A simplified method for stability analysis of multi-story frames considering vertical interactions between stories

Abstract

Effective length method is still widely used in engineering practice to evaluate the stability of compression members in frame structures. However, the conventional effective length method can be inaccurate in many cases as it considers columns in isolation from the stories above and below. This article proposes an improved method for simplified frame stability analysis that accounts for the vertical interaction effects of columns. The idealized sub-assemblage model includes the columns of all stories and their restraining beams. The stability matrix for a single-story unit is first derived, and then the system stability matrix is obtained by considering the compatibility of the story units. The governing equation for the elastic buckling load of the sub-assemblage is derived. The method is applicable to both sway-permitted and sway-prevented frames. The applicability and accuracy of the proposed method are demonstrated using a series of examples with a wide variation of parameters including numbers of story, boundary conditions, stiffness of beam-to-column connections, column length and stiffness, and axial force level.

Keywords

buckling columns effective length factors frames stability steel structures

Introduction

The effective length method has been widely used for stability evaluation and design of compression members for many years. Columns are considered in isolation with end restraints, and the effective length factor is evaluated based on the joint stiffness ratio at each end of the column (Julian and Lawrenee, 1959). The joint stiffness ratio, denoted by G, is defined by

G = \frac{\sum E_{c} I_{c} / L_{c}}{\sum E_{b} I_{b} / L_{b}}

(1)

where E = Young’s modulus, I = moment of inertia of the cross-section, L = length of the member, and the summations are over all the columns (beams) intersecting at the joint. The subscripts b and c indicate beam and column, respectively. The G-factor is intended to recognize the interaction between the column in question and its surrounding structure. Once the G-factors at the two ends of the column are determined, the column effective length factor can be determined from alignment charts (e.g. ANSI/AISC 360-05, 2005; AS4100-1990, 1990).

A basic assumption of the G-factor method is that the stiffness factors of the columns in each story are the same. Herein, the stiffness factor is defined as $L_{c} \sqrt{P / E_{c} I_{c}}$ , where P is the axial force of the column and L_c is the length of the column. This assumption generally cannot be fulfilled in a multi-story frame structure because of the changes in story height, axial force, and/or flexural rigidity of columns. In many cases, the conventional G-factor method cannot properly recognize the contribution of columns in other stories to the rotational end restraints of the column in question, thus leads to inaccurate results (Hellesland and Bjorhovde, 1997).

Efforts have been made to improve the accuracy of the G-factor method and extend its range of validity. Duan and Chen (1988, 1989) modified the G-factor approach by considering the difference in the boundary conditions of top and bottom columns. Kishi and Chen (1997) derived the governing equations of the effective length of columns for various boundary conditions with rigid or semi-rigid beam-to-column connections, incorporating the effects of nonlinear moment–rotation behavior of connections. Yura (1971) developed an approximate method for unbraced frames to consider the effect of adjacent columns in the same story on the rotational end restraints of the column under consideration. Recognizing that the rotational restraints imposed by top and bottom columns may have positive or negative effects on the effective length factor of the column considered, Bridge and Fraser (1987) proposed an iterative procedure to modify the effective length factor alignment chart to account for the effects of columns in other stories. Hellesland and Bjorhovde (1997) proposed a method which post-processes the effective length factor from conventional G-factor method to arrive at improved, weighted mean values. However, the improvement in accuracy relies on the results of conventional G-factor method, thus is limited in some cases due to the inaccuracy of the conventional G-factor method. Hellesland (2007) derived the exact effective length factor for a restrained, isolated compression member. However, the formulas for the effective lengths of columns in a multi-story frame were approximated. More recent advances on the simplified stability analysis of frames have focused on irregular frames (Girgin et al., 2006), the combined bending–shear deformation effects and the second-order axial load effects (Aristizabal, 1994), and the combined effects of bending and torsion (Adman and Saidain, 2013). All these works, however, did not incorporate explicitly the vertical interaction effects of columns in different stories.

This article proposes a new method for simplified system bifurcation instability analysis which considers the vertical interaction effects of all columns in different stories. The idealized column model in the conventional G-factor method is extended to include the columns of all stories and their restraining beams. The stability matrix for a single-story unit is first derived using the principle of moment–deflection relationship, and then the system stability matrix is obtained by considering the compatibility of story units. It can be used for sway-permitted and sway-prevented frames. The applicability and accuracy of the proposed method are examined by comparison with exact results for a series of examples.

