Investigation on elastic compound buckling of latticed columns considering eccentricity and geometric imperfections

Abstract

This article investigates the elastic compound buckling behavior of latticed columns with three lacing systems under two boundary conditions. Valid numerical method named SEM, that conducts buckling analysis using strain energy of bars and potential energy of chords, was built to address flexural, torsional, and flexural–torsional buckling loads along with their overall and local buckling deformations. Eurocode 3 and finite element software ABAQUS were used to validate the auxiliary of SEM. Through the research on X-lacing, E-lacing, and K-lacing latticed columns under two boundary conditions, the rigidity of bars was found to exceed a threshold value affecting the linear buckling load. When the cross-sectional area of lacing bars tends to 0 or threshold value, Eurocode 3 guidelines were imprecise in forecasting the linear buckling load. Eccentricity and geometric imperfections would decrease the buckling capacity of built-up columns substantially. The K-lacing columns respond more sensitively on local imperfections than the X-lacing and E-lacing columns under simply supported condition. However, for the columns under cantilever state, local imperfections have no significant impact for the three lacing columns. In addition, the numerical results of nonlinear buckling load and equilibrium path from SEM considering geometric imperfections are close to the output of finite element method, validating the accuracy of SEM.

Keywords

Compound buckling critical buckling load latticed columns eccentricity geometric imperfections nonlinear post-buckling analysis threshold stiffness

Introduction

Built-up columns are widely used in steel buildings and bridges. There mainly exist three types of built-up columns in structural engineering. One is the channel column strengthened by batten plates connecting with two flanges at the open side (Figure 1(a)). The other two are latticed columns with several chords connected by battens or lacing bars (Figure 1(b) and (c)). The battens or bars help to reduce the effective length of columns and enhance the ability of resistance on both the global and local buckling. The difference between them lies in the influence on the shear rigidity and the critical buckling load of columns.

Figure 1.

Built-up columns: (a) battened channel columns,(b) battened columns with two chords, and (c) laced columns with two chords.

The failure modes, for laced columns, include flexural buckling, torsional buckling, flexural–torsional buckling, and distortional buckling. Extended research has been done on the bending flexural buckling problems along the virtual axis of columns (flexural buckling around the real axis was assumed to coincide with the solid columns). The main issues focus on the influence of shear deformation of lacing bars on the columns’ buckling and the interaction between the global and local buckling modes. The shear effect on columns’ buckling, first proposed by Engesser,¹ was analyzed by Gjelsvik,² Timoshenko and Gere,³ Uhlman and Ramn,⁴ and then introduced by Eurocode 3.⁵ The battens or lacing bars were assumed to be a continuous web behaving as a shear panel. The interaction between global and local buckling of built-up columns was investigated by Koiter and Kuiken,⁶ Svensson and Kragerup,⁷ Bazant and Cedolin,⁸ and Duan et al.⁹ The buckling failure modes of chords between adjacent lacing joints were assumed to simple supports braced by lacing bars, with the shape functions of sinusoid in accordance with Euler beam. In recent study, the two issues were combined effectively using numerical analysis methods. Kalochairetis and Gantes¹⁰ investigated the collapse load of I-section-laced columns by replacing the effective bending rigidity by a reduced value. The initial imperfections of global and local columns were considered to reduce heavy computational work for practical engineering. Then, transverse loads and arbitrary supports of laced columns were studied in Gantes and Kalochairetis.¹¹ Approximate second-order analysis of imperfect Timoshenko member considering the above two issues was conducted for different boundary and load conditions of I-section-laced columns.

Torsional buckling and torsional–flexural buckling modes of built-up columns were concerned in few studies. The Euler elastic stability of pure flexural and twisting buckling was calculated in Razdolsky^12–15 using statically indeterminate method. The study showed the pure flexural buckling always preceded pure twisting buckling for I-section-laced columns, and Eurocode 3 (EC3) might get unsafe conclusions. The flexural–torsional buckling of evenly battened columns was analyzed theoretically by Wang¹⁶ using total potential energy equation solved by Rayleigh–Ritz method. Unevenly battened channel columns were studied by Ren et al.¹⁷ using piecewise cubic Hermit interpolation (PCHI) to express the rotational displacement function during flexural–torsional buckling deformations. It was found that flexural–torsional buckling was more likely to happen than pure flexural buckling for channel-section columns, which was affected by geometric characteristics and eccentricity. In addition, structural potential energy principle with Rayleigh–Ritz method was widely cited to solve nonlinear buckling modes.^18–20 Andrade et al.²¹ investigated in detail the effect of restraints stiffness, restraints types, and its interplay on the lateral–torsional buckling behavior of discretely restrained tapered beams. Some important conclusions for restraint characteristic were obtained, such as rigid restraint at the tip promoted the critical load of a cantilever and the distance between the rotation centers of the cross section is the main factor in increasing the buckling strength for translational restraint.

Experimental efforts were conducted for limited types of built-up columns. Battened columns with dimension parameters of batten spacing and chords distance were performed in Hashemi and Jafari^22,23 to evaluate the accuracy of the methods proposed in previous studies. The results showed the average value of Ayrton–Perry and ultimate strength curve methods led to best prediction of the column compressive strength, and the Engesser equation gave the least accurate results. Battened columns with four identical chords were conducted in Aghoury et al.,^24,25 accounting for both geometric and material nonlinearities. Both experimental and finite element models were used to investigate the interaction between slender outstanding width–thickness ratios, overall angle slenderness, and overall column slenderness on the strength of columns. More recently, Z-laced built-up columns with two chords were studied in Kalochairetis et al.²⁶ to consider the effect of initial geometric imperfections and thermally induced residual stresses. Battened built-up columns (buckling-restrained brace (BRB)) were conducted in Guo et al.²⁷ to investigate buckling modes and their buckling loads. All the results from the experiments showed a very good agreement with finite element method.

