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

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

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, *EI*1, while that of the middle segment may be different, say *EI*2. As long as the stiffness variation along the height of the member is symmetric about the mid-height, the buckled shape of the member is also symmetric about the same point as shown in Fig. 1. When such a symmetry exists, the buckling load of the three-segment member can be obtained by analyzing the simpler two-segment member shown in Fig. 2a. This "equivalent" two-segment member has a fixed (clamped) boundary condition at its bottom end whereas its top end is free. From comparison of Fig. 1 and Fig. 2a, one can also see that the length of the equivalent two-segment member equals to the half-length of the original three-segment member, i.e., *L*=*H*/2. Similarly, *L*2=*a*/2. Since the analysis of a two-segment column is much simpler than that of a three-segment column, the analytical study presented in this section is based on the equivalent two-segment member.

The undeformed and deformed shapes of the equivalent two-segment member under uniform compression are illustrated in Fig. 2a. The origin of *x*-*y* coordinate system is located at the bottom end of the column. Since the stiffnesses of two segments of the column can be different in general, each segment of the column has to be analyzed separately. Equilibrium equation at an arbitrary section in Segment I can be written from the free body diagram shown in Fig. 2b:

$$EI\_1 \frac{d^2 w\_1}{d\chi^2} \cdot P\left(\mathcal{S} \cdot w\_1\right) = 0 \tag{1}$$

Analytical, Numerical and Experimental

is also unknown, the

Studies on Stability of Three-Segment Compression Members with Pinned Ends 95

 (4)

*k L* (5)

/ *L* and prime denotes differentiation with

(6)

(7)

 

(9)

 

(10a)

(10b)

(8)

 2 2 1 11 1 *w w* 

where *x xL* / , 1 1 *w wL* / , 2 2 *w wL* / ,

functions have to be continuous, which requires

**2.2. Exact solution to buckling equations** 

11 1 2 1 *wC xC x* sin cos and

with

stability problem.

 

1 1 

solutions will contain four integration constants. Considering that

column, the end conditions can be written in nondimensional form as:

obtain their exact solutions, which can be written in the following form:

 

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

1

 

1

 

<sup>2</sup> <sup>0</sup> 0, *<sup>x</sup> <sup>w</sup>* <sup>2</sup>

and 2 2 *w w* 2 22 2

 *k L*and 2 2 

respect to *x* . Since both of the differential equations in Eq. (4) are in second order, 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 *x*=*L*2, the lateral displacement and slope

1 2 and *xs xs w w* 1 2 *x s x s*

where 2 *sL L* / . As far as the boundary conditions are concerned, for a clamped-free

0

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

Since the differential equations given in Eq. (4) are relatively simple, it is not too difficult to

where *Ci* (*i*=1-4) are integration constants to be determined from continuity and end

<sup>3</sup> *C* 0and and *C*<sup>4</sup>

1 2 1 21

2 21 2 1

*C ss ss* sin sin cos cos

*C ss ss* sin cos cos sin

<sup>2</sup>

<sup>2</sup>

conditions. From the first and second conditions given in Eq. (7), one can find that

*x*

*w*

0and

<sup>1</sup> *<sup>x</sup>* <sup>1</sup>

*w w*

*w*

 and 23 2 4 2 *wC xC x* sin cos 

 

> 

 

which can be expressed as

$$\frac{d^2 w\_1}{d\mathbf{x}^2} + k\_1^2 w\_1 = k\_1^2 \delta \text{ where} \quad k\_1^2 = \frac{P}{EI\_1} \tag{2}$$

In Eq. (1) and Eq. (2), *w*1 is lateral displacement of Segment I at any point, is the lateral displacement of the top end of the member, i.e., = *w*1 (*x* = *L*). Eq. (2) is valid for *L*<sup>2</sup> *x L*. Similarly, from Fig. 2c, the equilibrium equation at an arbitrary section in Segment II can be written as

$$\frac{d^2 w\_2}{d\mathbf{x}^2} + k\_2^2 w\_2 = k\_2^2 \delta \text{ where} \quad k\_2^2 = \frac{P}{EI\_2} \tag{3}$$

where *w*2 is the displacement of Segment II in *y* direction. Eq. (3) is valid for 0 *x L*2. For easier computations, the buckling equations in Eq. (2) and Eq. (3) can be written in nondimensional form as follows:

