**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, *Pcr,num,n=1* = 156.55 kN, of the uniform column. Exact value of the buckling load *Pcr* for this column can be computed from the well-known formula of Euler; 2 2 / *cr P EI L* , which gives *Pcr,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.

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 103

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 *s*=0.2, approximately 0.67 m in members with *s*=0.3333 and 1.0 m in members with *s*=0.5. This stiffening method increases the cross sectional area of the section about 1.56 times and major and minor axis flexural rigidities of the cross section, respectively, about 1.36 and 1.96 times. In the numerical analysis, the geometrical properties of the cross section for the stiffened region of the column have to be increased in these ratios. In SAP2000 (CSI, 2008), this step can easily be performed by using "property/stiffness modification factors" command (Fig. 6a). It is to be noted that axis-2 is still the minor axis of the member, so the buckling is expected to be observed about this axis, as in the uniform column case. Fig. 6b shows the buckled shape and buckling load (*Pcr,num,n=1.96,s=0.2* = 192.30 kN) of the stiffened members when one-fifth of the entire length of the member is stiffened as illustrated in Fig. 6a; i.e., when *n*=1.96 and *s*=0.2. Similar analyses on members with *s*=0.3333 and *s*=0.5 yield buckling loads of *Pcr,num,n=1.96,s=0.3333* = 220.42 kN and *Pcr,num,n=1.96,s=0.5* = 258.93 kN, respectively. If these values of buckling loads for stiffened elements are normalized with respect to the buckling load for the uniform member (*Pcr,num,n=1* = 156.55 kN), the amount of increase achieved in buckling load in each stiffening scheme is computed approximately as 1.23 when *s*=0.2, 1.41 when *s*=0.3333 and 1.65 when *s*=0.5. To compare numerical results with analytical results, buckling loads for three-segment symmetric stepped columns with *n*=1.96 are determined using VIM for various values of *s* and increase in buckling load with varying *s* is plotted in Fig. 7. It can be seen that the approximate results obtained through numerical analysis exactly match with VIM solutions. The effectiveness of the numerical analysis in solving this special buckling problem is examined further for different values of *n* and *s*. The results are presented in Table 4, which indicates very good agreement between the

**Table 3.** Comparison of numerical results with analytical (exact and approximate (VIM)) results for increase in buckling load for a three-segment compression member with pinned ends for various values

**n Exact VIM SAP2000 Exact VIM SAP2000 Exact VIM SAP2000** 1.5 1.18 1.18 1.18 1.37 1.37 1.38 1.48 1.48 1.48 2 1.30 1.30 1.30 1.68 1.68 1.68 1.95 1.95 1.93 2.5 1.38 1.38 1.38 1.93 1.93 1.92 2.40 2.40 2.38 3 1.44 1.44 1.44 2.13 2.13 2.13 2.85 2.85 2.80 5 1.56 1.56 1.56 2.69 2.69 2.67 4.48 4.48 4.35 7.5 1.63 1.63 1.63 3.06 3.06 3.03 6.22 6.22 5.94 10 1.66 1.66 1.67 3.27 3.27 3.24 7.65 7.65 7.20

**s=0.25 s=0.5 s=0.75**

of stiffness ratio (*n*=*EI*2/*EI*1) and stiffened length ratio (*s*=*a*/*H*)

analytical and numerical results.

a. cross sectional properties (in meters)

b. buckling load (in kN)

**Figure 5.** Geometric properties and buckling load of the uniform column (*n*=1) analyzed in numerical study

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 *s*=0.2, approximately 0.67 m in members with *s*=0.3333 and 1.0 m in members with *s*=0.5. This stiffening method increases the cross sectional area of the section about 1.56 times and major and minor axis flexural rigidities of the cross section, respectively, about 1.36 and 1.96 times. In the numerical analysis, the geometrical properties of the cross section for the stiffened region of the column have to be increased in these ratios. In SAP2000 (CSI, 2008), this step can easily be performed by using "property/stiffness modification factors" command (Fig. 6a). It is to be noted that axis-2 is still the minor axis of the member, so the buckling is expected to be observed about this axis, as in the uniform column case. Fig. 6b shows the buckled shape and buckling load (*Pcr,num,n=1.96,s=0.2* = 192.30 kN) of the stiffened members when one-fifth of the entire length of the member is stiffened as illustrated in Fig. 6a; i.e., when *n*=1.96 and *s*=0.2. Similar analyses on members with *s*=0.3333 and *s*=0.5 yield buckling loads of *Pcr,num,n=1.96,s=0.3333* = 220.42 kN and *Pcr,num,n=1.96,s=0.5* = 258.93 kN, respectively. If these values of buckling loads for stiffened elements are normalized with respect to the buckling load for the uniform member (*Pcr,num,n=1* = 156.55 kN), the amount of increase achieved in buckling load in each stiffening scheme is computed approximately as 1.23 when *s*=0.2, 1.41 when *s*=0.3333 and 1.65 when *s*=0.5. To compare numerical results with analytical results, buckling loads for three-segment symmetric stepped columns with *n*=1.96 are determined using VIM for various values of *s* and increase in buckling load with varying *s* is plotted in Fig. 7. It can be seen that the approximate results obtained through numerical analysis exactly match with VIM solutions. The effectiveness of the numerical analysis in solving this special buckling problem is examined further for different values of *n* and *s*. The results are presented in Table 4, which indicates very good agreement between the analytical and numerical results.

102 Advances in Computational Stability Analysis

study

**Figure 5.** Geometric properties and buckling load of the uniform column (*n*=1) analyzed in numerical

b. buckling load (in kN)

a. cross sectional properties (in meters)


**Table 3.** Comparison of numerical results with analytical (exact and approximate (VIM)) results for increase in buckling load for a three-segment compression member with pinned ends for various values of stiffness ratio (*n*=*EI*2/*EI*1) and stiffened length ratio (*s*=*a*/*H*)

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 105

**Figure 7.** Increase in critical buckling load for various stiffened length ratios (*s*) when stiffness ratio is *n*

**4. Experimental studies on elastic buckling of a three-segment stepped** 

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,

1.96 (VIM results)

B3-2 and B3-3.

**compression member with pinned ends** 

a. area/stiffness modifiers for the stiffened region of the column

b. buckling load (in kN)

1

110x3 plate

**Figure 6.** Geometric properties and buckling load a three-segment stepped column with stiffened

b. buckling load (in kN)

a. area/stiffness modifiers for the stiffened region of the column

2

RHCF 120x40x4

110x3 plate

length ratio *s*=0.2 and stiffness ratio *n*=1.96

**Figure 7.** Increase in critical buckling load for various stiffened length ratios (*s*) when stiffness ratio is *n* 1.96 (VIM results)