Recently, the buckling behavior of built-up columns under dynamic loads had been studied. Mirtaheri et al.²⁸ investigated the global and local buckling behavior of restrained brace under earthquake load. The research results acquired from finite element method and experiments show the flexural stiffness of restraining member has a significant influence on the global buckling behavior of brace. Jiang et al.²⁹ studied the effect of the axial and rotational restraint stiffness on dynamic buckling performance of columns under fire. The rotational restraint stiffness was found to be relatively insensitive to rotational stiffness ratio beyond a threshold value. Some researchers introduced the optimization issue based on the buckling constraint conditions. Ozbasaran and Yilmaz³⁰ conducted shape optimization for tapered I-beams based on constraints of lateral–torsional buckling, deflection, and stress. Schmidt and Curbach³¹ investigated the optimal cross section of concrete columns to increase the buckling capacity.

Yet, the solution and analysis of compound buckling modes for laced columns were rarely seen in the published literature. In this work, the deformations of compound buckling modes containing five low-order buckling modes were expressed by the superposition of global and local shape functions. Then, the total potential energy of chords was integrated and solved to get compound buckling loads. The five basic low-order buckling modes in Figure 3, obtained by linear buckling analysis, consist of flexural and torsional buckling deformations in local and global columns. The critical buckling mode of laced columns was found to transform among compound buckling modes, which were determined by geometric characteristics, type and size of lacings, eccentricity, and initial imperfections.

Three typical laced columns (X-lacing, K-lacing, and E-lacing) under two boundary conditions (simple supports and cantilever) were discussed in the article. Parametric studies were performed to consider the effect of eccentricity and initial geometric imperfections on critical buckling load. Design parameters were cross-sectional area $A_{d}$ of bars, eccentricity $e_{x}, e_{y}$ of compressive load, and initial imperfections $χ_{g} and χ_{l}$ of global and local columns, respectively. Moreover, threshold braced stiffness $E A_{h}$ of bars was analyzed in the situation of eccentricity. The critical buckling load $P_{m}$ varies merely after the rigidity $E A_{d}$ of bars exceeds to the threshold braced stiffness.

Structural model

A standardized latticed columns consisting of channel chords and lacing bars are plotted in Figure 2. As shown in the figure, O, C₁, and C₂ are the cross-sectional centroid of whole columns, chord 1, and 2, respectively. S₁ and S₂ are the shear centers of two channel chords. Functions $u_{i} (z)$ and $v_{i} (z)$ are the lateral displacements of chord i along the X_i-axis and Y_i-axis in local coordination X_i–Y_i–Z_i located at point C_i. The rotating functions of chords are denoted as $φ_{i} (z)$ about shear center S_i which offsets a distance $x_{0}$ from the centroid C_i for monosymmetric section. An axial compression load P offsetting the latticed columns centroid with $e_{x}$ and $e_{y}$ is applied to the structural model. Besides, some necessary structural dimensions for buckling analysis are marked in Figure 2.

Figure 2.

Latticed columns with lacing system: (a) measurements of cross-sectional A-A profile; (b) latticed columns; (c) twisting and bending deformations of chord 2 section, the original and buckling deformation position of section B-B are shown in dashed and solid lines, respectively; and (d) the braced loads produced by lacing bars at joints connecting chords and bars.

The structural material and characteristics conform to general standards in structural engineering. That means, (1) chords and lacing bars are made of homogeneous, isotropic, and linearly elastic material with Young’s modulus E and shears modulus G; (2) all lacing bars have identical cross sections with area $A_{d}$ and inertia moment $I_{b}$ outside the lattice plane.

To analyze the buckling failure modes of latticed columns with different lacing bars systems, the following important assumptions are set:

The cross sections of each chord remain a plane, and shear strain in the middle plane is zero before and after buckling deformations (two basic assumptions, Vlasov³²). Besides, the shear deformations, residual stresses, and distortional buckling of chords are neglected.

The lateral bending buckling of latticed column around the real axis (Y-axis) corresponds with the solid-web columns (EC3). As shown in Figure 2(d), that means the lacing bars provide no lateral braced load along the Y-axis.

Referring to AG Razdolsky,¹⁴ only axial rigidity $E A_{d}$ and bending rigidity $E I_{b}$ outside lattice plane were considered for lacing bars. Besides, bending rigidity $E I_{b}$ was ignored in the calculated example.

The braced loads produced by lacing bars are deemed to be constant before and after buckling deformations.

Based on above hypothesis, the critical buckling load relating to low-order buckling failure modes in Figure 3 were calculated with numeral analysis method SEM separately considering the strain energy of bars and total potential energy of chords.

Figure 3.

Low-order buckling failure modes of simply supported latticed columns obtained by finite element analysis: (a) global bending buckling mode in the X-direction, (b) local bending buckling mode in the X-direction, (c) global twisting buckling mode around shear center, (d) local twisting buckling mode around shear center, and (e) global bending buckling mode in the Y-direction.

Boundary conditions and shape functions

Two boundary conditions with simple supports (1) and cantilever conditions (2) are considered for latticed columns. The simply supported columns are restrained with two ends simply supported, while cantilever columns with one end fixed (warping deformation restrained) and the other end free. The global buckling displacements of latticed columns coincide with Euler columns under the two boundary conditions, while the local buckling deformations of columns were considered all along with simple supports between adjacent lacing joints.

