OUT OF PLANE BUCKLING OF

CROSS BRACING MEMBERS

Fathy Abdelmoniem Abdelfattah
Associate professor
Civil Engineering Department
Faculty of Engineering at Shoubra, Banha University

ABSTRACT
This study proposes simple technique for obtaining out of plane critical load of the
compression member in cross bracing members. Nonlinear large displacement finite
element analysis was performed using ANSYS program [10]. The results show that
the geometrical properties of the member’s cross-sections, the ratio of the tension to
compression loads induced in the members and the supporting conditions at their ends
affect significantly the out of plane critical load value. The intersection connection of
the two members is assumed to provide full continuity. An analytical simple model is
used and modified to deal with symmetrical and unsymmetrical cross bracing
members with different end conditions. The results obtained using this method show
good agreement with those obtained from the finite element analysis.

‫دت‬ ‫د د‬ ‫تقدذ ذدزا درذسد در قش قدر ة دحسر رح د ا نبدا دألعضاد ا ألربد درتدشةحة دربتق قادر ر دذ معضا‬
‫ تم م تخذد قش قر درا صش دربحذودا درد طسحدر و درتد تتبدبث تداألحش دألصدند‬. ‫تق قا‬ ‫ت‬ ‫درب د رل‬
‫مربد‬ ‫در ذ دحر رقس رد‬ ‫د من درخد د‬ ‫ در تد ج درب دتخش ر ت‬. ‫د‬ ‫رتبثحا ل ك مرب درتشةحة ر ذ معضا‬
‫منذ مربد درتدشةحة درد قد ا دربد ة د درابد دألطدش ة أل د ر درد‬ ‫درتشةحة و ع ضر ق ا درشذ درت قذ ت ت‬
‫من در صددلر ةددحث‬ ‫ تددم م تددشد‬.‫دألربد تددبألش تدداألحشد مضحددشد رلد قحبددر نبددا دألعضا د ا‬ ‫عد ا درشم د جض ر ددذ مقددشد‬
‫ تدم م دتخذد عبد را ة دحة رح د ا نبدا دألعضاد ا و تاذ لدر أل ن عحدر م دتخذد ر ألربد‬.‫تبشا‬ ‫مرب درتشةحة‬
‫ در تد ج درب دتخش ر دث ذدزا درسش قدر‬. ‫درتشةحة دربتب أللد و يحدش دربتب أللدر و درتد ر د سمد جض ختل در ر دذ مقشد د‬
.‫تت د ق ع تلل درب تخش ر ث قش قر درا صش دربحذودا‬

Keywords

Cross bracing members, out of plane buckling, compression member, critical load,
modeling of buckling,

1
INTRODUCTION
Diagonal cross bracing members are commonly used in structural steel works to resist
horizontal loads and/or to reduce effective unrestrained length of compression
members. This would subject one member to tension load and the other one to
compression. In design practice, the compression member is commonly assumed not
to be effective. Only the tension member is designed. The compression member is
taken typical to the tension member. The Egyptian code for steel construction and
design ECP [1] prohibit the use of rods and cables in bracing systems. For members in
buildings, designed on basis of tension, the ECP [1] and the American specification
AISC [2] specify that the maximum slenderness ratio λ should not exceed 300.

Diagonal cross bracing members are repeated many times in steel structures.
Including the contribution of the compression diagonal would produce economical
design. In seismic areas, it is important to predict in which plane the system will
buckle. The end connection should be detailed to permit ductile rotation in the
buckling plane. If buckling occurs in the perpendicular plane, the connection may
fracture prematurely. The compression diagonal should be designed against in plane
and out of plane buckling.

