Design of circular steel arches with hollow circular cross-sections according to EC3

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1 Journal of Constructional Steel Research ( ) Design of circular steel arches with hollow circular cross-sections according to EC3 C.A. Dimopoulos, C.J. Gantes Department of Civil Engineering, National Technical University of Athens, Greece Received 10 June 2007; accepted 15 September 2007 Abstract Design of either pin-ended or fixed circular steel arches with hollow circular cross-sections subjected to a uniformly distributed vertical load along the horizontal projection of the entire arch with the aid of the EC3 provisions is discussed. Appropriate modification factors are proposed that should be included in EC3 interaction equations, to improve their accuracy for the design of such arches. c 2007 Elsevier Ltd. All rights reserved. Keywords: Design; Circular arches; Fixed; Pin-ended; EC3; Steel; Strength 1. Introduction A great amount of work has been presented in the past concerning the behavior and the strength of arches. Timoshenko and Gere [1] calculated the linear buckling load of circular arches under uniformly distributed radial and vertical loads. Geometrically nonlinear buckling loads and load deflection curves for clamped arches have been given from Schreyer and Masur [2]. Accurate solutions and approximations for the nonlinear symmetric and antisymmetric buckling of both pin-ended and fixed shallow arches with arbitrary crosssection have been provided by Pi et al. [3]. The geometrically nonlinear buckling and postbuckling of elastic arches has been thoroughly investigated by Pi and Trahair [4]. Pi and Trahair [5] using a finite element program for the nonlinear inelastic analysis, investigated the in-plane inelastic buckling and strength of circular steel I-section arches. The effects of initial crookedness, rise-to-span ratio, residual stresses, and dead load to total load ratio on the in-plane inelastic stability of steel arches in uniform compression and in combined compression and bending were analyzed. A great effort has also been directed towards the design of steel arches. Kuranishi and Yabuki [6] proposed design criteria for parabolic steel arches that were expressed in terms of axial Corresponding author. Tel.: ; fax: address: chgantes@central.ntua.gr (C.J. Gantes). force and bending moment at the quarter point, but were only valid for slenderness ratio 0.5S/i, where S is the length of the arch and i is the radius of gyration of the cross-section about its major principal axis, in the range and riseto-span ratios f/l in the range However, using these actions for design seems doubtful because these are not always the maximum values. Verstappen et al. [7] proposed that the design rule for straight beam columns from Dutch codes, can be used for the check of the in-plane stability of pin-ended circular arches. However, a linear interaction equation for a straight beam column may be conservative because, among others, the favorable redistributions that take place in redundant arches after the first plastic hinge forms, is not taken into account. Pi and Trahair [8] found that the design equation of the Australian regulations for steel columns cannot be used directly for steel arches in uniform compression, nor can the design interaction equations for steel beam columns be used directly for steel arches under nonuniform compression and bending. Moreover, they proposed design equations for both uniform compression and nonuniform compression and bending that follow the provisions for the design of straight members of the Australian Steel Structures code. These proposed equations can be used for both shallow and non-shallow arches and are valid for slenderness ratio 0.5S/ i in the range and subtended angles 2Θ in the range In Eurocode 3 no design provision is given for the design of steel arches. To the authors knowledge, no proposal has been presented yet for the design of steel arches with X/$ - see front matter c 2007 Elsevier Ltd. All rights reserved. doi: /j.jcsr