Corresponding displacement functions of global and local buckling are listed in Table 1. The selection of shape functions aims to express corresponding buckling deformations exactly. In instance of simply supported boundary, the low-order buckling deformations in Figure 3 agree nicely with the corresponding shape functions in Table 1.

Table 1.

Boundary conditions of latticed columns and corresponding displacement functions (m = number of sub-elements in length L).

Boundary	Function $u_{i}$		Function $v_{i}$	Function $φ_{i}$
Boundary	GM	LM	Function $v_{i}$	GM	LM
1. Simple supports	$C_{1 (i)} \sin \frac{π z}{L}$	$C_{2 (i)} \sin \frac{π mz}{L}$	$C_{3 (i)} \sin \frac{π z}{L}$	$C_{4 (i)} \sin \frac{π z}{L}$	$C_{5 (i)} \sin \frac{π mz}{L}$
2. Cantilever	$C_{1 (i)} (1 - \cos \frac{π z}{2 L})$	$C_{2 (i)} \sin \frac{π mz}{L}$	$C_{3 (i)} (1 - \cos \frac{π z}{2 L})$	$C_{4 (i)} (1 - \cos \frac{π z}{2 L})$	$C_{5 (i)} \sin \frac{π mz}{L}$

GM: global buckling mode; LM: local buckling mode.

In this article, the shape functions of compound buckling were selected to consider all the low-order buckling failure modes. Based on assumption 3, the lacing bars only provide lateral braced loads around the X-axis and twisting moments for chords. Thus, for all boundary conditions, the shape functions $v_{i}$ of chords along the Y-axis just consist of global buckling mode. And, the lateral displacements $u_{i}$ and twisting displacements $φ_{i}$ can be expressed by the superposition of global and local buckling functions. For example, under simply supported boundary condition, the shape functions of chords are given in equation (1), which can express all the low-order buckling deformations of latticed columns

u_{i} = C_{1 (i)} \sin (\frac{π z}{L}) + C_{2 (i)} \sin (\frac{π mz}{L})

(1a)

v_{i} = C_{3 (i)} \sin (\frac{π z}{L})

(1b)

φ_{i} = C_{4 (i)} \sin (\frac{π z}{L}) + C_{5 (i)} \sin (\frac{π mz}{L})

(1c)

The strain energy of lacing bars system

According to assumptions 3 and 4, the lacing bars were considered to be separately deformable bodies with axial and bending deformation outside the lattice plane. Thus, the strain energy of lacing bars system consists of its own axial and bending strain energy.

The lateral displacements of chord i in connected joints k along the virtual axis X_i on the front and back planes are $w_{i} (k)$ and $w_{i}^{*} (k)$

w_{i} (k) = u_{i} (k) + \frac{h}{2} φ_{i} (k)

(2a)

{w_{i}}^{*} (k) = u_{i} (k) - \frac{h}{2} φ_{i} (k)

(2b)

The relative values of lateral displacements of chords induced lateral braced loads $f_{k} and f_{k}^{*}$ in Figure 2(d) from lacing bars. Furthermore, the work done by loads $f_{k} and f_{k}^{*}$ is load potential energy applied to chords, which is equal to axial strain energy of lacing bars system. However, the bending strain energy of lacing bars systems produced by bending moments $m_{k} and m_{k}^{*}$ is also ignored as AG Razdolsky.¹⁴ The loads $f_{k} and f_{k}^{*}$ are in equation (3)

f_{k} (k, j) = - c (k, j) * [w_{i} (k) - w_{3 - i} (j)]

(3a)

f_{k}^{*} (k, j) = - c (k, j) * [w_{i}^{*} (k) - w_{3 - i}^{*} (j)]

(3b)

where $c (k, j) = E A_{d} \cos^{2} θ (k, j) / l (k, j)$ ; $l (k, j)$ is the length of bars connecting point k of chord i and point j of chord $3 - i$ ; and $θ (k, j)$ is the inclined angle of corresponding bars.

For the three types of lacing bars systems shown in Figure 4, the lacing bars can be classified to transverse bars and diagonal bars. Thus, the axial strain energy of lacing bars counting front and back latticed planes is divided to the transverse bars in equation (4a) and diagonal bars in equation (4b)

\begin{matrix} W (k, k) = f_{k} (k, k) [w_{i} (k) - w_{3 - i} (k)] \\ + {f_{k}}^{*} (k, k) [{w_{i}}^{*} (k) - {w_{3 - i}}^{*} (k)] \end{matrix}

(4a)

\begin{matrix} W (k, k + 1) = f_{k} (k, k + 1) [w_{i} (k) - w_{3 - i} (k + 1)] \\ + {f_{k}}^{*} (k, k + 1) [{w_{i}}^{*} (k) - {w_{3 - i}}^{*} (k + 1)] \end{matrix}

(4b)

Figure 4.

Three typical lacing bars systems: (a) K-lacing, (b) X-lacing, and (c) E-lacing.