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 95

$$\alpha \left(\overline{w}\_1\right)'' + \beta\_1^2 \left(\overline{w}\_1\right) = \beta\_1^2 \left(\overline{\delta}\right) \\ \text{and} \quad \left(\overline{w}\_2\right)'' + \beta\_2^2 \left(\overline{w}\_2\right) = \beta\_2^2 \left(\overline{\delta}\right) \tag{4}$$

with

94 Advances in Computational Stability Analysis

which can be expressed as

nondimensional form as follows:

written as

**compression member with pinned ends** 

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

**2. Analytical studies on elastic buckling of a three-segment stepped** 

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, *EI*1, while that of the middle segment may be different, say *EI*2. As long as the stiffness variation along the height of the member is symmetric about the mid-height, the buckled shape of the member is also symmetric about the same point as shown in Fig. 1. When such a symmetry exists, the buckling load of the three-segment member can be obtained by analyzing the simpler two-segment member shown in Fig. 2a. This "equivalent" two-segment member has a fixed (clamped) boundary condition at its bottom end whereas its top end is free. From comparison of Fig. 1 and Fig. 2a, one can also see that the length of the equivalent two-segment member equals to the half-length of the original three-segment member, i.e., *L*=*H*/2. Similarly, *L*2=*a*/2. Since the analysis of a two-segment column is much simpler than that of a three-segment column, the analytical study presented in this section is based on the equivalent two-segment member.

The undeformed and deformed shapes of the equivalent two-segment member under uniform compression are illustrated in Fig. 2a. The origin of *x*-*y* coordinate system is located at the bottom end of the column. Since the stiffnesses of two segments of the column can be different in general, each segment of the column has to be analyzed separately. Equilibrium equation at an arbitrary section in Segment I can be written from the free body diagram shown in Fig. 2b:

> 2 1 1 1 <sup>2</sup> -- 0 *d w EI P w*

*dx*

1 2 2 <sup>2</sup> 11 1 where *d w kw k*

In Eq. (1) and Eq. (2), *w*1 is lateral displacement of Segment I at any point,

2 2 2 <sup>2</sup> 22 2 where *d w kw k*

2

*dx*

2

*dx*

displacement of the top end of the member, i.e.,

2

2

1

2

*<sup>P</sup> <sup>k</sup>*

*<sup>P</sup> <sup>k</sup>*

1

2

(1)

*EI* (2)

*EI* (3)

= *w*1 (*x* = *L*). Eq. (2) is valid for *L*<sup>2</sup> *x L*.

is the lateral

Similarly, from Fig. 2c, the equilibrium equation at an arbitrary section in Segment II can be

where *w*2 is the displacement of Segment II in *y* direction. Eq. (3) is valid for 0 *x L*2. For easier computations, the buckling equations in Eq. (2) and Eq. (3) can be written in

$$
\beta\_1 = k\_1 L \text{ and } \quad \beta\_2 = k\_2 L \tag{5}
$$

where *x xL* / , 1 1 *w wL* / , 2 2 *w wL* / , / *L* and prime denotes differentiation with respect to *x* . Since both of the differential equations in Eq. (4) are in second order, the solutions will contain four integration constants. Considering that 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 *x*=*L*2, the lateral displacement and slope functions have to be continuous, which requires

$$\left[\left.\overline{w}\_{1}\right\|\_{\overline{\mathfrak{X}}=s}=\left\[\;\overline{w}\_{2}\right\|\_{\overline{\mathfrak{X}}=s}\text{ and }\left[\left(\left.\overline{w}\_{1}\right)'\right\vert\_{\overline{\mathfrak{X}}=s}=\left[\left(\left.\overline{w}\_{2}\right)'\right\vert\_{\overline{\mathfrak{X}}=s}\right]\right] \tag{6}$$

where 2 *sL L* / . As far as the boundary conditions are concerned, for a clamped-free column, the end conditions can be written in nondimensional form as:

$$\left[\left(\overline{w}\_2\right)\right]\_{\overline{x}=0} = 0, \ \left[\left(\overline{w}\_2\right)'\right]\_{\overline{x}=0} = 0 \text{and} \ \left[\left(\overline{w}\_1\right)\right]\_{\overline{x}=1} = \overline{\delta} \tag{7}$$