Several studies in the literature show that the diagonal tension member provides
degree of restraint to the compression member against out of plane buckling.
Different expressions have been derived. Timoshinko and Gere [3] used differential
equations to find the relationship between the critical load and brace stiffness for a
column with mid height brace. They showed that there is a limit spring stiffness above
which the spring would behave as if it were a hinged support. Winter [4] proposed
simple model to find the value of this stiffness. Stoman [5], [6] and [7] employed
Raleigh – Ritz method of stationary potential energy to formulate closed form
stability criteria for evaluating the transverse stiffness provided by the tension brace.
Picard [8] concluded that the effective length of the compression diagonal is 0.5 times
the diagonal length for both out of plane and in plane buckling. The same conclusion
was drawn by El Tayem et. al. [9]. Most of these studies dealt with the problem as a
two dimensional problem. The members are assumed to have identical length, cross
sections and material properties. Further, one diagonal is under tension while the other
one is subjected to compression. The intersection of the two members is at half-length

2
and provides full continuity. The supporting conditions of each member are hinged at
one end and roller at the other one.

This study investigates the parameters that affect the out of plane critical load PC of
the compression member in cross bracing members. Buckling is assumed to occur
about one of the cross section principles’ axes. Bracing members of single angles are
not included in this study. A three-dimensional finite element analysis was performed.
An analytical simple model is used and modified to deal with cross bracing members
when they are symmetrical and unsymmetrical and having different end conditions.

FINITE ELEMENT ANALYSIS

(i) Mesh
Buckling behavior of cross bracing members is modeled using nonlinear large
displacement elastic finite element analysis. The ANSYS program [10] was used to
perform the analysis. Figure 1 shows two crossing members, B and C. Each member
was modeled using 20 uniaxial beam elements with tension, compression, torsion and
bending capabilities. The element has 6 degrees of freedom at each node: translations
in the directions of and rotations about the node X, Y and Z-axes. Large deflection
capabilities of the element are activated. Different supporting conditions were
considered. The material behavior was modeled to be elastic. The modulus of
elasticity E value is taken equal to 2100 t/cm2 according to the ECP [1]. Concentrated
compression load P was applied at point C1, figure 1. In some cases, tension load T
was applied as well at point B1. This is to simulate the behavior when compression
force induces in one member and tension force induces in the other one.

(ii) Modeling of buckling

Two cases of initial out of straightness were considered for the mesh of the
compression member. First, the imperfection displacement field was given a half sine
wave along member C in the Y - Z plane. The buckling in this mode represents the
case of a hinged – hinged column and named mode 1 of buckling. Second, a full sine
wave was superimposed along member C. The buckling in this mode represents the
case of a hinged – hinged column supported at its middle by a hinged support and
named mode 2 of buckling. The maximum initial imperfection of the mesh was made
equal to 1 / 500 of the member length L. This technique is used in the literature and

3
known as the seeding technique [11] and [12]. Incremental nonlinear solutions for the
two cases were obtained and evaluated for buckling. This is to guarantee that the
buckling occurred is the first buckling mode, which happens in practice. The critical
load PC is defined to occur at a load corresponding to very large deflection at which
the tangentt of the load – deflection relationship is smaller than a specified tolerance.
–3
This tolerance was set to 2 X 10 KN/mm. The Newton-Raphson technique was
used for equilibrium convergence with a tolerance limit of 0.01. The convergence
criteria were based on checking forces [10]. The model was verified against standard
cases.

PARAMETRIC STUDY
The parameters considered are the geometrical properties of the members’ cross-
sections, ratio of the tension to compression loads induced in the members and the
supporting conditions. The connection at the intersection of the two members is at
their half-length and assumed to provide full continuity to both of them. The effect of
this continuity is also examined.

1- Cross Section Geometrical Properties

Members B and C in figure 1 were given the same length and material properties.
Different values for the moments of inertia of member C about the X and Y-axes,
(IX)C and (IY)C, and those of member B about the Y and Z-axes, (IY)B and (IZ)B, were
considered. Hinged supports were provided at points B2 and C2. Roller supports are
provided at points B1 and C1 that allow transition in the directions of the X and Z axes
respectively. Rotations about the X, Y and Z-axes were allowed at all the supports.

