design of diagonal cross bracings_part 1 theoretical study.pdf

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Design of Diagonal Cross Bracings

Part 1: Theoretical Study

A. PICARD and D. BEAULIEU

Diagonal bracing is commonly used in steel structures toesist horizontal loads. In current practice, the design of this

ype of bracing system is based on the assumption the

ompression diagonal has negligible capacity and the tension

diagonal resists the total load (Fig. 1).

If the diagonals are connected at their intersection point

usual practice), this design procedure is conservative because

he effect of this connection on the out-of-plane buckling

apacity of the compression diagonal is ignored.1,2,3,4

The

estraint provided to the compression diagonal by the loaded

ension diagonal is generally sufficient to consider that the

ffective length of the compression diagonal is 0.50 times the

diagonal length (KL = 0.5L) for out-of-plane buckling as for

n-plane buckling. Analytical and experimental results1,2

have

lso shown the ultimate horizontal load on the bracing system

s much higher than the horizontal component of the yielding

trength of the tension member, because of load sharing

etween the diagonals. The assumption that the compression

diagonal has negligible capacity usually results in overdesign.

This paper presents the results of a theoretical study

imed at the determination of the transverse stiffness offered

y the tension diagonal in cross-bracing systems and at the

valuation of the effect of this stiffness on the out-of-plane

uckling resistance of the compression diagonal. The theory is

upported by seven transverse stiffness tests and 15 buckling

ests. The test results are reported in the second part of the

aper.

BUCKLING OF THE COMPRESSION DIAGONAL

n double diagonal cross bracing, the tension diagonal acts as

n elastic spring at the point of intersection of the compression

diagonal as shown in Fig. 2a, where is the spring stiffnesskips/in. or kN/mm). If = 0, then K= 1.0 (Fig. 2b) and thelastic critical load Cceis equal to:

C EIL

Cce c e= =2

2 (1)

. Picard is Professor of Civil Engineering, Laval University,

Quebec, Canada.

D. Beaulieu is Associate Professor of Civil Engineering, Laval

University.

Fig. 1. Usual design hypotheses

Fig. 2. Buckling modes for compression diagonal

22 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCT

3 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the


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In this equation, Ic is the moment of inertia of the

ompression diagonal considering out-of-plane buckling.

If = , then K = 0.5 and the elastic critical load isqual to:

CEI

LCce

ce= =

44

2

2

(2)

Therefore, the effective length factor is given by:

KC

C

e

ce

= (3)

Timoshenko and Gere5 demonstrated there is a limit

pring stiffness above which the elastic critical load is 4CeFig. 2c). In other words, the spring does not have to be

nfinitely stiff to obtain K= 0.5. The limit spring stiffness is

iven by:5

lim=16 2

3

EI

L

c

(4)

Let us define the non-dimensional spring stiffness as:

=

L

Ce

(5)

From Eqs. 1, 4 and 5, the limit value of the non-

dimensional spring stiffness is equal to:

lim lim= =L

Ce16 (6)

If we assume a linear relationship between Cceand , webtain:

C C Cce e e= +

13

164

(7)

Fig. 3. Relationship between elastic buckling load and non-

dimensional spring stiffness

The variation of the dimensionless elastic buckling loa

shown in Fig. 3 in terms of the dimensionless spring stiff

The exact solution obtained by Timoshenko and Gere is

shown in the figure. It can be seen that Eq. 7 is slig

conservative.

Using Eqs. 3 and 7, the effective length factor becom

K=+

4

16 305

.

The variation of K is plotted in Fig. 4 against

dimensionless spring stiffness. The exact solution, give

Refs. 5 and 6, is also shown in the figure.

TRANSVERSE STIFFNESS OF

THE TENSION DIAGONAL

To determine the parameters and , we will considprismatic member subjected to a transversely applied p

load Qand a concentrically applied tensile force T,as sh

in Fig. 5. For the problem studied here, Q represents

transverse force transmitted to the tension diagonal by

compression diagonal at buckling.

The general solution of the differential equation obta

from equilibrium of the tension member shown in Fig. classical, assuming elastic behavior. The elastic beha

assumption is easily satisfied for X-bracing, since

diagonals are identical. When the compression diag

buckles, the tension diagonal is still behaving elastically.