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

#### **2.2. Exact solution to buckling equations**

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:

$$
\overline{w}\_1 = \mathbb{C}\_1 \sin \left(\beta\_1 \overline{\mathbf{x}}\right) + \mathbb{C}\_2 \cos \left(\beta\_1 \overline{\mathbf{x}}\right) + \overline{\delta} \text{ and } \quad \text{and} \quad \overline{w}\_2 = \mathbb{C}\_3 \sin \left(\beta\_2 \overline{\mathbf{x}}\right) + \mathbb{C}\_4 \cos \left(\beta\_2 \overline{\mathbf{x}}\right) + \overline{\delta} \tag{8}
$$

where *Ci* (*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

$$\mathcal{C}\_3 = 0 \text{ and } \text{ and } \mathcal{C}\_4 = -\overline{\delta} \tag{9}$$

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

$$C\_1 = \overline{\delta} \left[ \frac{\rho\_2}{\rho\_1} \sin(\beta\_2 s) \cos(\beta\_1 s) - \cos(\beta\_2 s) \sin(\beta\_1 s) \right] \tag{10a}$$

$$\mathbf{C}\_2 = -\overline{\boldsymbol{\delta}} \left[ \frac{\beta\_2}{\beta\_1} \sin \left( \beta\_2 s \right) \sin \left( \beta\_1 s \right) + \cos \left( \beta\_2 s \right) \cos \left( \beta\_1 s \right) \right] \tag{10b}$$

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

$$\left\{ \tan \left[ \beta\_2 s \right] \tan \left[ \beta\_1 (1 - s) \right] - \frac{\beta\_1}{\beta\_2} \right\} \overline{\delta} = 0 \tag{11}$$

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 97

 

.

 

 

,

 (21a)

> 

, the iteration formulas for the buckling equations of

(21b)

 

and *n* as follows:

, <sup>4</sup>

 

(22a)

,

,

(18)

 (19)

> 

(20)

*x* (17)

0

*x w x w x Lw Nw d nn nn* 

iteration formula:

follows:

and

written as follows:

is decided to be used in this study.

 

Recalling that 1 2

 

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

<sup>1</sup>

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

> 1 0 0

2 1 <sup>1</sup>

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

2 2

2 2

 <sup>2</sup> ,2 ,1 ,1 , ,1 ,

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

1, 1 1,0 1, 0

*x w xwx x w d j j*

*w x w x xw w w w d i n i n i n in i in in*

 

(21c)

*w x w x xw w d i n i n in i in i* 

*x w xwx x w d i n <sup>i</sup> i in i* 

*w x w x Nw Nw d n n n n* 

0

,1 , , , 0

> , 1 ,0 , 0

> > 0

 4

/ *n* and <sup>2</sup>

the studied problem can be written in terms of

*x*

1

*x*

*x*

*x w x w x Nw d n n* 

 

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

$$
\tan\left[\beta\_2 s\right] \tan\left[\beta\_1 \left(1 - s\right)\right] = \frac{\beta\_1}{\beta\_2} \tag{12}
$$

Since 12 2 1 / / *EI EI* , if the stiffness ratio *n* is defined as 2 1 *n EI EI* / , Eq. (12) can be written in terms of 1 (square root of nondimensional buckling load of the equivalent twosegment element in terms of *EI*1), *n* (stiffness ratio) and *s* (stiffened length ratio) as follows:

$$
\tan\left[\mathcal{J}\_1(1-s)\right]\tan\left[\mathcal{J}\_1\frac{s}{\sqrt{n}}\right] = \sqrt{n}\tag{13}
$$

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 *L*=*H*/2 shown in Fig. 2a as

$$P\_{cr} = \mathcal{\lambda} \frac{EI\_1}{H^2} \text{where} \quad \mathcal{\lambda} = 4\mathcal{\beta}\_1^2 \tag{14}$$

In other words, is the nondimensional buckling load of the three-segment compression member *in terms of EI*1.