(i) Effect of (IZ)B

The moments of inertia (IX)C , (IY)C and (IY)B were made constant and given the same
value. The value of (IZ)B was varied. The critical load PC values corresponding to the
out of plane buckling were obtained. The results are presented in figure 2 in terms of
the ratios (IZ)B / (IX)C and PC / PE where PE is the Eular load of member C; i.e. PE = π2
E (IX)C / L2. The results show that the values of PC / PE are linearly proportional to
(IZ)B / (IX)C to a certain limit after which the value of PC / PE become nearly constant.
At (IZ)B / (IX)C = 0.1, the value of PC is equal 108% of PE . This means that member B
provides relatively low degree of support to member C against out of plane buckling.

4
For (IZ)B / (IX)C ranging between 3.4 to 4, the values of PC / PE are found to vary
between 3.92 and 3.95. The buckling in this range occurred in mode 2. Member B
restrains member C as if it were a hinged support. Increasing the value of (IZ)B
further did not cause significant change in the critical load and buckling mode. By
reviewing the values of the reactions at the different supporting points for the
different cases considered, the following notes were noticed. At buckling, the
reactions in the Z-axis direction at B1 and B2 were nearly negligible. Almost all the
applied compression load was transmitted to C2. The reaction in the X-axis direction
at point B2 did not exceed 0.6% of PC. However, the reactions at points B1 and B2 in
the Y-axis direction were nearly equal and ranging between 1.5% and 9.4% of PC.
Their values were proportional to the values of (IZ)B. This is explained as follows.
Increasing the value of (IZ)B would increase the flexural stiffness of member B about
the Z axis. This would require more force in the Y- axis direction to displace member
B and hence allow member C to buckle in Y-Z plane.

(ii) Effect of (IY)B

The moments of inertia (IX)C ,(IY)C and (IZ)B were made constant and given the same
value; i.e. (IY)C /(IX)C = (IZ)B /(IX)C =1.0. The value of (IY)B was varied. The results
obtained are presented in figure 3. The ratio of PC / PE is found to be proportional to
(IY)B / (IX)C. Unlike (IZ)B, increasing the value of (IY)B / (IX)C from 0.02 to 3 caused
limited increase in PC /PE that did not exceed 7.5%. It is noted that, when applying
compression load P on member C, the joint at the intersection of members B and C
displace down wards in the direction of the Z-axis. Part of this load is transmitted
through this joint to member B, and hence to the supports at B1 and B2 in turn. The
results show that the value of this part of the load is proportionally affected by the
value of (IY)B. At buckling, its value is nearly negligible in comparison to PC.

(iii) Effect of (IY)C

The cross section of member C was modeled having the geometrical properties of
rectangular hollow section RHS 203*102*4.8 complying with the Canadian Standard
Specification CSA [13]. In this case the value of (IY)C /(IX)C = 2.92. The value of
(IZ)B was made equal to (IX)C while the value of (IY)B was varied. The results
obtained are found typical to those of figure 3 for (IY)C / (IX)C = 1. The change in (IY)C
has no effect on the out of plane critical load.

5
2- Effect of Tension Load
Members B and C of figure 1 were modeled having the same cross-section, length,
and material properties. The moments of inertia of members B and C cross sections
were given the same values; i.e. (IY)C / (IX)C = (IY)B / (IZ)B, and hence (IZ)B / (IX)C =
(IY)B / (IX)C = 1. The restraining conditions are as in figure 1. Compression and
tension loads were applied at C1 and B1 respectively. Different values of T / P ratio
were considered. For each case, the values of T and P were increased, but keeping
their ratio T / P constant. The values of PC were obtained and presented in terms of the
ratio PC / PE, figure 4. The results in general show that PC / PE is bilinear proportional
to the ratio T / P to certain limit after which the value of PC / PE is constant. When no
tension load is applied, the value of PC / PE = 2.01. The value of PC / PE = 4.0 when T
/ P equals 0.628, figure 4. In this case, member B restrains member C against out of
plane buckling as it were a hinged support. Picard et al (8] found analytically that PC /
PE would equal 4.0 when T / P = 0.625 and increasing T /P further would not cause
any increase in PC / PE. The finite element results however show different behavior.
Increasing T / P ratio more than 0.628 elevated PC / PE value. This is valid up to T / P
= 0.8 after which PC / PE value is constant. It should be noted that the inclination of
part a b of the relation in figure 4 is different to that of b c.

