Torsional and flexural buckling of composite FRP columns with cruciform sections considering local instabilities

Transcription

1 Torsional and flexural buckling of composite FRP columns with cruciform sections considering local instabilities H.R. Naderian a, H.R. Ronagh b,, M. Azhari c a Department of Civil Engineering, Yazd University, Yazd, Iran b School of Civil Engineering, University of Queensland, Brisbane Qld 4072, Australia c Department of Civil Engineering, Isfahan University of Technology, Isfahan, Iran article info abstract Article history: Available online 1 May 2011 Keywords: Torsional buckling Flexural buckling Semi-analytical finite strip method Thin-walled Cruciform sections FRP composite columns In this paper, a semi-analytical finite strip method is developed for the prediction of torsional and flexural buckling stresses of composite FRP columns under pure compression. Numerical finite strip results will be compared with those obtained from closed-form equations for doubly symmetric open thin-walled FRP sections. The accuracy of the proposed finite strip method in determining critical flexural and torsional stresses of FRP columns will be assessed. Among the composite FRP columns with doubly symmetric open sections, buckling behavior of stiffened and unstiffened FRP cruciform sections will be evaluated and case studies performed. The effect of material properties and longitudinal stiffeners applied at the end of the web-plate and flange-plate on buckling modes of composite FRP cruciform sections is also reviewed. 1. Introduction Among the thin-walled sections made from orthotropic materials, FRP profiles are known to many engineers and researchers and are more popular. These members have wide applications in different industries, including aero structures, civil engineering, mechanical engineering, and the construction industry. There exists a growing demand for composite FRP structural members, which is owing to a combination of their structural efficiency, low fabrication costs and excellent behavior under aggressive environmental conditions. A large percentage of research in the field of orthotropic plate-structures is devoted to FRP sections. The mechanical properties of FRP sections clearly indicate that their behavior is strongly affected by instability phenomena. In fact, composite FRP members can be classified as thin-walled structures. Hence, it is necessary to investigate the buckling behavior of FRP sections in different boundary, dimension and loading conditions in order to achieve more accurate recognition of these structural members as well as optimum designs. In recent years, special attention has been paid to buckling of FRP sections. There exists a considerable amount of research work, both analytical and experimental, concerning the local and global buckling of thin-walled composite FRP members. Among these studies, research carried out by Pecce and Cosenza [1] on local Corresponding author. Tel.: ; fax: address: h.ronagh@uq.edu.au (H.R. Ronagh). buckling of FRP sections should be mentioned. Additionally, Qiao and colleagues have presented research about local buckling of FRP sections in different boundary and loading conditions [2 4]. They also succeeded in developing analytical solutions in the field of flexural torsional buckling of composite FRP members made of I and U sections [5,6]. Silvestre and Camotim extended GBT method to analyze the local-plate, distortional and mixed flexuraldistortional buckling modes of FRP-lipped channel members [7]. Among the various sections made from FRP, open sections have more applications. On the other hand, torsional buckling is usually critical for open thin-walled section columns due to their low torsion rigidity. Torsional buckling, generally, occurs under compressive loads, where the section of column twists on the rigid torsion. In fact, if the amount of torsion is too large, torsional instability will dominate. Displacements derived from torsional buckling are in-plane, and for the same reason, this buckling mode is regarded as a global mode. In this study, a semi-analytical finite strip method will be developed to analyze the torsional and flexural buckling of composite FRP columns. In conjunction with FSM, closed-form equations are used to study the torsional and flexural buckling behavior of composite FRP sections under pure compression stresses. The finite strip method can be considered as a special form of finite element procedure using the displacement approach. Unlike the standard finite element method which uses polynomial displacement functions in all directions, the finite strip method calls for use of simple polynomials in some directions and continuously differentiable smooth series in the other directions, with the stipulation that such