#### **2.3. VIM solution to buckling equations**

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

$$Lw(\mathbf{x}) + \mathcal{N}w(\mathbf{x}) = \mathcal{g}(\mathbf{x}) \tag{15}$$

where *L* is a linear operator and *N* is a nonlinear operator, *g*(*x*) is the nonhomogeneous term. Based on VIM, the "correction functional" can be constructed as

$$\mathbb{E}w\_{n+1}\left(\mathbf{x}\right) = \mathbb{w}\_n\left(\mathbf{x}\right) + \int\_0^\cdot \mathbb{A}\left(\boldsymbol{\xi}\right) \left(L\boldsymbol{w}\_n\left(\boldsymbol{\xi}\right) + N\tilde{\boldsymbol{w}}\_n\left(\boldsymbol{\xi}\right)\right) d\boldsymbol{\xi} \tag{16}$$

where is a general Lagrange multiplier that can be identified optimally via variational theory, *wn* is the n-th approximate solution and *wn* denotes a restricted variation, i.e., 0 *wn* (He, 1999). As summarized in He et al. (2010), for a second order differential equation such as the buckling equations given in Eq. (4), simply equals to

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 97

$$
\mathcal{A}\left(\xi\right) = \left(\xi - \infty\right) \tag{17}
$$

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

$$\operatorname{div}\_{n+1}\left(\mathbf{x}\right) = \operatorname{w}\_{n}\left(\mathbf{x}\right) + \int\_{0}^{\mathbf{x}} \lambda\left(\boldsymbol{\xi}\right) \left\{ L\boldsymbol{w}\_{n}\left(\boldsymbol{\xi}\right) + N\boldsymbol{w}\_{n}\left(\boldsymbol{\xi}\right) \right\} d\boldsymbol{\xi} \tag{18}$$

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:

$$\mathrm{div}\_{n+1}\left(\mathbf{x}\right) = \mathrm{w}\_{0}\left(\mathbf{x}\right) + \int\_{0}^{\mathbf{x}} \mathcal{A}\left(\boldsymbol{\xi}\right) \left\{ \mathrm{N}\boldsymbol{w}\_{n}\left(\boldsymbol{\xi}\right) \right\} d\boldsymbol{\xi}.\tag{19}$$

and

96 Advances in Computational Stability Analysis

Since 12 2 1

member with length *L*=*H*/2 shown in Fig. 2a as

**2.3. VIM solution to buckling equations** 

equation can be written in the following form:

Based on VIM, the "correction functional" can be constructed as

equation such as the buckling equations given in Eq. (4),

written in terms of

In other words,

where

0 *wn* 

 

member *in terms of EI*1.

 

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

characteristic equation for the studied buckling problem:

<sup>1</sup>

tan tan 1 *s s* 0

For a nontrivial solution, the coefficient term must be equal to zero, yielding the following

 

segment element in terms of *EI*1), *n* (stiffness ratio) and *s* (stiffened length ratio) as follows:

 

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

1 1 tan 1 tan *<sup>s</sup>*

1 <sup>2</sup> where *cr EI <sup>P</sup> <sup>H</sup>* 

 

2 1

tan tan 1 *s s*

2

<sup>1</sup>

/ / *EI EI* , if the stiffness ratio *n* is defined as 2 1 *n EI EI* / , Eq. (12) can be

*s n n*

 

<sup>2</sup>

According to the variational iteration method (VIM), a general nonlinear differential

where *L* is a linear operator and *N* is a nonlinear operator, *g*(*x*) is the nonhomogeneous term.

