Comparison of VIM predictions for nondimensional buckling load (* λ*) of a three-segment compression member with exact results for various values of stiffness ratio (

*=*n

EI

_{2}/

EI

_{1}) and stiffened length ratio (

*=*s

*/*a

*)*H

Open access

Submitted: 14 October 2011 Published: 01 August 2012

DOI: 10.5772/45807

In earthquake resistant structural steel design, there are two commonly used structural systems. “Moment resisting frames” consist of beams connected to columns with moment resisting (i.e., rigid) connections. Rigid connection of a steel beam to a steel column requires rigorous connection details. On the other hand, in “braced frames”, the simple (i.e., pinned) connections of beams to columns are allowed since most of the earthquake forces are carried by steel braces connected to joints or frame elements with pinned connections. The load carrying capacity of a braced frame almost entirely based on axial load carrying capacities of the braces. If a brace is under tension in one half-cycle of an earthquake excitation, it will be subjected to compression in the other half cycle. Provided that the connection details are designed properly, the tensile capacity of a brace is usually much higher than its compressive capacity. In fact, the fundamental limit state that governs the behavior of such steel braces under seismic forces is their global buckling behavior under compression.

After detailed evaluation, if a steel braced structure is decided to have insufficient lateral strength/stiffness, it has to be strengthened/stiffened, which can be done by increasing the load carrying capacities of the braces. The key parameter that controls the buckling capacity of a brace is its “slenderness” (Salmon et al., 2009). As the slenderness of a brace decreases, its buckling capacity increases considerably. In order to decrease the slenderness of a brace, either its length has to be decreased, which is usually not possible or practical due to architectural reasons, or its flexural stiffness has to be increased. Flexural stiffness of a brace can be increased by welding steel plates or by wrapping fiber reinforced polymers around the steel section. Analytical studies (e.g., Timoshenko & Gere, 1961) have shown that it usually leads to more economic designs if only the partial length, instead of the entire length, of the brace is stiffened. This also eliminates possible complications in connection details that have to be considered at the ends of the member.

Nonuniform structural elements are not only used in seismic strengthening and rehabilitation of existing structures. In an attempt to design economic and aesthetic structures, many engineers and architects nowadays prefer to use nonuniform structural elements in their structural designs. However, stability analysis of such nonuniform members is usually much more complex than that of uniform members (e.g., see Li, 2001). In fact, most of the design formulae/charts given in design specifications are developed for uniform members. Thus, there is a need for a practical tool to analyze buckling behavior of nonuniform members.

This study investigates elastic buckling behavior of three-segment symmetric stepped compression members with pinned ends (Fig. 1) using three different approaches: (i) analytical, (ii) numerical and (iii) experimental approaches. As already mentioned, such a member can easily be used to strengthen/rehabilitate an existing steel braced frame or can directly be used in a new construction. Surely, the use of stepped elements is not only limited to the structural engineering applications; they can be used in many other engineering applications, such as in mechanical and aeronautical engineering.

In analytical studies, first the governing equations of the studied stability problem are derived. Then, exact solution to the problem is obtained. Since exact solution requires finding the smallest root of a rather complex characteristic equation which highly depends on initial guess, the governing equation is also solved using a recently developed analytical technique by He (1999), which is called Variational Iteration Method (VIM). Many researchers (e.g., Abulwafa et al., 2007; Batiha et al., 2007; Coskun & Atay, 2007, 2008; Ganji & Sadighi, 2007; Miansari et al., 2008; Ozturk, 2009 and Sweilan & Khader, 2007) have shown that complex engineering problems can easily and successfully be solved using VIM. Recently, VIM has also been applied to stability analysis of compression and flexural members. Coskun and Atay (2009), Atay and Coskun (2009), Okay et al. (2010) and Pinarbasi (2011) have shown that it is much easier to solve the resulting characteristic equation derived using VIM. In this paper, by comparing the approximate VIM results with the exact results, the effectiveness of using VIM in determining buckling loads of multi-segment compression members is investigated.

The problem is also handled, for some special cases, using widely known structural analysis program SAP2000 (CSI, 2008). After determining the buckling load of a uniform member with a hollow rectangular cross section, the stiffness of the member is increased along its length partially in different length ratios and the effect of such stiffening on buckling load of the member is investigated. By comparing numerical results with analytical results, the effectiveness of using such an analysis program in stability analysis of multi-segment elements is also investigated.