2 series should satisfy a priori the boundary conditions at the end of strips. The philosophy of the finite strip method is similar to that of the Kantorovich method [8], which is used extensively for reducing a partial differential equation to an ordinary differential equation. In this method, especially for plates, Hermitian displacement functions are usually suitable for transverse direction [9]. Undoubtedly, the advantage of the finite strip method in comparison with other numerical methods is obvious in evaluating the buckling of plates. On the one hand, the simplicity of inputs and calculations, and on the other hand, reducing the problem to a standard eigenvalue problem (besides ease of programming) are factors which make the finite strip method superior. The first use of the finite strip method for buckling appears to be the work of Prezemieniecki [10], who showed how this method can be used to predict the initial elastic local buckling of plates and sections made of plates under biaxial compression. His approach utilized the approximate finite strip formulation of Cheung and Cheung [11]. Plank and Wittrick [12] employed the semi-analytical complex finite strip method to investigate buckling under combined loading of thin-walled structures. The advantage of their method over the formulations of the ordinary finite strip method is the ease with which it can handle shear forces. Wittrick [13] developed an exact finite strip method for buckling analysis of stiffened panels in compression. Azhari and Bradford [14] developed the bubble finite strip method, which augmented finite element formulations to obtain rapid convergence. They also extended the finite strip method to analyze the buckling of plates with different end conditions [15]. Adany and Schafer [16 18] derived a constrained finite strip method for decomposing the buckling modes of thin-walled open cross-section members. Azhari and Amoushahi [19] used the complex finite strip method for analyzing the buckling of composite FRP structural plates. In the present study, the finite strip method is extended in order to study the instability of the pultruded FRP section as an orthotropic material. For this purpose, a semi-analytical finite strip formulation is developed, validated and applied to solve the eigenvalue problem associated with the torsional and flexural buckling of doubly symmetric open thin-walled section FRP columns. The present explicit finite strip formulation can be applied effectively to determine the torsional and flexural buckling capacities of simply-supported composite FRP columns under axial compression. It will be shown that the semi-analytical predictions for torsional and flexural buckling of various FRP columns based on the present analysis are in excellent agreement with closed-form results. Numerical studies of unstiffened and stiffened FRP cruciform sections (which are among the weakest open sections in torsional buckling) will be carried out. Flexural buckling of these sections, in addition to torsional buckling, will be evaluated for prediction of dominant global buckling mode in different column lengths. It can be said about FRP cruciform sections under pure compression that torsional buckling usually occurs before flexural buckling. In addition, the results of an investigation concerning the influence of longitudinal stiffeners, section geometry, and material properties on the member buckling behavior are presented, which required the completion of a number of parametric studies. In these investigations, careful consideration is given to the occurrence and characterization of torsional and flexural buckling modes. methods is the ability to handle in-plane (membrane) as well as out-of-plane (flexural) displacements. On the other hand, displacements caused by torsional buckling and flexural buckling are of the in-plane type and occur in the long wavelengths. Therefore, the semi-analytical finite strip method can be a suitable tool for the prediction of torsional and flexural buckling of columns which are under compression loading. However, this method is only applicable for structures whose ends are simply supported. More complicated boundary conditions may be treated in the finite strip method developed by Bradford and Azhari [15]. Here, the purpose is studying the global buckling of FRP composite columns under pure compression, so the equations will be extracted only for compression loading and membrane (in-plane) displacements Definition of the problem In the finite strip method (FSM) a thin-walled member, such as the composite FRP cruciformsection of Fig. 1 is divided into longitudinal strips. The advantage of FSM over other methods, such as the finite element method which applies discreteness in both the longitudinal and transverse direction, is dependent on a judicious choice of the shape function for the longitudinal displacement field. In Fig. 1, a single strip is highlighted. The geometry, loading and degrees of freedom (DOF) for the strip are illustrated in Figs. 2 and 3. As it is shown in Fig. 2, the strip is loaded by a uniform longitudinal compressive stress r L. In this procedure, the elastic membrane stiffness and stability matrices of a composite FRP strip are obtained through standard finite element techniques based on the energy method Membrane stiffness and stability matrices In this derivation, the in-plane, or membrane, displacements of a strip u and v are assumed to be given by u ¼hf 1 ðgþifd I gcos npx ð1þ L and v ¼hf 2 ðgþifd I gsin npx L where {d I } is the nodal in-plane displacement in local coordinates shown in Fig. 3 and may be explicitly written as d I ¼½u 1 ;v 1 ; u 2 ; v 2 Š and f 1 (g) and f 2 (g) are linear shape functions employed in the transverse direction in the semi-analytical treatment given by 1 hf 1 ðgþi ¼ 0; 2ð1 gþ ; 0; 1 ð4þ 2ð1 þ gþ ð2þ ð3þ 2. Semi-analytical finite strip method for torsional and flexural buckling 2.1. General One of the most important advantages of the semi-analytical finite strip method in comparison with other kinds of numerical Fig. 1. Finite strip discreteness.