Stability matrix of frame structures

Figure 1 demonstrates the idealized sub-assemblage model adopted in this article, where Figure 1(a) is for the sway-permitted instability, and Figure 1(b) is for the sway-prevented instability. Different from the isolated column model used in the G-factor method, the present sub-assemblage consists of all the columns at each story (C-1 to C-n) and the relevant restraining beams (B-1-1, B-1-2 to B-n-1, B-n-2). Using this sub-assemblage model, it is possible to account for the vertical interaction effects of columns in different stories, and recognize the effects of changes in column length and/or stiffness, and axial force level.

Figure 1.

Idealized sub-assemblage models for (a) sway-permitted and (b) sway-prevented frames.

The assumptions adopted for the present method are as follows: (1) axial forces in beams are negligible; (2) the columns at the same story buckle simultaneously and the rotations at near and far ends of a beam are equal in magnitude, identical in direction for sway-permitted frame, and opposite in direction for sway-prevented frame (c.f. Figure 1(a) and (b) for the bending of beams); (3) the influence of beam shear force on the axial force of the column is negligible; and (4) the deformation of the member is elastic and small, as a result, the beam shear force is small compared with the column axial force, because large beam shear force is usually associated with large story drift and large deformation of columns.

We first derive the stability matrix for a single-story unit, referred to as story stability matrix, for the case of sway-permitted frame and sway-prevented frame, respectively. This story stability matrix describes the moment–rotation relationship between two vertically adjacent joints. Based on the story stability matrices, a recursive formula is used to compute the stability matrix of the whole frame, referred to as frame stability matrix, which describes the moment–rotation relationship between the top and bottom joints of a framed column. With the frame stability matrix established, the critical buckling load can be determined by imposing the boundary conditions.

The coordinate system and sign conventions of internal forces adopted in this article are shown in Figure 2. The internal forces include axial force, shear force, and bending moment. For the ith column, its bottom and top ends are referred to as node i − 1 and node i, respectively.

Figure 2.

Coordinate system and positive internal forces: (a) ith column in sway-permitted frame and (b) ith column in sway-prevented frame.

Stability matrix of sway-permitted frame

Consider the n-story idealized sub-assemblage model in Figure 1(a). For the ith column (Figure 2(a)), neglecting the shear deformation, its equilibrium is described by the differential equation

E I_{c, i} w_{i} (x) ″ = - M_{i} (x)

(2)

where EI_c,i = column flexural stiffness, M_i (x) =moment function, and w_i (x) = deflection function. The subscript i indicates the ith column. It should be noted that in classic bifurcation instability problems of sway-permitted frames, only vertical forces are applied to the frame; and moreover, the shear force is defined as remaining normal to the initial column axis, that is, it does not rotate during deflection and remains horizontal (Bazant and Cedolin, 1991). With this regard, and since the axial force in beams is neglected, the shear force in columns remains zero, and the variation of bending moment in a column is due to the effect of axial force.

The moment function can be written as

M_{i} (x) = P_{i} w_{i} (x) + M_{i, b}

(3)

where M_i,b is the bending moment at the bottom end of the column, and P_i is the axial force.

Substituting equation (3) into equation (2), the general solution of equation (2) is given by

w_{i} (x) = A \sin k_{i} x + B \cos k_{i} x - \frac{M_{i, b}}{P_{i}}, with k_{i} = \sqrt{\frac{P_{i}}{E I_{c, i}}}

(4)

By applying the boundary conditions w_i (0) = 0 and w_i ^’(0) = θ_i _− 1, the two parameters A and B are obtained as

A = \frac{θ_{i - 1}}{k_{i}}, B = \frac{M_{i, b}}{P_{i}}

(5)

where θ_i _− 1 represents the rotation of node i − 1.

With A and B obtained, equation (4) can be rewritten as

w_{i} (x) = \frac{θ_{i - 1}}{k_{i}} \sin k_{i} x + \frac{M_{i, b}}{P_{i}} \cos k_{i} x - \frac{M_{i, b}}{P_{i}}

(6)

Figure 3 demonstrates the moment equilibrium of node i, which is given by

- M_{i, t} + M_{i + 1, b} + R_{i} θ_{i} = 0

(7)

Figure 3.