2 2 C.A. Dimopoulos, C.J. Gantes / Journal of Constructional Steel Research ( ) Fig. 1. Geometry, loading and cross-section. the aid of Eurocode 3 provisions. This paper deals with the design of either pinned or fixed circular steel arches with hollow circular cross-sections, subjected to uniformly distributed vertical load on the entire arch (Fig. 1), under the provisions of EC3 concerning the design of uniform straight members in combined compression and bending. A modified interaction equation that provides good lower bounds for the in-plane strength of both shallow and non-shallow steel arches is proposed. Fig. 2. Material properties. 2. Methodology For the purposes of this paper, the nonlinear finite element analysis program, ADINA [9], has been used. A sufficient number of straight Hermite beam finite elements has been used for the analysis of the circular arches accounting for the effect of both geometric and material nonlinearities. For the solution of the nonlinear system of equations, the well-known displacement-control method (LDC) [9] is used in combination with the full Newton Raphson procedure and the use of line searches. An inelastic bilinear material with yield stress f y = 275 MPa, Young s modulus E = 210 GPa, Poisson s ratio ν = 0.30, without any hardening, is used for the inelastic analyses (Fig. 2). A hollow circular cross-section with a ratio of external diameter D to thickness t sufficiently small to avoid local buckling effects is used throughout this paper (Fig. 1). It is assumed that the cross-section has no residual stresses. In the analyses initial imperfections are taken into account, having the shape of the first antisymmetric eigenvector from a linearized buckling analysis (Fig. 3). The imperfection magnitude is equal to L/600, in accordance with the provisions of EC3 [10]. The behavioral characteristics of arches of this type under uniform vertical load have been discussed in [11]. Generally speaking, it was shown that the type of failure depends on the geometrical and the cross-sectional characteristics. When arches are extremely shallow, they suffer no buckling. However they are likely to violate serviceability tolerances. Shallow arches suffer from elastic snap-through. In the other cases, the strength of the arches is influenced by both the geometrical and the material nonlinearity. It is possible, however, for very deep arches to evaluate their strength considering only the material nonlinearity. A large number of geometrically and materially nonlinear analyses, including initial imperfection, was carried out in order to capture the real, highly nonlinear, behavior of the arches and Fig. 3. Antisymmetric and symmetric eigenvectors. to estimate their strength capacity. The results of these analyses were used in order to propose, an appropriate interaction factor in the linear interaction equation of EC3 for straight members, in order to include all the nonlinearity effects. As a result, it is possible to check the carrying capacity of arches by simply substituting the maximum axial and bending moment forces, which need not be situated at the same cross-section, obtained from a linear analysis. 3. EC3 linear interaction equation for beam column members The in-plane linear interaction equation for beam column members with class 1 or 2 cross-sections in combined compression and bending that is provided in EC3 [12] is: + k yy φ. (1) For the application of this equation for the case of arches, are taken as the design values of compression and bending actions, which are the maximum values of the internal actions appearing along the arch, not necessarily at the same cross-section, A is the area of the cross-section, W pl,y