3 where V is the volume of the strip. Hence, substituting Eqs. (6) and (8) into Eq. (10) yields the stiffness matrix from U ¼ 1 2 fd Ig T ½k m Šfd I g ð11þ Fig. 2. Basic state of stresses in a strip. in which [k m ] is the membrane stiffness matrix and may be obtained from Z ½k m Š¼ ½B m Š T ½D Composite Š½B m ŠdV ð12þ V Substitution and integration leads to a 4 4 symmetrical matrix [k m ]. The strip is subjected to the in-plane stresses r L shown in Fig. 2. The decrease in potential energy of these stresses V p during buckling is given by V P ¼ 1 Z fe N g T frgdv ð13þ 2 V where the strain matrix {e N } can be defined by the following nonlinear expression [15]: fe N g¼ 1 2 u2 ;x þ 1 v 2 ;x 2 þ 1 2 w2 ;x ; 1 T 2 w2 ;x ; w ;xw ;y ð14þ Hence, by appropriate substitution, the membrane stability matrix [g m ] may be obtained from Fig. 3. Strip DOFs. V P ¼ 1 2 fd Ig T ½g m Šfd I g ð15þ and hf 2 ðgþi ¼ 1 2ð1 gþ ; 0; 1 2ð1 þ gþ ; 0 in which g =2y/b and b is the width of the strip. Furthermore, cos npx L and sin npx L are shape functions in the longitudinal direction in which L is the buckling half-wavelength. The membrane strains {e m } at the mid-line of the strip are governed by plane stress assumptions and may be given by fe m g ¼hu ;x ; v ;y ; u ;y þ v ;x i T ð6þ The membrane strains {e m } may be written in terms of appropriate derivatives of the shape functions and the nodal displacements. On differentiating Eqs. (1) and (2), the membrane strains may be written fe m g¼½b m Šfd I g where [B m ] may be named as the strain matrix. The in-plane rigidity matrix of the composite plate D composite [19] defines the relationship between the buckling internal stress r and strains for membrane displacements by frg ¼½D Composite Šfe m g where 8 >< ½D Composite Š¼ >: E x ð1 v xyv yxþ v xye y ð1 v xyv yxþ v xye y ð1 v xyv yxþ 0 E y ð1 v xyv yxþ 0 9 >= >; 0 0 G in which, E x and v xy are the Young s modulus and Poisson s ratio in the longitudinal direction respectively, while E y and v yx are the Young s modulus and Poisson s ratio in the transversal direction respectively, and G is the shear modulus. The total strain energy U stored during buckling may be written as U ¼ 1 Z fe m g T frgdv 2 V ð5þ ð7þ ð8þ ð9þ ð10þ in which 2 Z *! ½g m Š¼r 2 2 sin npx L L 4 f V ðnpþ 2 1 ðgþf 1 ðgþ T +# npx sin L þ f 2 ðgþ T f 2 ðgþ ð16þ Once the strip stiffness and stability matrices [k m ] and [g m ], respectively, have been derived, and combined for each composite strip, they can be assembled into the respective global matrices [K m ] and [G m ] using standard procedures. The buckling problem can be solved by eigenvalue equations ð½k m Š ½G m ŠÞD ¼ 0 ð17þ where D is a scaling factor related to the critical load. It is worth noting that a similar method to that for the membrane matrices may be followed in order to obtain the flexural stiffness and stability matrices. 3. Closed form equations The critical flexural stress, or Euler critical stress, for an orthotropic member is given by p2 E x F cr ¼ ðkl=r min Þ 2 ð18þ where K is the effective length factor calculated with regard to the end conditions of column, while L is the un-braced length of column and r min is the radius of gyration of the section about the weak axis. The equation related to torsional critical stress for orthotropic columns with doubly symmetric open section can be formulated F cr ¼ GJ I P þ p2 E x C W ðklþ 2 I P ð19þ in which I p and C w are the polar moment of inertia and warping factor of the cross-section respectively and J is the equivalent

4 torsion rigidity of the section. It is noticeable that Eq. (19) is applicable when the critical stress F cr is lower than yield stress. In other words, Eq. (19) is limited to elastic buckling. Eq. (19) can be written in the form used for Euler critical stress p2 E x F cr ¼ ðkl=r t Þ 2 ð20þ in which r t is the equivalent radius of gyration for torsional buckling, and which for an orthotropic section is obtained by comparing Eqs. (19) and (20) r 2 t ¼ C w þ 0:1ðG=E x ÞJðKLÞ 2 I P ð21þ In fact, the only difference between Eqs. (18) and (20) is their radius of gyration. 4. Evaluation of the torsional and flexural buckling 4.1. General In order to illustrate the application of the semi-analytical finite strip method and, at the same time, provide a better grasp of the concepts and procedures just presented, the buckling behavior of unstiffened and stiffened cruciform section FRP columns is investigated next. As previously mentioned, one of the main objectives of this paper is evaluating the dominant global mode in buckling of composite FRP columns with doubly-symmetric open sections. For this purpose, the semi-analytical finite strip method developed in the previous section and the closed-form of analysis have been programmed on a desktop workstation and critical torsional and flexural stresses have been determined. Herein, the finite strip program can handle all buckling types of orthotropic sections, including local, distortional, flexural, and torsional modes. Comparisons between finite strip method and closed-form solutions are made. In addition, it is intended to assess how the different buckling modes of the FRP columns and corresponding bifurcation stress values are influenced by