The moment equilibrium at node i.

where M_i,t is the moment at the top end of column i, and R_i represents the equivalent rotational stiffness of node i due to the restraints provided by the connecting beams. The value of the equivalent rotational stiffness R has been given in the literature (Aristizabal, 1994; Liu, 2010)

R = \frac{Z ((m E_{b} I_{b}) / l_{b})}{Z + ((m E_{b} I_{b}) / l_{b})}

(8)

where Z is the rotational stiffness of the beam-to-column connection, Z = ∞ if the connection is rigid, and Z = 0 if the connection is hinged; m = the stiffness modifier depending on the boundary conditions at the far end of beams, and its value in different cases are listed in Table 1. It should be noted that equation (8) is for edge columns where the restraining beams are on one side. For columns whose joints are restrained by beams on both sides, both beams contribute to the rotational stiffness R.

Table 1.

Value of m in different cases.

Sway-permitted		Sway-prevented
Hinged	Rigid	Hinged	Rigid
3	6	3	2

From equations (2) and (6), we obtain that

\begin{matrix} M_{i, t} = - E I_{c, i} w ″_{i} (l_{c, i}) = \\ - E I_{c, i} (- \frac{M_{i, b}}{P_{i}} k_{i}^{2} \cos k_{i} l_{c, i} - θ_{i - 1} k_{i} \sin k_{i} l_{c, i}) \end{matrix}

(9)

and

θ_{i} = w'_{i} (l_{c, i}) = - \frac{M_{i, b}}{P_{i}} k_{b} \sin k_{b} l_{c, b} + θ_{i - 1} \cos k_{i} l_{c, i}

(10)

Substituting equation (9) and equation (10) in equation (7), we obtain

\begin{matrix} M_{i + 1, b} = (\frac{R_{i} k_{i} \sin k_{i} l_{c, i}}{P_{i}} + \cos k_{i} l_{c, i}) M_{i, b} \\ + (\frac{P_{i}}{k_{i}} \sin k_{i} l_{c, i} - R_{i} \cos k_{i} l_{c, i}) θ_{i - 1} \end{matrix}

(11)

It can be seen that θ_i and M_i ₊ _1,b are functions of θ_i _− 1 and M_i,b . Their relationship can be written in matrix form

{\begin{matrix} M_{i + 1, b} \\ θ_{i} \end{matrix}} = [s_{i}] {\begin{matrix} M_{i, b} \\ θ_{i - 1} \end{matrix}} = [\begin{matrix} s_{i, 11} s_{i, 12} \\ s_{i, 21} s_{i, 22} \end{matrix}] {\begin{matrix} M_{i, b} \\ θ_{i - 1} \end{matrix}}

(12)

where $[s_{i}]$ is referred to as story stability matrix, whose elements are

\begin{matrix} s_{i, 11} = \cos k_{i} l_{c, i} + \frac{R_{i} k_{i} \sin k_{i} l_{c, i}}{P_{i}} \\ s_{i, 12} = \frac{P_{i}}{k_{i}} \sin k_{i} l_{c, i} - R_{i} \cos k_{i} l_{c, i} \\ s_{i, 21} = - \frac{k_{i} \sin k_{i} l_{c, i}}{P_{i}} \\ s_{i, 22} = \cos k_{i} l_{c, i} \end{matrix}

(13)

With the story stability matrices [s _i] (i = 1–n) obtained, the relationship between {M_n ₊ _1,b, θ_n } and {M _1,b, θ ₀} can be calculated through a continued multiplication. For an n-story frame, M_n ₊ _1,b represents the moment restraint at the top end of column n, and M _1,b represents the moment restraints at the bottom of column 1. Replace M_n ₊ _1,b and M _1,b by M_t and M_b , respectively for the sake of simplicity, we obtain

{\begin{matrix} M_{t} \\ θ_{n} \end{matrix}} = Π_{i = 1}^{n} [s_{i}] {\begin{matrix} M_{b} \\ θ_{0} \end{matrix}} = [S] {\begin{matrix} M_{b} \\ θ_{0} \end{matrix}}

(14)

where $[S]$ is the so-called frame stability matrix, and is given by

[S] = Π_{i = 1}^{n} [s_{i}] = [\begin{matrix} s_{11} s_{12} \\ s_{21} s_{22} \end{matrix}]

(15)

Note that [S] is a 2 × 2 matrix. The [S] matrix defines the moment–deflection relationship between node 0 and note n.

