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International Specialty Conference on Cold-Formed Steel Structures

(1980) - 5th International Specialty Conference on Cold-Formed Steel Structures

Oct 18th, 12:00 AM

Lateral Bracing of Locally Buckled Columns Lateral Bracing of Locally Buckled Columns

Shien T. Wang

Hungy-Y Pao

Raghuram Ekambaram

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Recommended Citation Recommended Citation Wang, Shien T.; Pao, Hungy-Y; and Ekambaram, Raghuram, "Lateral Bracing of Locally Buckled Columns" (1980). International Specialty Conference on Cold-Formed Steel Structures. 2. https://scholarsmine.mst.edu/isccss/5iccfss/5iccfss-session5/2

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LATERAL BRACING OF LOCALLY BUCKLED COLUMNS

1 2 By Shien T. wang, Hung-Y. Pao, and

Raghuram Ekam baramJ

INTRODUCTION

The strength of thin-walled columns may be governed by overall buckling, crippling of the section or interaction of overall and local buckling. Extensive research work and design criteria are available for the first two types of column behavior (1, 2, J, 4). For the intermediate length thin-walled columns having large width-to-thickness ratio of the component plate elements, local buckling may occur before the overall buckling load is reached. The column is weakened by local buckling. The carrying capacity of the locally buckled column will be much reduced as compared to the column buckling load based on the full section. The effects of interaction of buckling modes on column strength will depend upon geometries of the column. A review of recent relevent research work on this subject was ~resented by Wang and Tlen (5) and more recently by Wang and Pao (6, 7).

In practipe, columns are often braced at discrete intermediate points along the length to increase the buckling strength. Braces must be designed to have sufficient stiffness in order to force the column to buckle into higher buckling mode. Braces must also be designed to have sufficient strength so that they will be able to resist the loads induced in the braces. Green, Winter, Cuykendall (8), Zuk (9) and Winter (10) investigated the above requirements for braces. Additional research was done by Horne and Ajmani (11), Toakley(12), and Urdall (1). More recently, the subject matter was investigated by Nethercot and Rockey (14, 15), Mutton and Trahair (16). In the aforementioned investigations, no local buckling was considered in the columns.

In reality, the column most likely will have initial imperfections. In order to investigate the behavior of the braced locally buckled column realistically, the effects of initial imperfection should be accounted for in the analysis. The interactive behavior involving local and overall buckling, lateral braCing restraints, and initial imperfections becomes an interesting problem to be investigated.

The purpose of this paper is to present a solution scheme based on the finite element method in conjuction with the concept of effective width for the analysis of braced locally buckled thin-walled columns. In order to account for the effects of initial imperfection and local buckling, an incremental Newton-Raphson solution procedure is proposed.

lAssociate Professor, Department of Civil Engineering, University of Kentucky, Lexington, KY. 40506

2')Graduate Assistants, Department of Civil Engineering, University of Kentucky, Lexington, KY. 40506

263

264 FIFTH SPECIALTY CONFERENCE

The effects of brace stiffness on the buckling strength of columns are studied. In this paper, the bracing restraints are only applied at the mid-height of the column. Both I and channel sections are considered. Results on some typical cases are presented and discussed.

POST-BUCKLING STRENGTH OF PLATE ELEMENT AND EFFECTIVE STIFFNESS OF LOCALLY BUCKLED SECTION

The available post-local-buckling strength of the buckled plate element can be accounted for by the concept of effective width. Based on this concept (3, 17), the idealized uniform stress distribution acts only on the effective portion of the buckled plate element. Parts of the buckled plates will then be deleted throughout the length of the column. The remaining area is the effective section. The effective cross-section is prismatic throughout the column length under a concentric axial load.

If a bending stress distribution is imposed on a locally buckled column, which may happen due to initial imperfections, the compressive stress on the concave side is increased with a further reduction in effective width. On the other hand, unloading takes place on the convex side of the column and the effective width of the elastically buckled plate element is increased. Since the bending stress is varied along the column length due to lateral displacements, the effective section of the locally buckled column becomes nonprismatic. Therefore, the laterally deflected, locally buckled column has to be treated as a nonprismatic beam-column (5).