several aspects, including the crosssection geometry and the composite material properties Composite FRP cruciform section columns Firstly, ten FRP columns with cruciform section under compression stresses are studied. The cross-section and the state of stress are shown in Figs. 4 and 5. The geometric data of the sections are summarized in Table 1, where b w and b f are web depth and flange width respectively, and t w and t f are the thicknesses of the web and the flange respectively. Critical torsional and flexural stresses F cr obtained from Fig. 4. FRP cruciform section. Fig. 5. State of compression stresses. Table 1 Geometric data of the cruciform FRP sections. Case number b f (cm) b w (cm) t f (cm) t w (cm) closed-form solutions, as well as critical stresses of finite strip method for three column-length L including 300, 500 and 1000 cm are derived and shown in Table 2. All the stresses are normalized with E x. Although both Eqs. (19) and (20) are related to torsional stresses, Eq. (20) is rather approximate. For the same reason, it is useful to compare the above equations to evaluate the accuracy of Eq. (20). The material properties adopted for each FRP column are E x = 21 GPa, E y = 8 GPa, G = 2.5 GPa, v xy = 0.3. Moreover, Poisson s ratio v yx in the transversal direction is given by v yx ¼ðE y =E x Þv xy ð22þ All of the solutions are of simply-supported columns subjected to uniform compression stresses. This means that the effective length factor K is equal to one. The simply-supported conditions applied here are in accordance with the basic assumption of the semianalytical finite strip method which increases the accuracy of results. Furthermore, with regard to the problem that torsional buckling occurs in columns under pure compression, this assumption is completely true. As can clearly be seen in Table 2, the dominant global buckling mode in the sections made of thinner plates is torsional, even for long lengths such as a column with length of 1000 cm. On the other hand, the dominant global buckling mode in the sections made of thicker plates is flexural, even for short lengths like a column with length of 300 cm. It is obvious that in all cases where torsional buckling occurs, the amount of equivalent radius of gyration r t is smaller than r min. The other result from Table 2 is that by increasing the column length L the possibility of torsional buckling

5 Table 2 Torsional and Flexural buckling stresses of FRP cruciform section columns in pure compression. Case number L (cm) r min r t F cr /(E x 10 3 ) (FSM) F cr /(E x 10 3 )(r min ) Eq. (18) F cr /(E x 10 3 )(r t ) Eq. (20) F cr /(E x 10 3 ) Eq. (19) Buckling type Error % Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Flexural Torsional Torsional Flexural Torsional Flexural Flexural Flexural Flexural Flexural Flexural Flexural Flexural Flexural Flexural Flexural Flexural Flexural Flexural 0.15 decreases and the probability of flexural buckling increases. In contrast, by decreasing the column length L the possibility of the torsional buckling increases and the probability of the flexural buckling decreases. In order to evaluate the accuracy and validity of the finite strip method for torsional and flexural buckling of FRP sections, the results obtained from this method are compared in Table 2 with those obtained from closed-form equations. In the last column of Table 2, the percentage of error is shown. This is the difference between critical stresses of Eq. (18) for flexural mode or Eq. (19) for torsional mode and finite strip method to critical stresses of Eq. (18) or Eq. (19) Error ¼ F cr FSM F creq: ð18þ or ð19þ F creq: ð18þ or ð19þ 100 ð23þ As the results show, the critical stresses obtained from the finite strip method are in excellent agreement with those obtained from closed-form solutions. Generally, as the columns lengthen and the sections become thinner, the amount of error indicates reductions ranging from 0.24% to 0% in the case of torsional buckling mode, while as the columns lengthen the amount of error indicates reductions ranging from 2.2% to 0.15% in the case of flexural buckling mode. In almost all the cases in which torsional buckling dominates, the percentage of error is positive which means that the critical stresses of finite strip method are greater than closed-form Eq. (19). On the other hand, in flexural buckling, in all of the columns, the percentage of error is negative which means that critical stresses of the finite strip method are smaller than closed-form Eq. (18). In other words, it can be concluded that the finite strip method is conservative in prediction of critical torsional stresses, while it is not conservative in prediction of critical flexural stresses. The non conservative nature of the critical flexural stresses obtained in FSM is due to the use of plate plane stress state for each wall of the cross-section, while Eq. (18) is based on Euler beam theory which considers a uni-axial stress state, disregarding the Poisson effects. It was predictable that critical stresses of Eqs. (19) and (20) are very close to each other; however, the results of Eq. (19) are naturally somewhat greater than those of Eq. (20). It is noteworthy that in all cases in which torsional