Stability matrix of sway-prevented frame

Similarly, the story stability matrix can be derived for sway-prevented frames. The existence of shear force in columns is the main difference between the sway-prevented and the sway-permitted frames. The equilibrium of a sway-prevented column is given by

M_{i} (x) = P_{i} w_{i} (x) + M_{i, b} - V_{i} x

(16)

where V_i is the shear force in the column, and

V_{i} = \frac{M_{i, b} - M_{i, t}}{P_{i} l_{c, i}}

(17)

Substituting equations (16) and (17) into equation (2), we have

w_{i} (x) = A \sin k_{i} x + B \cos k_{i} x + \frac{M_{i, b} - M_{i, t}}{P_{i} l_{c, i}} x - \frac{M_{i, b}}{P_{i}}

(18)

Considering the boundary conditions w_i (0) = 0 and w_i ^’(0) = θ_i ₋₁, the two parameters A and B are obtained as

A = \frac{θ_{i - 1}}{k_{i}} - \frac{M_{i, b} - M_{i, t}}{P_{i} k_{i} l_{c, i}}, B = \frac{M_{i, b}}{P_{i}}

(19)

Since the column is sway-prevented, we have

w_{i} (l_{c, i}) = 0

(20)

It follows from equation (20) that the relation between M_i,_t and M_i,b can be expressed as

M_{i, t} = α_{i} M_{i, b} - β_{i} P_{i} l_{c, i} θ_{i - 1}

(21)

where the two parameters α_i and β_i are given by

α_{i} = \frac{\sin k_{i} l_{c, i} - k_{i} l_{c, i} \cos k_{i} l_{c, i}}{\sin k_{i} l_{c, i} - k_{i} l_{c, i}}, β_{i} = \frac{\sin k_{i} l_{c, i}}{\sin k_{i} l_{c, i} - k_{i} l_{c, i}}

(22)

Using equations (18) and (19), the nodal rotation θ_i at node i is calculated as

\begin{matrix} θ_{i} = w'_{i} (l_{c, i}) = M_{i, b} \frac{1 - \cos k_{i} l_{c, i} - k_{i} l_{c, i} \sin k_{i} l_{c, i}}{P_{i} l_{c, i}} \\ + M_{i, t} \frac{\cos k_{i} l_{c, i} - 1}{P_{i} l_{c, i}} + θ_{i - 1} \cos k_{i} l_{c, i} \end{matrix}

(23)

Applying equation (21), equation (23) can be now rewritten as

\begin{matrix} θ_{i} = M_{i, b} (\frac{1 - \cos k_{i} l_{c, i} - k_{i} l_{c, i} \sin k_{i} l_{c, i}}{P_{i} l_{c, i}} + α_{i} \frac{\cos k_{i} l_{c, i} - 1}{P_{i} l_{c, i}}) \\ + θ_{i - 1} ((1 + β_{i}) \cos k_{i} l_{c, i} - β_{i}) \end{matrix}

(24)

Substituting equations (21) and (24) in equation (7), we obtain

\begin{matrix} M_{i + 1, b} = M_{i, b} \\ (α_{i} - \frac{R_{i}}{P_{i} l_{c, i}} ((1 - α_{i}) (1 - \cos k_{i} l_{c, i}) - k_{i} l_{c, i} \sin k_{i} l_{c, i})) \\ + β_{i} P_{i} l_{c, i} θ_{i - 1} - R_{i} θ_{i - 1} ((1 + β_{i}) \cos k_{i} l_{c, i} - β_{i}) \end{matrix}

(25)

Equations (24) and (25) can now be expressed using the same matrix form of equation (12), that is, ${\begin{matrix} M_{i + 1, b} \\ θ_{i} \end{matrix}} = [s_{i}] {\begin{matrix} M_{i, b} \\ θ_{i - 1} \end{matrix}} = [\begin{matrix} s_{i, 11} s_{i, 12} \\ s_{i, 21} s_{i, 22} \end{matrix}] {\begin{matrix} M_{i, b} \\ θ_{i - 1} \end{matrix}}$ , with the story stability matrix $[s_{i}]$ given by