Finally, buckling loads of uniform and three-segment stepped steel compression members with hollow rectangular cross section are determined experimentally. In the experiments, the “stiffened” columns are prepared by welding additional steel plates over two sides of the member in such a way that the addition of the plates predominantly increases the smaller flexural rigidity of the cross section, which governs the buckling behavior of the member. By changing the length of the stiffening plates, i.e., by changing the stiffened length ratio, the degree of overall stiffening is investigated in the experimental study. The experimental study also shows in what extent the * ideal* conditions assumed in analytical and numerical studies can be realized in a laboratory research.

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## 2. Analytical studies on elastic buckling of a three-segment stepped compression member with pinned ends

### 2.1. Derivation of governing (buckling) equations

E I 1 d 2 w 1 d x 2 - P ( δ - w 1 ) = 0 E1
d 2 w 1 d x 2 + k 1 2 w 1 = k 1 2 δ where k 1 2 = P E I 1 E2
d 2 w 2 d x 2 + k 2 2 w 2 = k 2 2 δ where k 2 2 = P E I 2 E3
( w ¯ 1 ) ′ ′ + β 1 2 ( w ¯ 1 ) = β 1 2 ( δ ¯ ) and ( w ¯ 2 ) ′ ′ + β 2 2 ( w ¯ 2 ) = β 2 2 ( δ ¯ ) E4
β 1 = k 1 L and β 2 = k 2 L E5
[ w ¯ 1 ] x ¯ = s = [ w ¯ 2 ] x ¯ = s and [ ( w ¯ 1 ) ′ ] x ¯ = s = [ ( w ¯ 2 ) ′ ] x ¯ = s E6
[ ( w ¯ 2 ) ] x ¯ = 0 = 0 , [ ( w ¯ 2 ) ′ ] x ¯ = 0 = 0 and [ ( w ¯ 1 ) ] x ¯ = 1 = δ ¯ E7
### 2.2. Exact solution to buckling equations

w ¯ 1 = C 1 sin ( β 1 x ¯ ) + C 2 cos ( β 1 x ¯ ) + δ ¯ and w ¯ 2 = C 3 sin ( β 2 x ¯ ) + C 4 cos ( β 2 x ¯ ) + δ ¯ E8
C 3 = 0 and C 4 = − δ ¯ E9
C 1 = δ ¯ [ β 2 β 1 sin ( β 2 s ) cos ( β 1 s ) − cos ( β 2 s ) sin ( β 1 s ) ] (a) C 2 = − δ ¯ [ β 2 β 1 sin ( β 2 s ) sin ( β 1 s ) + cos ( β 2 s ) cos ( β 1 s ) ] (b) E10
{ tan [ β 2 s ] tan [ β 1 ( 1 − s ) ] − β 1 β 2 } δ ¯ = 0 E11
tan [ β 2 s ] tan [ β 1 ( 1 − s ) ] = β 1 β 2 E12
tan [ β 1 ( 1 − s ) ] tan [ β 1 s n ] = n E13
P c r = λ E I 1 H 2 where λ = 4 β 1 2 E14
### 2.3. VIM solution to buckling equations