buckling dominates, critical stresses of the finite strip method are closer to Eq. (19) and more accurate than Eq. (20). In order to better understand the effect of geometric properties on buckling of cruciform section FRP columns under pure compression, in Fig. 6 the normalized buckling stresses F/E x are given as a function of the buckling half-wavelength L. As shown in Fig. 6, all the buckling curves exhibit two main distinct zones corresponding to torsional buckling (straight part of the curves) and flexural buckling (bent part of the curves). Furthermore, no minimum local stress can be seen in short half-wavelengths which mean that pure local buckling does not occur, however, mixed local-torsional mode might occur for very short columns. This is discovered by authors [20] using the deformed configuration of sections in buckling via the CUFSM software [21]. CUFSM is an open source software which was originally written by B. W. Schafer and colleagues [22] in order to explore elastic buckling behavior and to calculate the buckling stresses and buckling modes of arbitrarily shaped, simply supported, thin-walled members. The software employs the finite strip method to provide solutions for the cross-sectional stability of isotropic as well as orthotropic thinwalled members. It is worthy of mention that in a recent study carried out by Dinis and colleagues [23] on buckling modes of steel cruciform section columns using Generalized Beam Theory (GBT)

6 Fig. 6. Buckling stresses in pure compression for different flange widths. Torsional Buckling Torsional Buckling Flexural Buckling Fig. 7. Buckling mode shapes of FRP section in different lengths. method, the fact that local-torsional mode might occur for very short columns with cruciform section was also confirmed. In Fig. 7, deformed configuration (buckling mode shapes) of the three FRP cruciform section columns with the same geometry as well as material properties and different lengths L are shown. The material properties and section dimensions are E x = 21 GPa, E y = 8 GPa, G = 2.5 GPa, v xy = 0.3, b f = 30.5 cm, b w = 30.5 cm, t w = 0.95 cm, t f = 0.95 cm. It is obvious that the buckling mode in FRP columns with lengths of 300 cm and 500 cm is torsional and for the length of 1000 cm, it is flexural. It is noticeable that torsional and flexural buckling modes which occurred here are absolutely uncoupled. In other words, they are not coupled flexural torsional buckling modes, but are pure modes. In Fig. 6, b w /t w = 32 and t f /t w = 1 which means b w = 30.5 cm, t w = 0.95 cm, t f = 0.95 cm and buckling curves are given for different ratios of b f /b w. In fact, the web slenderness factor b w /t w is constant but the flange slenderness factor b f /t f is varied. It can be seen from Fig. 6 that when the ratio of flange width to web depth b f /b w or flange slenderness factor b f /t f increases, the length L zone related to torsional buckling increases, but that the length L zone related to flexural buckling decreases. Furthermore, for high ratios of b f /b w torsional stresses decrease and for all cases, flexural stresses increase; however, there is an exception for b f /b w = 1 as it has the maximum flexural stresses. With regard to designing of structures, it can be shown that the optimum flexural stability of sections under pure compression is where the slenderness factor KL/r is equal for both main x and y coordinate axes. Here the maximum flexural buckling strength has also occurred when the width and thickness of web and flange are the same as each other (b f /b w = 1) in which slenderness factor KL/r for both main x and y coordinate axes are identical. These kinds of sections are usually named the optimum sections. Here, the finite strip method results for buckling of simplysupported steel cruciform section columns are compared with those obtained through Generalized Beam Theory analysis carried out by Dinis and colleagues [23]. The material properties and section dimensions are E x = E y = 210 Gpa, v xy = v yx = 0.3, G = GPa, and b f = 14 cm, b w = 14 cm, t w = 0.24 cm, t f = 0.24 cm. Fig. 8 shows GBT and semi-analytical FSM results concerning the buckling behavior of simply-supported cruciform section steel columns. The curves displayed provide the variation of column buckling load R b with its length L in logarithmic scale. It can clearly be seen that there is a virtual coincidence between the buckling loads and wave-lengths obtained through semi-analytical FSM and GBT analyses. As stated, according to the GBT method the mixed localtorsional mode might occur for very short steel cruciform section columns [23] Composite FRP stiffened cruciform section columns In recent decades, the use of columns with stiffened cruciform section as shown in Fig. 9 has been standardized in construction