A generalized equation for effective width may be expressed as (17)1

b It = 0.95 J KEI a (1-O.95E; i JKE/ a ) e max w max (1)

for wit > 0.64 ~KE/a ,in which b :: effective width of compression element; t :: thiCkne~~ E = moduluseof elasticity; w :: flat width of the compression element; a :: maximum edge stress; K :: coefficient determined by boundary con~~ions and aspect ratio of compression element; and E; = modification factor. For values of wit smaller than 0.64 JKE/a ,b = w. Equation (1) has been shown through experimental verificati~ftXto ~e applicable to both stiffened and unstiffened elements if K is appropriately adjusted. For practical purpose, howeVer, E; may be taken as 0.22 and K may be taken as 0.5 and 4.0 for unstiffened and stiffened elements, respectively.

Based on the effective width concept, the effective stiffness of the buckled plate elements can be computed if the stresses in the corresponding plates are known. Knowing the effective width and the location of the shifted neutral axis, the effective moments of inertia about z and y axes, (I ) ff and (I) ,respectively, can be computed. The effective warpi~geconstant \Ce,f and St. Venant's torsion constant (J)eff can also be obtainedw(~;f7).

METHODS OF ANALYSIS

Bifurcation Analysis By ignoring the interaction between axial force and the lateral

LOCALLY BUCKLED COLUMNS

displacement, the upper bound solution of the buckling load can be obtained by a bifurcation analysis. Based on the finite element method, the critical load may be obtained from the following determinantal equation

(2 )

in which.[~J ~ ~tiffness matrix; [KG] = geometric stiffness_matrix; and:\ = ~nstab~ll.ty parameter or load factor. The matrices LKJ and LK ] presented by Barsoum and Gallagher (18) are used in this ingestigation. There are three translations, three rotations and a warping degree of freedom for the element. Therefore, a 14x14 element stiffness matrix is resulted. The stiffness matrices for the elements are assembled by the direct stiffness method to give the complete structure stiffness matrix. The stiffness coefficients of the bracing restraints (translational, rotational or torsional) are incorporated into the assembled structure stiffness matrix ac appropriate locations. It has been found that four elements are required between supports in order to obtain good results with less than 0.1% error in general.

In the post-local-buckling range the stiffness of the column depends upon the effective widths in the buckled plates. Furthermore, the applied load and the effective stiffnesses of the column are interdependent. Therefore, an iterative solution procedure is necessary. Consequently, the coefficients in C~J and LK ] must be revised continuously according to the current applied load and the corresponding effective section during the process of iteration.

265

A determinant search method which was proposed by the authors and was used successfully in the previous investigations, has also been used in this investigation. The details of the method and the solution procedure were reported elsewhere (6, 7, 19).

Incremental Analysis In order to account for the effects of initial imperfections and

lateral displacements of braces, an incremental analytical procedure has been developed in conjuction with Newton-Raphson's procedure for convergence at each step of load increment. The proposed solution procedure which is also applicable to other complex framed structures includes the following steps I (1) For the first load increment, a linear analysis is performed

in which C~J = stiffness matrix; [r] = displacement matrix; and LP J = load matrix. (2) Determine member forces from calculated displacements. (3) From calculated member forces, determine stresses in plate

components and the corresponding effective widths by iteration. (4) Repeat steps (1) through (J) until the p~oces~ is converg~d. The

updated stiffness matrix and the geometr~c st~ffness matr~x are obtained.

(5) The unbalanced loads are calculated by

266 FIFTH SPECIALTY CONFERENCE

[~Q] = [p] - ([~] + ~ [~]) [r] (4 )

in which [~Q] = unbalance~ loads; [K.".] = updated stiffness mat~ix for the buckled section; L~] = updated geometric matrix; and Lr] = updated displacement matrix. _

(6) Calculate additional displacements LAr] due to unbalanced loads by

(5)

(78) D:ltermine additional member forces LAF] from calculated CAr]. ( ) The total member forces, [F] + [~F], and the total displacements of

the structure, [r] + [~r] are obtained. (9) Calculate stresses in plate components and the corresponding _

effective width by iteration based on the total member forces, [F] + CAF].

