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Numerical Studies on the Buckling Behaviour of Cable-stiffened Steel Columns With Pin-connected Crossarm Systems
Accepted Manuscript Numerical Studies on the Buckling Behaviour of Cable-stiffened Steel Columns With Pin-connected Crossarm Systems
Pengcheng Li, Ce Liang, Jun Yuan, Ke Qiao PII: DOI: Reference:
S2352-0124(18)30091-2 doi:10.1016/j.istruc.2018.08.008 ISTRUC 317
To appear in:
Structures
Received date: Revised date: Accepted date:
26 April 2018 21 August 2018 22 August 2018
Please cite this article as: Pengcheng Li, Ce Liang, Jun Yuan, Ke Qiao , Numerical Studies on the Buckling Behaviour of Cable-stiffened Steel Columns With Pin-connected Crossarm Systems. Istruc (2018), doi:10.1016/j.istruc.2018.08.008
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ACCEPTED MANUSCRIPT Numerical studies on the buckling behaviour of cable-stiffened steel columns with pin-connected crossarm systems Pengcheng Li*1,2, Ce Liang3, Jun Yuan3, Ke Qiao3 1
Key Laboratory of New Technology for Construction of Cities in Mountain Area (Chongqing University), Ministry of Education, Chongqing 400045, China 2 3
School of Civil Engineering, Chongqing University, Chongqing 400045, China
College of Engineering and Technology, Southwest University, Chongqing 400715, China
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Abstract In this study, the buckling behaviour of a cable-stiffened steel column with a four-branch pin-connected crossarm system is investigated. A new geometric imperfection distribution is constructed with a nonlinear buckling analysis to trigger interactive buckling, and a method to obtain the actual load carrying capacity of the cable-stiffened steel column is proposed. The effects of different parameters, including the stay diameter, crossarm length, pretension in cables, and imperfection size, have also been analysed and discussed in this article. The findings of this study show that the interactive buckling of steel column stiffened by the four-branch pin-connected crossarms must be taken into account for accurate analysis. It is also demonstrated that the cable-stiffened steel columns are imperfection sensitive, particularly when the critical buckling mode is located at the transition between symmetric and antisymmetric buckling, though the corresponding configuration is proved to be optimal with respect to the material utilisation rate.
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Keywords: cable-stiffened column; stability behaviour; buckling analysis; geometric imperfection; imperfection sensitive
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1. Introduction Pretensioned cables are commonly introduced in lightweight or slender structures for enhancing the stiffness [1-4]. A cable-stiffened steel column, also known as prestressed stayed steel column, is one of the simplest prestressed steel structures adopting pretensioned cables. In cable-stiffened steel columns, the pretensioned cables are used in conjunction with crossarms to stiffen the main column (see Figure 1 [5]). This structural type has been widely used when slender support is required; other practical applications of cable-stiffened steel columns are discussed in reference [5].
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Corresponding author E-mail address: [email protected] (P.C. Li)
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Figure 1 Practical application of a cable-stiffened steel column
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Research work on cable-stiffened steel columns has been conducted intensively since it was proposed. Maunch and Felton developed an analytical foundation for designing cable-stiffened steel structures [4]. Hafez et al. divided the effect of pretension on the column buckling load into three zones, and formulated an equation describing this effect [6]. Liew and Li investigated the effect of different structural parameters on the stability behaviour of cable-stiffened columns numerically, and extended this stiffened column to three-dimensional (3D) frames [7]. Chan et al. analysed the buckling load of cable-stiffened steel columns using pointwise equilibrium polynomial element, and the geometric imperfection was input following the symmetric or anti-symmetric buckling modes [8]. However, the interactive stability behaviour of cable-stiffened steel columns has not been included in the above-mentioned research work. Wadee et al. conducted a series of investigations on the interactive behaviour of stiffened steel columns to determine their actual load carrying capacity and post-buckling behaviour [9–11]. Bai et al. numerically investigated the stability behaviour of a stiffened column with a welded I-section, taking into account the geometric and material nonlinearity [12]. In addition, experimental studies including those focused on the stability behaviour under a concentrated load [13–15] or fire [16] have also been conducted. Note that the above-mentioned studies were primarily focussed on determining the behaviour of cable-stiffened steel columns with rigidly-connected crossarms, whereas research on stiffened columns with pin-connected crossarms is limited. In fact, pin-connected crossarms are simpler for construction, and it has been proved that the using them in conjunction with pretensioned cables is effective and improves the stability behaviour of ordinary columns [17]. Thus, the objective of this study was to investigate the stability behaviour of cable-stiffened steel columns with a pin-connected crossarm system. In this current work, the principle of inputting geometric imperfections in nonlinear buckling analysis to capture the actual buckling strength was proposed based on finite element (FE) analysis performed by ABAQUS. Moreover, a parametric analysis and an imperfection sensitive analysis were also conducted. 2. Model development As stated earlier, a cable-stiffened steel column comprises a main column and pretensioned cables in conjunction with crossarms. Figure 2 illustrates the configuration of the cable-stiffened steel column, and as it can be seen, a four-branch crossarm system is adopted in this work. The main column is simply-supported at the two ends with a concentric load axially applied to the upper end. The four crossarms are pin-connected to the main column.
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Figure 2 Analytical model of the cable-stiffened steel column
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In this study, steel tubes with outside and inside diameters of 38.1 mm and 25.4 mm, respectively, are selected for both the main column and crossarms. The material properties of the cable-stiffened steel columns listed in Table 1, are generally the same as those in the model of Hafez [6]. The element type selected for the main column and crossarms is the Timoshenko beam element, whereas the cables are modelled with a truss element. Noting that the cable slackens when the axial forces change from tension to compression, the truss element is also adopted without compression to simulate the slackening phenomenon.