\begin{matrix} s_{i, 11} = \frac{\sin k_{i} l_{c, i} - k_{i} l_{c, i} \cos k_{i} l_{c, i}}{\sin k_{i} l_{c, i} - k_{i} l_{c, i}} \\ + \frac{R_{i}}{P_{i} l_{c, i}} (\frac{{(1 - \cos k_{i} l_{c, i})}^{2}}{\sin k_{i} l_{c, i} - k_{i} l_{c, i}} + k_{i} l_{c, i} \sin k_{i} l_{c, i}) \\ s_{i, 12} = - P_{i} l_{c, i} \frac{\sin k_{i} l_{c, i}}{\sin k_{i} l_{c, i} - k_{i} l_{c, i}} \\ - R_{i} (\frac{\sin k_{i} l_{c, i} - k_{i} l_{c, i} \cos k_{i} l_{c, i}}{\sin k_{i} l_{c, i} - k_{i} l_{c, i}}) \\ s_{i, 21} = - \frac{1}{P_{i} l_{c, i}} (\frac{{(1 - \cos k_{i} l_{c, i})}^{2}}{\sin k_{i} l_{c, i} - k_{i} l_{c, i}} + k_{i} l_{c, i} \sin k_{i} l_{c, i}) \\ s_{i, 22} = \frac{\sin k_{i} l_{c, i} - k_{i} l_{c, i} \cos k_{i} l_{c, i}}{\sin k_{i} l_{c, i} - k_{i} l_{c, i}} \end{matrix}

(26)

Similar to the sway-permitted column, the frame stability matrix [S] for the sway-prevented frame can be calculated using equation (15).

Buckling resistance of multi-story frames

Critical buckling load of framed columns

The previous section established the moment–rotation relationship between node 0 and node n. Next, we consider the boundary conditions at node n and node 0 as shown in Figure 4. Let R_t and R_b denote the equivalent rotational stiffness provided by the connecting beams at node n and node 0, respectively. When R_t (R_b ) is 0, it represents a hinged restraint; when R_t (R_b ) is ∞, it represents a fixed restraint.

Figure 4.

Moment restraints at the top and bottom ends of continuous column: (a) node n and (b) node 0.

Now impose the boundary conditions. At node n, we have

M_{t} - R_{t} θ_{n} = 0

(27)

and at node 0

M_{b} + R_{b} θ_{0} = 0

(28)

Applying equations (27) and (28) in equation (14), we obtain

s_{12} - R_{t} s_{22} - R_{b} s_{11} + R_{t} R_{b} s_{21} = 0

(29)

where s ₁₁, s ₁₂, s ₂₁, and s ₂₂ are elements of the frame stability matrix given by equation (15). Note that the frame stability matrix depends on the axial forces in the columns. The axial force in the ith column P_i can be expressed as

P_{i} = η_{i} P_{1}

(30)

where P ₁ is the axial force in the ground level column (C-1), and η_i represents the ratio of the axial force in column i to the axial force in C-1. For a given sub-assemblage, η_i can be easily identified.

Substituting equation (30) in equation (29), we obtain an equation with only one unknown, P ₁. The solution of equation (29) gives the elastic buckling load (P_cr ) for the whole sub-assemblage, that is, P_cr = P ₁. Note that equation (29) is a transcendental equation, but it can be easily solved by mathematical software like MATLAB. With P_cr solved, the effective length factor (K_i ) for the ith column can be calculated by

K_{i} = \frac{π}{l_{i}} \sqrt{\frac{E I_{c, i}}{η_{i} P_{cr}}}

(31)

Procedures for simplified elastic buckling analysis of frame structures

For a given multi-story frame structure, the following parameters are known, including, column stiffness (EI_c,i ), beam stiffness (EI_b,i ), story height (l_c,i ), beam length (l_b,i ), rotational stiffness of the beam-to-column connection (Z_i ), rotational restraints at the top and bottom (R_t and R_b ), and the column axial force ratio (η_i ). The procedure of the proposed simplified elastic buckling analysis can be summarized as follows.

Determine the equivalent rotational stiffness for each node according to equation (8).

Calculate the story stability matrices by equation (13) for sway-permitted frame or equation (26) for sway-prevented frame.

Calculate the frame stability matrix using equation (15).

Solve equation (29) for the buckling resistance P_cr .

Examples

To demonstrate the accuracy and applicability of the proposed method, five examples were selected from references to compare the proposed method with the existing methods and the exact results.

Example 1

Figure 5 shows a continuous column with different story height. The example has been considered in Bridge and Fraser (1987) and Column Research Committee of Japan (1971). This example is chosen to illustrate the application of the proposed method to frames with varied story height. The exact solution for the buckling load, P_cr , of the whole structure is $P_{cr} = ((8.18 π^{2} EI) / L^{2})$ (Bridge and Fraser, 1987; Column Research Committee of Japan, 1971).

Figure 5.

Example 1: a continuous column with different story height (adopted from Liu (2010)).