(10) Update [~] and [KG] and repeat steps (6) through (9) until the process converges.

(11) ~culate unbalanced loads due to ulldated total displacements Lr] in step (5) and repeat steps (6) through (10) until the additional displacements due to unbalanced loads are within acceptable tolerance.

(12) Use the tangent stiffness matrix ([K.".] + [~]) obtained at the end of last load increment for the next toad increment. Start from step (6) using the load increment as the unbalanced load. The process is continued until the determinant of the tangent stiffness matrix becomes singular.

Computer programs for the IEM370/165 and DEC-IO computer systems following the procedures outlined have been prepared. Results have been obtained for both bifurcation and incremental analyses for some typical cases. It appears that the analytical solution schemes developed are well suited for the problem considered.

RESULTS FROM BIFURCATION ANALYSIS

Column with Doubly Symmetric I Section Figs. 1 and 2 show the ratio of increase of overall buckling load

for the braced column over the unbraced column. The column is 84" (2.1)4m) long and simply supported at both ends. The lateral restraints are applied at the mid-height of the column. In the figure, p' and (pI )1 are defined as the buckling loads for the braced column witnout and cr with consideration of local buckling, respectively. P and (p )1 are the buckling loads for the unbraced column without andc~ith cr consideration of local buckling, respectively. ~or this particular case, P and (p )1 are 44.01 kips (195.84 x 10 N) and 39.91 kips (177.60c~ 10) N)~rre5pectivelY. Both values are first mode buckling loads about y axis, i.e. P and (p )1' respectively. In the figure, B is defined as the ratio orYthe com~ined stiffness of the external restraint and the column to the column stiffness alone. Therefore, there is no restraint applied if B = 1. In the figure, Bland B 2 are refered to the translational restraint in the z direction and the torsional restraint about the x axis, respectively.

LOCALLY BUCKLED COLUMNS 267

The upper curve in Fig. 1 is for the fully effective column ignoring local buckling while the lower curve is for the locally buckled column. The overall buckling load increases considerably with a brace only possessing a very small fraction of the column stiffness. With only a translational restraint applied in the the Z direction at the mid height of the column, the buckling mode is forced to ch~e from flexural buckling (p', (pI )1) to torsional buckling (pI, (pI) ) for the locally unbuckled an~ bucIied columns with Bl = 1.006 ~d l~O!, respectively. Beyond these values, the translational stiffness of the brace does not affect the column buckling load. However, the buckling load is increased further if an additional torsional restraint is also applied at the midspan of the column. The buckling mode is changed from torsional buckling (p~'(~~)l) to the 2nd mode of flexural buckling (P~, (P~)l) as shown in ~~g. 2 with B2 = 1.00}5 and 1.001, respectively, wh1le holaing Bl =1.1.

It is of interest to point out that there is no difficulty to provide these required stiffnesses for the brace since they are only small fractions of the column being braced. On the other hand, it is also important to point out that the 2nd mode flexural buckling load for the locally buckled column is only twice that of the unbraced locally buckled column.

Column with Singly Symmetric Channel Section Fig. ) shows the increase of buckling load with an increase of

torsional restraint at the midspan for a channel column. The coupled torsional-flexural buckling mode is possible since the channel section is singly symmetric. The lowest torsional-flexural buckling load is defined as Pl' If tje column is unbraced, the buc~ling loads are 14.82 kips (55.95 x 10 N) and 9.90 kips (44.06 x 10 N), respectively, for the cases without and with consideration of local buckling. They are corresponding to Ptfl and (p )1' With Bl = 1.0) and 132 = 1.00, the buckling mode for the case with rocal buckling changes from flexural mode, (p )1' to torsional-flexural mode, (Ptf1 )1 with a value of 10.80 kips (48~05 x 10) N). Since the coupled torsional-flexural buckling governs, any additional translational restraint in the z direction will not further increase the buckling load. The only way to further increase the buckling load is to apply a torsional restraint at midspan. Again, only a small amount of torsional stiffness is required to force the column to buckle into 2nd mode. It is of interest to note in Fig. } that the governing buckling mode for the locally buckled case is again switched back to flexural mode (2nd) with a modest increase in buckling load. For the case ignoring the effects of local buckling, the torsional-flexural buckling mode continues to govern the behavior. The buckling load can be as high as }. 5 P cr'