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Table 1 Material properties of the cable-stiffened steel columns. Ec and Ea are the Young’s modulus of the main column and crossarms, respectively. f yc and f ya denote the nominal yield strength of the main column and crossarms, respectively. Es is the Young’s modulus of the cable.
f yc （MPa）
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345
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202000
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3. Structural parameters 3.1 Geometric configuration According to previous studies [9-10], the post-buckling behaviour of cable-stiffened steel columns is significantly affected by the critical buckling modes which are primarily determined by the structural stiffness (crossarm length and cable diameter). Thus, different crossarm lengths and cable diameters must be selected to activate different critical buckling modes. In this study, to simplify the analysis, the column length was fixed to 3000 mm; however, the crossarm length and cable diameter were varied according to Table 2. A following analysis will demonstrate that different critical buckling modes can be activated by varying the crossarm length and cable diameter, as per Table 2, even if the column length is fixed. Table 2 Crossarm length and cable diameter Crossarm length Cable diameter
a (mm)
150
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s ( mm)
1.6
3.2
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3.2 Pretension in cables
ACCEPTED MANUSCRIPT Pretension in cables could certainly affect the nonlinear buckling behaviour of the cable-stiffened steel columns, and thus, determining the initial pretension in the cables prior to performing the nonlinear buckling analysis is essential. According to the research conducted by Hafez [6], the initial pretension that corresponds to a critical buckling load can be calculated by equation (1) Topt
C11 c P Zone3,T 0 C22
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where P c Zone3,T 0 is the critical buckling load obtained from FE analysis, and C11 and C22 are
2cos 2 1 2sin 2 Kc ( ) Ks Ka
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two parameters which can be calculated from equations (2) and (3) cos C11 1 2sin 2 2cos 2 2Kc ( ) Ks Ka Kc
(2)
(3)
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where is the angle between the main column and cable, which is substantially determined by the crossarm length; and K c , Ks , and K a are the stiffness of the main column, cable, and
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Note that pretension Topt is generally adopted as the benchmark magnitude in the numerical
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analysis in this work. 4. Buckling analysis 4.1 Critical buckling analysis Figure 3 presents the buckling modes of the cable-stiffened steel columns with pin-connected crossarm systems. To compare the effect of the joint type on the critical buckling modes, the buckling modes of a stiffened column with both pin-connected and rigid-connected crossarm systems are presented in Figure 3 and Figure 4. Overall, the buckling modes of the stiffened column are symmetric (see Figures 3 (a) and 4(a)) or anti-symmetric (see Figures 3 (b) and 4(b)) with both pin and rigidly-connected crossarms. However, it should be noted that the crossarms remain straight when the connection is pinned, implying that rotational stiffness is not provided to the main column by the crossarms. This is a notable characteristic to distinguish pin-connected stiffened columns from those with rigid connections. Consequently, it is essential to conduct a nonlinear buckling analysis to examine the interactive buckling behaviour of the stiffened columns with pin-connected crossarms, even if stiffened columns with rigidly-connected crossarms have been investigated previously. For simplification, in the following analysis, the symmetric and antisymmetric buckling modes are respectively referred to Mode 1 and Mode 2.
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(a) Symmetric buckling
(b) Anti-symmetric buckling
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Figure 3 Buckling modes of stiffened columns with pin-connected crossarms (Front view)
(a) Symmetric buckling
(b) Anti-symmetric buckling
Figure 4 Buckling modes of stiffened columns with rigidly-connected crossarms (Front view)
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Figure 5 presents the buckling loads of cable-stiffened steel columns with different cable diameters and crossarm lengths. Obviously, the buckling mode varies from Mode 1 to Mode 2 as the cable diameter and crossarm length increase. It must be noted that the effects of the cable diameter and crossarm length on the buckling loads are significantly different depending on the buckling mode. When the buckling mode is Mode 1, increasing the cable diameter or crossarm length within a moderate range commonly enhances the buckling load, but it decreases when the crossarm is extremely long. However, the effect of the cable diameter and crossarm length on the buckling load corresponding to Mode 2 is limited or adverse (see Figure 5(b)). This can be explained from the buckling deformation of Mode 2, as shown in Figure 3(b). For Mode 2 buckling, only the axial stiffness of the crossarm can be activated to stiffen the main column because it remains straight during the deformation. Therefore, it is inadvisable to achieve a high buckling load by increasing the crossarm length when Mode 2 occurs because it will decrease the crossarm’s axial stiffness.