<sup>1</sup>

theory, *wn* is the n-th approximate solution and *wn* denotes a restricted variation, i.e.,

(He, 1999). As summarized in He et al. (2010), for a second order differential

 

is a general Lagrange multiplier that can be identified optimally via variational

0

*x w x w x Lw Nw d nn nn* 

 4

 

2

(12)

1 (square root of nondimensional buckling load of the equivalent two-

1

*Lw x Nw x g x* (15)

 

simply equals to

(16)

 

is the nondimensional buckling load of the three-segment compression

(11)

(13)

(14)

2 1

$$\operatorname{div}\_{n+2}\left(\mathbf{x}\right) = \operatorname{\boldsymbol{w}}\_{n+1}\left(\mathbf{x}\right) + \int\_{0}^{\mathbf{x}} \mathcal{A}\left(\boldsymbol{\xi}\right) \left(\operatorname{N}\mathbf{w}\_{n+1}\left(\boldsymbol{\xi}\right) - \operatorname{N}\mathbf{w}\_{n}\left(\boldsymbol{\xi}\right)\right) d\boldsymbol{\xi} \tag{20}$$

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

$$
\Delta \overline{w}\_{i,n+1} \left( \mathbf{x} \right) = \overline{w}\_{i,n} \left( \mathbf{x} \right) + \int\_0^\mathbf{x} \left( \boldsymbol{\xi} - \mathbf{x} \right) \left( \overline{w}\_{i,n}^{\top} \left( \boldsymbol{\xi} \right) + \boldsymbol{\beta}\_i^2 \, \overline{w}\_{i,n} - \boldsymbol{\beta}\_i^2 \, \overline{\boldsymbol{\delta}} \right) d\boldsymbol{\xi}, \tag{21a}
$$

$$\overline{w}\_{i,n+1}\left(\boldsymbol{\chi}\right) = \overline{w}\_{i,0}\left(\boldsymbol{\chi}\right) + \int\_{0}^{\boldsymbol{\chi}} \left(\boldsymbol{\xi} - \boldsymbol{\chi}\right) \left\{\boldsymbol{\beta}\_{i}^{2}\overline{w}\_{i,n} - \boldsymbol{\beta}\_{i}^{2}\overline{\boldsymbol{\delta}}\right\} d\boldsymbol{\xi} \,\tag{21b}$$

$$
\overline{w}\_{i,n+2}\left(\mathbf{x}\right) = \overline{w}\_{i,n+1}\left(\mathbf{x}\right) + \int\_0^\mathbf{x} \left(\xi - \mathbf{x}\right) \left| \left(\overline{w}\_{i,n+1}^\cdot \left(\xi\right) - \overline{w}\_{i,n}^\cdot \left(\xi\right)\right) + \mathcal{J}\_i^2 \left(\overline{w}\_{i,n+1} - \overline{w}\_{i,n}\right) \right| d\xi \,\tag{21c}
$$

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 1 2 / *n* and <sup>2</sup> 1 4 , the iteration formulas for the buckling equations of the studied problem can be written in terms of and *n* as follows:

$$\overline{w}\_{1,j+1}\left(\boldsymbol{x}\right) = \overline{w}\_{1,0}\left(\boldsymbol{x}\right) + \bigwedge\_{0}^{\scriptscriptstyle \mathsf{x}}\left(\boldsymbol{\xi} - \boldsymbol{x}\right) \left\{\frac{\mathcal{A}}{4}\left(\overline{w}\_{1,j} - \overline{\boldsymbol{\delta}}\right)\right\} d\boldsymbol{\xi} \,\tag{22a}$$

$$
\overline{w}\_{2,j+1}\left(\mathbf{x}\right) = \overline{w}\_{2,0}\left(\mathbf{x}\right) + \int\_0^\mathbf{x} \left(\xi - \mathbf{x}\right) \left\{\frac{\mathcal{A}}{4n} \left(\overline{w}\_{2,j} - \overline{\mathcal{S}}\right)\right\} d\xi \tag{22b}
$$

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 99

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

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 (*s*) ratios. The results listed in this table can directly be used by design engineers who design/strengthen three-segment symmetric stepped compression members with pinned