7 Fig. 8. Buckling behavior of simply supported steel cruciform section columns. (a) FSM results; (b) GBT results [23]. Fig. 9. FRP stiffened cruciform section. of tall steel buildings. Fortunately, it is possible to make this kind of section using composite FRP plates. With regard to the geometric shape of these sections, they can also be named as double-i sections. The ease of connecting beams to stiffened cruciform sections in two directions is their practical advantage. As far as the authors are aware, there are not enough basic studies on the buckling strength and behavior of stiffened cruciform sections. Therefore, it is necessary to evaluate their stability more deeply. With regard to the complexity of the geometric shape and material properties, different parameters influence the buckling behavior of FRP stiffened cruciform sections. For the same reason, firstly, using the finite strip method, the bucking modes of these composite sections under uniform compression stresses and the effect of variation of section geometry as well as material properties are evaluated. In the next step, it will be shown that in many cases the torsional buckling dominates in FRP stiffened cruciform sections. In other words, the equivalent radius of gyration r t is smaller than r min and the slenderness factor used in design affairs will be KL/r t instead of KL/r min Buckling behavior under uniform compression stresses The semi-analytical finite strip method has been applied to study the buckling modes of FRP stiffened cruciform section columns under uniform compression stresses. The state of stress is shown in Fig. 10. The material properties adopted for each FRP column are E x = 21 Gpa, E y = 8 Gpa, G = 2.5 GPa, v xy = 0.3. In Fig. 11, the normalized buckling stresses F/E x of FRP stiffened cruciform section columns with b wv = b wh, t wv = t wh = t fw = t fh, b wv / t wv = 32, b fv = b fh = alpha b wv, are given as a function of the dimensionless buckling half-wavelength L/b wv. For all curves, the plate thickness is kept constant. Several curves are given for different ratios of stiffener width (b fv = b fh ) to vertical web depth (b wv ) Fig. 10. State of compression stresses. alpha. Figs. 12 and 13 show the normalized buckling stresses of FRP stiffened cruciform section columns with b fv /t fv = 10.75, b fv = b fh, t wv = t wh = t fw = t fh,infig. 12, b wv = b wh and several curves are given for different ratios of web depth (b wv = b wh ) to stiffener width (b fv = b fh ) beta, while in Fig. 13, b wv /t wv = 32, and b wh is varied with b wv. All half-wavelengths shown in the figures in this section are in logarithmic scale. The curves for all columns exhibit three main distinct zones, the first of which has a minimum value at L/b wv between 0.5 and 1.5. In this region, the buckling mode is local and it may occur in the flange plate, web plate or stiffeners. As the wavelength increases, the curves rise to peak, and buckling mode becomes a mixed local and flexural torsional type, called a coupled buckling mode [24]. Beyond the peak, the buckling stresses decrease with increasing half-wavelength L until the mode is predominantly a global buckling mode that can be named as flexural or torsional. It can be seen from Fig. 11 that when the ratio of stiffener width (b fv = b fh ) to vertical web depth (b wv ) alpha increases, the local buckling stress decreases and the related half-wavelength increases, but that flexural and torsional stresses increase. In this case, the geometric section properties related to flexural and torsional buckling resistances, such as the equivalent torsion rigidity J and warping factor c w increases, but the resistance of the section to local buckling decreases. In fact, with the increase in the width of stiffeners they become thinner and consequently the stiffness of the conjunction of the flange-plate or web-plate to stiffeners and

8 Fig. 11. Buckling curves for different ratios of alpha. Fig. 12. Buckling curves for different ratios of beta. Fig. 13. Buckling curves for different horizontal web depths. the local buckling strength of plates decrease. It is obvious that the strength of FRP stiffened cruciform sections in all buckling modes is greater than that of unstiffened ones. It is noticeable that in stiffened cruciform sections, boundary conditions of the half-web plate and half-flange plate are between those of simply supported and clamped because of adopting stiffeners at the end of the plates, while in unstiffened cruciform sections, boundary conditions of the half-web plate and half-flange plate are between simple-free and clamped-free supports. Fig. 12 shows that by increasing the ratio of web depth (or flange width) to stiffener width beta, local buckling stress and half-wavelength related to this and flexural and torsional buckling stresses decrease. The decrease of the half-wavelength related to local buckling is due to the increase in the slenderness of web and flange, and for this case, the local buckling transforms from stiffeners to web or flange plates. According to Fig. 13, by increasing the ratio of b wh /b wv, local buckling stress decreases and half-wavelength related to that increases. However, flexural buckling and torsional buckling are not sensitive to the variation of b wh /b wv. In Figs , the influence of material properties of composite FRP, including Young s modulus in the transversal direction E y, shear modulus G and Poisson s ratio in the longitudinal direction v xy respectively, on buckling behavior of stiffened cruciform