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Figure 5 Buckling loads with different cable diameters and crossarm lengths
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4.2 Post-buckling analysis 4.2.1 Imperfection construction For an ordinary steel column without cable-stiffened systems, the geometric imperfection is affined to the critical buckling mode in the nonlinear buckling analysis. However, it has been shown that the load carrying capacity of a cable-stiffened steel column with a geometric imperfection distribution obtained following only the critical buckling mode would be overestimated because interactive buckling cannot be considered by this method, though it could determine the actual buckling strength in some cases [9]. To consider interactive buckling, a complex imperfection distribution has been suggested for cable-stiffened columns with rigidly-connected crossarms [9, 10]. However, the above critical buckling analysis shows that the anti-symmetric buckling mode of cable-stiffened columns with pin-connected crossarms is substantially different from that with rigidly-connected crossarms. Thus, applicability of the method of inputting the imperfection distribution for stiffened columns with rigidly-connected crossarms can be simply to the pin-connected case needs to be examined because the connection stiffness would certainly affect the buckling mode of the stiffened columns. Based on this consideration, in this work, a new imperfection distribution is constructed that includes the components of both the symmetric and antisymmetric buckling modes to activate interactive buckling. The constructed imperfection can be described by equation (4): W ( x) 1 sin
x L
2 sin
2 x L
(4)
where and L are the imperfection size and main column length, respectively. The first and second terms of the right side represent the contribution of Mode 1 and Mode 2, respectively. Thus,
1 and 2 denote the weights of the symmetric and antisymmetric components, respectively, in the constructed imperfection distribution. To determine the magnitudes of 1 and 2 , it is assumed that the axially compressive of the main column owing to the different imperfection distributions, are the same to leading order, and thus, equation (5) is derived to be
12 +422 =1
(5)
ACCEPTED MANUSCRIPT Table 3 presents five different combinations of 1 and 2 , which would be adopted in the nonlinear buckling analysis. In fact, different magnitudes of 1 and 2 represent different geometric imperfection shapes. Thus, five different imperfection distributions were inputted for each case in the buckling analysis. Table 3 Selected weights of symmetric and antisymmetric buckling modes 0
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4.2.2 Interactive buckling Having constructed the geometric imperfection distribution, in this section we examine via nonlinear buckling analysis a possible domination of interactive buckling on the load carrying capacity of the stiffened column. It should be mentioned that an imperfection magnitude of L/300 (L is the main column length) is generally adopted in this study. For the nonlinear buckling analysis, it is comprehensible to consider the geometrical nonlinearity because the column is slim. In addition to the geometrical nonlinearity, material nonlinearity has also been considered in this work to investigate the post-buckling behaviour. Though it has been demonstrated that elastic analysis is sufficient to capture the buckling strength of a cable-stiffened column, the material plasticity can also be observed beyond the instant the buckling occurs. Based on this, both the geometrical and material nonlinearities were taken into account in the buckling analysis. However, it must be pointed out that the cables are completely considered as elastic even though the plastic behaviour of steel is taken into account. Figures 7 and 8 illustrate the axial loads and compressions of the cable-stiffened steel columns with different crossarm lengths and cable diameters. When the combination of crossarm length and cable diameter size is 150 mm and 4.8 mm or 300 mm and 1.6 mm, respectively, the actual load carrying capacity of the stiffened column is determined by the imperfection following the symmetric buckling mode (Mode 1). Otherwise, it is determined by the interactive buckling which is triggered by the asymmetric imperfection. It is interesting to note that irrespective of the crossarm length and cable diameter being 150 mm and 4.8 mm or 300 mm and 1.6 mm, respectively, the corresponding critical buckling mode is symmetric (see Figure 5). Thus, the principle to input the geometric imperfection shape can be correlated to the critical buckling modes, i.e. if the critical buckling mode is symmetric, the imperfection shape should just follow the critical buckling mode in the nonlinear buckling analysis. Conversely, if the critical buckling mode is antisymmetric, the imperfection shape is suggested to be asymmetric to consider the interactive buckling behaviour. This conclusion, which has also been obtained in cable-stiffened column with rigidly-connected crossarms [9], is the principle to input geometric imperfection in the following parametric studies.
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Figure 7 Load versus end-shortening curves with different crossarm lengths ( s =4.8mm )
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Figure 8 Load versus end-shortening curves with different cable diameters (a = 300 mm)
5. Parametric studies Having determined that the principle of introducing the geometric imperfection to obtain the actual buckling strength by conducting nonlinear buckling analysis. In this section, the effect of different parameters (crossarm length, cable diameter, pretension in the cables, and imperfection size) on the buckling strength of the cable-stiffened steel column would be discussed. 5.1 Effect of the geometry configuration For the cable-stiffened steel columns, the buckling strength of the main column is enhanced by pretensioned cables and crossarms. Thus, the crossarm length and cable diameter would certainly affect the buckling strength of the column. Figure 9 illustrates the load carrying capacities of cable-stiffened steel columns with different cable diameters and crossarm lengths. Overall, the
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Figure 9 Load carrying capacities with different cable diameter and crossarm length
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Table 4 presents the critical buckling modes of the cable-stiffened steel columns with different parameters. It can be seen from Figure 9 and Table 4 that the effect of the crossarm length and cable diameter is governed by the critical buckling mode. When the critical buckling mode is Mode 1, the buckling strength of the stiffened column is drastically increased by the crossarm length or cable diameter; however, when it is transferred to Mode 2, their effect on the buckling strength is limited or even adverse. From this perspective, the material utilisation rate is the highest when the critical buckling mode is antisymmetric and close to the transition between the symmetric and antisymmetric modes. Thus, the optimal structural configuration for the cable-stiffened steel column corresponds to a critical buckling mode located at the transition from Mode 1 to Mode 2. Cable diameter (mm)
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Table 4 Summary of the critical buckling modes
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5.2 Effect of pretension in cables The pretension in cables is another important factor that affects the buckling behaviour of cable-stiffened steel columns. So far, the pretension in the cables was fixed to Topt derived from a linear buckling analysis. However, it is unclear whether pretension Topt corresponds to the maximum buckling strength. Thus, in this section, we investigate the effect of pretension in the cables on the buckling strength. Figure 10 illustrates the load carrying capacities of the cable-stiffened steel columns with pretensions in cables vary from Topt to 5Topt . When the
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T / Topt 5 . When the cable diameter is fixed to 4.8 mm (see Figure 10(b)), the buckling strength increases with pretension in the range of T / Topt 2 ; however, the buckling strength starts to decrease for the cases a = 375 mm and 450 mm when T / Topt 2 . Thus, a pretension level higher
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5.3 Effect of imperfection size In the above analysis, L/300 (L is the main column length) is adopted as the imperfection size. However, it has been shown in previous studies that the buckling strength of slim columns is significantly affected by the imperfection size [18-19]. In this section, we quantitatively examine the effect of the imperfection size on the load carrying capacities of cable-stiffened steel columns with four pin-connected crossarms. Five different imperfection sizes including L/5000, L/1000, L/500, L/300, and L/100 are selected. In fact, L/5000 is quite small as the imperfection size; thus, the corresponding load carrying capacity is used to represent the actual capacity of a perfect stiffened column. Figure 11 presents the load carrying capacities of the cable-stiffened steel columns with different imperfection magnitudes. Generally, the capacities show an obvious decrease with increasing imperfection size.
