EUROSTEEL 8, 3-5 September 8, Graz, Austria 785 DESIGN OF FIXED CIRCULAR ARCHES WITH TUBE CROSS-SECTIONS UNDER CONCENTRATED LOADS ACCORDING TO EC3 C.A. Dimopoulos a, C.J. Gantes a a National Technical Universit of Athens, Department of Civil Engineering, Athens, Greece. INTRODUCTION The design of arches has been addressed b a number of scientists in the past. Kuranishi and Yabuki [] were the first to propose design criteria for parabolic steel arches. The criteria were expressed in terms of the axial force and the bending moment at quarter point, but were onl valid for slenderness ratio.5s/i, where S is the length of the arch and i is the radius of gration of the crosssection about its major principal axis, in the range 5-5 and rise to span ratios f/l in the range.-.3. However, using these actions for design seems doubtful because these are not alwas the maximum values. Verstappen et al [] proposed that the design rule for straight beam-columns from Dutch codes can be used for the check of the in-plane stabilit of pin-ended circular arches. The use of linear interaction equations for beam-columns without an appropriate modification ma lead to conservative results since the beneficial redistributions that take place in redundant arches after the first plastic hinge forms is not taken into account. Pi and Trahair [3] proposed appropriatel modified design equations, based on the Australian interaction equation for steel beam-columns, for both uniform compression and nonuniform compression and bending. These equations can be used for both shallow and non-shallow arches and are valid for slenderness ratio.5s/i in the range - 7 and subtended angles Θ in the range -8. Dimopoulos and Gantes [4] found that the EC3 design equation for steel beam-columns ma lead to either unsafe or conservative results, in the case of circular steel arches subjected to uniform vertical loads, depending on the shallowness of the arch and the estimation method of the interaction factor that takes into account the bending effects. Moreover, the proposed modified interaction equations for both pin-ended and fixed circular arches that give sufficientl close to lower bound predictions of strength.. METHODOLOGY In this paper the design of circular arches with a hollow cross-section under a concentrated load at the crown is investigated (Fig. ). The used hollow cross-section is assumed to have a ratio of external diameter D to thickness t sufficientl small to avoid local buckling effects. It is assumed that the cross-section has no residual stresses. An inelastic bilinear material with ield stress f 75MPa, Young s modulus E GPa, Poisson s ratio.3 without an hardening, is used for the inelastic analses. In the analses initial imperfections are taken into account, having the shape of the first antismmetric eigenvector from a linearized buckling analsis. The imperfection magnitude is equal to L/6 in accordance to the provisions of EC3 [5]. For the numerical analses, carried out in this paper, the general purpose finite element program ADINA [6] is used. The arches are modeled with a sufficient number of straight Hermite beam finite elements. The effect of both geometric and material nonlinearities is taken into account. The displacement-control method (LDC) [6] is used for the solution of the nonlinear sstem of equations, in combination with the full Newton-Raphson procedure and the use of line searches. The results of the analses are used in order to propose an appropriate interaction factor in the linear interaction equation of EC3 for straight members. Using the proposed modified equation, it is possible to check the carring capacit of arches b simpl substituting the maximum axial and bending moment forces, obtained from a linear analsis. These maximum forces need not be situated at the same cross-section.
