ML20206T239
| ML20206T239 | |
| Person / Time | |
|---|---|
| Site: | Comanche Peak  |
| Issue date: | 02/28/1986 |
| From: | Dermitzakis S, Desmond T, Eidinger J ABB IMPELL CORP. (FORMERLY IMPELL CORP.) |
| To: | |
| Shared Package | |
| ML20206T195 | List:
|
| References | |
| 01-0210-1470, 01-0210-1470-R0, 1-210-1470, 1-210-1470-R, NUDOCS 8610060139 | |
| Download: ML20206T239 (45) | |
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EFFECTIVE-LENGTH FACTORS FOR BUCKLING OF CABLE-TRAY SUPPORTS Prepared for:
O Texas Utilities Generating Company P.O. Box 1002 Glen Rose, Texas 76043 Prepared by:
Impe11 Corporation 350 Lennon Lane Wainut Creek, California 94598 Report No. 01-0210-1470 Revis1,qn 0.s,,, p *)
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O 8610060139 860717 PDR ADOCK 05000445 A
EFFECTIVE-LENGTH FACTORS FOR BUCKLING OF CABLE-TRAY SUPPORTS Impe11 Corporation 350 Lennon Lane Walnut Creek, California 94598 Prepared for:
l Texas Utilities Generating Cogany Post Office Box 1002 Glen Rose, Texas 76043 Date Issued: February, 1986 This work was performed in accordance with the Impell Quality Assurance Program. The signatures below verify the accuracy of this report and its compliance with applicable quality assurancy requirements.
O Authors:
74 T.P. Desmond F&
5.h. Mrmitf akis Reviewedby:[J.M.Eidinger
/
l V
Approved by:
M.5.' 5watta O
Report No. 01-0210-1470 Revision 0
EFFECTIVE-LENGTH FACTORS FOR BUCKLING OF CABLE-TRAY SUPPORTS Page
1.0 INTRODUCTION
)
1.1 AISC Design Procedure 1
1.2 Method for Calculating Effective-Length Factors 3
1.3 Sumary 4
2.0 STRUCTURAL MODELS AND ANALYSIS METHOD 5
2.1 Trapeze Support 5
2.2 L-Shaped Support 5
2.3 Cantilever Support 6
2.4 Analysis Method 6
3.0 BUCKLING RESPONSE OF FRAMES AND SENSITIVITY STUDIES 7
3.1 vertical Load Distribution 7
3.2 Transverse Load 7
3.3 Longitudinal Restraint 8
3.4 Rotational Restraint 9
4.0 EFFECTIVE-LENGTH FACTORS 11 4.1 Trapeze Support 11 4.2 L-Shaped Support 12 4.3 Cantilever Support 12 5.0 DESIGN RECOMMENDATIONS 13 5.1 Recomended Effective-Length Factors 13 5.2 Conservatisms in Reconmended Effective-Length Factors 14 APPENDIX A Calculation of Longitudinal Restraint 15 APPENDIX 8 Design Example 17 REFERENCES 18 l
l TABLES FIGURES O
i 4
Report No. 01-0210-1470 Revision 0
LIST OF TABLES 1.1 Theoretical and AISC Recomended Effective-Length Factors 1.2 Effective-Length Factors for Design of Cable-Tray Supports--Fixed Anchorage 1.3 Effective-Length Factors for Design of Cable-Tray Supports--Pinned Anchorage 4.
O l
ii Report No. 01-0210-1470 Revision 0
LIST OF FIGURES 1.1 Cable-Tray System and Local Coordinate System t
O 1.2 Trapeze Support l.3 L-Shaped Support 1.4 Cantilever Support 2.1 Structural Model of Trapeze Support 2.2 Structural Model of L-Shaped Support 2.3 Structural Model of Cantilever Support 3.1 Parameters which Influence the Effective-Length Factor 3.2 Effect of Vertical Load Distribution on Effective-Length Factor 6
3.3 Effect of Transverse Loads on Effective-Length Factor.
3.4 Effect of Longitudinal Restraint on Effective-Length Factor 1
3.5 Effect of Rotational Restraint at Support Anchorage on Effective-Length Factor 4.1 Effective-Length Factors for Unbraced Trapeze Supports--Vertical Load Case 4.2 Effective-Length Factors for Unbraced Trapeze Supports--Transverse and Vertical Load Case 4.3 Thrust Diagrams for Braced Trapeze
_4.4 Comparison of Effective-Length Factors for Braced and Unbraced Trapeze Supports 4.5 Effective-Length Factors for L-Shaped Supports--Vertical Load Case 4.6 Effective-Length Factors for Cantilever Supports--Load Collinear with Support Axis B.1 Trapeze Support O
iii Report No. 01-0210-1470 Revision 0
1.0 INTRODUCTION
Cable tray systems at the Comanche Peak Steam Electric Station are supported by structural steel supports.
The support types include structural frames, cantilever beams, and vertical columns. Many of these are suspended from the ceiling. Figure 1.1 schematically shows a cable-tray system, and Figures 1.2 through 1.4 show details of typical cable-tray supports. Because vertical and horizontal earthquake motion may induce compressive loads on some supports, these supports must be design-verified to meet compressive load allowables.