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Figure 11 Load carrying capacities with different imperfection sizes
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denote the buckling strengths of perfect and imperfect cable-stiffened steel columns, respectively. (6)
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Figure 12 shows the imperfection sensitivity coefficient of the stiffened steel columns with different cable diameters and crossarm lengths. Obviously, the maximum coefficient is approximately 0.4, implying that the buckling strength is decreased by 40% in this case. It can also be observed that the maximum coefficient corresponds to the structural configurations of a =
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Recalling the critical buckling modes shown in Table 4, it is interesting to note that the above two structural configurations are close to the transition between the symmetric and antisymmetric buckling modes. Thus, attention must be paid to the effect of imperfection size when designing a stiffened column with a configuration that leads to the simultaneous occurrence of symmetric and antisymmetric buckling, though this configuration is optimal in respect to improving the material utilisation rate. 0.5
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6. Critical buckling loads of stiffened columns different crossarm connections The buckling behaviour of cable-stiffened steel column with pin-connected crossarms has been
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numerically investigated in above sections, the method to determine the initial geometrical imperfections in nonlinear buckling analysis has also been proposed. However, the buckling loads of stiffened columns with pin-connected and rigidly-connected crossarm systems have not been comparatively analysed. This section aims to compare the critical buckling loads of stiffened steel columns with the above two crossarm connections. The critical buckling loads and buckling modes of the cable-stiffened steel columns are presented in Figure 13. As it can be seen, when the critical buckling modes of stiffened column with pin-connected crossarms is symmetric, the critical buckling loads correspond to pin and rigid connected crossarm structural schemes are the same. In contrast, when the critical buckling modes of stiffened column with pin-connected crossarms is anti-symmetric, the critical buckling loads corresponds to pin connected crossarm structural schemes are much lower than those of rigidly-connected cases.
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Figure 13 Critical buckling loads of stiffened columns with different crossarm connections
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7. Conclusions To the best knowledge of the authors, this is the first study on the interactive buckling behaviour of cable-stiffened steel columns with four pin-connected crossarms. The following conclusions can be drawn from the numerical analysis in this study: (1) The typical buckling modes of the cable-stiffened steel columns with four pin-connected crossarms can be classified as symmetric and antisymmetric, similar to columns with rigidly-connected crossarms. However, their antisymmetric buckling modes are substantially different. For the stiffened column with pin-connected crossarms, the rotational stiffness of the crossarms cannot be activated during buckling; however, both the axial and rotational stiffness of the crossarms can be activated when they are rigid-connected to the main column. (2) It has been demonstrated that adopting distinct bucking modes in the geometric imperfection distribution in the nonlinear buckling analysis could overestimate the buckling strength in some cases, and thus, interactive buckling must be taken into account for accurate analysis. When the critical buckling mode is symmetric, it is sufficient to adopt the geometric imperfection shape to which it is affine. In contrast, the combination of symmetric and anti-symmetric buckling modes must be adopted to construct a new imperfection shape to consider the interactive buckling behaviour when the critical buckling mode is antisymmetric. (3) The parametric study shows that the buckling strength of the cable-stiffened steel columns is significantly affected by the crossarm length or cable diameter; however, these effects are significantly governed by the critical buckling modes. For the case when the critical buckling mode is symmetric, the buckling strength can be strongly enhanced by increasing the crossarm
ACCEPTED MANUSCRIPT length or cable diameter. However, this effect on the buckling strength is limited or adverse if the critical buckling mode is antisymmetric. For the optimal design of cable-stiffened steel columns, critical buckling mode should be Mode 2 and close to the transition between Mode 1 to Mode 2. This is because this structural configuration corresponds to a maximum material utilisation rate. (4) The nonlinear buckling analysis demonstrates that the maximum load carrying capacity of cable-stiffened steel columns cannot be achieved by introducing the initial pretension of Topt ,
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though this pretension value corresponds to the maximum critical buckling load. A higher pretension level, depending on the critical buckling modes and structural stiffness, is suggested by the authors if the designers aim to obtain the maximum buckling strength. (5) The imperfection sensitivity exhibits that the buckling strength of a cable-stiffened steel column can be drastically decreased by the imperfection size, implying that this steel column is imperfection sensitive. It has also been illustrated that the imperfection sensitivity coefficient, which quantifies the reduction of the imperfection, reaches a maximum value when the critical buckling mode is located at the transition between the symmetric and antisymmetric buckling modes.
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Acknowledgement The research was supported by the National Natural Science Foundation of China (No. 51808070), the Fundamental and Frontier Research Project of Chongqing (No. cstc2018jcyjA2698), the Fundamental Research Funds for the Central Universities (No. 106112017CDJXY200008 and No. 106112016CDJRC000088). These financial support are gratefully acknowledged. References
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[1] Masao S, Kurasiro T. A study on structural behaviors of beam string structure. Summaries of technical papers of annual meeting architectural institute of Japan. Tokyo, Japan, B, 1985; 1: 280-284. [2] Kawaguchi M, Abe M, Tatemichi I. Design, tests and realization of ‘suspen-dome’ system. Journal of the International Association of Shell and Spatial Structures 1990; 40(131): 179-192.
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[3] Chu K, Berge S. S. Analysis and design of struts with tension ties. J Struct Div (ASCE) 1963; 89 (1): 127-163. [4] Mauch H.R., Felton L P. Optimum design of columns supported by tension ties. J Struct Div (ASCE) 1967; 93(3): 201-220.