L w ( x ) + N w ( x ) = g ( x ) E15
w n + 1 ( x ) = w n ( x ) + ∫ 0 x λ ( ξ ) { L w n ( ξ ) + N w ˜ n ( ξ ) } d ξ E16
λ ( ξ ) = ( ξ − x ) E17
w n + 1 ( x ) = w n ( x ) + ∫ 0 x λ ( ξ ) { L w n ( ξ ) + N w n ( ξ ) } d ξ E18
w n + 1 ( x ) = w 0 ( x ) + ∫ 0 x λ ( ξ ) { N w n ( ξ ) } d ξ . E19
w n + 2 ( x ) = w n + 1 ( x ) + ∫ 0 x λ ( ξ ) { N w n + 1 ( ξ ) − N w n ( ξ ) } d ξ E20
w ¯ i , n + 1 ( x ) = w ¯ i , n ( x ) + ∫ 0 x ( ξ − x ) { w ¯ i , n ′ ′ ( ξ ) + β i 2 w ¯ i , n − β i 2 δ ¯ } d ξ , (a) w ¯ i , n + 1 ( x ) = w ¯ i , 0 ( x ) + ∫ 0 x ( ξ − x ) { β i 2 w ¯ i , n − β i 2 δ ¯ } d ξ , (b) w ¯ i , n + 2 ( x ) = w ¯ i , n + 1 ( x ) + ∫ 0 x ( ξ − x ) { ( w ¯ i , n + 1 ′ ′ ( ξ ) − w ¯ i , n ′ ′ ( ξ ) ) + β i 2 ( w ¯ i , n + 1 − w ¯ i , n ) } d ξ , (c) E21
w ¯ 1 , j + 1 ( x ) = w ¯ 1 , 0 ( x ) + ∫ 0 x ( ξ − x ) { λ 4 ( w ¯ 1 , j − δ ¯ ) } d ξ , (a) w ¯ 2 , j + 1 ( x ) = w ¯ 2 , 0 ( x ) + ∫ 0 x ( ξ − x ) { λ 4 n ( w ¯ 2 , j − δ ¯ ) } d ξ (b) E22
w ¯ 1 , 0 = C 1 x ¯ + C 2 and w ¯ 2 , 0 = C 3 x ¯ + C 4 E23
[ F ( λ ) ] δ ¯ = 0 E24
### 2.4. Comparison of VIM results with exact results

### 2.5. VIM results for various stiffness and stiffened length ratios

Consider a three-segment symmetric stepped compression member subjected to a compressive load * P* applied at its top end, as shown in Fig. 1. Assume that both ends of the member are pinned; i.e., free to rotate. Also assume that the top and bottom segments of the member have identical flexural stiffness,

The undeformed and deformed shapes of the equivalent two-segment member under uniform compression are illustrated in Fig. 2a. The origin of * x*-

which can be expressed as

In Eq. (1) and Eq. (2), _{1} is lateral displacement of Segment I at any point, * δ* is the lateral displacement of the top end of the member, i.e.,

where _{2} is the displacement of Segment II in * y* direction. Eq. (3) is valid for 0

with

where* δ* is also unknown, the solution of these buckling equations requires five conditions to determine the resulting five unknowns. Two of these conditions come from the continuity conditions where the flexural stiffness of the column changes and the remaining three conditions are obtained from the boundary conditions at the ends of the column. At

where

Thus, Eq. (4) with Eq. (6) and Eq. (7) constitutes the governing equations for the studied stability problem.

Since the differential equations given in Eq. (4) are relatively simple, it is not too difficult to obtain their exact solutions, which can be written in the following form:

where _{i} (* i*=1-4) are integration constants to be determined from continuity and end conditions. From the first and second conditions given in Eq. (7), one can find that

Then, using Eq. (6), the other integration constants are obtained as:

Finally, the last condition given in Eq. (7) results in

For a nontrivial solution, the coefficient term must be equal to zero, yielding the following characteristic equation for the studied buckling problem:

Since* n* is defined as

One can show that the buckling load of the three-segment stepped compression member with length * H* shown in Fig. 1 can be written in terms of that of the equivalent two-segment member with length

In other words, * λ* is the nondimensional buckling load of the three-segment compression member

According to the variational iteration method (VIM), a general nonlinear differential equation can be written in the following form:

where * L* is a linear operator and

where

The original variational iteration algorithm proposed by He (1999) has the following iteration formula:

In a recent paper, He et al. (2010) proposed two additional variational iteration algorithms for solving various types of differential equations. These algorithms can be expressed as follows:

and

Thus, the three VIM iteration algorithms for the buckling equations given in Eq. (4) can be written as follows:

where * i* is the segment number and can take the values of one or two. It has already been shown in Pinarbasi (2011) that all VIM algorithms yield exactly the same results for a similar stability problem. For this reason, considering its simplicity, the second iteration algorithm is decided to be used in this study.