Using the conventional G-factor method, the critical buckling load was found to be $P_{cr} = ((4 π^{2} EI) / L^{2})$ , which underestimates the buckling resistance by 51%. Using the proposed method, the critical buckling load was obtained as $P_{cr} = ((8.18 π^{2} EI) / L^{2})$ , the same as the exact solution. This demonstrates the accuracy of the proposed method. The detailed calculation procedures can be found in Appendix 1.

Example 2

A continuous column, as shown in Figure 6, is studied to demonstrate the application of the proposed method to framed columns with varied flexural rigidity and axial force level. This problem has been analyzed by Hellesland and Bjorhovde (1997) and Bridge and Fraser (1987). Utilizing the proposed method, the buckling load for the system was found to be $P_{cr} = ((π^{2} EI) / {(0.96 h)}^{2})$ . The MATLAB script for this example is included in Appendix 2. It can be seen that the implementation of the method with MATLAB is simple and straightforward.

Figure 6.

Example 2: a continuous column with varied stiffness and axial force (adopted from Hellesland and Bjorhovde (1997)).

Table 2 compares the effective length factors obtained from the conventional G-factor method, the proposed method, those from Bridge and Fraser (1987) and Hellesland and Bjorhovde (1997) and the exact results. To compare the accuracy of different methods, the relative error of each method is calculated as

Err = \frac{K_{calct} - K_{exact}}{K_{exact}} \times 100 %

(32)

where K_calct is the calculated effective length factor, and K_exact is the exact effective length factor (reported in Bridge and Fraser (1987)). The relative errors of each method are also presented in Table 2. In the conventional G-factor method, the effective length factors are equal to unity for all five columns. The error for C-5 is −40.5%. This demonstrates that the conventional G-factor method can introduce significant errors for cases where drastic changes of column stiffness and axial force level occur. The results from Hellesland and Bjorhovde (1997) and Bridge and Fraser (1987) agree reasonably well with the exact solution, with relative errors on the order of 5%. The method in Bridge and Fraser (1987) yields overestimated effective length factors while the method in Hellesland and Bjorhovde (1997) underestimated the effective length factors. On the other hand, the proposed method gives exact effective length factors for all columns.

Table 2.

Example 2: effective length factors by different methods.

Column	Effective length factor (relative error)
Column	Conventional	Bridge and Fraser (1987)	Hellesland and Bjorhovde (1997)	Proposed	Exact (Bridge and Fraser, 1987)
C-1	1.0 (4.2%)	1.02 (6.3%)	0.92 (−4.2%)	0.96 (0)	0.96
C-2	1.0 (−9.1%)	1.17 (6.4%)	1.05 (−4.6%)	1.10 (0)	1.10
C-3	1.0 (20.5%)	0.88 (6.0%)	0.79 (−4.8%)	0.83 (0)	0.83
C-4	1.0 (−9.1%)	1.17 (6.4%)	1.05 (−4.6%)	1.10 (0)	1.10
C-5	1.0 (−40.5%)	1.73 (3.0%)	1.62 (−3.6%)	1.68 (0)	1.68

Example 3

Example 3 is a continuous column with nodal rotational restraints, as shown in Figure 7. The example has been studied in Hellesland and Bjorhovde (1997). Various combinations of G-factors at node A and B (G_A and G_B ) were considered. Two cases of boundary conditions were studied: the top and bottom of the continuous column are both fixed or both hinged. Since the axial force, length, and flexural stiffness of all columns are identical, the effective length factors are the same for all columns according to equation (31).

Figure 7.

Example 3: continuous columns with nodal rotational restraints (adopted from Hellesland and Bjorhovde (1997)): (a) fixed and (b) hinged.

Table 3 presents the effective length factors obtained by Hellesland and Bjorhovde (1997), the proposed method, and the exact results from finite element analysis (FEA) for various combinations of G_A and G_B and the two boundary conditions. It can be seen that Hellesland and Bjorhovde (1997) gave reasonable results for all cases, with the effective length factor underestimated by about 5% in certain cases. The proposed method is more accurate than Hellesland and Bjorhovde’s (1997) method. In all cases, the proposed method essentially agrees with the FEA, with relative errors of less than 0.3%, showing that the proposed method works for frames with beams and columns having different flexural rigidities.

Table 3.

Example 3: effective length factors by different methods.