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[5] Lapira L, Wadee MA and Gardner L. Stability of multiple-crossarm prestressed stayed columns with additional stay systems. Struct 2017; 12: 227-241. [6] Hafez HH, Temple MC, Ellis JS. Pre-tensioning of single-crossarm stayed columns. J Struct Div (ASCE) 1975; 105(2): 359-375. [7] Liew JYR, Li JJ. Advanced analysis of pre-tensioned bowstring structures. Int J Steel Struct 2006; 6(3): 153-162. [8] Chan SL., Shu GP, Lü ZT. Stability analysis and parametric study of pre-stressed stayed columns. Eng Struct 2002; 24(1): 115-124. [9] Saito D, Wadee MA. Numerical studies of interactive buckling in prestressed steel stayed column. Eng Struct 2009; 31(2): 432-443. [10] Li PC, Wadee MA, Yu JL, Christie NG, Wu ME. Stability of prestressed stayed steel columns with a three branch crossarm system. J Constr Steel Res 2016; 122: 274-291.
ACCEPTED MANUSCRIPT [11] Zschernack C, Wadee MA, Völlmecke C. Nonlinear buckling of fiber-reinforced unit cells of lattice materials. Compos Struct 2016; 136: 217-228. [12] Bai ZX, Chen MF, Hou XM, Xu KR, Liu YM, Dai CS, et al. Study on stability behavior of cable-stayed ultra-high strength steel column with welded I-section. Appl Mech Mater 2012; 193-194: 1352-1356. [13] De Araujo RR, de Andrade SAL, da S Vellasco PCG, da Silva JGS, de Lima LRO. Experimental and numerical assessment of stayed steel columns. J Constr Steel Res 2008; 64: 1020-9. [14] Serra M, Shahbazian A, da S Simões L, Marques L, Rebelo C, da S Vellasco PCG. A full scale experimental study of prestressed stayed columns. Eng Struct 2015; 10: 490-510. [15] Osofero AI, Wadee MA, Gardner L. Experimental study of critical and post-buckling behaviour of prestressed
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[16] Zhou HT, Kodur VKR, Nie HB, Wang YZ, Naser MZ. Behavior of prestressed stayed steel columns under fire
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conditions. Int J Steel Struct 2017; 17(1): 195-204.
[17] Wu ME, Sasaki M, Ohmori H. Geometrically nonlinear analysis and experiment on pin joint stayed column.
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[18] Aristizabal-Ochoa JD. Stability of imperfect slender columns with non-linear connections. Int J Nonlin Mech 2013; 54: 66-75.
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[19] Dou C, Pi YL. Effects of geometric imperfections on flexural buckling resistance of laterally braced columns.
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Pengcheng Li, Ce Liang, Jun Yuan, Ke Qiao PII: DOI: Reference:
S2352-0124(18)30091-2 doi:10.1016/j.istruc.2018.08.008 ISTRUC 317
To appear in:
Structures
Received date: Revised date: Accepted date:
26 April 2018 21 August 2018 22 August 2018
Please cite this article as: Pengcheng Li, Ce Liang, Jun Yuan, Ke Qiao , Numerical Studies on the Buckling Behaviour of Cable-stiffened Steel Columns With Pin-connected Crossarm Systems. Istruc (2018), doi:10.1016/j.istruc.2018.08.008
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ACCEPTED MANUSCRIPT Numerical studies on the buckling behaviour of cable-stiffened steel columns with pin-connected crossarm systems Pengcheng Li*1,2, Ce Liang3, Jun Yuan3, Ke Qiao3 1
Key Laboratory of New Technology for Construction of Cities in Mountain Area (Chongqing University), Ministry of Education, Chongqing 400045, China 2 3
School of Civil Engineering, Chongqing University, Chongqing 400045, China
College of Engineering and Technology, Southwest University, Chongqing 400715, China
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Abstract In this study, the buckling behaviour of a cable-stiffened steel column with a four-branch pin-connected crossarm system is investigated. A new geometric imperfection distribution is constructed with a nonlinear buckling analysis to trigger interactive buckling, and a method to obtain the actual load carrying capacity of the cable-stiffened steel column is proposed. The effects of different parameters, including the stay diameter, crossarm length, pretension in cables, and imperfection size, have also been analysed and discussed in this article. The findings of this study show that the interactive buckling of steel column stiffened by the four-branch pin-connected crossarms must be taken into account for accurate analysis. It is also demonstrated that the cable-stiffened steel columns are imperfection sensitive, particularly when the critical buckling mode is located at the transition between symmetric and antisymmetric buckling, though the corresponding configuration is proved to be optimal with respect to the material utilisation rate.
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Keywords: cable-stiffened column; stability behaviour; buckling analysis; geometric imperfection; imperfection sensitive
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1. Introduction Pretensioned cables are commonly introduced in lightweight or slender structures for enhancing the stiffness [1-4]. A cable-stiffened steel column, also known as prestressed stayed steel column, is one of the simplest prestressed steel structures adopting pretensioned cables. In cable-stiffened steel columns, the pretensioned cables are used in conjunction with crossarms to stiffen the main column (see Figure 1 [5]). This structural type has been widely used when slender support is required; other practical applications of cable-stiffened steel columns are discussed in reference [5].