3 C.A. Dimopoulos, C.J. Gantes / Journal of Constructional Steel Research ( ) 3 is the plastic section modulus, f y is the yield stress, is a reduction factor due to in-plane flexural buckling, k yy is an interaction factor due to combined compression and bending. The expression on the left part of inequality (1), defined as the utilization factor, should be bounded for design purposes by a maximum allowable value, denoted with φ, where φ = 1 according to [12]. The reduction factor is obtained from = 1 [ ] (2) Φ + Φ 2 λ 2 y where [ Φ = α ( λ y 0.20 ) ] + λ 2 y (3) λ y = = L cr,y 1 (4) N cr,y i y λ 1 E λ 1 = π (5) f y L cr,y is the equivalent buckling length. For in-plane buckling of arches, this length can be taken as L cr,y = γ S, where γ is a factor that takes the value 0.50 in the case of pin-ended arches and 0.35 in the case of fixed arches [13]. For a hot-rolled circular hollow section the a buckling curve is the appropriate one, so the imperfection factor α is equal to The interaction factor k yy can be computed with two alternative, methods, outlined in Annexes A and B of Part 1-1 of EC3 [12] Method 1 According to method 1 [14], the interaction factor is: k yy = C my C mlt µ y 1 1 N. (6) Ed C N yy cr,y Since the in-plane strength of the arches is studied, λ 0 = 0, where λ 0 is a non-dimensional slenderness factor for lateraltorsional buckling due to uniform bending moment. This means that the flexural buckling factor C my = C my,0, the lateraltorsional buckling factor C mlt = 1 and the lateral-torsional factor b LT = 0. For the computation of the above interaction factor the following variables must first be computed. ( ) π 2 E I y δ N C my,0 = 1 + Ed 1 (7) My,Ed N cr,y L 2 cr C yy = 1 + ( w y 1 ) [( Cmy 2 w λ max 1.6 Cmy 2 y w λ 2 max ] y n pl b LT W el,y (8) W pl,y ) N cr,y = π 2 E I y L 2, µ y = cr,y 1 N cr,y w y = W pl,y W el,y, n pl = N Rk /, N 1 χ Ed y N cr,y where I y is the in-plane second moment of area of the crosssection, δ is the value of the vertical displacement of the crown of the arch obtained from a linear analysis, is the maximum bending moment, W el,y and W pl,y are the in-plane elastic and plastic section moduli, λ max = λ y, N Rk =. The variable C my,0 depends on the bending diagram of the arch and is chosen to be equal to the expression of Eq. (7), because the corresponding bending diagram from [12] is similar and closer to the bending diagram exhibited by one-half of an arch, either fixed or pinned Method 2 The modification of this method for the case of arches lies in the appropriate computation of the factor C my, as explained next. According to this method [15] the interaction factor is: ( k yy = C my 1 + ( λ y 0.2 ) ) N Rk / ) C my ( (10) N Rk / where α s 0.40, 0 α s 1, 1 ψ α C my = s 0.40, 1 α s < 0, 0 ψ 1 (11) 0.1 (1 ψ) 0.8α s 0.40, 1 α s < 0, 1 ψ < 0 α s = M s /M h, M s is the bending moment at the quarter point of the arch, = 1.00 is a partial factor, M h = max [ M (0), M (S/2) ] and (12) { M (S/2) /M(0), M (S/2) < M(0) ψ = (13) M(0)/M (S/2), M(0) < M (S/2) where M(0), M(S/2) are the bending moments that appear at the support and the crown of the arch, respectively. For pinended arches the parameter ψ is always equal to zero because M(0) = 0. The corresponding parameter ψ for fixed arches is always different from zero. 4. Use of EC3 linear interaction equation for straight beam column members for checking in-plane inelastic stability of circular arches 4.1. Pin-ended arches A representative sample of two hundred and fifty two arches were used to investigate the in-plane strength of pin-ended steel arches subjected to uniformly distributed vertical load along the entire arch. They were divided into 14 groups with slenderness ratio 0.5S/i in the range from 40 to 170 and the subtended angle 2Θ in the range from 10 to 180. (9)

4 4 C.A. Dimopoulos, C.J. Gantes / Journal of Constructional Steel Research ( ) Fig. 4. Load displacement curve for a shallow pin-ended steel arch. Fig. 6. Strength of pin-ended steel arches method 1. Fig. 5. Load displacement curve for a deep pin-ended steel arch. In Figs. 4 and 5 some typical load displacement curves for the studied arches are given. On the horizontal axis the dimensionless vertical displacement v c /f of the arch crown is plotted, where f is the height of the arch. The dimensionless load q/q L B is plotted on the vertical axis, where q L B is the linearized buckling load. These curves were obtained either through a geometrically nonlinear