EFFECTS OF INITIAL IMPERFECTION

It should be noted that in the preceding analyses, the locally buckled column is kept straight throughout the whole loading range until the column buckling load of the effective cross-section is reached. Due to column initial imperfections, the straight form equilibrium position may not be possible. 'Ibis could also happen for a perfectly straight column with initial imperfections in the component plate elements or

268 FIFTH SPECIALTY CONFERENCE

with any distu~bance during loading process. It is, therefore, necessary to study the post-local-buckling behavior of thin-walled columns containing imperfections since such imperfections always exist.

Utilizing the incremental procedure developed, results are obtained for columns with various initial imperfections and various brace stiffnesses. Fig. 4 shows load-displacement curves for the column shown with an initial imperfection and various A /A ratios. A and A are cross-sectional areas for the brace at mid~pag and the cofumn berng braced, respectively. The load-deformation curves approach the buckling loads based on the bifurcation analyses as upper bound limiting values. Again, two sets of curVes are presented, i.e. with or wi thout consideration of local buckling ,"ffects. Due to local buckling effects, the buckling loads . (p' \ are much smaller than the buckling loads P' • This is especiallyCfio~eable for columns with braces with stiffneg~es adequate enough to provide full support.

Fig. 5 shows load displacement curves for the same column with various initial imperfections. The ratio of A /A is kept as a constant value, i.e. 0.00017. Again two sets ~f gurves ~re provided for the column with or without consideration of local buckling effects. The effects of the magnitude of initial imperfection on the behavior of columns can be seen from these curves.

EFFECTS OF RESTRAINT STIFFNESS

In Fig. 6, the increase of buckling load for a braced I section column (Figs .'~ 2-nd 5hs plotted against the increase of cross-sectional area of the brace at midspan. Only the translational restraint ~n the z direction is provided. The cross-sectional area A required for providing full lateral support for the column ignori~ the local buckling effects is only around 0.070% of the area of the column being braced. If local buckling is considered, the increase of buckling load for the braced column is much less pronounced. The brace provided for the case without considering local buckling effects appears to be more than adequate for the case considering local buckling effects.

The required value for A for full lateral support recommended by Winter (10, 20) can be obtain&d from the following equation

2 Is --..;1=--_::-A/Ac = 21f (1) «1 /r )2) (6)

c c c in which r = radius of gyration of the column; 1 , 1 = lengths for the brace Rnd the column, respectively. Using th~ diI\iensions shown in Figs. 4 and 5, the calculated ratio A /A from Eq. 6 is 0.0713% which is almost identical with the value ob~aiRed. Eq. 6 can also be rewritten as follows

EI .....:J... I)

c

(7)

in which I = moment of inertia of the column about y axis. This value is only l/~ of that recommended by Mutton and Trahair (16) for full lateral support.

LOCALLY BUCKLED COLUMNS 269

Fig. 7 shows the lateral displacements corresponding to the various As/Ac ratios for the column considered. These displacements are corresponding to the 95% of the calculated bifurcation buckling loads. With adequate bracing stiffness, for instance A/A = 0.1%, the lateral displacement for the column is correspoRdiRg to the 2nd buckling mode. With inadequate lateral bracing stiffness, the lateral displacement will result at the bracing point and the magnitude will depend upon the initial imperfection and the brace stiffness.

CONCLUSIONS

Methods on bifurcation and incremental analyses for locally buckled columns have been developed and presented. An incremental Newton-Raphson's procedure has been used for convergence at each step of load increment. The analytical solution schemes developed are..rell suited for the problem considered, i.e. interactive behavior of columns involving local and overall buckling, lateral bracing restraints, and initial imperfections. The methods can be applied to other complex structural systems.