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Corresponding author E-mail address: [email protected] (P.C. Li)
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Figure 1 Practical application of a cable-stiffened steel column
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Research work on cable-stiffened steel columns has been conducted intensively since it was proposed. Maunch and Felton developed an analytical foundation for designing cable-stiffened steel structures [4]. Hafez et al. divided the effect of pretension on the column buckling load into three zones, and formulated an equation describing this effect [6]. Liew and Li investigated the effect of different structural parameters on the stability behaviour of cable-stiffened columns numerically, and extended this stiffened column to three-dimensional (3D) frames [7]. Chan et al. analysed the buckling load of cable-stiffened steel columns using pointwise equilibrium polynomial element, and the geometric imperfection was input following the symmetric or anti-symmetric buckling modes [8]. However, the interactive stability behaviour of cable-stiffened steel columns has not been included in the above-mentioned research work. Wadee et al. conducted a series of investigations on the interactive behaviour of stiffened steel columns to determine their actual load carrying capacity and post-buckling behaviour [9–11]. Bai et al. numerically investigated the stability behaviour of a stiffened column with a welded I-section, taking into account the geometric and material nonlinearity [12]. In addition, experimental studies including those focused on the stability behaviour under a concentrated load [13–15] or fire [16] have also been conducted. Note that the above-mentioned studies were primarily focussed on determining the behaviour of cable-stiffened steel columns with rigidly-connected crossarms, whereas research on stiffened columns with pin-connected crossarms is limited. In fact, pin-connected crossarms are simpler for construction, and it has been proved that the using them in conjunction with pretensioned cables is effective and improves the stability behaviour of ordinary columns [17]. Thus, the objective of this study was to investigate the stability behaviour of cable-stiffened steel columns with a pin-connected crossarm system. In this current work, the principle of inputting geometric imperfections in nonlinear buckling analysis to capture the actual buckling strength was proposed based on finite element (FE) analysis performed by ABAQUS. Moreover, a parametric analysis and an imperfection sensitive analysis were also conducted. 2. Model development As stated earlier, a cable-stiffened steel column comprises a main column and pretensioned cables in conjunction with crossarms. Figure 2 illustrates the configuration of the cable-stiffened steel column, and as it can be seen, a four-branch crossarm system is adopted in this work. The main column is simply-supported at the two ends with a concentric load axially applied to the upper end. The four crossarms are pin-connected to the main column.
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Figure 2 Analytical model of the cable-stiffened steel column
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In this study, steel tubes with outside and inside diameters of 38.1 mm and 25.4 mm, respectively, are selected for both the main column and crossarms. The material properties of the cable-stiffened steel columns listed in Table 1, are generally the same as those in the model of Hafez [6]. The element type selected for the main column and crossarms is the Timoshenko beam element, whereas the cables are modelled with a truss element. Noting that the cable slackens when the axial forces change from tension to compression, the truss element is also adopted without compression to simulate the slackening phenomenon.
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Table 1 Material properties of the cable-stiffened steel columns. Ec and Ea are the Young’s modulus of the main column and crossarms, respectively. f yc and f ya denote the nominal yield strength of the main column and crossarms, respectively. Es is the Young’s modulus of the cable.
f yc （MPa）
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202000
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3. Structural parameters 3.1 Geometric configuration According to previous studies [9-10], the post-buckling behaviour of cable-stiffened steel columns is significantly affected by the critical buckling modes which are primarily determined by the structural stiffness (crossarm length and cable diameter). Thus, different crossarm lengths and cable diameters must be selected to activate different critical buckling modes. In this study, to simplify the analysis, the column length was fixed to 3000 mm; however, the crossarm length and cable diameter were varied according to Table 2. A following analysis will demonstrate that different critical buckling modes can be activated by varying the crossarm length and cable diameter, as per Table 2, even if the column length is fixed. Table 2 Crossarm length and cable diameter Crossarm length Cable diameter
a (mm)
150
225
300
375
450
s ( mm)
1.6
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3.2 Pretension in cables
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C11 c P Zone3,T 0 C22
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2cos 2 1 2sin 2 Kc ( ) Ks Ka
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(3)
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where is the angle between the main column and cable, which is substantially determined by the crossarm length; and K c , Ks , and K a are the stiffness of the main column, cable, and
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Note that pretension Topt is generally adopted as the benchmark magnitude in the numerical
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analysis in this work. 4. Buckling analysis 4.1 Critical buckling analysis Figure 3 presents the buckling modes of the cable-stiffened steel columns with pin-connected crossarm systems. To compare the effect of the joint type on the critical buckling modes, the buckling modes of a stiffened column with both pin-connected and rigid-connected crossarm systems are presented in Figure 3 and Figure 4. Overall, the buckling modes of the stiffened column are symmetric (see Figures 3 (a) and 4(a)) or anti-symmetric (see Figures 3 (b) and 4(b)) with both pin and rigidly-connected crossarms. However, it should be noted that the crossarms remain straight when the connection is pinned, implying that rotational stiffness is not provided to the main column by the crossarms. This is a notable characteristic to distinguish pin-connected stiffened columns from those with rigid connections. Consequently, it is essential to conduct a nonlinear buckling analysis to examine the interactive buckling behaviour of the stiffened columns with pin-connected crossarms, even if stiffened columns with rigidly-connected crossarms have been investigated previously. For simplification, in the following analysis, the symmetric and antisymmetric buckling modes are respectively referred to Mode 1 and Mode 2.
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(a) Symmetric buckling
(b) Anti-symmetric buckling
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Figure 3 Buckling modes of stiffened columns with pin-connected crossarms (Front view)
(a) Symmetric buckling
(b) Anti-symmetric buckling
Figure 4 Buckling modes of stiffened columns with rigidly-connected crossarms (Front view)
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Figure 5 presents the buckling loads of cable-stiffened steel columns with different cable diameters and crossarm lengths. Obviously, the buckling mode varies from Mode 1 to Mode 2 as the cable diameter and crossarm length increase. It must be noted that the effects of the cable diameter and crossarm length on the buckling loads are significantly different depending on the buckling mode. When the buckling mode is Mode 1, increasing the cable diameter or crossarm length within a moderate range commonly enhances the buckling load, but it decreases when the crossarm is extremely long. However, the effect of the cable diameter and crossarm length on the buckling load corresponding to Mode 2 is limited or adverse (see Figure 5(b)). This can be explained from the buckling deformation of Mode 2, as shown in Figure 3(b). For Mode 2 buckling, only the axial stiffness of the crossarm can be activated to stiffen the main column because it remains straight during the deformation. Therefore, it is inadvisable to achieve a high buckling load by increasing the crossarm length when Mode 2 occurs because it will decrease the crossarm’s axial stiffness.