The total strain energy of the three typical lacings systems in Figure 4 was calculated according to equations (2)–(4). The results were shown in equation (5). For X-lacing type

W = - \sum_{1}^{m} \begin{matrix} 2 c [u_{1} (k) - u_{2} (k + 1)]^{2} + \frac{h^{2} c}{2} [φ_{1} (k) - φ_{2} (k + 1)]^{2} \\ + 2 c [u_{2} (k) - u_{1} (k + 1)]^{2} + \frac{h^{2} c}{2} [φ_{2} (k) - φ_{1} (k + 1)]^{2} \end{matrix}

(5a)

For K-lacing type

W = - \sum_{1}^{m} {\begin{matrix} 2 c [u_{1} (k) - u_{2} (k + 1)]^{2} + \frac{h^{2} c}{2} [φ_{1} (k) - φ_{2} (k + 1)]^{2}, k = even \\ 2 c [u_{2} (k) - u_{1} (k + 1)]^{2} + \frac{h^{2} c}{2} [φ_{2} (k) - φ_{1} (k + 1)]^{2}, k = odd \end{matrix}

(5b)

For E-lacing type

\begin{matrix} W = - \sum_{1}^{m} 2 c {[u_{1} (k) - u_{2} (k)]}^{2} + \frac{h^{2} c}{2} {[φ_{1} (k) - φ_{2} (k)]}^{2} \\ - \sum_{1}^{m} {\begin{matrix} 2 c [u_{1} (k) - u_{2} (k + 1)]^{2} + \frac{h^{2} c}{2} [φ_{1} (k) - φ_{2} (k + 1)]^{2}, k = even \\ 2 c [u_{2} (k) - u_{1} (k + 1)]^{2} + \frac{h^{2} c}{2} [φ_{2} (k) - φ_{1} (k + 1)]^{2}, k = odd \end{matrix} \end{matrix}

(5c)

Buckling analysis of chords using energy method with strain energy of lacing bars (SEM)

Subjected to an eccentric axial compression load P offsetting the center $e_{x}$ and $e_{y}$ , the distributions of applied load P to each identical chord are expressed in equation (6)

P_{1} = \frac{(\frac{d}{2} - e_{x}) P}{d}, P_{2} = \frac{(\frac{d}{2} + e_{x}) P}{d}

(6a)

M_{xi} = \frac{1}{2} P e_{y}, i = 1, 2

(6b)

where $P_{i} and M_{xi}$ are the axial compression and bending moment in the X-direction of chord i, respectively. The total potential energy function of chords can be described as

Π = \sum_{i} (Π_{1}^{i} + Π_{2}^{i}) - W

(7)

where $Π_{1}^{i}$ is flexural strain energy and load potential energy of chord i caused by $u_{i}$ and $v_{i}$

Π_{1}^{i} = \frac{1}{2} \int_{L} [E I_{y} u_{i}''^{2} + E I_{x} v_{i}''^{2} - P_{i} (u_{i}'^{2} + v_{i}'^{2})] dz

(8)

And, $Π_{2}^{i}$ is the torsional–flexural strain energy and load potential energy of chord i caused by $u_{i}$ , $v_{i}$ , and $φ_{i}$

\begin{matrix} Π_{2}^{i} = \frac{1}{2} \int_{L} [E I_{w} φ_{i}''^{2} - (P_{i} r^{2} - G I_{d}) φ_{i}'^{2} + 2 {(- 1)}^{3 - i} P_{i} x_{0} v_{i} φ_{i}' + 2 M x_{i} u_{i}' φ_{i}'] dz \end{matrix}

(9)

W is the load potential energy of chords caused by the total strain energy of lacing bars system which has been derived in equations (5a)–(5c). $I_{y} and I_{x}$ are the second moments of cross section about the Y-axis and X-axis of chords, respectively. $I_{d} and I_{w}$ are the twisting and warping moments of inertia, respectively. And, $r^{2} = {x_{0}}^{2} + {(I_{x} + I_{y}) / A}_{c}$ , where r is the polar rotating radius about shear center for both chords.

An X-lacing type of laced columns with two ends simply supported was selected to illustrate the computational process of SEM in detail. The total potential function $Π$ was calculated in equations (10)–(12). While the calculation program under the cantilever boundary condition was showed in Appendix 2

\begin{matrix} Π_{1}^{i} = \frac{π^{2}}{4 L} {C_{1 (i)}}^{2} (P_{Ey} - P_{i}) + \frac{{(m π)}^{2}}{4 L} {C_{2 (i)}}^{2} (P_{Eym} - P_{i}) \\ + \frac{π^{2}}{4 L} {C_{3 (i)}}^{2} (P_{Ex} - P_{i}) \end{matrix}

(10a)

And

\begin{matrix} Π_{2}^{i} = {C_{4 (i)}}^{2} \frac{{(r π)}^{2}}{4 L} (P_{Ew} - P_{i}) + {C_{5 (i)}}^{2} \frac{{(rm π)}^{2}}{4 L} (P_{Ewm} - P_{i}) \\ + {(- 1)}^{3 - i} P_{i} x_{0} C_{3 (i)} C_{4 (i)} \frac{π^{2}}{2 L} \\ + M x_{i} C_{1 (i)} C_{4 (i)} \frac{π^{2}}{2 L} + M x_{i} C_{2 (i)} C_{5 (i)} \frac{m^{2} π^{2}}{2 L} \end{matrix}

(10b)

And

\begin{matrix} W = - 2 c [({C_{1 (1)}}^{2} + {C_{1 (2)}}^{2}) K_{1} + C_{1 (1)} C_{1 (2)} K_{2}] \\ - \frac{h^{2} c}{2} [({C_{4 (1)}}^{2} + {C_{4 (2)}}^{2}) K_{1} + C_{4 (1)} C_{4 (2)} K_{2}] \end{matrix}

(10c)

In equation (10), $P_{Ey}, P_{Ex}$ , and $P_{Ew}$ are Euler buckling loads of solid-web columns over the length L. $P_{Eym}$ and $P_{Ewm}$ are local Euler buckling loads between the connection joints

P_{Eym} = E I_{y} \frac{{(m π)}^{2}}{L^{2}}, P_{Ewm} = \frac{1}{r^{2}} [G I_{d} + E I_{w} \frac{{(m π)}^{2}}{L^{2}}]