elastic analysis with initial imperfections (GNIA), or through a material and geometry nonlinear analysis with initial imperfections (GMNIA). The equilibrium curve in Fig. 4 is representative of a shallow arch characterized by snap-through buckling, while that in Fig. 5 is representative of a deep arch characterized by material failure. The ultimate load, denoted as q GMNIA, is applied in a subsequent linear elastic analysis in order to obtain the maximum compression force and the maximum bending moment that are developed along the arch. These maximum values do not necessarily appear at the same cross-section and are the design values for the compression force and the bending moment that are encountered in the following linear interaction equations. Fig. 7. Strength of pin-ended steel arches with λ s /λ T method Method 1 In Figs. 6 and 7 the predictions of the interaction equation (1) with φ = 1 are given, and they are compared with the finite element results. It is concluded that in most cases the predictions of Eq. (1) are significantly conservative. In some cases of very shallow arches, the predictions are not on the safe side. So, it is seen that the interaction Eq. (1) is not adequate enough to be used for the design of pin-ended circular arches. A modified interaction equation is proposed for the design of circular arches: + k yy p 1 φ. (14) In Fig. 8 the three-dimensional graphs of p 1 and p1 are given. p1 is the factor that results in φ = 1. p 1 is an approximation of the exact factor p1 that is obtained using the optimization tools of Matlab [16], given by the following

5 C.A. Dimopoulos, C.J. Gantes / Journal of Constructional Steel Research ( ) 5 Fig. 8. Graphic representation of factors p 1 and p 1. Fig. 10. Strength of pin-ended steel arches with λ s /λ T modified method 1. Fig. 9. Strength of pin-ended steel arches modified method 1. Fig. 11. Strength of pin-ended steel arches method 2. equation: (λ s /λ T ) (0.5S/i) (λ s /λ T ) (0.5S/i), p 1 = 15 λ s /λ T 50 (15) 4.10ln 1.59 ( (λ s /λ T )), 50 < λ s /λ T 140 where λ s = S 2 /4i R, λ T = (γ S/i) f y / ( π 2 E ). In Figs. 9 and 10, the strengths of steel arches using modified Eq. (14) are given. It is verified that the modified Eq. (14) provides close to lower bound predictions of strength. For design purposes it is suggested that the maximum allowable value of the utilization factor φ is taken equal to Moreover, the modified Eq. (14) should be used for λ s /λ T Method 2 In Figs. 11 and 12 the predictions of the interaction Eq. (1) with φ = 1 and the interaction factor calculated according to method 2 are given, and they are compared with the finite element results. In contrast to the results of method 1, these results are in many cases not on the safe side. Thus, it is concluded that neither interaction Eq. (1) using method 2 for the determination of the interaction factor is appropriate to be used for the design of pin-ended circular arches. A modified interaction equation is also proposed: + k yy p 2 φ. (16) In Fig. 13 the three-dimensional graphs of p 2 and p2 are given. p2 is that factor that results in φ = 1. p 2 is an approximation of the exact factor p2 obtained by Matlab [16] and given by the following equation: p 2 = (λ s /λ T ) (0.5S/i) (λ s /λ T ) (0.5S/i), 15 λ s /λ T < (λ s /λ T ) (0.5S/i) (λ s /λ T ) (0.5S/i), 50 λ s /λ T 140. (17)

6 6 C.A. Dimopoulos, C.J. Gantes / Journal of Constructional Steel Research ( ) Fig. 12. Strength of pin-ended steel arches with λ s /λ T method 2. Fig. 14. Strength of pin-ended steel arches modified method 2. Fig. 13. Graphic representation of factors p 2 and p 2. In Figs. 14 and 15, the strengths of steel arches using modified Eq. (16) are given. It is seen that the modified Eq. (16) provides sufficiently close to lower bound predictions. For design purposes it is suggested that the maximum allowable value of the utilization factor φ is taken equal to 0.90 for 15 λ s /λ T < 60 and 0.95 for λ s /λ T Fixed arches A representative sample of one hundred and ninety eight arches is used to investigate the in-plane strengths