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Figure 5 Buckling loads with different cable diameters and crossarm lengths
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4.2 Post-buckling analysis 4.2.1 Imperfection construction For an ordinary steel column without cable-stiffened systems, the geometric imperfection is affined to the critical buckling mode in the nonlinear buckling analysis. However, it has been shown that the load carrying capacity of a cable-stiffened steel column with a geometric imperfection distribution obtained following only the critical buckling mode would be overestimated because interactive buckling cannot be considered by this method, though it could determine the actual buckling strength in some cases [9]. To consider interactive buckling, a complex imperfection distribution has been suggested for cable-stiffened columns with rigidly-connected crossarms [9, 10]. However, the above critical buckling analysis shows that the anti-symmetric buckling mode of cable-stiffened columns with pin-connected crossarms is substantially different from that with rigidly-connected crossarms. Thus, applicability of the method of inputting the imperfection distribution for stiffened columns with rigidly-connected crossarms can be simply to the pin-connected case needs to be examined because the connection stiffness would certainly affect the buckling mode of the stiffened columns. Based on this consideration, in this work, a new imperfection distribution is constructed that includes the components of both the symmetric and antisymmetric buckling modes to activate interactive buckling. The constructed imperfection can be described by equation (4): W ( x) 1 sin
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1 and 2 denote the weights of the symmetric and antisymmetric components, respectively, in the constructed imperfection distribution. To determine the magnitudes of 1 and 2 , it is assumed that the axially compressive of the main column owing to the different imperfection distributions, are the same to leading order, and thus, equation (5) is derived to be
12 +422 =1
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4.2.2 Interactive buckling Having constructed the geometric imperfection distribution, in this section we examine via nonlinear buckling analysis a possible domination of interactive buckling on the load carrying capacity of the stiffened column. It should be mentioned that an imperfection magnitude of L/300 (L is the main column length) is generally adopted in this study. For the nonlinear buckling analysis, it is comprehensible to consider the geometrical nonlinearity because the column is slim. In addition to the geometrical nonlinearity, material nonlinearity has also been considered in this work to investigate the post-buckling behaviour. Though it has been demonstrated that elastic analysis is sufficient to capture the buckling strength of a cable-stiffened column, the material plasticity can also be observed beyond the instant the buckling occurs. Based on this, both the geometrical and material nonlinearities were taken into account in the buckling analysis. However, it must be pointed out that the cables are completely considered as elastic even though the plastic behaviour of steel is taken into account. Figures 7 and 8 illustrate the axial loads and compressions of the cable-stiffened steel columns with different crossarm lengths and cable diameters. When the combination of crossarm length and cable diameter size is 150 mm and 4.8 mm or 300 mm and 1.6 mm, respectively, the actual load carrying capacity of the stiffened column is determined by the imperfection following the symmetric buckling mode (Mode 1). Otherwise, it is determined by the interactive buckling which is triggered by the asymmetric imperfection. It is interesting to note that irrespective of the crossarm length and cable diameter being 150 mm and 4.8 mm or 300 mm and 1.6 mm, respectively, the corresponding critical buckling mode is symmetric (see Figure 5). Thus, the principle to input the geometric imperfection shape can be correlated to the critical buckling modes, i.e. if the critical buckling mode is symmetric, the imperfection shape should just follow the critical buckling mode in the nonlinear buckling analysis. Conversely, if the critical buckling mode is antisymmetric, the imperfection shape is suggested to be asymmetric to consider the interactive buckling behaviour. This conclusion, which has also been obtained in cable-stiffened column with rigidly-connected crossarms [9], is the principle to input geometric imperfection in the following parametric studies.
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5. Parametric studies Having determined that the principle of introducing the geometric imperfection to obtain the actual buckling strength by conducting nonlinear buckling analysis. In this section, the effect of different parameters (crossarm length, cable diameter, pretension in the cables, and imperfection size) on the buckling strength of the cable-stiffened steel column would be discussed. 5.1 Effect of the geometry configuration For the cable-stiffened steel columns, the buckling strength of the main column is enhanced by pretensioned cables and crossarms. Thus, the crossarm length and cable diameter would certainly affect the buckling strength of the column. Figure 9 illustrates the load carrying capacities of cable-stiffened steel columns with different cable diameters and crossarm lengths. Overall, the
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Figure 9 Load carrying capacities with different cable diameter and crossarm length
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Table 4 presents the critical buckling modes of the cable-stiffened steel columns with different parameters. It can be seen from Figure 9 and Table 4 that the effect of the crossarm length and cable diameter is governed by the critical buckling mode. When the critical buckling mode is Mode 1, the buckling strength of the stiffened column is drastically increased by the crossarm length or cable diameter; however, when it is transferred to Mode 2, their effect on the buckling strength is limited or even adverse. From this perspective, the material utilisation rate is the highest when the critical buckling mode is antisymmetric and close to the transition between the symmetric and antisymmetric modes. Thus, the optimal structural configuration for the cable-stiffened steel column corresponds to a critical buckling mode located at the transition from Mode 1 to Mode 2. Cable diameter (mm)
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Table 4 Summary of the critical buckling modes
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5.2 Effect of pretension in cables The pretension in cables is another important factor that affects the buckling behaviour of cable-stiffened steel columns. So far, the pretension in the cables was fixed to Topt derived from a linear buckling analysis. However, it is unclear whether pretension Topt corresponds to the maximum buckling strength. Thus, in this section, we investigate the effect of pretension in the cables on the buckling strength. Figure 10 illustrates the load carrying capacities of the cable-stiffened steel columns with pretensions in cables vary from Topt to 5Topt . When the
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5.3 Effect of imperfection size In the above analysis, L/300 (L is the main column length) is adopted as the imperfection size. However, it has been shown in previous studies that the buckling strength of slim columns is significantly affected by the imperfection size [18-19]. In this section, we quantitatively examine the effect of the imperfection size on the load carrying capacities of cable-stiffened steel columns with four pin-connected crossarms. Five different imperfection sizes including L/5000, L/1000, L/500, L/300, and L/100 are selected. In fact, L/5000 is quite small as the imperfection size; thus, the corresponding load carrying capacity is used to represent the actual capacity of a perfect stiffened column. Figure 11 presents the load carrying capacities of the cable-stiffened steel columns with different imperfection magnitudes. Generally, the capacities show an obvious decrease with increasing imperfection size.
