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

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 (*n*2) while their stiffened length ratios (*s*) are varied in each set. Such stiffening is attained 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 sections of the test specimens, in different lengths. The length of the stiffening plates is 0.4 m for the members with stiffened length ratio *s*=0.2, which are named B1-1, B1-2 and B1-3, approximately 0.67 m for the members with *s*=0.3333, named B2-1, B2-2 and B2-3, and 1.0 m for the members with *s*=0.5, named B3-1, B3-2 and B3-3.

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 107

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 *ideal* columns, however, test columns have *not* buckled suddenly during the tests. This is mainly due to the fact that all test specimens have unavoidable initial crookedness. Even though the amount of these imperfections remain within the tolerances specified by the specifications, they cause bending of the specimens with the initiation of loading. This is also apparent from the graphs presented in Fig. 11. These graphs plot strain gage measurements taken at the opposite sides of the column faces (SG1 and SG2) during the test of each specimen with respect to the applied load values. The divergence of strain gage readings (SG1 and SG2) from each other as the load increases clearly indicates onset of the bending under axial compression. This is compatible with the expectations since as stated by Galambos (1998), "geometric imperfections, in the form of tolerable but unavoidable out-of-straightness of the column and/or eccentricity of the axial load, will introduce bending from the onset of loading". Even though the test columns start to bend at smaller load levels, they continue to carry additional loads until they reach their "buckling" capacities, which are characterized as the peak values of their load-strain

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 *partially* to the initial out-of-straightness. Unlike uniform columns, stiffened columns have additional initial imperfections due to the welding process of the stiffeners. It is now well known that welding cause unavoidable residual stresses to develop within the cross section of the member, which, in turn, can change the behavior of the member significantly. Since the columns with larger stiffened length ratios have longer welds, they are expected to have more initial imperfection. The effects of initial imperfections can also be seen from the last column of Table 4, where the ratios of experimental results to the analytical results which are

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

curves.

curve.

obtained for *ideal* columns are presented.

**Figure 8.** Test setup

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 *ideal* columns, however, test columns have *not* buckled suddenly during the tests. This is mainly due to the fact that all test specimens have unavoidable initial crookedness. Even though the amount of these imperfections remain within the tolerances specified by the specifications, they cause bending of the specimens with the initiation of loading. This is also apparent from the graphs presented in Fig. 11. These graphs plot strain gage measurements taken at the opposite sides of the column faces (SG1 and SG2) during the test of each specimen with respect to the applied load values. The divergence of strain gage readings (SG1 and SG2) from each other as the load increases clearly indicates onset of the bending under axial compression. This is compatible with the expectations since as stated by Galambos (1998), "geometric imperfections, in the form of tolerable but unavoidable out-of-straightness of the column and/or eccentricity of the axial load, will introduce bending from the onset of loading". Even though the test columns start to bend at smaller load levels, they continue to carry additional loads until they reach their "buckling" capacities, which are characterized as the peak values of their load-strain curves.

106 Advances in Computational Stability Analysis

**Figure 8.** Test setup

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 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 *partially* to the initial out-of-straightness. Unlike uniform columns, stiffened columns have additional initial imperfections due to the welding process of the stiffeners. It is now well known that welding cause unavoidable residual stresses to develop within the cross section of the member, which, in turn, can change the behavior of the member significantly. Since the columns with larger stiffened length ratios have longer welds, they are expected to have more initial imperfection. The effects of initial imperfections can also be seen from the last column of Table 4, where the ratios of experimental results to the analytical results which are obtained for *ideal* columns are presented.

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

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 109

**Figure 10.** Buckled shapes of stiffened test specimens with *s*=0.3333 and *s*=0.5

b. Stiffened columns with s=0.5

a. Stiffened columns with s=0.3333

b. Stiffened columns with s=0.2

**Figure 9.** Buckled shapes of unstiffened and stiffened (with *s*=0.2) test specimens

a. Stiffened columns with s=0.3333

**Figure 9.** Buckled shapes of unstiffened and stiffened (with *s*=0.2) test specimens

b. Stiffened columns with s=0.2

a. Unstiffened columns

b. Stiffened columns with s=0.5

**Figure 10.** Buckled shapes of stiffened test specimens with *s*=0.3333 and *s*=0.5

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 111

192.98

**Table 4.** Experimental buckling loads for uniform and stiffened columns compared with the analytical

0.5 260.10

0.3333 221.78

0.2

**Specimen s Pcr,exp (kN) Pcr,analy (kN) Pcr,exp / Pcr,analy** B0-1 129.60 0.823 B0-2 128.49 0.816 B0-3 150.18 0.954 B1-1 166.31 0.862 B1-2 177.44 0.919 B1-3 176.32 0.914 B2-1 190.23 0.858 B2-2 153.52 0.692 B2-3 188.56 0.850 B3-1 241.96 0.930 B3-2 194.12 0.746 B3-3 172.43 0.663