This study has evaluated the stability of typical cable tray supports an' '.tas developed effective-length factors for vertical posts (or columns) of cable-tray supports.
The effective-lengths are for weak-axis flexural buckling *,
These factors account for the interaction of the posts with the remainder of the frame and can be used in routine design to predict the capacity of such Although most of these effective-length factors have been developed supports.
for ceiling-mounted supports, they are applicable to floor-mounted frames as well.
This study has employed rigorous analysis methods as suggested by current design codes (Ref. 1).
The rigorous analysis methods used here are an acceptable method for calculating the buckling strength of structures.
Results recomended for design are given in Tables 1.2 and 1.3.
They incorporate a suitable margin of conservatism as described in Section 5.2.
Sections 1.0 through 4.0 summarize analyses on which the results given in Section 5.0 are based.
1.1 AISC Design Procedure i
O The American Institute of Steel Construction (AISC) Specification (Ref.1) has adopted an allowable stress method for design of beam-columns.
The method uses an effective-length concept for calculating the buckling capacity of structural steel elements. Equation 1.6-la of the Specification is used to proportion compression elements subjected to combined axial compression and bending stresses.
I a
mx bx my by
{ * (1 - f,/Fg)F
+
.0 bx
( ~ T /F'
}F W
a where F,
axial stress that would be permitted if axial force alone existed.
=
For inelastic buckling, an equation for Fa which is based largely on experimental evidence is used (Eq.1.5-1 of Ref.1).
- The effective-length factors developed here are for flexural (or Euler) buckling. Effective-length factors for other forms of buckling such as torsional-flexural and lateral-torsional buckling may differ. Therefore, other forms of buckling should be considered by the designer and appropriate effective-length factors selected for them.
O Report No. 01-0210-1470 Revision 0 1
1
For elastic buckling, the Euler buckling equation with an appro-priate factor of safety is used (Eq.1.5-2 of Ref.1). Both equations are expressed as functions of the effective-length 1
factor k.
Fb
= compressive bending stress about the x or y axis that would be permitted if bending moment alone existed.
2 w E/[FS(kL /rb) ), where Lb is the actual unbraced F ',
=
b length in the plane of bending and rb is the corresponding i
radius of gyration, k is the effective-length factor in the plane of bending, and FS is the factor of safety, g
f
= computed axial stress, a
fb
= computed compressive bending stress about the x or y axis at the point under consideration.
C
= a reduction factor which depends upon the magnitude and sign of m
the end moments.
Because the effective-length factor appears in all terms in Equation 1 above (Eq.1.6-la of the AISC Specification), it is an important parameter in evaluating the capacity of compressive elements. The concept of the effective-length factor is best explainec. by interpreting the Euler buckling equation where the effective-length factor k is used to account for column boundary conditions other than pinned ends.
The i
Euler equation gives the critical load for a centrally loaded column.
,2 EI P
=
(kl)2 cr (2) where E = Young's modulus I = the bending moment of inertia 1 = the length of the column (or vertical post) k = the effective-length factor which accounts for the boundary conditions of the column.
For a column pinned at both ends, k equals one. Columns with other l
boundary conditions may have effective-length factors larger than, or less than, one, depending upon the rotational and translational restraint at the ends of the column. For idealized boundary conditions, theoretical effective-length factors are provided in the literature and have been conservatively modified for design purposes by the AISC (Ref. 1).
These are given in Table 1.1.
For a structural frame, the same AISC equation (Eq. 1.6-la) is used to evaluate the structural integrity of each element of the frame. However, the effective-length factor now implicitly accounts for the interaction of the compression element with the remainder of the structure. The interaction equation must also account for the potential of sideway O Report No. 01-0210-1470 Revision 0
buckling. These situations are not adequately covered by the guidelines for idealized boundary conditions given in Table 1.1.
These shortcomings of effective-length factors for idealized boundary conditions when used O
for frames have been recognized for quite some time (Refs. 2 through 5).
Therefore, the AISC recommends that a rational analysis be performed to calculate an effective-length factor. Some approximate analysis methods i
have been developed to obtain effective lengths. One approximate method (Ref. 6) is currently described in the AISC Specification. More accurate, although more rigorous, methods are available and are permitted by the AISC. One rigorous method is a nonlinear large-deflection finite element analysis. However, the drawback of this method is that it is not well suited for production-oriented work.
This study has performed nonlinear large-deflection analyses for repre-g sentative cable-tray supports. The k-factors which account for the interaction of the compression element with the remainder of the structure are provided for a family of structures. The k-factors developed under this study, therefore, allow an accurate prediction of Fa and F',.
1.2 Method for Calculating Effective-Length Factors An effective (or equivalent) length is obtained by calculating (from a stability analysis) the maximum compressive load (P r) in a vertical e
~
column and then solvinD the Euler equation for an effective-length factor k.
The method of calculating the effective-length factor for a frame is best explained through a simple example.
A simply supported column loaded at its end and at its midheight can be represented as an equivalent system, as shown below, where kl is a reduced or effective column length.