**0.1 0.2 0.25 0.3333 0.5 0.75 0.9999**

**s**

1 9.8696 9.8696 9.8696 9.8696 9.8696 9.8696 9.8696 1.5 10.5592 11.3029 11.6881 12.3342 13.5322 14.6186 14.8044 2 10.9332 12.1571 12.8290 14.0255 16.5379 19.2404 19.7392 2.5 11.1676 12.7211 13.6051 15.2433 19.0149 23.7328 24.6740 3 11.3282 13.1202 14.1651 16.1557 21.0707 28.0942 29.6088 4 11.5338 13.6465 14.9165 17.4239 24.2442 36.4193 39.4784 5 11.6599 13.9775 15.3962 18.2587 26.5469 44.2105 49.3480 7.5 11.8311 14.4372 16.0711 19.4641 30.1728 61.3848 74.0220 10 11.9181 14.6750 16.4240 20.1076 32.2453 75.4700 98.6960 20 12.0504 15.0419 16.9731 21.1249 35.6828 109.4880 197.3920 50 12.1307 15.2680 17.3139 21.7652 37.9220 138.1940 493.4800 100 12.1577 15.3444 17.4295 21.9836 38.6944 148.2010 986.9600

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

) of a three-segment column for various

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

available for the problem, as in our case.

**Table 2.** VIM predictions for nondimensional buckling load (

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

ends.

**n**

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

$$
\overline{w}\_{1,0} = \mathbb{C}\_1 \overline{\mathfrak{x}} + \mathbb{C}\_2 \quad \text{and} \quad \overline{w}\_{2,0} = \mathbb{C}\_3 \overline{\mathfrak{x}} + \mathbb{C}\_4 \tag{23}
$$

where *Ci* (*i*=1-4) are to be determined from continuity and end conditions. After conducting seventeen iterations, *w*1,17 and *w*2,17 are obtained. Substituting these approximate solutions to the continuity equations in Eq. (6) and to the end conditions in Eq. (7), five equations are obtained. Four of them are used to determine the unknown coefficients in terms of , while the remaining one is used to construct the characteristic equation for the studied problem:

$$
\left[\left[F\left(\mathcal{k}\right)\right]\overline{\delta} = 0\right.\tag{24}
$$

where *F* is the coefficient term of . For a nontrivial solution *F* must be equal to zero. The smallest possible real root of the characteristic equation gives the nondimensional buckling load ( <sup>2</sup> 1 *PH EI* / ) of the three-segment compression member in the first buckling mode.

#### **2.4. Comparison of VIM results with exact results**

For various values of stiffness ratio (*n*=*EI*2/*EI*1) and stiffened length ratio (*s*=*a*/*H*), nondimensional buckling loads of a three-segment compression member with pinned ends are determined both by using Eq. (13) and VIM. VIM results are compared with the exact results in Table 1.


**Table 1.** Comparison of VIM predictions for nondimensional buckling load () 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*)

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.

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

98 Advances in Computational Stability Analysis

with unknown coefficients is used:

where *F*

buckling mode.

results in Table 1.

length ratio (*s*=*a*/*H*)

buckling load ( <sup>2</sup>

is the coefficient term of

1

**2.4. Comparison of VIM results with exact results** 

**Table 1.** Comparison of VIM predictions for nondimensional buckling load (

compression member with exact results for various values of stiffness ratio (*n*=*EI*2/*EI*1) and stiffened

2, 1 2,0 2, <sup>0</sup> 4

As an initial approximation for displacement function of each segment, a linear function

where *Ci* (*i*=1-4) are to be determined from continuity and end conditions. After conducting seventeen iterations, *w*1,17 and *w*2,17 are obtained. Substituting these approximate solutions to the continuity equations in Eq. (6) and to the end conditions in Eq. (7), five equations are

obtained. Four of them are used to determine the unknown coefficients in terms of

the remaining one is used to construct the characteristic equation for the studied problem:

zero. The smallest possible real root of the characteristic equation gives the nondimensional

For various values of stiffness ratio (*n*=*EI*2/*EI*1) and stiffened length ratio (*s*=*a*/*H*), nondimensional buckling loads of a three-segment compression member with pinned ends are determined both by using Eq. (13) and VIM. VIM results are compared with the exact

**Exact VIM Exact VIM Exact VIM Exact VIM**

**s**

100 15.344 15.344 27.052 27.052 59.843 59.843 225.706 225.706 10 14.675 14.675 24.006 24.006 44.978 44.978 85.880 85.880 5 13.978 13.978 21.109 21.109 33.471 33.471 46.651 46.651 2.5 12.721 12.721 16.694 16.693 21.275 21.275 24.186 24.186 1.67 11.632 11.632 13.642 13.642 15.406 15.406 16.306 16.306 1.25 10.689 10.689 11.471 11.471 12.039 12.039 12.297 12.297