0 157.42

**Figure 12.** Experimental results compared with analytical and modified analytical buckling loads

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-ofstraightness 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

predictions

d. Stiffened columns with s=0.5

**Figure 11.** Load versus strain gage measurements for the test specimens

Analytical, Numerical and Experimental Studies on Stability of Three-Segment Compression Members with Pinned Ends 111


110 Advances in Computational Stability Analysis

**Figure 11.** Load versus strain gage measurements for the test specimens

a. Unstiffened columns

Strain (m/m) Strain (m/m)

Strain (m/m)

b. Stiffened columns with s=0.2

Strain (m/m) Strain (m/m) Strain (m/m)

c. Stiffened columns with s=0.3333

Strain (m/m) Strain (m/m) Strain (m/m)

d. Stiffened columns with s=0.5

Strain (m/m) Strain (m/m) Strain (m/m)

**Table 4.** Experimental buckling loads for uniform and stiffened columns compared with the analytical predictions

**Figure 12.** Experimental results compared with analytical and modified analytical buckling loads

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-ofstraightness 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 *ideal* three-segment compression members, a *more realistic* analytical curve is drawn. This curve is plotted in Fig. 12 with a label '0.877 *Pcr,analy*'. From Fig. 12, it is seen that the "modified" analytical curve *almost* "averages" most of the test results. The larger discrepancies observed in stiffened specimens with *s*=0.3333 and *s*=0.5 are believed to be resulted from the residual stresses locked in the specimens during welding of the steel stiffening plates, which highly depends on quality of workmanship. For this reason, while calculating the buckling load of a multi-segment compression member formed by welding, not only the initial out-ofstraightness of the member, but also the effects of welding have to be taken into account. Considering that stiffened columns will always have more initial imperfections than uniform columns, it is suggested that a smaller modification factor be used in the design of multisegment columns. Based on the limited test data obtained in the experimental phase of this study, the following modification factor is proposed to be used in the design of three-segment symmetric steel compression members formed by welding steel stiffening plates:

$$\text{MFF} = \left(0.877 - 0.2s\right) \tag{25}$$

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 113

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

Abulwafa, E.M.; Abdou, M.A. & Mahmoud, A.A. (2007). Nonlinear fluid flows in pipe-like domain problem using variational iteration method. *Chaos Solitons & Fractals*, Vol.32,

American Institute of Steel Construction (AISC). (2010). *Specification for Structural Steel* 

Atay, M.T. & Coskun, S.B. (2009). Elastic stability of Euler columns with a continuous elastic restraint using variational iteration method. *Computers and Mathematics with* 

Batiha, B.; Noorani, M.S.M. & Hashim, I. (2007). Application of variational iteration method

Computers and Structures Inc. (CSI) (2008) *SAP2000 Static and Dynamic Finite Element* 

to heat- and wave-like equations. *Physics Letters A*, Vol. 369, pp. 55-61.

*Analysis of Structures* (Advanced 12.0.0), Berkeley, California.

stiffened regions.

**Author details** 

**6. References** 

*Kocaeli University, Turkey* 

No.4, pp. 1384–1397.

*Buildings (AISC 360-10)*, Chicago.

*Applications*, Vol.58, pp. 2528-2534.

Seval Pinarbasi Cuhadaroglu, Erkan Akpinar,

Fuad Okay, Hilal Meydanli Atalay and Sevket Ozden

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 (*Pcr,proposed*) for such a member can be computed by modifying the analytical buckling load (*Pcr,analy*) as in the following expression:

$$P\_{cr,proposed} = MF \times P\_{cr,analy} \tag{26}$$

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 '*Pcr,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.

## **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.