The transverse stiffness or spring stiffness provide

the tension diagonal is obtained from the solution of

differential equation and is given by:1,2

=

163

3EI

L

v

v vt

( )tanh

Fig. 4 Relationship between effective length factor and non-

dimensional spring stiffness

THIRD QUARTER / 1987



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where vis defined as:

vL T

EIt

2

2

2=

(10)

In Eqs. 9 and 10, E is the elastic modulus of the

material and It is the moment of inertia of the tension

diagonal, considering out-of-plane bending of the X-

racing.

Combining 1, 5 and 9, we obtain the following equation

or the evaluation of the dimensionless spring stiffness:

=

16

2

3I

v tanh

t

cI

v

v( )(11)

n X-bracing systems, both diagonals are identical and It =

c= I.However, in X-trusses supporting loads which always

ct in the same direction, the compression diagonal could be

tiffer. Since this paper deals with double diagonal bracings,

Eq. 11 can be rewritten as:

=

162

3v

v v( )tanh(12)

Fig. 5. Tension member subjected to transverse force at mid-span

The following approximation for Eq. 12 is proposed

= +16

3 1092

2( . )v

Eqs. 12 and 13 are compared in Fig. 6. It can be seen Eq

very closely approximates Eq. 12 when 0 v2 80. practical range for parameter v

2was evaluated as follow

Let us first assume the out-of-plane slenderness

of the compression diagonal, i.e., both diagonals, should

exceed 200. Since the diagonals are connected at intersection, we get:

L

r

2200

L I

A240 000

2

The maximum force in the tension diagonal being AFy,

10 together with Eq. 14 andE= 29 000 ksi leads to:

v

I

A

AF

EI Fy

y

2

40 000 138

= .

Considering a maximum value forFyof 58 ksi, then

v280.

In the previous equation, the constant 1.38 is in (ksi)1

.

units are usedE= 200 000MPaand the constant is equ

0.20 (MPa)1

. ThenFyis inMPa.

Fig. 6. Comparison of Eqs. 12 and 13

24 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTI



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Eq. 13, being an acceptable approximation (Fig. 6), will

e used in the theoretical analysis. Since Ic = It = I,

ombining Eqs. 1, 5, 10 and 13 gives:

= = +C

L

EI

L L

e 48 4 363

.T

(15)

t is interesting to note the first term in Eq. 15 represents the

lexural stiffness of the tension diagonal. Then T = 0, the

pring stiffness is equal to the flexural stiffness. Therefore s never equal to zero and Kis always smaller than 1.0 (min

= 48 EI/L3; from Eq. 5 or Eq. 18, min = 4.86; from Eq. 8,Kmax= 0.72 (see Fig. 4).

The results of the transverse stiffness tests reported in the

econd part of the paper demonstrate the validity of Eqs. 13

nd 15.

Eq. 10 can be rewritten as:

eC

T

EI

TLv 47.2

2 2

2

2 =

= (16)

Introducing Eq. 16 into Eq. 13 gives:

= +4 86 4 36. .T

Ce

(17)

From Eq. 3, Ce = K2Cce. Therefore, Eq. 17 can be

ewritten as:

= +

4 86 4 36

2. .

T

K Cce

(18)

Combining 8 and 18 gives:

KC Tce

= 05230428

050..

. (19)

During the buckling tests reported in the second part of

he paper, the force in the tension diagonal was kept constant.Therefore, the value of T is known. If the compression

diagonal buckles in the elastic range, Eq. 19 can be used to

determine the effective length factor (Cce= Ccrwhere Ccr is

he measured buckling load). For inelastic buckling, Eq. 19 is

not valid. However, the measured inelastic buckling load can

e introduced into AISC equations7 (safety factor removed)

r into CSA equations8 (resistance factor removed) to

determine the effective length factor. With this value of K,

wo values of Ccecan be computed. One is obtained from Eq.

9 and the other from the following equation: Cce = 2

EI/(KL)2. These two values are compared to evaluate the

ccuracy of Eq. 19.Two final comments should be made concerning the

revious equations. These equations are valid for continuous

diagonals which are connected at their intersection point. If

ne diagonal is interrupted at the intersection point, constant

tiffness is not maintained and the connection at this point

may become a weak link when the interrupted diagonal is

ompressed.3

In the derivations the rotational restraint of the

connections as ignored. In Ref. 4, the rotational restraint

relatively important, and test results suggest use o

effective length factor of 0.85 times the half diagonal len

i.e., K = 0.425 if the total length of the diagona

considered.