Note that the effective-length factor for this simply supported column can be obtained by a closed-form solution (Ref.11). However, a more detailed frame structure would require a computer analysis.
P P+P
,hg g
2 I
is y
I L
s
(
s
'2 k = 0.87 1
when P = P g
g 2
Q W
P+P g
2 Actual System Equivalent System The critical load can be obtained by analysis for the actual configura-tion (P1 + P )ce. Then an effective-length factor can be obtained 2
for the equivalent configuration from the Euler equation by solving for k.
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,2 EI 0.5 (P) + P I I
2 cr The effective-length factor is an efficient tool for designers to predict buckling capacities of other columns which are similarly loaded. Knowing k, the equivalent system is easily evaluated by hand calculations.
In a similar manner, partial rotational and partial translational restraints at the ends of the column segments can be accounted for by an effective-length factor k.
1.3 Summary Effective-length factors were developed for trapeze, L-shaped, and cantilever supports.
The influences of vertical load distribution, transverse load, longitudinal restraint of the cable tray, and rotational restraint at the support anchor were considered in developing these effective-length factors. For each support type, effective-length factors were shown not to be significantly sensitive to the support's length.
Therefore, support length was not a parameter in those effective-length factors recomended for design.
Table 1.2 gives effective-length factors for trapeze, L-shaped, and cantilever supports which have a fixed boundary condition (i.e., an infinite rotational restraint) at the anchorage, and Table 1.3 gives effective-length factors for supports which have a pinned boundary O.
condition (i.e., no rotational restraint) at the anchorage. Guidelines for determining when an anchorage is fixed or flexible are given in Section 5.1 O Report flo. 01-0210-1470 Revision 0
2.0 STRUCTURAL f10DELS AND ANALYSIS METHOD Cable-tray, supports evaluated were the trapeze, L-shaped, and canti. lever O
supports. These support types must be checked to prevent buckling under seismic loads.
Figures 2.1, 2.2, and 2.3 show structural models for these supports and show the tray configurations considered in these analyses.
Models for each support type are described below.
2.1 Trapeze Support Figure 2.1 shows the structural model developed for analysis of the trapeze support.
Weak axis buckling occurs out of the plane of the frame (i.e., in the longitudinal direction as shown in Fig.1.1). Buckling analyses were perfomed for several lengths
'L', and for several configurations of trays.
For all analyses, tier spacing 'a' (vertical spacing of trays) and width 'w' were constants.
Supports with one, two, three, and four tiers were considered.
Based on a review of trapeze support drawings, the minimum tier spacing is about 16 inches.
In some cases, the tier spacing exceeds 16 inches.
However, the effective-length factors calculated here can he conservatively used for tier spacings larger than 16 inches.
The vertical posts were C6x8.2 structural channels, and the horizontal beams were C4x7.25 structural channels. Selections of these structural sections were based on a review of typical trapeze supports.
Both unbraced and braced trapeze supports were evaluated.
(Bracing is in the plane of the structural frame as shown in Fig. 2.1c. ).
O For braced trapeze supports, L3x3x3/8 angles were used as in-plane diagonal braces.
Tray configurations considered are shown in Figure 2.1 The boundary condition at the anchorage point for all models was fixed for the analyses perfonned.
Because some supports may have more flexible boundary conditions (see Figs.1.2,1.3, and 1.4), the analysis results obtained here for a fixed boundary condition will be modified to account for a flexible anchorage (see Section 3.4).
Effective-length factors for both a fixed and a flexible anchorage will be provided for design purposes.
2.2 L-Shaped Support Figure 2.2 shows the structural model developed for analysis of the L-shaped support.
As shown in the figure, weak-axis buckling occurs out of the plane of the page (which is in the longitudinal direction).
Similar to the trapeze support, buckling analyses were performed for several lengths
'L' where tier spacing 'a' for the trays was 16 inches.
Supports with one, two, three, and four tiers were considered.
The vertical posts were C6x8.2 structural channels, and the horizontal beams were C4x7.5 structural channels. These structural shapes were based on a review of L-shaped support drawings.
O Report No. 01-0210-1470 Revision 0
2.3 Cantilever Support Figure 2.3 shows the structural model developed for analysis of the cantilever support. Weak axis buckling occurs in the longitudinal direction (i.e., along the axis of the tray). Buckling analyses were performed for several lengths 'L' and for several configurations of trays. For all analyses, the tray spacing (center-to-center) was 18 inches (see Fig. 2.3).
This spacing was based on a review of cantilever support drawings for 12-inch-wide trays. This spacing is conservative for other situations where trays are wider than 12 inches.
Therefore, the results developed here can be used for cantilever supports with multiple 12-inch or wider cable trays. For multiple trays less than 12 inches, results for a single tray can be conservatively used.
2.4 Analysis Method A geometric nonlinear (large-deflection) buckling analysis was performed i
to obtain the critical buckling load. As described in Section 1.0, knowing the critical load, the Euler buckling equation can be used to obtain an effective-length factor for the equivalent system. The AISC Specification (Ref. 1) accepts alternate " rational" analysis methods for predicting effective-length factors; therefore, the method described here is acceptable to the AISC. For these analyses, the nonlinear analysis computer program IMSNAP (Ref'. 7) was used.