of fixed steel arches subjected to uniformly distributed vertical load along the entire arch. They are divided into 11 groups with the slenderness ratio 0.5S/i in the range from 40 to 170 and the subtended angle 2Θ in the range from 10 to Method 1 In Figs. 16 and 17 the predictions of the interaction Eq. (1) with φ = 1, compared with the finite element results are given. It is concluded that in all cases the predictions of Eq. (1) are significantly conservative. Thus, it is observed that the Fig. 15. Strength of pin-ended steel arches with λ s /λ T modified method 2. interaction Eq. (1) is not adequate to be used in the case of fixed circular arches. A modified interaction equation is proposed for the design of circular fixed steel arches: + k yy f 1 φ. (18) In Fig. 18 the three-dimensional graphics of f 1 and f1 are given. f1 is the factor that results in φ = 1. f 1 is an approximation of the exact factor f1 obtained by Matlab and given by the following equation: (λ s /λ T ) (0.5S/i) (λ s /λ T ) (0.5S/i), f 1 = 20 λ s /λ T < 70 (19) ln ( (λ s /λ T )), 70 λ s /λ T < 195. In Figs. 19 and 20, the strengths of fixed steel arches using modified Eq. (18) are given. It is verified that the modified Eq. (18) provides close to lower bound predictions. For design purposes it is suggested that the maximum allowable value of

7 C.A. Dimopoulos, C.J. Gantes / Journal of Constructional Steel Research ( ) 7 Fig. 16. Strength of fixed steel arches method 1. Fig. 19. Strength of fixed steel arches modified method 1. Fig. 17. Strength of fixed steel arches with λ s /λ T method 1. Fig. 20. Strength of fixed steel arches with λ s /λ T modified method Method 2 In Figs. 21 and 22 the predictions of the interaction Eq. (1) with φ = 1 and the interaction factor calculated according to method 2, compared with the finite element results are given. In contrast to the results of method 1, these results are in the case of small values of the ratio λ s /λ T not on the safe side. Furthermore, the results are generally less conservative than the corresponding results of method 1. A modified interaction equation is proposed for the design of circular arches: Fig. 18. Graphic representation of factors f 1 and f 1. the utilization factor φ be taken equal to 0.90 for 20 λ s /λ T < k yy f 2 φ. (20) In Fig. 23 the three-dimensional graphics of f 2 and f2 are given. f2 is that factor that results in φ = 1. f 2 is an approximation of the exact factor f2 obtained by Matlab and

8 8 C.A. Dimopoulos, C.J. Gantes / Journal of Constructional Steel Research ( ) Table 1 Geometry, material and cross-sections of the three pinned example arches θ ( ) R (m) S (m) f y (MPa) E (GPa) D (m) t (m) 0.5S/i P P P Fig. 21. Strength of fixed steel arches method 2. Fig. 23. Graphic representation of factors f 2 and f 2. Fig. 22. Strength of fixed steel arches with λ s /λ T method 2. given by the following equation: f 2 = (λ s /λ T ) (0.5S/i) (λ s /λ T ) (0.5S/i), 20 λ s /λ T < (λ s /λ T ) (0.5S/i) (λ s /λ T ) (0.5S/i), 70 λ s /λ T < (λ s /λ T ) (0.5S/i) (λ s /λ T ) (0.5S/i), 120 λ s /λ T < 195. (21) Fig. 24. Strength of fixed steel arches modified method 2. In Figs. 24 and 25, the strengths of fixed steel arches using modified Eq. (20) are given also. It is seen that the modified Eq. (20) provides sufficient close to lower bound predictions. For design purposes it is suggested that the maximum allowable value of the utilization factor φ be taken equal to 0.90 for 20 λ s /λ T < Examples For the demonstration of the validity of the two modified interaction Eqs. (14) and (16), three pin-ended and three fixed arches with a variety of geometries and cross-sectional characteristics, listed in Tables 1 and 2, respectively, have been