COMPRESSION-TENSION RATIO

Since the behavior of the braced frame shown in Fig.

elastic up to buckling of the compression diagonal, the Cce/T is equal to the ratio of the force in the compres

diagonal Cto that in the tension diagonal Tobtained from

elastic analysis of the frame. Eq. 19 can thus be rewritten

KC T

= 05230428

050..

. (

Eq. 20 is plotted in Fig. 7. As seen, when the C/Trat

larger than 1.6, the effective length factor increase

reaches its maximum value when the C/Tratio is equal to

Fig. 7. Effective length factor vs. C/T ratio

Fig. 8. Variation in C/T ratio with A/Abratio (Eq. 22)

THIRD QUARTER / 1987



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nfinity, which corresponds to the minimum spring stiffness,

qual to the flexural stiffness of the tension diagonal (T = 0

n Eq. 15 or Eq. 17).

By considering the elastic deformations of the braced

rame shown in Fig. 1, Vickers3 derived the following

quation to compute the C/Tratio:

C

T A

A

A

A

b

c

= ++

1

1

3

3

cos

( sin )

(21)

The parameters of this equation are defined in Fig. 1. Eq. 21

was derived for a load pushing against one side of the frame

s in Fig. 1. This is the most critical loading condition for the

design of the compression diagonal because the C/T ratio

xceeds one.3

In a typical braced frame, the area of the cross section of

he columns Ac is usually much greater than the area of the

ross section of the diagonals A. Moreover, sin3 is always

maller than one. Consequently, the term Asin3/Acin Eq. 21

s small compared to 1 and can be neglected with only a

minor loss in accuracy resulting in errors which are on the

onservative side. Eq. 21 becomes:

C

T

A

Ab= +

1 3cos (22)

Eq. 22 is plotted in Fig. 8. It can be seen that large values of

Ab, i.e. small values of the A/Ab ratio, tend to equalize the

orces in the diagonals. In practical situations, the A/Ab ratio

s much smaller than one. Therefore, as shown in Fig. 8, the

C/Tratio is usually smaller than 1.6 and the Kvalue is equal

o 0.5 (Fig. 7).

CONCLUSION

The theoretical study reported in this paper shows that in

double diagonal bracing systems the effective length of the

ompression diagonal is 0.5 times the diagonal length when

he diagonals are continuous and attached at the intersection

oint.

Tests were done to demonstrate the validity of

equations used to determine the transverse stiffness or sp

stiffness provided by the tension diagonal and the validit

the equations used to determine the effective length fa

The results of these tests are reported in the second pa

the paper.

ACKNOWLEDGMENTS

A part of the results presented in the paper are drawn f

the Master's thesis of Pierre Moffet. The writers are than

to Moffet and to the Natural Sciences and Enginee

Research Council of Canada for its financial support.

REFERENCES

1. DeWolf, J. T. and J. F. Pellicione Cross-Bracing DesignA

Journal of the Structural Division, Vol. 105, ST7, July

(pp. 1,379-1,391).

2. Kitipornchai, S. and D. L. Finch Double Diagonal

Bracings in BuildingsResearch Report No. CE56, Depar

of Civil Engineering, University of Queensland, Austr

November 1984.3. Vickers, D. G. Tension-Compression Cross Bracing Usin

AnglesMechanical Engineering Project Report, Departme

Civil Engineering, McGill University, May 1982, Montral

4. El-Tayem, A. A. and S. C. GoelEffective Length Factor f

Design of X-bracing SystemsAISC Engineering Journal

Qtr., 1986, Chicago, Ill. (pp. 41-45).

5. Timoshenko, S. P. and J. M. Gere Theory of Elastic Sta

McGraw-Hill Book Co., 1961, New York,

(pp. 70-73).

6. Moffet, P. and D. BeaulieuRsistance au flambement de p

comprimes supportes en divers points par des pices te

Report GCT-06-03 (in French), Department of

Engineering, Laval University, April 1986.

7. American Institute of Steel Construction, Inc. Specificatiothe Design, Fabrication and Erection of Structural Ste

Buildings 1978, Chicago, Ill.

8. Canadian Standards Association Steel Structures

BuildingsLimit States Design Standard CAN3-S16.1,

Rexdale, Ontario.

26 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTI

3 by American Institute of Steel Construction Inc All rights reserved This publication or any part thereof must not be reproduced in any form without the written permission of the

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