The effective-length factor is a function of the boundary condition and
+
is independent of material nonlinearity. Note that the same effective-length factors are used in both the inelastic and elastic buckling O
equations of the AISC Specification (Egns.1.5-1 and 1.5-2 of Ref.1).
For this reason, linear elastic material laws were used for these analyses.
Because a geometric nonlinear analysis method was used, an initial
'out-of-straightness' or imperfection was imposed to induce an instability in the compression element.
Initial imperfections were based on maximum allowable sweep tolerances as specified in Reference 1.
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3.0 BUCKLING RESPONSE OF FRAMES AND SEhSITIVITY STUDIES For the cable-tray supports evaluated in this study, there are four influences which effect the evaluation of buckling strength.
They are:
(a) distribution of the vertical load along the column's length (at locations of the cable trays)
(b) transverse loads which induce additional (non-uniform) compression in t
one vertical post while reducing compression in the other post (c) longitudinal restraint provided by the cable tray (d) finite rotational restraint at the support anchorage.
These are schematically shown in Figure 3.1 and are discussed in more detail below.
3.1 Vertical Load Distribution When the compressive load is distributed along the length of the column, the critical buckling load is increased as compared to when all of the load is applied at the ends of the column. The effective-length (as calculated by the procedure described in Section 1.0) is therefore less for the distributed load case compared to the case where the total load is applied at the column ends.
To demonstrate this for the trapeze supports, buckling analyses of four i
trapezes were performed.
Trapezes with one, two, three, and four trays were analyzed, each having a length L (i.e., total length of the vertical post) of 66 inches.
Tier spacing (i.e., vertical spacing of the cable trays) was 16 inches.
Figure 3.2 plots the effective-length factor k versus a nondimensional length L /L, where L1 is the vertical post 1
length from the anchorage to the first cable tray and where L is the total length of the vertical post.
When L /L is one, all the load is applied 1
at the ends of the vertical posts, and when L /L is less than one, the 1
load is distributed vertically at the tray locations (Fig. 3.2).
Reductions in effective-length factors can be realized when the load is distributed for the multi-tier trays.
(Note that the effective-length factors plotted in Fig. 3.2 ignore longitudinal restraint, transverse load, and rotational restraint effects.
These effects will be discussed later.)
In Section 4.0, the effect of vertical load distribution will be included when oeveloping effective-length factors for design purposes.
r 3.2 Transverse Load When trapeze frames are subjected to transverse loads (which are in the plane of the frame), compressive forces are induced in one vertical post,
' O
- Report No. 01-0210-1470 Revision 0
and tensile forces
- are induced in the other post. The tensile forces (or reduction in net compressive force) in one vertical post have a stabilizing effect on the other (critical) post. Consequently, a higher O
compressive stress is required in the critical post to induce instability than would be required if the frame were loaded with vertical loads only.
To demonstrate this response for the trapeze supports, buckling analyses were performed for several trapeze frames of various lengths. For some analyses, only vertical loads were applied, and for other analyses, both vertical and transverse loads were applied. Figure 3.3 shows results for analyses of double tier trapezes. For the longer length trapezes (which are more susceptible to instability), transverse loads have a significant influence in reducing the effective-length factor.
In Section 4.0, transverse load effects will be accounted for when developing effective-length factors for design purposes.
For supports other than frames such as L-shaped and cantilever supports, frame action will not induce compressive forces. Therefore, only trapeze frame supports were evaluated for transverse loads in this study. This is not to say, however, that transverse and longitudinal loads are not an important consideration when evaluating the stability of L-shaped and cantilever supports. Transverse and longitudinal loads will induce bending moments in the L-shaped and cantilever supports which may
~
significantly reduce a support's buckling strength. This effect is addressed at the design stage. The interaction of axial stress (due to vertical loads) and the bending stress (due to transverse and longitudinal loads) is addressed by the designer when allowable axial and bending stresses are checked by an interaction equation.
3.3 Longitudinal Restraint l
Cable trays provide longitudinal restraint to the cable tray supports, I
thereby increasing the support's buckling strength. For example, a full effective longitudinal restraint will modify the fixed-free (flag pole) y buckling mode to a more stable fixed-pinned mode as shown.
-11//11 s /f / ///
I 1
i I
/
k2 l
g,o,7
/
t
/
\\
N f
n
- If compressive vertical loads are applied to the support simultaneously with the transverse loads, both vertical posts may remain in compression; however, the net compressive force in one vertical post will be reduced due to the transverse loads.
O l
8-Report No. 01-0210-1470 Revision 0 i
,-,w
,----n-
,,-,--.,,a-
,,,,,,.,--...a
,-w,
,m,-,,,n-w-
,,w,wwa__,,,-wo-,-->m.,a--w.aww-,.,s.,e- _,., - -.
To evaluate this for trapeze supports, buckling analyses of single tier l
trapeze supports were performed for different magnitudes of longitudinal restraining forces. An elastic-perfectly-plastic truss element was used to model the restraining force. The elastic stiffness of the truss models the longitudinal stiffness of the cable tray, and the yield force of the truss models the force at which the static frictional resistance is overcome.