3 - Effect of Supporting Conditions

Members B and C in figure 1 were modeled having the same length and material
properties. They were given cross section of RHS 203*102*4.8 complying with CSA
[13]. Six cases of different out of plane supporting conditions; i.e. in the Y-Z plane,
were considered. Table 1 shows the obtained values of PC / PE ratio. The results show
that the type of supporting conditions at the members’ ends affects the critical load
significantly. Increasing the fixity degree at the ends of member C is more effective
than doing this to member B. When changing the supporting conditions of members B
and C in case 1 to those in case 6, the critical load increased by nearly 4 times. In this
case, member B provided support as if it were a hinge and each part of member C
behaved as if it were hinged-fixed column.

6
Table 1 Values of Pc /PE at different supporting conditions

Supporting conditions* Pc /PE

Cas Properties of members B
Member C Member B F.E. Analytical Ratio**
e &C
C1 C2 B1 B2 analysis model
1 the same length, material R H 2.06 1.91 92.7%
2 properties R H R F 3.22 3.04 94.4%
cross section of
3 FR F 3.94 4.0 101.5%
RHS 203*102*48
4 (IY)B/(IZ)C= (IY)C/(IX)C=2.92 R H 5.14 4.97 96.6%
5 (IZ)B/(IX)C=(IY)B/(IY)C=1 FR F R F 6.47 6.16 95.2%
6 NO tension is induced in B FR F 8.05 7.87 97.7%

Notes
* Symbols used for supporting conditions means:
R = roller support that allows transition in the direction of the member length,
H = hinged support, F = fixed support and FR = fixed support which allows transition
in the direction of the member length.
** Ratio = % of simple model results to the F. E. results

4 - Intersection Connection
In practice, cross bracing members are usually made co-planner. One member is
interrupted and the other one is continuous. It is usual practice to connect the
interrupted member to the continuous one by means of gusset plate connection. The
finite element analysis was used to model case 1 of table 1 when member B is
interrupted and connected to member C by hinges as shown in figure 5. The results
show that member B in this case does not provide any degree of restraining to C
against out of plane buckling and the value of PC / PE = 1.0. The study in reference
[14] considered the cases of semi-rigid intersection connection for cross bracing
members with pinned end connections.

ANALYTICAL MODEL
Cross bracing members are modeled as follows. The compression member is
supported by a spring at the two members intersection. Winter [4] modified this
model by introducing a fictitious hinge as shown in figure 6. This is to find the spring
stiffness value after which the spring would restrain the compression member against
buckling as if it were a hinged support. The spring is assumed to be elastic. At
buckling, the equilibrium at the hinge O is given as follows:

PC Δ = ( K Δ / 2) * ( L / 2 ) (1)

7
Where PC is the critical load and K is the transitional stiffness of the spring. At
buckling, each part of the compression member would buckle individually and the
critical load of the system would equal:

PC = π2 E I / ( L / 2 )2 = 4 PE (2)

Where PE is the Eular load of the compression member. By substituting PC of

equation 2 into equation 1, the spring would behave as if it were a hinged support
when:

K = 16 PE / L (3)

For the cross bracing members B and C in figure 1, K represents the transitional
stiffness of member B that provide restraining to member C against out of plane
buckling. From the finite element results, the values of Fy and δy are related by
equation 4. The symbols Fy and δy are used for the force induced and the deflection
occurred at members B and C intersection in the Y-axis direction.