Recalling that * λ* and

As an initial approximation for displacement function of each segment, a linear function with unknown coefficients is used:

where _{i} (* i*=1-4) are to be determined from continuity and end conditions. After conducting seventeen iterations,

where

For various values of stiffness ratio (* n*=

As it can be seen from Table 1, VIM results perfectly match with exact results, verifying the efficiency of VIM in this particular stability problem. It is worth noting that it is somewhat difficult to solve the characteristic equation given in Eq. (13) since it is highly sensitive to the initial guess. While solving this equation, one should be aware of that an improper initial guess can result in a buckling load in higher modes. On the other hand, the characteristic equations derived using VIM are composed of polynomials, all roots of which can be obtained more easily. This is one of the strength of VIM even when an exact solution is available for the problem, as in our case.

Table 2 tabulates VIM predictions for nondimensional buckling load of a three-segment stepped compression member for various values of stiffness (* n*) and stiffened length (

At this stage, it can be valuable to investigate the amount of increase in buckling load due to partial stiffening of a compression member. Fig. 3 shows variation of increase in critical buckling load, with respect to the uniform case, with stiffened length ratio for different values of stiffness ratio. From Fig. 3, it can be inferred that there is no need to stiffen entire length of the member to gain appreciable amount of increase in buckling load especially if * n* is not too large. For

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## 3. Numerical studies on elastic buckling of a three-segment stepped compression member with pinned ends

In order to obtain directly comparable results with the experimental results that will be discussed in the following section, in the numerical analysis, the reference “unstiffened” member is selected to have a hollow rectangular cross section, namely RCF 120x40x4, the geometric properties of which is given in Fig. 5a. The length of the steel (with modulus of elasticity of E=200 GPa) columns is chosen to be 2 m., which is the largest height of a compression member that can be tested in the laboratory due to the height limitations of the test setup. Elastic stability (buckling) analysis is performed using a well-known commercial structural analysis program SAP2000 (CSI, 2008).

Fig. 5b shows numerical solutions for the buckled shape and buckling load, _{cr,num,n=1} = 156.55 kN, of the uniform column. Exact value of the buckling load _{cr} for this column can be computed from the well-known formula of Euler;_{cr,exact,n=1} = 157.42 kN. The error between the numerical and exact analytical result is only 0.5 %, which encourages the use of this technique in determining the buckling load of “stiffened” members.

In the experimental study, in addition to the unstiffened members, three different types of stiffened columns are tested. In these specimens, the stiffness ratio is kept constant (* n*2) while the stiffened length ratio is varied. The stiffnesses of the three-segment members are increased by welding rectangular steel plates, with 100 mm width and 3 mm thickness as shown in Fig. 6a, to the wider faces of the hollow cross section. The length of the stiffening plates is 0.4 m in members with

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## 4. Experimental studies on elastic buckling of a three-segment stepped compression member with pinned ends

M F = ( 0.877 − 0.2 s ) E25
P c r , p r o p o s e d = M F × P c r , a n a l y E26

The experimental part of the study is conducted in the Structures Laboratory of Civil Engineering Department in Kocaeli University. Test specimens are subjected to monotonically increasing compressive load until they buckle about their minor axis in a test setup specifically designed for such types of buckling tests (Fig. 8). Due to the height limitations of the test setup, the length of the test specimens is fixed to 2 m. To observe elastic buckling, “unstiffened” (uniform) * reference* specimens are selected to have a rather small cross section; hollow rectangular section with side dimensions of 120 mm x 40 mm and wall thickness of 4 mm, as shown in Fig. 5a. In addition to the three unstiffened specimens, named B0-1, B0-2 and B0-3, three sets of “stiffened” specimens, each of which consists of three columns with identical stiffening, are tested. To obtain comparable results, the stiffness ratio of the stiffened specimens is kept constant (

As shown in Fig. 8, the test specimens are placed between the top and bottom supports in the test rig, which is rigidly connected to the strong reaction wall. To ensure minor-axis buckling of the test columns, the supports are designed in such a way that the rotation is about a single axis, resisting rotation about the orthogonal axis. In other words, the supports behave as pinned supports in minor-axis bending whereas fixed supports in major-axis bending. The compressive load is applied to the columns through a hydraulic jack placed at the top of the upper support. During the tests, in addition to the load readings, which are measured by a pressure gage, strains at the outermost fibers in the central cross section of each column are recorded via two strain gages (SG1 and SG2) (see Fig. 8).