3. EC3 LINEAR INTERACTION EQUATION FOR BEAM-COLUMN MEMBERS The in-plane linear interaction equation for beam-column members with class or cross-sections in combined compression and bending that is proposed in EC3 [7] is: Q 786 f t D Θ L Fig.. Geometr, loading and cross-section N M, k () Af Wpl,f M M For the application of this equation for the case of arches N, M 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 necessaril at the same cross section, A is the area of the crosssection, W pl, is the in-plane plastic section modulus, f is the ield stress, χ is a reduction factor due to in-plane flexural buckling, and k is an interaction factor due to combined compression and bending. The expression at the left part of inequalit (), defined as the utilization factor, should be bounded for design purposes b a maximum allowable value, denoted with, where according to [7]. The reduction factor χ is obtained from: χ Φ Φ λ (), λ Af Ncr, Lcr, iλ Φ.5 α λ. λ cr, Lcr,, λ π Ef (3) L is the equivalent buckling length. For in plane buckling of arches, this length can be taken as γs, where γ is a factor that takes the value.5 in the case of pin-ended arches and.35 in the case of fixed arches [8]. 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 can be computed with two alternative methods, outlined in Annexes A and B of Part - of EC3 [7]. 3. Method According to method [9], the interaction factor is: μ k CmCmLT N C N cr, Because the in-plane strength of the arches is studied, λ, where λ is a non-dimensional slenderness factor for the lateral-torsional buckling due to uniform bending moment. This means that the flexural buckling factor Cm Cm,, the lateral-torsional buckling factor CmLT and the lateral-torsional factor blt. For the computation of the above interaction factor the following variables must first be computed. (4)
π EI δ N Cm, (5) L cr M, N cr,.6.6 W el, C w Cmλmax Cmλmax npl blt (6) w w Wpl, EI N N W pl, N Ncr, π, μ L χ, w cr, N cr, N, n pl cr, W el, NRk γ (7) M where I is the in-plane second moment of area of the cross-section, δ is the value of the vertical displacement of the crown of the arch obtained from a linear analsis, M, is the maximum bending moment, W el, and W pl, are the in-plane elastic and plastic section moduli, λ max λ, NRk Af. The variable C m, depends on the bending diagram of the arch and is chosen to be equal to the expression of Eq. (5), because the corresponding bending diagram from [7] is similar and closer to the bending diagram exhibited b one half of an arch, either fixed or pin-ended. 3. Method The modification of this method for the case of arches lies in the appropriate computation of the factor C, as explained next. According to this method [] the interaction factor is: m N N k Cm λ. Cm (8) χ NRk γ M χ NRk γ M where. αs, αs, ψ Cm.αs, - αs, ψ (9).ψαs, - αs, ψ αs Ms Mh, M s is the bending moment at the quarter point of the arch, γ M. is a partial factor, M max M, M S () h M, MS M,MS M ψ () M M S, M M S where M S are the bending moments that appear at the support and the crown of the arch, respectivel. For fixed arches the parameter ψ is alwas different from zero. 4. USE OF EC3 LINEAR INTERACTION EQUATION FOR STRAIGHT BEAM-COLUMN MEMBERS FOR CHECKING IN-PLANE INELASTIC STABILITY OF CIRCULAR ARCHES A representative sample of two hundred and fift two arches was used to investigate the in-plane strength of fixed arches subjected to a concentrated load at the top. The were divided in 4 groups with slenderness ratio.5s i in the range from 4 to 7 and subtended angle Θ in the range from º to 8º. In Figs. and 3 some tpical load-displacement curves are given. On the horizontal axis the dimensionless vertical displacement vc f of the arch crown is plotted, where f is the height of the arch. The dimensionless load Q Q is plotted on the vertical axis, where Q is the linearized buckling load. Besides the curves based on material and geometr nonlinear analses with initial 787
imperfections (GMNIA), the curves based on nonlinear elastic analses with initial imperfections (GNIA) are also given for comparison purposes. The ultimate load, denoted as Q GMNIA, is applied in a subsequent linear elastic analsis in order to obtain the maximum compression force and the maximum bending moment that are developed along the arch. These maximum values do not necessaril 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. 