δy = Fy L3 / 48 E ( IZ )B (4)

Equation 4 can be rearranged as follows:

K = Fy / δy = SB E ( IZ )B / L3 (5)

Where SB = 48. Equation 5 is substituted into equation 3. In figure 7, equation 3 is

represented by the dashed line o e f in terms of the ratios PC / PE and (IZ)B / (IX)C . PE
in this case is the Eular load for out of plane buckling of member C and taken as
follows:

PE = π2 E ( IX )C / L2 (6)

The results show that member B would behave as if it were a hinged support at ( IZ )B
/ ( IX )C = 3.29. This relation is modified by the line d e f for the original proposed
model; i.e. with out the fictitious hinge. For comparison, the finite element results of

8
figure 2 are superimposed on figure 7. The two relations are coinciding up to (IZ)B /
(IX)C = 1.85 and then deviates. The maximum difference in PC / PE values obtained
from the two analyses equals 4% at ( IZ )B / ( IX )C = 3.29.

EVALUATION OF OUT OF PLANE Critical LOAD

(i) Symmetrical members
Figure 8 can be used for the evaluation of out of plane critical load for cross bracing
members as members B and C in figure 1. The relationship d e f of figure 7 is
implemented for the cases when no tension is considered. The finite element results at
different values of T/PC are superimposed. Member B is assumed to provide a hinged
support at T / PC = 0.628 and the increase in T / PC is assumed to cause no increase in
PC/PE. Figure 9 is used when members B and C have fixed supports at their ends as
shown in the figure. Winter model [4] is not valid for this case. This relation is
obtained as follows. When member C is supported only at its ends by fixed supports,
its critical load PC = 4 PE. When member B is considered, it is modeled as an elastic
spring. This spring would restrain member C against buckling as if it were a hinged
support when:

K = 21 PE / L (7)

And the critical load in this case would equal:

PC = 8.184 PE (8)

These values are obtained using the stability functions Φ and Ψ in reference [3]. By
equating equation 7 to 5 and using SB = 192, member B would restrain C as if it were
a hinged support at (IZ)B / (IX)C = 1.08.

(ii) Unsymmetrical members

Figures 8 and 9 can still be used when cross bracing members are not symmetrical;
i.e. having different lengths, supporting conditions and/or cross sections. In this case,
a fictitious member having supporting conditions and length typical to those of
member C is used instead of member B. The transitional stiffness provided by that
member should equal that of member B and calculated as follows:

9
 SB E ( IZ )B / LB3 = SF E (IZ)F / LC3 (9)

The symbols B, C and F are used for members B, C and the fictitious one
respectively. The value of SF would depend on the supporting conditions of member
C. SF equals 48 and 192 when dealing with figures 8 and 9 respectively. Similarly, the
value of SB would equal 48, 107.3 or 192 when the supporting conditions of member
B are hinged-roller, fixed-roller or fixed-fixed respectively. The value of (IZ)F is
obtained in terms of ( IZ )B. The ratio of (IZ)F / (IX)C is calculated and used instead of
(IZ)B / (IX)C to get the value of PC/PE from figures 8 or 9. It should be noted that this
method do not include the effect of (IY)B / (IX)C. Figures 8 and 9 are used to obtain the
values of PC/PE for cases 1 to 6 of table 1. The results are presented in table 1 and
show good agreement.

PRACTICAL CONSIDRATIONS
In practice, designers normally use typical cross sections for diagonal cross bracing
members. In this case, the values of (IZ)B / (IX)C = (IY)B /(IY)C = 1. The value of (IY)B /
(IX)C = (IY)B / (IZ)B which is the ratio of the moments of inertia of the member’s cross
section about its principle axes. This ratio equals unity for circular and square hollow
sections. By reviewing the Canadian specification CSA [13], it is found that (IY)B /
(IZ)B ranging between 1.5 and 3.1 for rectangular hollow sections. For two angles
back to back, the value of (IY)B / (IZ)B would depend on the thickness of the gusset
plate and either the short or the long leg is parallel to the Y axis. The values of (IY)B /
(IZ)B are calculated using the data provided in the CSA [13] considering the global
coordinate system in figure 1. The values of (IY)B / (IZ)B are found ranging between
0.125 to 0.47 for equal leg angles. For unequal leg angles, (IY)B / (IZ)B values are
ranging between 0.46 to1.25 when the short legs are parallel to the Y axis and 0.09 to
0.2 when the long legs are parallel to the Y axis. This in turn limits the effect of (IY)B /
(IX)C on the value of PC /PE.