(11)

K ₁ and K₂ are the influence coefficients of sub-elements spacing in the length L

\begin{matrix} K_{1} = \sum_{1}^{m} \sin^{2} \frac{π z (k)}{L} + \sin^{2} \frac{π z (k + 1)}{L} \\ K_{2} = 4 * \sum_{1}^{m} \sin \frac{π z (k)}{L} \sin \frac{π z (k + 1)}{L} \end{matrix}

(12)

Applying the principle of minimum potential energy, the non-zero solution of buckling loads P can be obtained by setting the first-order derivative of total potential energy $Π$ about coefficient $C_{1 (i)}, \dots, C_{5 (i)}$ to zero

\begin{matrix} \frac{\partial Π}{\partial C_{1 (i)}} = \frac{π^{2}}{2 L} C_{1 (i)} (P_{Ey} - P_{i}) + M x_{i} C_{4 (i)} \frac{π^{2}}{2 L} \\ + 4 c C_{1 (i)} K_{1} + 2 c C_{1 (3 - i)} K_{2} \end{matrix}

(13a)

\frac{\partial Π}{\partial C_{2 (i)}} = C_{2 (i)} \frac{{(m π)}^{2}}{2 L} (P_{Eym} - P_{i}) + M x_{i} C_{5 (i)} \frac{{(m π)}^{2}}{2 L}

(13b)

\frac{\partial Π}{\partial C_{3 (i)}} = C_{3 (i)} \frac{π^{2}}{2 L} (P_{Ex} - P_{i}) + P_{i} x_{0} C_{4 (i)} \frac{π^{2}}{2 L}

(13c)

\begin{matrix} \frac{\partial Π}{\partial C_{4 (i)}} = C_{4 (i)} \frac{π^{2}}{2 L} r^{2} (P_{Ew} - P_{i}) + (- 1)^{3 - i} P_{i} x_{0} C_{3 (i)} \frac{π^{2}}{2 L} \\ + M x_{i} C_{1 (i)} \frac{π^{2}}{2 L} \\ + \frac{h^{2} c}{2} [2 C_{4 (i)} K_{1} + C_{4 (3 - i)} K 2] \end{matrix}

(13d)

\frac{\partial Π}{\partial C_{5 (i)}} = C_{5 (i)} \frac{{(rm π)}^{2}}{2 L} (P_{Ewm} - P_{i}) + M x_{i} C_{2 (i)} \frac{{(m π)}^{2}}{2 L}

(13e)

Equation (13) can be expressed in the form of matrix by $[A] [C] = 0$ . Because of the interaction between chord 1 and chord 2, the vector [C] is

[C] = {[\begin{matrix} C_{1 (1)}, \dots, C_{5 (1)} & C_{1 (2)}, \dots, C_{5 (2)} \end{matrix}]}^{T}

(14)

The solutions of equation (15) are buckling loads relating to all possible buckling failure modes. To compare the compound buckling loads with Euler critical loads, a parabolic function $f (P)$ was applied. And, $P_{m}$ is the minimum buckling load (critical buckling load) among buckling loads P of the corresponding failure modes

f (P) = {| A |}_{10 \times 10} = 0

(15)

Comparison with EC3 and FEM

The existing specification EC3⁵ and Tong and Chen³³ provided detailed expressions for global and local buckling loads of compressive latticed columns with several uniform types of lacing bars. The global bending buckling loads $P_{mx} and P_{my}$ and local buckling load $P_{ml}$ were cited in equation (16), whereas the other buckling failure modes, such as torsional and flexural–torsional buckling, were ignored in the literature

P_{mx} = \frac{1}{\frac{{(μ L)}^{2}}{π^{2} E I_{eff}} + \frac{1}{S_{v}}}, I_{eff} = 2 I_{y} + 0.5 d^{2} A_{c}

(16a)

P_{my} = \frac{2 π^{2} E I_{x}}{{(μ L)}^{2}}

(16b)

P_{ml} = \frac{2 π^{2} E I_{y}}{{(L / m)}^{2}}

(16c)

where $μ$ is the effective length factor (equal to 1 for simply supported column and 2 for cantilever column) and $S_{v}$ is the shear rigidity produced by lacing bars system.

Three types of lacing bars system in Figure 4 are selected to validate the correctness of SEM proposed in this article. For all types, the columns are made of structural steel Q235 (elastic modulus E = 210 MPa and shear modulus G = 79 MPa). The chords have bending moments of inertia $I_{x} = 1520 c m^{4}$ and $I_{y} = 71.3 c m^{4}$ , and twisting and warping moments of inertia $I_{d} = 3.9 c m^{4}$ and $I_{w} = 9530 c m^{6}$ , respectively. Meanwhile, the geometric properties are listed in Table 2.

Table 2.

Geometrical characteristics used in Figure 2.

L (mm)	l (mm)	A_c (mm²)	h (mm)	t₁ (mm)	t₂ (mm)	x₀ (mm)	d (mm)	A_d (mm²)
8100	1350	2420	200	7.3	7.3	41	500	Varies

Three lacing bars systems under two boundary conditions have been analyzed, corresponding to the varying values of non-dimensional ratio $d^{2} A_{d} / I_{y} = (0, 20]$ . For comparison, the critical buckling load $P_{m}$ referring to local bending buckling load $P_{ml}$ from SEM is plotted in Figure 5. The results from EC3 method and linear buckling of finite element method (FEM) (ABAQUS) are also displayed in the graphs. In particular, there exist three possible critical buckling loads for EC3 method, $P_{mx}, P_{my}$ , and $P_{ml}$ . The lowest one is chosen as critical buckling load $P_{m}$ of $P_{m} / P_{ml}$ in Figure 5.

