Nonlinear RC beam element model under combined action of axial, bending and shear

Transcription

1 icccbe 21 Nottingham University Press Proceedings of the International Conference on Computing in Civil and Building ngineering W Tiani (ditor) Nonlinear RC beam element model under combined action of aial, bending and shear Yunpeng Zhang, Bo Diao & Yinghua Ye Beihang University, China Shaohong Cheng University of Windsor, Canada bstract Based on higher-order shear deformation theory, a nonlinear RC beam element model is developed, the element deformation at any point is determined by ais displacements and transverse compressive deformation is considered here. The compatibility equations and equilibrium equations of plane stress are introduced into the derived beam element model to take into account stirrup effect. The material constitutive law from Stevens et al. is used to the element model, which is modified constitutive law of MCFT from Vecchio and Collins. The program is being debugged with no data illustrated here. Keywords: higher-order shear deformation, modified compression field, combined action, RC beam 1 Introduction Structure concrete members are subjected to comple loading combinations, such as combined aial, bending and shear loading. Due to the comple stress state, researches on the mechanism analysis and calculation accuracy have been dissatisfactory during the last decades. Beam and pillar as very important members of the structure have been under the interaction between aial force, bending moment, shear force and torsional moment and so on. However, there are no suitable beam element theoretical models to deal with the comple stress state. t the present time, all of the beam element models such as uler-bernoulli beam theory, Timoshenko beam theory (Timoshenko, 1921) and higher-order shear deformation theories (Heyliger and Reddy, 1988; Mechab et al., 28) have the same assumed condition that there is no etrusion between transverse fibers. The beam deformations along stirrup direction have been ignored on this assumption, and the contribution by the stirrups can not be considered. nd then, these beam elements are not suitable to analye the nonlinear reaction of reinforced concrete beam when stirrups needed to be considered. In this paper, a 2d nonlinear RC beam element model under combined action of aial, bending and shear is presented. This RC beam element model is based on higher-order shear deformation theories, taking the lateral etrusion of the cross section restricted by the stirrups into account. The compatibility equations and equilibrium equations of plane stress state are simultaneously introduced into the proposed beam element, so that the geometry compatibility and the transformation of concrete rigidity for shear can be reasonable. The contribution of stirrups is ignored before the inclined cracks appear and is considered after the inclined cracks appear. The Stevens material model (Stevens et al., 1991) which is closely based on Modified Compression Field Theory (MCFT) is used to describe the constitutive material property for concrete.

2 2 Model formulation The Finite lement Method and the displacement formulation are used in the proposed model. It is described through 6 parts as following: basic assumption, element formulation, section formulation, regional analysis, stirrups consideration and constitutive model. 2.1 Basic assumption (1) Only plane beam element with uniform section is considered; (2) Material nonlinearity is considered and geometric nonlinearity is ignored; (3) The contribution of stirrups is considered after incline cracks appears but ignored before incline cracks appears; (4) To follow the basic assumptions of the MCFT. 2.2 lement formulation 3-node element with three dofs (degrees of freedom) per node is adopted and the details of the analytical formulation are available according to Navarro (Navarro et al., 27). Fig 1 shows a straight uniform cross section beam of constant thickness of h with width b and length l. The displacement components along the and directions can be denoted as u and w respectively with coordinate system, y,. Two translations and one rotation are used to describe the planar response of the element in the - plane. O denotes the origin of the local coordinate system. These displacements at each node can be epressed as a three-component vector: ( ) u = u w θ y (1) T The following displacement field for the beam is assumed on the basis of the general higher-order theory: u (, ) = u w +Φ( ) ( w ) (, ) = w and w (, ) = w (2), where u and w represent aes displacement components along the and directions and is the transverse shear measured on the aes: () = w, θy (3) and shape function Φ( ) employed in the proposal model is the same with parabolic shear deformation beam theory of Reddy (Reddy, 1984). It is as follows: 2 2 Φ ( ) = (1 4 /3 h ) (4) so the displacement field for the beam (q.2) can be written as: u (, ) = u w + (1 4 /3 h) and 2 2, w (, ) = w (5) 2.3 Section formulation The proposed beam element is composed of several integration sections which must permit a load combination including aial force N, homotaial bending moment M and homotaial shear force V (Fig 2).