(See Appendix A for a discussion of frictional forces.)
Figure 3.4 shows results of these analyses. For no longitudinal restraint, the effective-length f actor of all supports is 2.
As the restraining force increases, the factor asymptotically approaches 0.7 These are the theoretical effective-length factors for a fixed-free (flag pole) column and a fixed-pinned column. Note also that for the longer length columns, g
the effective-length f actor is closer to its asymptote at much smaller restraining forces compared to those for shorter columns. This was expected since the shorter columns buckle at higher cogressive loads.
As shown in Figure 3.4, a 12-foot-long support requires a frictional force of only one hundred pounds to provide sufficient restraint to simulate a pinned-fixed column (k = 0.7).
For analyses performed to develop effective-length factors for design purposes, frictional forces will be accounted for, and they will be developed based on the normal force (vertical or transverse load) at incipient buckling. Justification of frictional forces are discussed later in this report.
(See Appendix A.)
3.4 Rctational Restraint Cable tray supports which are suspended from the ceiling are anchored with O
varying rotational stiffnesses. Some supports are welded to embedment plates and can be considered as a fixed boundary condition; whereas others are (rotationally) flexible, as shown in Figures 1.2 and 1.3.
Therefore, modeling the anchorage boundary condition as fixed (zero rotation) may not be conservative for all supports. For example, a pinned-free column has an infinite effective-length when the rotational stiffness k and it has an effective-length factor of 2 when k is infinite. g = 0, g
"tt, "g g "I 8
k==
when k
=0 g
. P l'.
k=2 when k,= =
n O
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To address this, buckling analyses tore perfomed in which the anchorage rotational stiffness was varied.
Figure 3.5 shows results for buckling analyses perfonned for the L-shaped support. The effective-length factor k O
is plotted versus the rotational stiffness km of the support anchorage.
Results are presented for both single and triple tier supports. Two cases for each are considered. One case assumes no longitudinal restraint, and the other assumes a longitudinal.estraint.
(The method for detennining the magnitude of the longitudinal restraint is described in Appendix A. )
Figure 3.5 shows that as the rotational restraint approaches zero, the effective-length factor as expected becomes infinite when there is no longitudinal restraint at the load application (flag pole column), and the effective length factor goes to one when there is longitudinal restraint (pinned-pinned column). However, of more importance, the figure shows L
that the effective-length factor is roughly constant for rotational stiffnesses which bound those of flexible anchors at Comanche Peak *. For single tier supports with longitudinal restraint, k is one. This is the theoretical effective-length factor for a pinned-pinned compression element.
For triple tier supports, k is 0.82.
This reduction is due to the vertical load distribution as described in Section 3.1.
Based on these results, the flexibly anchored supports behave more like a pinned r
boundary condition than a fixed boundary condition. For a fixed boundary condition, the effective-length factor of the single tier support would be theoretically 0.7 rather than 1.0.
Based on these observations, one effective-length factor is needed for rigid or welded anchorages (fixed boundary condition), and another is needed for flexible anchorages. However, since the k-factor is' insensitive to the range of km values for supports at Comanche Peak, only O
one k-factor is needed for filxible anchorages.
(The effective-length factor k will not be a function of k.)
g To minimize the analyses performed in Section 4.0, a fixed boundary condition was used to model the anchorage.
For the flexible boundary condition, the fixed (or rigid anchorage) k-factors were multiplied by 1.43 (=1.0/0.7) to account for reduced rotational stiffness.
l l
- The range of k6 for flexibly anchored supports at Comanche Peak was obtained from nonlinear analyses of anchorages.
The analyses included the nonlinear effects of baseplate flexibility (Ref. 8).
O Report No. 01-0210-1470 Revision 0 L
4.0 EFFECTIVL-l.ENGTH FACTORS O
Nonlinear buckling analyses were performed to develop effective-length factors for trapeze, L-shaped, and cantilever supports. These effective-length factors include the effects of vertical load distributions, transverse load effects, longitudinal restraint, and rotational end connections.
Results of these analyses are presented here.
The method for predicting effective-length factors was described in Section 1.0; the structural models and analysis methods were described in Section 2.0; and the method for predicting the longitudinal restraint was given in Appendix A.
Refer to those sections for details of the models and analyses.
A-4.1 Trapeze Support Because the load distribution in unbraced trapeze and (in-plane) braced trapeze supports are slightly different, effective-length factors were developed for both support types.
(Fig. 2.1c schematically shows the unbraced trapeze and braced trapeze.)
Unbraced Trapeze.
Figure 4.1 shows effective-length factors of trapeze supports for the vertical load case.
Effective-length factors are given for supports with no longitudinal restraint (Fig. 4.la) and for supports with longitudinal restraint (Fig. 4.1b).
For the case of no longitudinal restraint, the support's effective-length factors are sensitive to the length of the support.
In contrast, for the case where longitudinal restraint is effective, the effective-length factors are smaller, and they are not as sensitive to the support length.