Case	G_A	G_B	Effective length factor (error)
Case	G_A	G_B	Hellesland and Bjorhovde (1997)	Proposed	Exact
(a) Fixed boundary conditions
1	0.1	0.3	0.559 (−1%)	0.573 (0.3%)	0.571
2	0.2	0.6	0.600 (−2.6%)	0.617 (0.1%)	0.616
3	0.3	0.4	0.596 (−2.7%)	0.607 (0.2%)	0.606
4	0.4	0.8	0.633 (−2.2%)	0.648 (0.2%)	0.647
5	0.5	2	0.674 (−2.7%)	0.693 (0)	0.693
6	1	0.1	0.604 (−4.7%)	0.635 (0.1%)	0.634
7	25	50	0.805 (−0.5%)	0.809 (0)	0.809
(b) Hinged boundary conditions
1	0.1	0.3	0.706 (−5.4%)	0.746 (0)	0.746
2	0.2	0.6	0.752 (−3.6%)	0.780 (0)	0.780
3	0.3	0.4	0.747 (−2.4%)	0.766 (0.1%)	0.765
4	0.4	0.8	0.790 (−1.7%)	0.804 (0)	0.804
5	0.5	2	0.839 (−1.6%)	0.853 (0)	0.853
6	1	0.1	0.759 (−5.4%)	0.803 (0.1%)	0.802
7	25	50	0.992 (0)	0.992 (0)	0.992

Example 4

Example 4 considers the multi-story frames shown in Figure 8. Both sway-permitted and sway-prevented frames were considered. The frames are one-bay and n-story, with n varying from 1 to 8. For simplicity, the column stiffness (EI_c ) is assumed to equal to column stiffness (EI_b ). Two cases of boundary conditions were examined, that is, (1) bottom columns are fixed and (2) bottom column are hinged.

Figure 8.

Examples 4 and 5: multi-story frames: (a) sway-permitted and (b) sway-prevented.

Table 4 compares the effective length factors obtained from the proposed method and the exact solutions. For one-story and two-story sway-prevented frames with bottom fixed, the exact solutions for effective length factor are available (Che and Liu, 1987); it is 0.626 for one-story frame and 0.752 for two-story frame. Table 4 shows that the proposed method gives exact solution for these two cases. For other cases, finite element models of the frames were built using the commercial software SAP2000. The “straight-frame element” in SAP2000 was used to model the behaviors of beams and columns, which incorporates flexural, shear, and torsion deformation of the members. With the system buckling resistance obtained using FEA, the effective length factors for columns were calculated using equation (31), and they are the “exact” solutions in Table 4.

Table 4.

Example 4: effective length factors by the proposed method and FEA.

No. of stories	Sway-permitted				Sway-prevented
	Bottom fixed		Bottom hinged		Bottom fixed		Bottom hinged
	FEA	Proposed	FEA	Proposed	FEA	Proposed	FEA	Proposed
1	1.159	1.158	2.336	2.330	0.626^a	0.626	0.882	0.875
2	1.394	1.385	2.336	2.330	0.752^a	0.752	0.882	0.875
3	1.490	1.474	2.336	2.330	0.805	0.797	0.882	0.875
4	1.541	1.517	2.336	2.330	0.825	0.817	0.882	0.875
5	1.575	1.540	2.336	2.330	0.836	0.828	0.882	0.875
6	1.601	1.554	2.336	2.330	0.843	0.835	0.882	0.875
7	1.626	1.563	2.336	2.330	0.847	0.840	0.882	0.875
8	1.648	1.570	2.336	2.330	0.850	0.843	0.882	0.875

From theoretical analysis (Column Research Committee of Japan, 1971), rather than FEA.

Figure 9 plots the relative errors for the proposed method. It can be seen that in most cases, the error is no more than 1%, demonstrating the high accuracy of the proposed method. Note that in the proposed model, the effects of axial force in beams and shear deformation in columns are neglected. These factors, however, were accounted for in the FEA.

Figure 9.

Example 4: relative errors versus frame’s number of stories.

Figure 9 also shows that for the bottom fixed sway-permitted frame, the error of the proposed method grows gradually with increasing number of stories. For other cases, the errors have no correlation with the number of stories. This is because that the proposed method assumes that the influence of beam shear on the axial force of columns is negligible. In reality, the shear forces in beams are transmitted to the beam-to-column connections and contribute to the axial force of the column. As more stories are included in the frame, more shear forces are transmitted to the columns, resulting in more error in the calculated buckling load. The error, however, is still below 5% even for eight-story sway frame. In sway-prevented frames, there is no shear force in beams, thus the accuracy of the proposed method is not affected by the number of stories. For bottom hinged sway-permitted frames, the buckling occurs mainly in the bottom story, thus the accuracy of the proposed method is not affected by the number of stories.