CONCLUSIONS
Nonlinear large displacement finite element analysis was used to model the buckling
behavior of cross bracing members using ANSYS program. One member is subjected
to compression load and the other one to tension load in some cases, members B and

11
C in figure 1. The results show that member B provides degree of restrain to member
C against out of plane buckling. The value of (IZ)B / (IX)C effects significantly this
degree of restraining. At (IZ)B / (IX)C = 3.4, member B would restrain C against out of
plane buckling as if it were a hinged support. Another effective parameter is the
tension load induced in B. At T/PC = 0.628, the value of PC / PE = 4.0. The supporting
conditions at the ends of members C and B are other parameters that affect the out of
plane critical load, table1. Further, an analytical simple model is implemented and
developed to evaluate the out of plane critical load. Figures 8 and 9 can be used for
symmetrical and unsymmetrical cross bracing members. The results obtained are
compared to those of the finite element. The results show good agreement.

NOMENCLATURE

E modulus of elasticity
FY force in the direction of the Y axis
( IN )M moment of inertia of member M about N-N axis
K spring transitional stiffness
L member length
P compression load
PC critical load
PE Eular load
Sn numerical factor of member N
T tension load
Δ deflection at mid length of member
Δb deflection at buckling
δy deflection in the direction of the Y-axis

REFERENCES

1. Egyptian Code of Practice For Steel Constructions And Bridges, Code No

205, Ministerial Decree No. 279 – 2001, Housing and Building Research
Center, Cairo, Egypt, 2001.
2. Specification for Structural Steel Buildings ANSI / AISC 360 - 05, American
Institute of Steel Construction AISC, Chicago, 9 March 2005.
3. Timoshinko, S. and Gere, J.M., “Theory of Elastic Stability,” Mc. Graw-Hill
Book Company, New York, 1970.
4. Winter, G., “Lateral Bracing of Columns and Beams,” Trans. ASCE, Vol.
125, pp. 807 – 845, 1960.

11
5. Stoman, S. H., “Stability Criteria for X-Bracing Systems,” ASCE, J. Engrg.
Mech., Vol. 114, 8, pp 1426 –1434, (1988).
6. Stoman, S. H., “Effective Length Spectra for Cross Bracings,” ASCE, J.
Struct. Engrg., Vol.115 (12), pp. (1989)
7. Stoman, S. H., ”Design of Diagonal Cross Bracings – Discussion,” Engrg. J.,
AISC, Vol.26 (4), pp. 155-159, Fourth Quarter (1989)
8. Picard, A. and Beaulieu, D., ”Design of Diagonal Cross Bracing – Part1:
Theoretical Study,” Engrg. J., AISC, Vol.24 (3), pp. 122-126, Third Quarter
(1987)
9. El-Tayem, A. A. and Goel, S.C., “Effective Length Factor for the Design of
X-bracing Systems,” Engrg. J., AISC, Vol.23 (1), pp. 41-45, First Quarter
(1986).
10. ANSYS User’s Manual for Revision 5.0, Vol. 1, Procedures 1992.
11. Gil, H. and Yura, A.J., “Bracing Requirements of Inelastic Columns ,”
Journal of constructional steel Research, Elsevier Science, Vol 51, (1), pp 1-
19, (1999)
12. Tch, L. H. and Clarke, M. J., “Tracing Secondary Equilibrium Paths of
Elastic Framed Structures,” ASCE, J. Engrg. Mech., Vol. 125, 12, pp 1358 –
1364, (1999).
13. National Standard of Canada CAN / CSA – S16. 1-94, Limit States Design of
Steel Structures, Candian Standards Association, Ontario, Canada, (1994).
14. Davaran,A. “ Effective Length Factor for Discontinuous X-bracing
Systems,” ASCE, J. Engrg. Mech., Vol. 127, 2, pp 106 –112, (2001).

12