Figure 4.2 shows effective-length factors for the simultaneous vertical and transverse load case.
(The ratio of transverse to vertical load was one.) Again, effective-length factors are given for supports with no longitudinal restraint and for supports with longitudinal restraint. As for the vertical load case, the effective-length factor is not as sensitive to the length parameter when there is longitudinal restraint compared to when there is no longitudinal restraint.
Effective-length factors were also calculated for trapeze supports where the ratio of transverse to vertical loads was four. Although those results are not reported here, the predicted effective-length factors were not significantly different from those predicted when the ratio of transverse to vertical loads was one.
For design applications, the effective-length factors for supports with longt-tudinal restraint will be used.
For trapeze supports which have longitudinal re:traint, the effective-length factor varies between 0.77 (single tier) and
[
0.28 (four tier), as shown in Figures 4.1(b) and 4.2(b).
These effective-1ength factors are for the range of support lengths and the number of tiers considered in this study.
They are representative of supports at Comanche Peak.
Recommended k-factors for design purposes are given in S_ection 5.0.
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v Braced Trapeze. The load distribution of the trapeze support may be altered when an in-plane diagonal brace is added. When only vertical loads are applied, the compressive forces in the vertical posts are not significantly altered by the diagonal brace; however, when transverse loads are applied, the compressive forces in the vertical posts are altered by the diagonal brace.
Also, the direction of the transverse load has a significant influence on the compressive forces in the vertical post. Figure 4.3 shows this by comparing thrust diagrams for several unit load cases. For buckling analyses performed for the braced trapeze support, transverse loads were applied in the direction which gave the worst-case (i.e., largest) effective-length factor.
Figure 4.4 shows effective-length factors determined for braced and unbraced trapezes where no longitudinal restraint was applied. The ratio of transverse load to vertical load was one. The comparison shows that results for the unbraced trapeze envelop those for the braced trapeze; therefore, to minimize computational effort, effective-length factors calculated for the unbraced trapeze shall be (conservatively) used for the braced trapeze.
Reconnended k-factors for design purposes are given in Section 5.0.
4.2 L-Shaped Support Figure 4.5 shows effective-length factors of L-shaped supports for the vertical load case.
Effective-len longitudinal restraint (Fig. 4.5a)gth factors are given for supports with no and for supports with longitudinal restraint (Fig. 4.5b). Results are similar to those obtained for the unbraced trapeze. For the case of no longitudinal restraint, the effective-O length factors are sensitive to the length of the support.
In contrast, for the case where longitudinal support is effective, the effective-length factors are smaller, and they are not as sensitive to the support length.
For design applications, the effective-length factors for supports with longitudinal restraint will be used. For L-shaped supports which have longitudinal restraint, the effective-length factor varies between 0.77 (single tier) and 0.53 (four tier) as shown in Figure 4.5b.
Recommended k-factors for design purposes are given in Section 5.0.
4.3 Cantilever Support Figure 4.6 shows effective-length factors of cantilever supports.
For canti-lever supports, horizontal loads (i.e., loads which are collinear to the support's axis) induce compressive buckling loads. Effective-length factors are given for supports with no longitudinal restraint (Fig. 4.6a) and for
~
supports with longitudinal restraint (Fig. 4.6b). Again, results are similar" to those obtained for the unbraced trapeze.
For design applications, the effective-length factors for supports with longitudinal restraint will be used. For cantilever supports which have longitudinal restraint, the effective-length factor varies between 0.63 (single tray) and 0.31 (double tray) as shown in Figure 4.6b.
Reconnended k-factors for design purposes are given in Section 5.0. Report No. 01-0210-1470 Revision 0
5.0 DESIGN REC 0miENDATIONS 5.1 Recommended Effective-Length Factors Buckling analyses performed in Section 3.0 showed that vertical load distribution, transverse loads, longitudinal restraint, and rota-tional stiffness at the anchorage point were important factors in determining buckling loads of cable-tray supports.
In Section 4.0, these influences were accounted for in developing effective-length factors for design.
i Based on those results, it is reconnended that:
(a) the effect of vertical load distribution be included in determining effective-length factors.
(b) the effect of transverse loads be conservatively neglected in determining effective-length factors for trapeze supports, (c) the effect of longitudinal restraint be included in determining effective-length factors, and (d) the effect of rotational stiffness be included in determining effective-length factors.
Figures 4.lb, 4.2b, 4.5b, and 4.6b show that effective-length factors are not very sensitive to the support length when longitudinal O
restraint is present. Therefore, support length was not a parameter in the reconnended effective-length f actors for design purposes.
Table 1.2 gives envelopes of effective-length factors calculated in Section 4.0 for trapeze, L-shaped, and cantilever supports. To account for other uncertainties such as, joint fixity, the calculated factors given in the figures are conservatively increased by ten percent. The effective length factors given in Table 1.2 reflect the ten percent increase and should be used for design of cable tray supports with fixed boundary conditions. An example of a fixed i
I boundary condition is a vertical post welded to an embedment plate as shown in Figures 2.1, 2.2, and 2.3.