9 Fig. 14. Buckling curves for different Young s modulus in the transversal direction. Fig. 15. Buckling curves for different shear modulus. Fig. 16. Buckling curves for different Poisson s ratio in the longitudinal direction. section columns in different buckling modes are investigated. In Fig. 14, by increasing the ratio of E y /E x local buckling stress increases but the related half-wavelength decreases. Fig. 15 shows that by increasing the shear modulus G, local buckling stress increases while the related half-wavelength does not change. According to Fig. 16, by increasing Poisson s ratio in the longitudinal direction v xy, both local buckling stress and half-wavelength related to that do not change. It can be concluded from Figs that variation of E y, G and v xy have no effect on flexural and torsional buckling stresses of FRP stiffened cruciform section columns Torsinal and flexural buckling The procedure is exactly similar to that presented in the previous part for cruciform sections. Firstly, ten FRP columns with double-i sections under compression stresses as shown in Fig. 10. are studied. The geometric data of the sections are summarized in Table 3, in which Suffixes v and h denote the vertical and horizontal parts of section respectively. The geometric data related to the unstiffened part of the sections (section without stiffeners) are the same as given in Table 1. Critical torsional and flexural stresses F cr obtained from closed-form solutions, as well as critical stresses of finite strip method for three column-length L, including 300, 500

10 Table 3 Geometric data of the stiffened cruciform FRP sections. Case number Vertical section Horizontal section b fv (cm) b wv (cm) t fv (cm) t wv (cm) b fh (cm) b wh (cm) t fh (cm) t wh (cm) and 1000 cm, are obtained and shown in Table 4. All stresses are normalized with E x. The material properties adopted for each FRP column are E x = 21 GPa, E y = 8 GPa, G = 2.5 GPa, V xy = 0.3. All of the solutions are of simply-supported columns subjected to uniform compression stresses, so the effective length factor K is equal to one. Warping factor C w for a double-i section is obtained by C w ¼ I yðivertical Þ b 2 wv 4 þ I b 2 wh y ðihorizental Þ 4 ð24þ The results summarized in Table 4 show that the dominant global buckling mode in the majority of cases is torsional. However, for some columns with longer length and for sections made of thicker plates, critical stress is flexural. Considering Tables 2 and 4, it can be concluded that the possibility of torsional buckling in FRP stiffened cruciform sections is, generally, higher than that in unstiffened sections. On the contrary, the possibility of flexural buckling in FRP stiffened cruciform sections is, in general, less than that in FRP unstiffened cruciform sections. Taking into account of all these features, one can reach the conclusion that stiffeners in FRP cruciform sections cause an increase in the flexural rigidity and a decrease in the torsion rigidity of the section. In the last column of Table 4, the percentage of error using Eq. (23) between finite strip method and closed-form solutions in predicting of the critical torsional and flexural stresses of FRP stiffened cruciform sections is shown. As the results show, the critical stresses obtained from the finite strip method are in perfect agreement with those obtained from closed-form solutions. In general, as the columns lengthen, in the case of torsional buckling mode the amount of error indicates reductions ranging from 4% to 0.11%, while in the case of flexural buckling mode this value is between 3.62% and 0.42%. Here, in all cases, the percentage of error is negative, which means that critical stresses of the finite strip method are lower than closed-form Eqs. (18) and (19) for flexural and torsional buckling modes respectively. It can be concluded that the finite strip method is not conservative in prediction of both critical torsional and flexural stresses of FRP stiffened cruciform sections. Similarly to Table 2, the critical stresses of Eqs. (19) and (20) are very close to each other, as the results of Eq. (19) are naturally Table 4 Torsional and Flexural buckling stresses of FRP stiffened cruciform section columns in pure compression. Case number L (cm) r min r t F cr /(E x 10 3 ) (FSM) F cr /(E x 10 3 )(r min ) Eq. (18) F cr /(E x 10 3 )(r t ) Eq. (20) F cr /(E x 10 3 ) Eq. (19) Buckling type Error % Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Torsional Flexural Torsional Flexural Flexural Torsional Flexural Flexural Torsional Flexural Flexural Torsional Flexural Flexural 0.63