The re~uired stiffness for braces in order to provide full bracing effects is only a small fraction of that for the column being braced. The same conclusion has also been found for the locally buckled columns although the benefits are relatively less pronuounced than that for the column without the effects of local buckling. The governing buckling mode may change with the effects of local buckling, bracing stiffness, and type and combination of restraints applied.

Using the incremental procedure, the displacement of the buckled column can be determined. The load-displacement curve approaches the buckling load based on the bifurcation analysis as upper bound limiting value. With such displacements, forces developed in the braces can be determined. Further investigations on the related subjects are being conducted.

270

(I)

(2)

(3)

(4 )

(5)

(6)

(7)

(8)

(10)

(11)

(12 )

(13)

(14)

(15)

(16)

(17)

(18)

FIFTH SPECIALTY CONFERENCE

APPENDIX-REFERENCES

Column Research Committee of Japan, Handbook of Structural Stability, 1971. Johnston, B. G., editor, Guide to Stability Desi~ Criteria for Metal Structures, John Wiley and Sons, 3rd Edition, 197 • Winter, G., Cold Formed Steel Design Manual, Commentary, American Iron and Steel Institute, New York, 1971. Timoshenko, S. T. and Gere, J. M. Theory of Elastic Stability, McGraw-Hill Book Co., 1961. Wang, S. T., and Tien, Y. L., "Pos t-Local-Buckling Behavior of Thin-Walled Columns," Proceedings, Second Specialty Conference on Cold-Formed Structures, St. Louis, Missouri, October, 1973, pp 53-81. Wang, S. T., and Pao, H. Y., "Stability Analysis of Locally Buckled Singly Symmetric Columns," Thin Walled Structures, edited by J. Rhodes and A. C. Walker, Granada Publishing Limited, London, 1980, pp 14-23. Wang, s. T., and Pao, H. Y., "Torsional-Flexural Buckling of Locally Buckled Columns," International Journal of Computers and Structures, in press. Green, G. G., Winter, G., Cuykendall, T. R., "Light Gauge Steel Columns in Wall Braced Panels," Bulletin No. 35, Engineering Experimental Station, Cornell UniVersity, Ithaca, New York, October, 1947. Zuk, W., "Lateral Bracing Forces on Beams and Columns," Journal of the Engineering Mechanics Division, ASCE, Vol. 82, No. EM 3, July 1956. Winter, G., "Lateral Bracing of Columns and Beams," Transactions, ASCE, Vol. 125, Paper No. 3044, 1960, pp 807-845. Horne, M. R., and Ajmani, J. L., "Stability of Columns Supported Laterally By Side-Rails," International Journal of Mechanical Sciences, Vol. II, No.2, February, 1969. Toakley, A. R., "Buckling Loads for Elastically Supported Struts," Journal of the Engineering Mechanics Division, ASCE, Vol. 91, No. EM3, June 1965. Urdall, T. B., "Bracing of Continuous Columns," Engineering Journal, AISC, Vol. 6, No.3, July 1969. Nethercot, D. A., and Rockey, K. C., "Finite Element Solutions for the Buckling of Columns and Beams," International Journal of Mec.'.1c,nical Sciences, Vol. 13, 1971. Nethercot, D. A., and Rockey, K. C., "The Lateral Buckling of Beams Having Discrete Intermediate Restraints," Journal of the Institution of Structural Engineers, Vol. 50, No. 10, October, 1972. Mutton, B. R., and Trahair, N. S., "Stiffness Requirements for Lateral Bracing," Journal of the Structural DiVision, ASCE, Vol. 99, No. ST 10, October 1973. Wang, S. T., Errera, S. J. and Winter, G., "Behavior of Cold-Rolled Stainless Steel Members," Journal of Structural Division, ASCE, Vol. 101, No. ST II, 1975. Barsoum, R. S., and Gallagher, R. H., "Finite Element Analysis of Torsional and Torsional-Flexural Stability Problems," International Journal of Numerical Methods in Engineering, Vol. 2, 1970

LOCALLY BUCKLED COLUMNS

(19) Wang, S. T., and Wright, R. S., "Torsional-flexural Buckling of Locally Buckled Beams and Colwnns," Proceedings, ASCE, International Colloquium on Stability of Structures Under Static and Dynamic Loads, Washington, D. C. 1977.