Since the effective-length fac' tors developed in the analyses reported here assumed a fixed boundary condition, additional effective-length factors are provided which should be used for supports with flexible anchorages. An example of flexible anchorages are those shown in Figures 1.2,1.3, and 1.4.
Effective-length factors for supports with a flexible anchorage were obtained by multiplying the factors of Table 1.2 by 1.43, (which equals 1.0/0.7). Table 1.3 gives effective length factors for trapeze, L-shaped, and cantilever supports which have flexible boundary conditions.
These values reflect (about) a ten percent increase to a: count for uncertainties in the actual boundary conditions.
O Report No. 01-0210-1470 Revision 0
5.2 Conservatisms in Recomended Effective-Length Factors O
Conservatisms which have been incorporated in the determination of the effective-length factors recommended for design are:
(a) an envelope of effective-length f actors predicted for each support type was recommended for design, (b) available longitudinal restrainina forces were calculated conservatively, as described in Appendix A, (c) an overall factor of safety of 1.1 was applied to the recomended factors, and u
(d) the stabilizing effect of horizontal loading (for frame supports) was ignored.
h O
i O Report No. 01-0210-1470 i
Revision 0
._,,n,.
b APPENDIX A O
CALCULATIOh 0F LONGITUDINAL RESTRAINT J
For the buckling analyses performed in this report, the magnitude of the longi-tudinal restraint was based on the frictional force developed between the cable tray and the support. The frictional forces are a function of the normal force between the tray and support, which in turn are due to the seismic inertial loads of the cable tray system. This appendix explains how the frictional force was conservatively calculated. This is a realistic and a conservative manner to estimate the longitudinal restraint.
It is realistic because the system must develop inertial loads of sufficient magnitude for the support to buckle.
Therefore, the maximum normal force between the tray and support must equal the critical load at incipient buckling. As calculated here, it is conservative because the normal force will be based on the incipient buckling load for an unrestrained support. Actually, the normal force (and frictional force) will be higher since the buckling load will increase with nominal increases in longitudinal restraint. Also, the ' thermal-lag' (fire-protection) material provides some longitudinal restraint, although no credit was taken for it.
The frictional force F is a function of the normal force N and the coefficient
_of friction u.
F=uN The methods for estimating the normal force and coefficient of friction are described below.
hormal Force. The normal force is the incipient buckling load. For the sup-ports evaluated here, the normal forces on each support tier were lower bound normal forces at incipient buckling of a support assuming no longitudinal restraint.
This is conservative since higher normal forces occur if there is any longitudinal restraint.
Coefficient of Friction. A Coulomb-friction force generally results from the relative motion of two solids held together under pressure.
In general, the static coefficient of friction tends to be greater than the dynamic coeffi-cient; that is, the resistance offered by friction decreases somewhat after a relative velocity has been attained.
In steady-state vibration, the relative velocity goes to zero twice during each cycle.
The effective coefficient of friction thus falls between the extremes of static and dynamic coefficients of friction.
From Reference 9. static and dynamic coefficients for lubricated steel on steel are u = 0.23 s
e = 0.108 u
Reference 10 recomends a good effective ccefficient as u = 0.15 O Report No. 01-0210-1470 Revision 0 1
_ - -- - n._ - - - - - -, - -,,,. _. _,, _,,,. _
-,.._-,-n,_____.,--_,,,.,,n--,-,-..,---n---am
For these analyses, a f actor of safety was used for uncertainties in the coefficient of friction (FS = 1.5). The factored coefficient used in the m
buckling an'alysis was u = u/FS j
= 0.10 The frictional force was F=uN
= 0.1 Per 6
where P represents the load on each tier of a multitier support.
er Conservatisms. Conservative lower bound estimates were made for the normal forca N and the coefficient of friction u.
An estimate of the overall factor of safety in the frictional force is given below.
The actual normal force on the vertical post will be increased when frictional restraint is present.* Therefore, the factor of safety on the normal force is FSN = (2/0.7)2
= 8.16 where 2 and 0.7 are the effective-length factors for an unrestrainted (flag O
pole) column and a restrained column.
As noted above, the factor of safety on the coeffient of friction is FSu"I5 The overall f actor of safety is FS = FSN FSu
= 12.2 f
- For the calculation of the frictional force, the normal force is conserva-tively estimated as the critical load of a vertical post with no longitudinal restraint.
O Report No. 01-0210-1470 Revision 0
APPENDIX B l
DESIGN EXAMPLE Recomended effective-length factors for design were presented in Tables 1.2 4
and 1.3.
Examples of their use are provided here.
)
Design Example. A sample design calculation demonstrates the use of the effective-length factors.
The example is based on the support shown in Figure B.1, which is a braced trapeze with three tiers. The anchorage of the support is fixed.
It has an overall height of 12 feet and tier width of 5 feet. Note that the top two trays are separated by a distance of 32 inches. Therefore, the effective-length factor for 3 tiers is conservative for this application.
To obtain the buckling capacity of this support, we calculate the critical buckling load Per.
From Table 1.2, the effective-length factor for a three tier, trapeze support is k = 0.51.
From Section 1.5.1.3 of the AISC Specification, it can be shown that this column would buckle elastica 11y.