Example 5

We consider the same frames in Example 4 again, except that the beam-to-column connections are now semi-rigid. The frame is one-bay four-story. Two cases are considered: (1) the frame is sway-permitted and (2) the frame is sway-prevented. Various values of the equivalent rotational stiffness (Z) of the semi-rigid beam-to-column connections were considered. Similar to Example 4, FEA models for the frames were established using the software SAP2000 to make a benchmark. Table 5 compares the results by the proposed method and those by FEA.

Table 5.

Example 5: effective length factors by the proposed method and FEA.

Case	Rotational stiffness Z (unit: EI_b/l_b )				Effective length factor		Error (%)
Case	Floor 1	Floor 2	Floor 3	Roof	FEA	Proposed	Error (%)
Sway-permitted
1	0	0	0	0	0.971	0.964	−0.682
2	0.2	0.2	0.2	0.2	0.951	0.945	−0.699
3	0.5	0.5	0.5	0.5	0.930	0.924	−0.730
4	1	1	1	1	0.908	0.901	−0.765
5	2	2	2	2	0.883	0.876	−0.797
6	5	5	5	5	0.855	0.848	−0.856
7	10	10	10	10	0.842	0.834	−0.882
8	100	100	100	100	0.827	0.819	−0.908
9	0	0	1	1	0.927	0.921	−0.744
10	1	1	0	0	0.955	0.948	−0.702
11	0.2	5	1	2	0.888	0.883	−0.602
12	5	1	2	0.2	0.911	0.904	−0.759
13	10	1	100	1	0.871	0.864	−0.829
14	1	10	10	0	0.891	0.884	−0.797
15	2	2	2	100	0.867	0.860	−0.835
Sway-prevented
1	0	0	0	0	8.000	8.000	0
2	0.2	0.2	0.2	0.2	5.045	5.043	−0.035
3	0.5	0.5	0.5	0.5	3.792	3.786	−0.156
4	1	1	1	1	3.016	3.005	−0.347
5	2	2	2	2	2.444	2.429	−0.640
6	5	5	5	5	1.972	1.951	−1.067
7	10	10	10	10	1.774	1.750	−1.365
8	100	100	100	100	1.568	1.540	−1.790
9	0	0	1	1	3.934	3.928	−0.153
10	1	1	0	0	5.434	5.431	−0.067
11	0.2	5	1	2	2.570	2.555	−0.575
12	5	1	2	0.2	3.005	2.994	−0.388
13	10	1	100	1	2.146	2.127	−0.884
14	1	10	10	0	2.538	2.523	−0.594
15	2	2	2	100	2.313	2.298	−0.641

As seen in Table 5, excellent results were obtained from the proposed method; the relative errors are less than 1% for most cases with the maximum error less than 2%. Cases 1–8 represent a gradual increase in equivalent rotational stiffness of semi-rigid beam-to-column connections. Table 5 shows that the relative error increases from cases 1 to 8, especially for sway-permitted cases. This is because the proposed method neglects the axial forces in beams. As the rotational stiffness of beam-to-column connection becomes larger, the restraining beams have more influences on the stability of frame columns, and the effect of axial forces in beams on the buckling resistance of the frame becomes more significant, resulting in slightly larger errors of the proposed method.

Conclusion

This article proposes an effective simplified method to analyze the buckling resistance of multi-story frames. The method is based on general principles of elastic mechanics and accounts for the vertical interaction between columns on adjacent stories. The method provides an explicit equation for solving the buckling resistance of multi-story frames. The method is computationally simple and can be easily implemented using mathematical software like MATLAB. Under the assumption of simultaneous buckling of columns in the same story, it can be used for sway-permitted and sway-prevented frames with wide variation of parameters including numbers of story, boundary conditions, stiffness of beam-to-column connections, column length and stiffness, and axial force level.

The accuracy of the proposed method is demonstrated through five examples. It was found that (1) the proposed method yields exact solutions for the continuous column examples, as seen in Examples 1–3; (2) the proposed methods give very accurate results for the two multi-story frame examples (Examples 4 and 5), with maximum relative errors of less than 5%.

Footnotes

Appendix 1

Appendix 2 Acknowledgements

The support from the China Road & Bridge Corporation is acknowledged.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research described in this paper was supported, in part, by Major Projects Foundation of Chinese Ministry of Transport under grants 201332849A090, and the International Program Development Fund from the University of Sydney.

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