(20) McGuire,W., Steel Structures, Prentice-Hall, 1968.

271

272

~ .......

I-t "",0

'-'

---....... ~ -

I-t 0

"'" I-t 0

I-t 0

"'" ---- I-t 0

"'"

....... ~ I-t U

"'" '-"

---....... ~ -

"'" '-'

I-t o

I-t 0

3.0

2.0

1.0

4.0

3.0

I-t 2.0 "",0

---

1.0

FIFTH SPECIALTY CONFERENCE

0'1".J~. "I .2" 7.8

0.1 "T t.---..I B = 1 4" 2

~-o----~----------~~~~

p'

pI

" B 2 = 1

'h--~----~'\--O---4.~-~ I ,--,p')

pZ(Py)l "t __ without local buckling

/ --- with local buckling

1.00

Fig. 1.

1.02 1.05

Column Buckling Load Ratio vs. Various Lateral Restraints in z Direction at Mid-span ( B 2= 1.0)

p' (II mode) y

0.1" I.L~x 0.2" 7.8" z

0.1" rL--J

1.1

_0~"'--(p~(P?lII mo~)----- ----- 61 = ?l ,,:e. without local buckling

with local buckling

1.00

Fig. 2.

1.002 1.004 1.006 1.008 1.010 6

2

Column Buckling Load Ratio vs. Various Torsional RestraintS at Mid-span ( 6 1= 1.1)

..... III .....

k o

4.0

LOCALLY BUCKLED COLUMNS

B 1 1.03 ~~~~.~, --------~~---0

P tfl (II mode)

1.0 .L-______ -4 ________ L-______ ~ ______ ~ __________ ~

60

1.00

Fig. 3.

1.01 1.02 1.03 1.04 1.05

Column Buckling Load Ratio vs. Various Torsional Restraints at Mid-span ( B 1= 1.03)

p' = 74.00kips r-----------------------------------------~ cr (A /A = 0.0005 )

8 cy

P'- >.

50"f.g'~075"-I:E: 0.15"

50" 7 .85" 0.075".~

= 35.30 kips

29.20 kips 24.60 kips

4 4l.05kifs (0.0003 33.90kips (0.0001)

~ 20 18.97 kips 23.40kips (0.0)

o 0.00

Fig. 4.

0.02 0.04 0.04

---without local buckling _____ with local buckling

0.06 0.08 0.10

Mid-span Displacement in z Direction, in.

- Axial Load vs. Lateral Displacement - with Various Support Stiffnesses (1 kip = 4.45 x 103 N, 1 in. = 2.54 em)

273

274

., P-.....

..>j

-0 01 0

...:l

.... 01 ..... ~

,.... .... '"' U p..

'-' "-

,.... .... -

'"' p..u '-'

'"' 0

'"' p..u ...... -

'"' p..u

FIFTH SPECIALTY CONFERENCE

pI = 41.05 kips 40.0 cr

20 •

O.

4.0

3.0

2.0

1.0

0.00 0.02

- - - --SOli 7.85" Z

---,._--- -~i'~O }075

11.l£:Y

P50" 0" I XO.lS'

-- without local buckling .075'~~

---- with local buckling

0.04 0.06 0.08

A fA = 0.00017 s c

0.10 0.12 0.14

Mid-span Displacement in z Direction, in.

Fig. 5. - Axial Load vs. Lateral Displacement with Various Initial Imperfections ( 1 kip = 4.45 x 103 N, 1 in. = 2.54 cm)

0.0

----without local buckling

---- with local buckling

0.0002 0.0004 0.0006 0.0008 A fA

0.0010 0.0012 0.0014

0.0 0.1

s c

fA -= 0.00010 s c 0.00025

0.2

0.00050 0.00069 0.00100

Lateral Displacement in z Direction, in.

Fig. 6. - Column Buckling Load Ratio vs. Support Cross-sectional Area Ratio

Fig. 7. - Lateral Displacement vs. Various Support Stiff­nesses (1 in. = 2.54 cm)