Expressing Equation 1.5-2 of the Specification in tems of load, P
EI er =
(FS) (kl)2
,2(29x10 )(0.693) 6 1.92 [(0.51) (144)]2
= 19,180 lbs.
To obtain the applied compressive load, a linear dynamic analysis is performed.
The maximum compressive load in the vertical post Pmax is then obtained from that analysis.
If Pmax is less than Per, the support does not buckle and meets the requirements of AISC.
l l
l i
O Report No. 01-0210-1470 Revision 0 i
_.____,_._,_..._m..-,.
REFERENCES O
1.
Manual of Steel Construction, 7th Edition American Institute of Steel l
construction Inc., New York, New York,1970 (" Specification for the Design.FabrIcationandErectionofStructuralSteelforBuildings,"Feb.
12, 1969, including Supplements 1, 2, and 3 through 1973).
2.
McGuire, W., Steel Structures, p. 574, Prentice Hall, Englewood Cliffs, NJ, 1968.
- 3. -Yura, J.A., "The Effective Length of Columns in Unbraced Frames". AISC Engineering Journal, Vol. 8, No. 2, April,1971.
~
4.
Kavanaugh, T.C., " Effective Length of Framed Columns" Transactions.
American Society of Civil Engineers, Vol. 127, 1962.
5.
Guide to Stability Design Criteria for Metal Structures, p. 410, 3rd Ed.,
edited by B.G. Johnson, John Wiley, New York, NY,1976.
6.
- Chajes, A., Principles of Structural Stability Theory, p.190, Prentice Hall, NJ, 1974.
7.
"IMSNAP: Structural Nonlinear Analysis Program", User's Guide, Version 1-1-84, Impell Corporation, Walnut Creek, California.
l 8.
Impell Calculation No. M-04, Rev. O, " Base Angle Stiffness," Job Number 0210-04 0, Impell Corporation, Walnut Creek, Calif., January 1986.
9.
Baumeister, T., Avallone E.A., Baumeister III, T., Marks' Standard Handbook for Mechanical Engineers, 8th Ed., McGraw-Hill, New York, NY, IW/U.
- 10. Harris, C.M., and Crede. C.E., Shock and Vibration Handbook, 2nd Ed.,
McGraw-Hill, New York, NY,1976.
- 11. Timoshenk, S.P., and Gere, J.M., Theory of Elastic Stability, 2nd Ed.,
McGraw Hill, New York, NY,1961.
f O Report ho. 01-0210-1470 Revision 0
Table 1.1 THE0RETICAL ANP AISC REC 0lEENDED EFFECTIVE-LENGTH FACTORS a) 4)
(c)
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T Rotaties Amed and transinusa tend y
Rotaties free and tasslaties Amed 9
Rotaties Ausd and treamlaties fue f
Rotation free and translaties free 1
i Sourca: AISC Specification (Ref.1)
O Report No. 01-0210-1470 Revision 0
Table 1.2 EFFECTIVE-LENGTH FACTORS FOR DESIGN OF CABLE-TRAY SUPPORTS--Fixed hnchorage BRACED AND UN8 RACED TRAPEZE SUPPORTS NO. OF LOADED tttECTIVE-LtNGTH TIERS FACTOR I
0.85 2
0.69 3
0.51 4 or more o,40 L-SHAPE SUPPORTS NO. OF LOADED EFFECTIVE-LENGTH TIERS FACTOR l
0.85 2
o,74 3
0.70 4 or more 0.64
- O l
CANTILEVER SUPPORTS NO. OF LOADED EFFECTIVE-LENGTH TRAYS FACTOR l
1 0.69 l
2 or more 0.51 i
Notes:
(1) For cantilever supports with multiple trays of width less than 12 inches, use k = 0.69.
(2) Weak-axis buckling occurs in the longitudinal direction (i.e., along the axis of the cable tray).
(3) Bracing is in the plane of the trapeze support.
O 0618a/01-30-86 Report No. 01-0210-1470 Revision 0
Table 1.3 EFFECTIVE-LENGTH FACTORS FOR DESIGN OF CABLE TRAY StPPORTS--Pinned Anchorage UNBRACED AND BRACED TRAPEZES NO. OF LOADED EFFECTIVE-LENGTH TIERS FACTOR 1
1.21 2
0.99 3
0.73 u
4 or more 0.57 L-SHAPE SUPPORTS NO. OF LOADED EFFECTIVE-LENGTH TIERS FACTOR 1
1.21 2
1.06 3
1.00 4 or more 0.91 CANTILEVER SUPPORTS NO. OF LOADED EFFECTIVE-LENGTH TRAYS FACTOR 1
0.99 2 or more 0.73 Notes:
(1 ) For cantilever supports with multiple trays of width less than 12 inches, use k = 0.99.
(2) Weak-axis buckling occurs in the longitudinal direction (i.e., along the axis of the cable tray).
(3) Bracing is in the plane of the trapeze support.
O Report No. 01-0210-1470 Revision 0
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Report No. 01-0210-1470 Revision 0
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Report No. 01-0210-1470 Revision 0
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