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Figure 11 Load carrying capacities with different imperfection sizes
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Figure 12 shows the imperfection sensitivity coefficient of the stiffened steel columns with different cable diameters and crossarm lengths. Obviously, the maximum coefficient is approximately 0.4, implying that the buckling strength is decreased by 40% in this case. It can also be observed that the maximum coefficient corresponds to the structural configurations of a =
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6. Critical buckling loads of stiffened columns different crossarm connections The buckling behaviour of cable-stiffened steel column with pin-connected crossarms has been
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numerically investigated in above sections, the method to determine the initial geometrical imperfections in nonlinear buckling analysis has also been proposed. However, the buckling loads of stiffened columns with pin-connected and rigidly-connected crossarm systems have not been comparatively analysed. This section aims to compare the critical buckling loads of stiffened steel columns with the above two crossarm connections. The critical buckling loads and buckling modes of the cable-stiffened steel columns are presented in Figure 13. As it can be seen, when the critical buckling modes of stiffened column with pin-connected crossarms is symmetric, the critical buckling loads correspond to pin and rigid connected crossarm structural schemes are the same. In contrast, when the critical buckling modes of stiffened column with pin-connected crossarms is anti-symmetric, the critical buckling loads corresponds to pin connected crossarm structural schemes are much lower than those of rigidly-connected cases.
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80
c
80
120
Anti-symmetric buckling
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Symmetric buckling
P max (kN)
Symmetric buckling
40
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c
P max(kN)
Symmetric buckling
120
Anti-symmetric buckling
40
Symmetric buckling
0 2
4
6
8
s (mm)
(a) a = 300 mm
10
0 100
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0
200
300
400
500
a (mm)
(b) s =4.8mm
Figure 13 Critical buckling loads of stiffened columns with different crossarm connections
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7. Conclusions To the best knowledge of the authors, this is the first study on the interactive buckling behaviour of cable-stiffened steel columns with four pin-connected crossarms. The following conclusions can be drawn from the numerical analysis in this study: (1) The typical buckling modes of the cable-stiffened steel columns with four pin-connected crossarms can be classified as symmetric and antisymmetric, similar to columns with rigidly-connected crossarms. However, their antisymmetric buckling modes are substantially different. For the stiffened column with pin-connected crossarms, the rotational stiffness of the crossarms cannot be activated during buckling; however, both the axial and rotational stiffness of the crossarms can be activated when they are rigid-connected to the main column. (2) It has been demonstrated that adopting distinct bucking modes in the geometric imperfection distribution in the nonlinear buckling analysis could overestimate the buckling strength in some cases, and thus, interactive buckling must be taken into account for accurate analysis. When the critical buckling mode is symmetric, it is sufficient to adopt the geometric imperfection shape to which it is affine. In contrast, the combination of symmetric and anti-symmetric buckling modes must be adopted to construct a new imperfection shape to consider the interactive buckling behaviour when the critical buckling mode is antisymmetric. (3) The parametric study shows that the buckling strength of the cable-stiffened steel columns is significantly affected by the crossarm length or cable diameter; however, these effects are significantly governed by the critical buckling modes. For the case when the critical buckling mode is symmetric, the buckling strength can be strongly enhanced by increasing the crossarm
ACCEPTED MANUSCRIPT length or cable diameter. However, this effect on the buckling strength is limited or adverse if the critical buckling mode is antisymmetric. For the optimal design of cable-stiffened steel columns, critical buckling mode should be Mode 2 and close to the transition between Mode 1 to Mode 2. This is because this structural configuration corresponds to a maximum material utilisation rate. (4) The nonlinear buckling analysis demonstrates that the maximum load carrying capacity of cable-stiffened steel columns cannot be achieved by introducing the initial pretension of Topt ,
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though this pretension value corresponds to the maximum critical buckling load. A higher pretension level, depending on the critical buckling modes and structural stiffness, is suggested by the authors if the designers aim to obtain the maximum buckling strength. (5) The imperfection sensitivity exhibits that the buckling strength of a cable-stiffened steel column can be drastically decreased by the imperfection size, implying that this steel column is imperfection sensitive. It has also been illustrated that the imperfection sensitivity coefficient, which quantifies the reduction of the imperfection, reaches a maximum value when the critical buckling mode is located at the transition between the symmetric and antisymmetric buckling modes.
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Acknowledgement The research was supported by the National Natural Science Foundation of China (No. 51808070), the Fundamental and Frontier Research Project of Chongqing (No. cstc2018jcyjA2698), the Fundamental Research Funds for the Central Universities (No. 106112017CDJXY200008 and No. 106112016CDJRC000088). These financial support are gratefully acknowledged. References
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