Figure 5.

Relationship between bars’ cross-sectional area $A_{d}$ and critical buckling load $P_{m}$ of latticed columns with (1) simple supports and (2) cantilever: (a) X-lacing system; (b) E-lacing system and (c) K-lacing system.

As shown in Figure 5(a)–(c), the forecasts of elastic critical buckling load from SEM have a good match with EC3 and FEM for the three typical lacing bars systems. When the cross-sectional area $A_{d}$ of lacing bars increases from a very small value, the critical buckling load $P_{m}$ of channel latticed columns rises sharply, whereas the critical buckling load $P_{m}$ increases barely after $A_{d}$ reaches a specific value. The extreme value of $A_{d}$ coincides with the concept of threshold braced stiffness in Zhang et al.³⁴

Compared with SEM and FEM, EC3 method is found to have two questions in forecasting critical buckling load of latticed columns. The calculating results are both inaccurate, whether cross-sectional areas of lacing bars $A_{d}$ are rather small or large. When $A_{d} \to 0$ , the value of $P_{m} / P_{ml}$ from EC3 tends to 0. But, the $P_{m} / P_{ml}$ from SEM and FEM tends to $P_{mx} / P_{ml}$ , which conforms to the property of engineer structures. When $A_{d} \to max (A_{d})$ , the critical buckling load from EC3 is behind in compound buckling load from SEM and FEM. That means the EC3 may get unsafe conclusions for conduction of critical load of latticed columns.

Influence of eccentric compressive load

As shown in equation (6a) and (6b), the eccentricity $e_{x}$ only influences the distribution of total compressive load P, which brings out inequality pressure in each chord. However, the eccentricity $e_{y}$ leads to the lateral moment $M_{xi}$ of chords.

An X-lacing latticed column under simply supported boundary was selected to illustrate the influence of eccentric axial compression the critical buckling load. Three different sizes of lacing bars with cross-sectional area $A_{d}$ varying uniformly from 10 to 50 mm² were analyzed as shown in Figure 6.

Figure 6.

Critical buckling load $P_{m}$ of simply supported latticed columns with the X-lacing system versus the eccentricity $e_{x}$ and $e_{y}$ : (a) $e_{y} = 0$ , (b) $e_{x} = 0$ , and (c) 3D curved surface with variable $e_{x}$ and $e_{y}$ .

When the eccentricity $e_{x} = e_{y} = 0 mm$ , the critical buckling failure mode is flexural buckling mode around the X-axis (Figure 3(a)) and critical buckling load $P_{m}$ reaches the maximum. The eccentricity of axial load leads to rapid decline of columns’ compressive carrying capacity. Comparing Figure 6(a) and (b), the critical buckling failure mode of latticed column was found to correspond with Figure 3(a) and remain untransformed with variety of eccentricity $e_{x}$ . However, the increment of off-center $e_{y}$ changes the critical buckling mode from Figure 3(a) to (e).

For example, in the case of $A_{d} = 50 m m^{2}$ , when $| e_{y} | ⩽ 120 mm$ , the critical buckling load changes extremely little with $P_{m} / P_{l} = 0.583$ , which was quite close to buckling load $P_{mx} / P_{l}$ in EC3 method and corresponds to lateral bending buckling mode around the X-axis (Figure 3(a)). When $| e_{y} | > 120 mm$ , the critical buckling load $P_{m} / P_{l}$ drops from 0.583 to a very small value, which coincides with the buckling load $P_{my} / P_{l}$ relating to lateral buckling mode around the Y-axis (Figure 3(e)).

Comparing Figure 6(a) and (b), when the cross-sectional area $A_{d}$ of lacing bars increases, the increments of critical buckling load become smaller and smaller. Thus, the tensile or compressive threshold stiffness $E A_{h}$ of lacing bars was defined in the article, which can be used to select size of lacing bars for latticed columns in engineering. Set function $P_{m} / P_{l} = f (A_{d}, e_{x}, e_{y})$ to express the variables’ relationship in Figure 6. The braced stiffness $E A_{d}$ increases from a rather little value (close to 0) adding Δ (0.5 mm²) until equation (17) is satisfied. Then, the braced stiffness $E A_{d}$ is threshold stiffness $E A_{h}$

\frac{f (A_{d} + Δ, e_{x}, e_{y}) - f (A_{d}, e_{x}, e_{y})}{x (A_{d} + Δ) - x (A_{d})} ⩽ 0.01

(17)

The variable curves between the eccentricity $e_{x}, e_{y}$ and $A_{h}$ of threshold stiffness $E A_{h}$ are plotted in Figure 7. The data results can be referred as vital guidance for selection of lacing bars.

Figure 7.

Relationship between the eccentricity $e_{x}, e_{y}$ and $A_{h}$ of threshold braced stiffness $E A_{h}$ : (a) the eccentricity $e_{x}$ and (b) the eccentricity $e_{y}$ .

Influence of initial geometric imperfections

There are always initial geometric imperfections in built-up columns that reduce their buckling ability of resisting external loads. In the previous research works of laced columns,^10,11,25 the shapes of initial lateral imperfections in global and local locations were chosen to be consistent with global and local flexural buckling modes (Figure 3(a), (b), and (e)), respectively. Similarly, the initial twisting imperfections should also coincide with the shapes of torsional buckling modes (Figure 3(c) and (d)).