**n 0.2 0.4 0.6 0.8**

 *F* 

. For a nontrivial solution *F*

*PH EI* / ) of the three-segment compression member in the first

 (22b)

*w Cx C* 1,0 1 2 and *w Cx C* 2,0 3 4 (23)

<sup>0</sup> (24)

) of a three-segment

, while

must be equal to

 

*x w xw x x w d j j <sup>n</sup>*

> 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 (*s*) ratios. The results listed in this table can directly be used by design engineers who design/strengthen three-segment symmetric stepped compression members with pinned ends.


**Table 2.** VIM predictions for nondimensional buckling load () of a three-segment column for various values of stiffness ratio (*n*=*EI*2/*EI*1) and stiffened length ratio (*s*=*a*/*H*)

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 *n*=2, increase in buckling load when only half length of the member is stiffened is more than 80 % of the increase that can be gained when the entire length of the member is stiffened. Fig. 3 also shows that if *n* increases, to get such an enhancement in buckling load, *s* has to be increased. For example, when *n*=10, the stiffened length of the member has to be more than 75% of its entire length if similar enhancement in member behavior is required. In fact, this can be seen more easily from Fig. 4 where the increase in buckling load is plotted in terms of stiffness ratio for various stiffened length ratios. Fig. 4 shows that if the stiffened length ratio is small, there is no need to increase the stiffness ratio too much. As an example, if only one-fifth of the entire length of the member is to be stiffened, increase in buckling load when *n*=2 is more than 80% of that when *n*=10. On the other hand, if 75 % of the entire length is allowed to be stiffened, increase in buckling load when *n*=2 is approximately 25% of that when *n*=10.

Analytical, Numerical and Experimental

Studies on Stability of Three-Segment Compression Members with Pinned Ends 101

**Figure 4.** Variation of increase in buckling load with stiffness ratio (*n*) for various values of stiffened

1 2 3 4 5 6 7 8 9 10

s=0.2 s=0.3333 s=0.5 s=0.75

**n=EI2/EI1**

**3. Numerical studies on elastic buckling of a three-segment stepped** 

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

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

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"

, which gives *Pcr,exact,n=1* =

**compression member with pinned ends** 

structural analysis program SAP2000 (CSI, 2008).

computed from the well-known formula of Euler; 2 2 / *cr P EI L*

length ratio (*s*)

1

10

**Pcr,stepped/Pcr,uniform(n=1)**

members.

**Figure 3.** Variation of increase in buckling load with stiffened length ratio (*s*) for various values of stiffness ratio (*n*)

stiffness ratio (*n*)

1

10

**Pcr,stepped/Pcr,uniform(n=1)**

when *n*=2 is approximately 25% of that when *n*=10.

length of the member to gain appreciable amount of increase in buckling load especially if *n*  is not too large. For *n*=2, increase in buckling load when only half length of the member is stiffened is more than 80 % of the increase that can be gained when the entire length of the member is stiffened. Fig. 3 also shows that if *n* increases, to get such an enhancement in buckling load, *s* has to be increased. For example, when *n*=10, the stiffened length of the member has to be more than 75% of its entire length if similar enhancement in member behavior is required. In fact, this can be seen more easily from Fig. 4 where the increase in buckling load is plotted in terms of stiffness ratio for various stiffened length ratios. Fig. 4 shows that if the stiffened length ratio is small, there is no need to increase the stiffness ratio too much. As an example, if only one-fifth of the entire length of the member is to be stiffened, increase in buckling load when *n*=2 is more than 80% of that when *n*=10. On the other hand, if 75 % of the entire length is allowed to be stiffened, increase in buckling load

**Figure 3.** Variation of increase in buckling load with stiffened length ratio (*s*) for various values of

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

n=2 n=3 n=5 n=10

**s=a/H**

**Figure 4.** Variation of increase in buckling load with stiffness ratio (*n*) for various values of stiffened length ratio (*s*)