788 Q/Q GNIA GMNIA Q GMNIA Initial ielding..5s/i=7, Θ=, (Fixed arch).5 v /f c.5 Q/Q.5S/i=7, Θ=8 (Fixed arch) GNIA GMNIA. Q GMNIA Initial ielding. v c /f Fig.. QQ vs. vc f for a shallow arch Fig. 3. QQ vs. vc f for a deep arch 4. Method In Figs. 4 and 5 the predictions of the interaction Eq. () with are given and the are compared with the finite element results. It is found that in all cases the predictions of Eq. () are significantl conservative. So, it is seen that the interaction Eq. () is not adequate enough to be used for the design of fixed arches. A modified interaction equation is proposed for the design of fixed circular arches: N k M, () Af f W pl,f M M The graphics of f and f * are given in Fig. 6, where f * is the factor that results in. f is equal to:.38.96λs λt.99.5s i.5λs λt.5s i, λs λt 45 f.498 (3).677 ln 4.659λs λ T,45 λs λt 95 where λs S 4iR, λt γsi f π E In Fig. 7 the strengths of steel arches using the modified Eq. () are given. It is verified that the modified Eq. () provides close to lower bound predictions of strength. For design purposes it is suggested that the maximum allowable value of the utilization factor is taken equal to.9. 4. Method In Figs. 8 and 9 the predictions of interaction Eq. () with and the interaction factor calculated according to method are given and the are compared with the finite element results. In contrast to the results of method, these results are in all cases not on the safe side. Thus, it is concluded that neither interaction Eq. () using method for the determination of the interaction factor is appropriate to use for the design of fixed circular arches. A modified interaction equation is also proposed:
N k M, Af f W f M pl, M 789 (4) /χ. Eq.(), method (φ=) FEM (Eq.()).5S/i=4 7,. Θ= 8,. k. /χ +k.6.4..5s/i=4 7, Θ= 8 Eq.(), method (φ=) FEM (Eq.()) 5 λ /λ s T 5 Fig. 4. Strength method Fig. 5. Strength vs. λs λ T - method f *, f.5.5 f * f ( λ s 45) f * f (45<λ s 95) λ s.5s/i /χ. Method Modified method Eq.()(φ=) Eq.()(φ=.95) Eq.()(φ=.9) FEM (Eq.()) FEM (Eq.()).. k Fig. 6. Factors f and * f Fig. 7. Strength modified method The graphics of f and f * are given in Fig., where f * is the factor that results in. f is equal to:.75.47λs λt.9.5s i.455λs λt.5s i, λs λt 3 f.76 (5).6ln 5.98λs λ T,3λs λt 95 In Fig. the strengths of steel arches using the modified Eq. (4) are given. It is seen that the modified Eq. (4) provides sufficientl close to lower bound predictions. For design purposes it is suggested that the maximum allowable value of the utilization factor is taken equal to.9. 5. CONCLUSIONS In this paper an investigation of the effectiveness of the EC3 linear interaction equation for beamcolumn members for the design of steel circular arches subjected to a concentrated load at the crown was carried out. It has been found that when the interaction factor is estimated according to method, the results are significantl conservative. When method is used for the determination of the interaction factor, the results are not on the safe side. Thus, modified interaction equations are proposed for the design of fixed steel arches. The linear interaction equation for beam-column members is modified through factors f and f when using method and method, respectivel, for the determination of the interaction factor. The proposed modified interaction equations give sufficientl close to lower bound predictions of strength.
79 /χ. Eq.(), method (φ=) FEM (Eq.()).5S/i=4 7, Θ= 8,. k M /M max Rk /χ +k Eq.(), method (φ=) FEM (Eq.()).5S/i=4 7, Θ= 8 5 5 λ s Fig. 8. Strength method Fig. 9. Strength vs. λs λ T - method f *, f.5 f ( λ /λ 3) f * s T f (3<λ s 95) λ s.5s/i f * /χ.. Eq.(4)(φ=) Eq.(4)(φ=.95) Eq.(4)(φ=.9) FEM (Eq.()) FEM (Eq.(4)) Modified method Method.. k Fig.. Factors f and * f Fig.. Strength modified method 6. REFERENCES [] Kuranishi S, Yabuki T. Some numerical estimation of the ultimate in-plane strength of twohinged steel arches. Proceedings of JSCE 979;87:55-58. [] Verstappen I, Snijger H, Bijlaard FSK, Steenbergen HMGM. Design rules for steel arches Inplane stabilit. Journal of Constructional Steel Research 998;46(-3):5-6. [3] Pi YL, Trahair NS. In-plane buckling and design of steel arches. Journal of Structural Engineering 999;5(). [4] Dimopoulos CA, Gantes CJ. Design of circular steel arches with hollow circular cross-sections according to EC3. Journal of Constructional Steel Research 8; [accepted for publication]. [5] European Standard. Eurocode 3: Design of steel structures. Part : Steel bridges. Annex D [informative]. 4. p. 9-. [6] ADINA Sstem 8.3. Release Notes. ADINA R & D Inc, 7 Elton Avenue, Watertown, MA 47 USA. October 5. [7] Eurocode 3: Design of steel structures. Part -: General rules and rules for buildings, pren 993--.4 [8] Pi YL, Trahair NS. In-plane inelastic buckling and strength of steel arches. Journal of Structural Engineering 996;(7) [9] Boissonnade N, Jaspart JP, Museau JP, Villette M. New interaction formulae for beamcolumns in EC3 : The French-Belgian approach. Journal of Constructional Steel Research 4;6:4-3. [] Greiner R, Lindner J. Interaction formulae for member subjected to bending and axial compression in EUROCODE 3-the method approach. Journal of Constructional Steel Research 6;6:757-7.