Also, an X-lacing laced column under simply supported boundary was chosen to discuss the influence of initial imperfections on critical buckling load. Namely, the maximum global and local imperfections X_g, X_l are L/500 and l/500, respectively. So, the imperfections of chords can be expressed in equation (18)

u_{0 i} = \frac{L}{500} \sin \frac{π z}{L} + \frac{l}{500} \sin \frac{π mz}{L}

(18a)

v_{0 i} = \frac{L}{500} \sin \frac{π z}{L}

(18b)

φ_{0 i} = \frac{L}{500} \sin \frac{π z}{L} + \frac{l}{500} \sin \frac{π mz}{L}

(18c)

In theory of stability of structure,³⁵ the precise solution of nonlinear equilibrium paths is the convergence of all the linear buckling modes. But in reality, only finite modes can be guaranteed due to calculated problem of the nonlinearity. In equation (19), the five low-order modes were considered to get approximate nonlinear solutions, compared with nonlinear post-buckling analysis of FEM

u_{i} = C_{1 (i)} \sin (\frac{π z}{L}) + C_{2 (i)} \sin (\frac{π mz}{L})

(19a)

v_{i} = C_{3 (i)} \sin (\frac{π z}{L})

(19b)

φ_{i} = C_{4 (i)} \sin (\frac{π z}{L}) + C_{5 (i)} \sin (\frac{π mz}{L})

(19c)

where $u_{i}, v_{i}, and φ_{i}$ represent additive deformations on the basis of initial imperfections, and the total displacements of chords are $u_{i} + u_{0 i}$ , $v_{i} + v_{0 i}$ , and $φ_{i} + φ_{0 i}$ , respectively. Then, the total potential energy of chords with initial geometric imperfections was expressed in equation (20). Flexural strain energy and its load potential energy

\begin{matrix} Π_{1}^{i} = \frac{1}{2} \int_{L} [E I_{y} u_{i}''^{2} + E I_{x} v_{i}''^{2} - P_{i} (u_{i} + u_{0 i})'^{2} + P_{i} (v_{i} + v_{0 i})'^{2}] dz \\ = \frac{π^{2}}{4 L} [{C_{1 (i)}}^{2} P_{Ey} - (C_{1 (i)} + X_{g})^{2} P_{i}) + \frac{{(m π)}^{2}}{4 L} ({C_{2 (i)}}^{2} P_{Eym} - (C_{2 (i)} + X_{l})^{2} P_{i}) \\ + \frac{π^{2}}{4 L} ({C_{3 (i)}}^{2} P_{Ex} - (C_{3 (i)} + X_{g})^{2} P_{i}) \end{matrix}

(20a)

Torsional–flexural strain energy and its load potential energy

\begin{matrix} Π_{2}^{i} = \frac{1}{2} \int_{L} [E I_{w} φ_{i}''^{2} - (P_{i} r^{2} - G I_{d}) φ_{i}'^{2} - P_{i} r^{2} φ_{i 0}'^{2} + 2 P_{i} x_{0} (v_{i} + v_{0 i})' (φ_{i} + φ_{0 i})' + 2 M x_{i} (u_{i} + u_{0 i})' (φ_{i} + φ_{0 i})'] dz \\ = \frac{{(r π)}^{2}}{4 L} ({C_{4 (i)}}^{2} P_{Ew} - ({C_{4 (i)}}^{2} - {X_{g}}^{2}) P_{i}) + (C_{3 (i)} + X_{g}) (C_{4 (i)} + X_{g}) \frac{P_{i} x_{0} π^{2}}{2 L} \\ + (C_{1 (i)} + X_{g}) (C_{4 (i)} + X_{g}) \frac{M x_{i} π^{2}}{2 L} + (C_{2 (i)} + X_{l}) (C_{5 (i)} + X_{l}) \frac{M x_{i} m^{2} π^{2}}{2 L} \\ + \frac{{(rm π)}^{2}}{4 L} ({C_{5 (i)}}^{2} P_{Ewm} - ({C_{5 (i)}}^{2} - {X_{l}}^{2}) P_{i}) \end{matrix}

(20b)

The strain potential energy of lacings system (unchanged)

W = \sum_{1}^{m} \begin{matrix} - 2 c [u_{1} (k) - u_{2} (k + 1)]^{2} - \frac{h^{2} c}{2} [φ_{1} (k) - φ_{2} (k + 1)]^{2} \\ - 2 c [u_{2} (k) - u_{1} (k + 1)]^{2} - \frac{h^{2} c}{2} [φ_{2} (k) - φ_{1} (k + 1)]^{2} \end{matrix}

(20c)

Similar solution method was applied to acquire the load–displacement curves of nonlinear deformation with initial imperfections. The numerical results of SEM and FEM for X-lacing, E-lacing, and K-lacing columns under two boundary conditions are all plotted in Figures 8 –10, respectively. It can be observed that, for all the lacing types and boundary conditions, the critical buckling loads reduce largely due to the initial imperfections existing. In addition, linear buckling analysis always provides an upper value in forecasting critical buckling loads of laced columns, which are in accordance with the theory of Bazant and Cedolin.³⁵

Figure 8.

Load–displacement curve of elastic equilibrium paths for X-lacing columns under (1) simple supports and (2) cantilever state: (a) A_d = 10 mm², (b) A_d = 15 mm², (c) A_d = 20 mm², (d) A_d = 25 mm², and (e) A_d = 30 mm².

Figure 9.

Load–displacement curve of elastic equilibrium paths for E-lacing columns under (1) simple supports and (2) cantilever state: (a) A_d = 10 mm², (b) A_d = 15 mm², (c) A_d = 20 mm², (d) A_d = 25 mm², and (e) A_d = 30 mm².

