 PAGEst(oOF 3D 12 of deflection with respect to distance along the beam. This, in turn, k

can be used to calculate moment in the beam using equation (1). Results are shown in Table V.

TABIE IV MODE 1 MODE 2 Sample Problem Computer Sample Problem Computer 16.1654 Hertz 16.1658 Hertz 104.158 Hertz 104.154 Hertz 4.66866 x 10-31n 4.66856E-3 in 2.89908 x 10 *in 2.89933E4 in 1.65$3 x 10-2 1.655%E-2 5 73832 x 10 5 77892E4 1

3 2 M 39 x 10-2 3 26421E-2 3 12079 x 10 "

312081E4

~

-2 5 04577 x 10 5 045L8E-2 4.16989 x 10 4.17014E4 MODE 3 MODE 4 Sa. ole Problem Comipter Sample Problem Computer 294.072 Hertz 294.171 Hertz 529 744 Hertz 529 952 Hertz 3 98469 x 10-5

-6 1n 3 98327-5 in 7.03353 x 10 1n 7.03382E-6 in

-5

-6 1 33112 x 10 1 3294E-5

-6.81390 x 10

-6.81624E-6

-5

-6 I

-3.875% x 10

-3 87259E-5 4 35122 x 10 4 3$2E-6

-5

-6 1 70173 x 10 1.70068E-5

-1.27468 x 10

-1.23153E-6 TEE V Sample Problem

-509 272 in-lbs

-31.6241 4.$662

.767239

-393 625 326377 3 62061 1.13888

-229 290 29 7621 1 39239

-1 3M 22

-94.0406 25 4881

-5 88133 913627 0

0 0

0 Computer 509 261 in-ibs 31.6268 4 34502 767271

' 3 61994

-1.13904 393 587

.325831

'29 7671

-1 38981 1 3& 66 229 285 94.02 %

-25 48 %

5 87689

.913908 0

0 0

0 1(

I 25

ATTACHMENT A ST.HL AE I PAGE,7? O 13 A review of Table V shows very close correspondence between the

(

l computer results and the sample problem except the signs are reversed.

Since the sign is arbitrary, depending on the sense of the deflection and i

the moments will be added in quadrature the sign will drop out. This is 4

considered to be a complete correlation between the sample problem and the computer program ".3 TRESS".

Finally, the modal moments were added vectorily and the total moment f

and maximum beam stress calculated. Table VI gives these results.

TABIE VI Sample Problem Computer M

S M

S 510.243 in-lb 5197.29 psi 510.261 in-lb 5197.48 Psi 393 619 4009 37 393.606 4009 23' 231.218 2355 16 231.217 2355 16 f

97 5987 994.132 97.6038 994.184 0

0 0

0 The close correspondence between the above results and the sample problem, even though an extensive series of computations have been made, i

justifies our assertion that the programs "BMDAT", "CA}3M", "MDIDF" and

" STRESS" are validated.

In an effort to validate the program "C013M" as an alternate to "CANBM" I

in the sequence an example problem was sought. No problem which would exercise the multiple support feature could be found. However in the book Mechanical Design and Systens Handbook (7) on page 6-58 there appear tables taken from Vibration Design Charts (8) which give frequency data for con-(7) Rothbart, Harold A. Ed., Eechanical Design and Systens Handbook, McGraw-((

Mill, 1964.

(8) MacDuff and Felgar, " Vibration Design Charts", Trans. AS?2, Vol. 79, PP. 1453-1475 f

26

ATTACHMENT G -

ST HL AE- /93 gg PAGE c,25'0F tinuous beams over multiple supports. The beams have a distributed load

(

equal to the beam weight. The values in the table are for C which is g

related to the modal frequencies by the equation f" = C" lo" I'

(18) 1,8

% I **

~

where I,

is the length of each span, I is the moment of inertia of the span and A is the cross-section area of the beam.