The buckled shapes of the tested columns are presented in Fig. 9 and Fig. 10. As shown in Fig. 9a, uniform columns buckle in the shape of a half-sine wave, which is in agreement with the well-known Euler’s formulation for * ideal* pinned-pinned columns. In contrast to

The buckling loads of all test specimens are tabulated in Table 4. When the buckling loads of three uniform columns are compared, it is observed that the buckling load for Specimen B0-3 (150.18 kN) is larger than those for Specimens B0-1 (129.60 kN) and B0-2 (128.49 kN). When Fig. 11a is examined closely, it can be observed that strain gage measurements start to deviate from each other at larger loads in Specimen B0-3 than B-01 and B0-2. Thus, it can be concluded that the capacity difference among these specimens occurs * most probably* due to the fact that the initial out-of-straightness of Specimen B0-3 is much smaller than that of B-01 and B-02. When the load-strain plots of the stiffened specimens (Fig. 11b-d) are examined, similar trends are observed for specimens with larger load values in their own sets, e.g., B2-1 and B2-3 in the third set, B3-1 in the forth set. These differences can also be attributed

For better comparison, experimental (_{cr,exp}) and analytical (_{cr,analy}) buckling loads are also plotted in Fig. 12. As shown in the figure, all test results lay below the analytical curve.

It is important to note that most design specifications modify the buckling load equations derived for * ideal* columns to take into account the effects of initial out-of-straightness of the columns in the design of compression members. As an example, to reflect an initial out-of-straightness of about 1/1500, AISC (2010) modifies the “Euler” load by multiplying with a factor of 0.877 in the calculation of compressive capacity of elastically buckling members (Salmon et al., 2009). By applying a similar modification to the analytical results obtained in this study for

where * s* is the stiffened length ratio of the compression member, which equals to the weld length in the stiffened members. Thus, the proposed buckling load (

The proposed buckling loads for the multi-segment columns tested in the experimental part of this study are computed using Eq. (26) with Eq. (25) and plotted in Fig. 12 with a label ‘_{cr,proposed}’. For easier comparison, a linear trend line fitted to the experimental data is also plotted in the same figure. Fig. 12 shows perfect match of design values of buckling loads with the trend line. While using Eq. (25), it should be kept in mind that the modification factor proposed in this paper is derived based on the limited test data obtained in the experimental part of this study and needs being verified by further studies.

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## 5. Conclusion

In an attempt to design economic and aesthetic structures, many engineers nowadays prefer to use nonuniform members in their designs. Strengthening a steel braced structure which have insufficient lateral resistant by stiffening the braces through welding additional steel plates or wrapping fiber reinforced polymers in partial length is, for example, a special application of use of multi-segment nonuniform members in earthquake resistant structural engineering. The stability analysis of multi-segment (stepped) members is usually very complicated, however, due to the complex differential equations to be solved. In fact, most of the design formulae/charts given in design specifications are developed for uniform members. For this reason, there is a need for a practical tool to analyze buckling behavior of nonuniform members.

In this study, elastic buckling behavior of three-segment symmetric stepped compression members with pinned ends is analyzed using three different approaches: (i) analytical, (ii) numerical and (iii) experimental approaches. In the analytical study, first the governing equations of the studied stability problem are derived. Then, exact solution is obtained. Since exact solution requires finding the smallest root of a rather complex characteristic equation which highly depends on initial guess, the governing equations are also solved using a recently developed analytical technique, called Variational Iteration Method (VIM), and it is shown that it is much easier to solve the characteristic equation derived using VIM. The problem is also handled, for some special cases, by using widely known structural analysis program SAP2000 (CSI, 2008). Agreement of numerical results with analytical results indicates that such an analysis program can also be effectively used in stability analysis of stepped columns. Finally, aiming at the verification of the analytical results, the buckling loads of steel columns with hollow rectangular cross section stiffened, in partial length, by welding steel plates are investigated experimentally. Experimental results point out that the buckling loads obtained for * ideal* columns using analytical formulations have to be modified to reflect the initial imperfections. If welding is used while forming the stiffened members, as done in this study, not only the initial out-of-straightness, but also the effects of welding have to be considered in this modification. Based on the limited test data, a modification factor which is a linear function of the stiffened length ratio is proposed for three-segment symmetric steel compression members formed by welding steel plates in the stiffened regions.

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Submitted: 14 October 2011 Published: 01 August 2012

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.