Figure 10.

Load–displacement curve of elastic equilibrium paths for K-lacing columns under (1) simple supports and (2) cantilever state: (a) A_d = 10 mm², (b) A_d = 15 mm², (c) A_d = 20 mm², (d) A_d = 25 mm², and (e) A_d = 30 mm².

It can also observed that, compared with cantilever state, the three lacing columns under simple supports have a more obvious response for local imperfections. The local imperfections have almost no impact on the critical load of the three lacing columns under cantilever state. However, the K-lacing columns under simply supported conditions respond more sensitively on local imperfection than the X-lacing and E-lacing columns when the cross-sectional area of bars ranges from 10 to 20 mm². Once the rigidity of bars exceed to the threshold value, such as A_d = 25 and 30 mm² in Figure 10, the local imperfections would no longer affect the critical loads intuitively.

Although not deeper discussion about the threshold braced stiffness, as Figure 7, in the procedure of nonlinear buckling with imperfections, the threshold braced stiffness A_h of lacing bars still exists. Note that the critical load of the three lacing columns increases in smaller and smaller values, when the cross-sectional area A_d of bars varies from 10 to 30 mm² in Figures 8 –10.

Example verification with shell-link element models of FEM

Three finite element models for X-lacing, E-lacing, and K-lacing built-up columns are established in Figure 11 to verify the numerical results of proposed SEM in this article. The chords are modeled with four-node shell element, while the lacing bars two-node link element by commercial software ABAQUS. All freedoms of lacing bars were coupled to the corresponding nodes of chords. To avoid the local warping at the columns ends, the master–slave nodes technique was used in FEM to simulate loading and boundary conditions more realistically as shown in Figure 11(b). Both the analytical modules of linear buckling analysis and nonlinear post-buckling with initial imperfections were performed to validate the relationship between critical buckling loads with corresponding design parameters in the article.

Figure 11.

Modeling procedures for X-lacing, E-lacing, and K-lacing columns using ABAQUS: (a) interaction between chords and lacing bars; (b) boundary and loading conditions under simple supports and cantilever state.

From the equilibrium paths of nonlinear post-buckling analysis in Figure 12, it can be observed that the nonlinear deformations mainly consist of global bending buckling and global twisting buckling of each chord. Local bending and twisting buckling are less important than the global buckling in the post-buckling path for the analytical model.

Figure 12.

Equilibrium path of the three lacing columns under two boundary conditions of simple supports and cantilever state acquired through nonlinear post-buckling analysis of ABAQUS (A_d = 30 mm²).

To better compare the results of FEM and SEM, equally spaced test points A–F were selected from chord 1 of X-lacing, E-lacing, and K-lacing columns (under simple supports and cantilever state) as shown in Figure 13. The nonlinear equilibrium paths of SEM are the superposition of global bending and twisting buckling functions, which are very close to the results of FEM. It validates that the SEM is very concise and accurate in calculating the linear buckling and nonlinear buckling capacity of latticed columns.

Figure 13.

Comparing the equilibrium path of method SEM with FEM (A_d = 30 mm²): (a) X-lacing built-up columns, (b) E-lacing built-up columns, and (c) K-lacing built-up columns.

Summary and conclusion

Considering total potential energy of bars and chords individually, the compound buckling loads of latticed columns were discussed with three types of lacings under two boundary conditions. The present analysis highlights the critical buckling load affected by design parameters (size of bars, eccentricity, and initial geometric imperfections). The threshold stiffness EA_h was calculated on conditions of eccentricity in linear buckling analysis. Based on EC3, FEM, SEM, and their output, the following main conclusions were drawn:

The calculation of elastic buckling loads from EC3 has a shortcoming. Whether the cross-sectional area of bars A_d tends to 0 or A_h, the forecasting of linear buckling load from EC3 was inaccurate. When A_d tends to 0, the minimum buckling load tends to 0, unconfirmed to the property of engineering structure. When A_d tends to A_h, the minimum buckling load from EC3 was about 20% higher than that of FEM, getting unsafe result for laced columns.

Through the research on linear buckling load of X-lacing, E-lacing, and K-lacing columns under two boundary conditions, the eccentricity of compressive load would obviously decrease the buckling capacity of built-up columns, especially for bidirectional eccentricity e_x, e_y. And, the cross-sectional area of bars has a threshold value A_h. The elastic critical buckling load P_m would almost not increase after the cross-sectional areas of lacing bars A_d exceed to A_h. The threshold stiffness was affected by eccentricity e_x, e_y and initial geometric imperfection X_g, X_l, besides the structural characteristics.

Based on the nonlinear buckling load considering geometric imperfections, the existing of initial imperfections X_g and X_l would reduce the critical buckling load of all the lacing columns. But, the local imperfections X_l have almost no impact on the critical load of the three lacing columns under cantilever state. However, the K-lacing columns under simple supports respond more sensitively on local imperfection than the X-lacing and E-lacing columns when the cross-sectional area of bars varied below the threshold value.

Based on the nonlinear buckling deformation using methods SEM and FEM, the equilibrium paths of laced columns were found to consist of the low-order linear buckling modes. The proposed method SEM, expressing the nonlinear buckling deformation by the superposition low-order mode functions, has a reliable accuracy as FEM in forecasting the nonlinear buckling load and deformation. However, the method SEM is more simple and formulaic than FEM.

Footnotes

Appendix 1

Appendix 2

Handling Editor: James Baldwin

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: This work was supported by the National Natural Science Foundation of China (NSFC) under grant numbers 51675450 (working) and 51175442 (finished).

ORCID iD

Yinqi Li

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