11 Table 5 Comparison of critical local buckling stresses and lengths. Case number CUFSM Proposed FSM Length ratio Critical stress ratio L cr (cm) F cr (MPa) L cr (cm) F cr (MPa) L cr (FSM)/L cr (CUFSM) F cr (FSM)/F cr (CUFSM) somewhat greater than Eq. (20). It is worth noting that in all columns with length of 1000 cm, where torsional buckling is principal, critical stresses of finite strip method are closer to Eq. (19) in comparison with Eq. (20) Local buckling As the further investigation and in order to evaluate the accuracy and validity of the method for local buckling, the first four FRP columns in Table 3 are studied. The material properties are E x = 21 GPa, E y = 8 GPa, G = 2.5 GPa, v xy = 0.3. The local buckling stresses and related half-wave lengths obtained by the proposed finite strip method are compared in Table 5 with those obtained from the CUFSM software. The minimum stress F cr (critical local buckling stress) is determined by fitting a quadratic interpolation function through three points close to the local nadir of the garlandshape curve. As the results show, the critical local buckling stresses and related half-wave lengths obtained from the proposed finite strip method are in excellent agreement with those obtained from the CUFSM software. 5. Discussion In Section 4, closed-form equations in cases of torsional and flexural buckling of FRP columns with doubly symmetric open thin-walled sections were compared with those of the semianalytical finite strip method. This investigation was carried out through two case studies, including unstiffened and stiffened FRP cruciform sections. It was demonstrated that in many cases, especially in stiffened cruciform sections, torsional buckling is the dominant global buckling mode. Results showed that finite strip method is more accurate than flexural buckling in predicting torsional buckling of FRP columns. One of most important items in calculating the equivalent radius of gyration r t is the ratio of shear modulus to Young s modulus in the longitudinal direction G/E x (Eq. (21)). The amount of this ratio in composite FRP in comparison with another material, such as steel, is lower. This matter decreases the equivalent radius of gyration r t in composite FRP sections even in long-length columns. Consequently, it can be concluded that with the same geometry conditions, torsional buckling in FRP sections is more likely than in steel sections, even in long columns. However, this demands closer studies. This further investigation has recently been performed by the authors [20] and will be the focus of a subsequent paper. 6. Conclusions This paper began with the development and presentation of a semi-analytical finite strip method capable of predicting the torsional and flexural buckling stresses of composite FRP columns under uniform compressive loads. The semi-analytical finite strip results of torsional and flexural buckling of composite FRP columns were validated with closed-form formulas. The study also includes a set of results concerning the dominant global buckling of composite FRP columns with open thin-walled sections. To achieve this, closed form solutions, as well as semi-analytical finite strip method, were used to predict flexural and torsional stresses. Among doubly symmetric sections, numerical studies have been carried out on FRP cruciform sections in two cases, including unstiffened and stiffened sections. In addition, an investigation was reported which required several parametric studies of the influence, on the member buckling behavior, of the applied stress distribution, the cross-section geometry, and the material properties. It was concluded that in FRP cruciform section columns with short length as well as thinner sections, torsional buckling is more likely to occur than flexural buckling. On the other hand, with lengthening of the columns and decreasing the slenderness of the section, flexural buckling will be more likely than torsional buckling. Furthermore, the use of stiffeners in FRP cruciform sections shows that the possibility of torsional buckling increases in comparison with unstiffened sections. However, stiffeners increase the stability of FRP cruciform sections in all buckling modes. This paper has also shown the semi-analytical finite strip method to be suitable for accurately predicting the torsional and flexural buckling of FRP columns in which the percentage of error is at worst 4% and for a majority of cases is below 1.5%. Among the results obtained for the buckling behavior of pultruded FRP cruciform section columns, the following deserves to be emphasized: In all cases where torsional buckling occurs, the equivalent radius of gyration r t is lower than the minimum radius of gyration r min. The optimum flexural stability of FRP cruciform sections under pure compression occurs when the widths and the thicknesses of flange and web are equal to each other. Torsional and flexural buckling modes occurring in unstiffened FRP cruciform sections are absolutely pure modes and uncoupled. With increasing slenderness of the section plates, the half-wave lengths range related to torsional buckling increases, while torsional stresses decrease. By increasing the slenderness of stiffener plates in stiffened FRP cruciform sections, local buckling strength decreases but flexural and torsional buckling strength increases. Torsional buckling and flexural buckling of FRP columns are not sensitive to different values of material properties of FRP sections, such as Young s modulus in the transversal direction E y, shear modulus G and Poisson s ratio in the longitudinal direction v xy. Finite strip method is generally more accurate in predicting torsional buckling of FRP columns than flexural buckling. Acknowledgments The authors are grateful to Professor B.W. Schafer from Johns Hopkins University for his precious help in verifying the results with CUFSM software.

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