X, is a material

constant, 1.000 for steel. In order to set up a sample problem a beam 120 inches long supported at the ends and the middle was assumed. The lumped masses were 100 pounds each at 11 places including the control

~

supports. Thiscorrespondsto10lb/indistributedloading. The value of I was taken as 2 5 in. Table VII shows a comparison between calculated

(

results from the Vibration Table and the computer run.

TABIE VII From Design TabIe Computer c = 31 73 fy = 23.M Hertz fy = 23.4878 Hertz y

c - 49 59 f2 - 36.64 f2 - 36.6884 2

3, 93 85@

l c) - 126.94 3f - 93 79 f

l c - 160.66 f4 - 118.70 f4 - 118.642 4

l f ~ *11' 3

  1. - 2 9.856 c) - 285 61 5

5 The correspondence between the two methods is considered remarkable l

when one considers that the desigh table is for a distributed load, whereas the computer results are for a lumped mass model of the beam. Since the other programs have been validated by other example problems it is con-sidered that these results validate the program " CON 3M" and its use in 27

An ACHMENT,1 ST HL-AE IS'6D b7 PAGE,M OF 15

(

the computer pro 6 ram sequence.

When an effort was made to check the program '3EAM" no comparable problems with known solutions were found. The first test problem posed was that of a cantilever beam having a weight of 10 pounds 4 inches from -

6 the support which had a torsional spring rate of 2 x 10 in.-lb/ rad.and a second weight of 15 pounds 8 inches from the support. The value of 6

2 E x I for the entire beam was assumed to be 1 x 10 lb-in. Table VIII gives the results of hand calculations compared with the computer run.

TABLE VIII Hand Calculations Computer 4

4 2 93333 x 10 6.93333 x 10 2 93333E4 6.93333E4

-3

~3 1.040 x 10 3 040 x 10

.00104

.00304

.(

('

The second problem was a cantilever beam with a sp' ring support 8 inches

here were 2 wei hts on the beam, one at 3 inches from the built-in end.

6 of 10 pounds and a second at 12 inches of 12 pounds. For this case values of ExI were 4 x 10 for the first three inches of the beam, 2 x 10 9

for the remainder. The support spring rate used was 1 x 10 lb/in. The deflection was calculated three ways: using '3EAM", using "CAN3M" and by hand. Table IX compares results.

TABIE IX Comouter "CANBM" bonputer " BEAM" Hand Calculations

-5

-2.10166 x 10' 1.08178E-5

-2.10109E-5 1.08175E-5

-2.10024E-5 1.08173 x 10

-5 3 04350 x 10'

-2 52192E-5 3 04353E4

-2 52166E-5 3 04375E4

-2 52198 x 10

,(

A study was made to understand the descrepancies at the second wei ht.

6 L

It was found that this results from roundoff errors in the solution of the 28

~ '

ATTACHMENT A ST-HL-AE- /f3J PAGE A00Fdd 16 matrices. As values of the spring rate are reduced this error is reduced but comparison with a fixed support becomes less accurate. We consider this pro 6 ram 0

checket, but the support spring rate should be limited to 1 x 10 lb/in.

The computer programs described herein and validated by comparison with the several check solutions quoted are validated, not only by comparison with example problems but by comparison with engineering experience. The solutions obtained in analysing a large number of vertical pumps using these programs have always been reasonable and check with our experience'. There is very little numerical data on the vibration of vertical pumps; however, there is a great deal of experience with vibration problems associated with them. The analytical results of the use of these computer problems check with these observations.

,()

i I

9 6

9 e

e D

9 U

29

.,-----r----

-w._,.

.-,,-_.,,-,,,,._n,,

O 4

1 1

4 1

l l

., _ _ _ _ _ _ _ _ _ _ _ _ _ _.., _. _ _ _..., _ _ _ _ _ _. _ _

Retrieved from "https://kanterella.com/w/index.php?title=ML20211Q873&oldid=4003183"