9 Table 2 Geometry, material and cross-sections of the three fixed example arches ARTICLE IN PRESS C.A. Dimopoulos, C.J. Gantes / Journal of Constructional Steel Research ( ) 9 θ ( ) R (m) S (m) f y (MPa) E (GPa) D (m) t (m) 0.5S/i F F F Table 3 Utilization factors for the three pinned example arches Eq. (1) Method 1 Eq. (1) Method 2 Eq. (14) Eq. (16) P P P Table 4 Utilization factors for the three fixed example arches Eq. (1) Method 1 Eq. (1) Method 2 Eq. (18) Eq. (20) F F F studied. The utilization factors presented in Table 3 show that the proposed Eqs. (14) and (16) provide improved accuracy compared to methods 1 and 2 of EC3 and could be used for the design of pinned arches under the aforementioned loading pattern. The same conclusions are drawn for fixed arches from the results presented in Table Conclusions In this paper, an investigation of the effectiveness of the linear interaction equation for straight beam column members according to the EC3 provisions for the design of circular steel arches subjected to a uniformly distributed load is carried out. Modified interaction equations for the design of such arches are proposed, making use of the two methods provided in EC3 for the determination of the interaction factor k yy. It has been concluded that the linear interaction equation for the beam column members provided by EC3 is not sufficient for the design of circular steel arches. Particularly, using method 1 for the determination of the interaction factor k yy leads, for the majority of arches, except some very shallow arches, to very conservative estimations. Using method 2 for the determination of the interaction factor k yy may lead either to conservative or to unsafe estimations, depending on the case. Thus, modified interaction equations are proposed for the design of both pin-ended and fixed circular steel arches. The linear interaction equation for beam column members is modified through a factor p 1 for pin-ended arches ( f 1 for fixed arches) when using method 1 for the determination of the interaction factor, or p 2 ( f 2 ) when using method 2 for the determination of the interaction factor. The proposed modified interaction equations give sufficiently close to lower bound predictions of strength. Fig. 25. Strength of fixed steel arches with λ s /λ T modified method 2. References [1] Timoshenko SP, Gere JM. Theory of elastic stability. 2nd ed. New York (NY): McGraw-Hill Book Co., Inc.; [2] Schreyer HL, Masur EF. Buckling of shallow arches. In: Proceedings of the American society of civil engineers. Journal of the Engineering Mechanics Division 1996;92(EM4). [3] Pi YL, Bradford MA, Uy B. In-plane stability of arches. International Journal of Solids and Structures 2002;39: [4] Pi YL, Trahair NS. Nonlinear buckling and postbuckling of elastic arches. Engineering Structures 1998;20(7): [5] Pi YL, Trahair NS. In-plane inelastic buckling and strengths of steel arches, ASCE. Journal of Structural Engineering 1996;122(7). [6] Kuranishi S, Yabuki T. Some numerical estimation of the ultimate inplane strength of two-hinged steel arches. Proceedings of JSCE 1979;287: [7] Verstappen I, Snijger H, Bijlaard FSK, Steenbergen HMGM. Design rules for steel arches in-plane stability. Journal of Constructional Steel Research 1998;46(1 3): [8] Pi YL, Trahair NS. In-plane buckling and design of steel arches. Journal of Structural Engineering 1999;125(11). [9] ADINA System 8.3. Release notes. ADINA R & D Inc, 71 Elton Avenue, Watertown, MA USA. October [10] European standard. Eurocode 3: Design of steel structures. Part 2: Steel bridges. Annex D [informative] p [11] Dimopoulos CA, Gantes CJ. Nonlinear in-plane behaviour of circular steel arches with hollow circular cross-section. Journal of Constructional Steel Research [in press]. [12] Eurocode 3: Design of steel structures. Part 1-1: General rules and rules for buildings, pren [13] Pi YL, Trahair NS. In-plane inelastic buckling and strengths of steel arches. Journal of Structural Engineering 1996;122(7). [14] Boissonnade N, Jaspart JP, Muzeau JP, Villette M. New interaction formulae for beam columns in Eurocode 3: The French Belgian approach. Journal of Constructional Steel Research 2004;60: [15] Greiner R, Lindner J. Interaction formulae for member subjected to bending and axial compression in EUROCODE 3-the method 2 approach. Journal of Constructional Steel Research 2006;62: [16] MATLAB. Version 7 (R14), Copyright , The MathWorks Inc.