3 Figure 1. Geometrical definition. Figure 2. Section loads t each integration point of the sections, the following two strain components must be considered: = ( u, ) = u w +Φ( ),,,, = u (, ), + w (, ), =Φ( ), (6) where is the normal strain along -ais, is the shear strain and, is the high order shear strain. Bring q.3 into q.6, then = ( u, ) = u θ + [ Φ( ) ],, y,, = u (, ), + w (, ), =Φ( ), During the derivation of finite element equation, the high order shear strain can be ignored (Levinson, 198). So the translations of the strain vector at any point of the section % to the generalied strain vector of the section% can be epressed as follows: u, 1 1 θ, φ = ( ) = ( ) (8) Φ, Φ, T 1 T that is, % = S %, % = [ ], S = = φ (,% Φ ), where is the longitudinal strain at the section centroid, φ is the curvatures about the y-ais and = is the generalied shear strains as q.3. nd the stress vector at any pointσ% corresponding to% is written as [ ] T % σ = σ τ ccording to the principle of virtual work, the section is in equilibrium if the virtual work density of the stress field in the section equals the virtual work density of the generalied section forces. Hence, the section forces F can be epressed as: T T δ % % σ d = δ % F (9) where, % % and F [ N M V] T % d % S d T T T δ σ = δ σ =, so, % T S σ d= F In order to get the tangent stiffness of beam cross-section, the differential form above function is (7)

4 F D% σ σ S σ d S % % d S % = = = = Sd = S DSd % % % % % % T T T T % σ where D is the Jacobian matri epression of %. (1) 2.4 Regional analysis ccording to assumption (1), only the deformations in --plane are considered. That is, the stress state is in the --plane and the contribution of the reinforcement along y-ais is ignored. Thus the section can be divided into several regions along -ais with different amount, distribution and orientation of the reinforcement. n eample of this subdivision is illustrated in Fig.3. The height of each region is not fied. Usually the range of influence of the rebar on the concrete is used to refer to the heights of 1-region and 2-region. Figure 3. Subdivision of cross-section nalysis with uncracked concrete ccording to assumption (3), the contribution of the stirrups is ignored before the concrete cracks, so = and d =. Then the following can be gotten from q.22 and q.23: σ c c ρs s τ = cτ + cτ (11) c c d c σ c ρs D s d dτ = + τc τ c d c c where subscript c and s means concrete and steel respectively,, D and ρ means secant modulus, tangent modulus and rebar ratio respectively. For 2-region, ρ s = and for 1-and3-regions, ρs nalysis with cracked concrete fter the concrete cracks, stirrups will contribute to the shear resistance of the beam, so. Then from q.21, q.22 and q.23: σ c c c ρs s τ = cτ c τ + c (13) (12)

5 c c c % σ c ρs D s = + % τ c c η ρs D + s c c where η is adjustable factor which reflects the relationship between the -ais normal strain and the shear strain Section response It is possible to calculate the section forces and tangent stiffness matri for every region with uncracked concrete and cracked concrete according to q.9 and q.1, and then to sum them up to get the forces and tangent stiffness matri of the entire section. It can be epressed as follows: F = F1 + F2 + F3 and D% = D% 1+ D% 2 + D% 3 (15) 2.5 Stirrups consideration When cracks appear and cross the stirrups, the strained condition of the stirrups is comple and irregular (Baumann and Rüsch, 197). It s also difficult to calculate the eact stress of every stirrup. s equilibrium equations of plane stress shows: + = (16) it can be epanded as: = (17) ccording to the Stevens constitutive model,,,,, and are all complicated nonlinear functions about, and. It s impossible to derive the eplicit epression of from q.17. For the proposed model, the contribution of the stirrups is analyed from macro point of view. polynomial function is adopted to describe the strain distribution of the stirrups along the -ais, 2, = h h 3 ( ) where = η is nominal -ais normal strain, η is adjustable factor. It is obvious that q.18 satisfies the compatibility equation of plane stress (q.19) when the degrees of the element shape functions are smaller than 3. nd it will make the translation of stirrup effect to concrete shear rigidity simple (q.21) = (19) 2 2 Before stirrup effect is considered into the rigidity matri (q.14), q.17 can be simplified as q.2 on the assumption that σ and τ are mainly influenced by and respectively, and the influence of other strain components have on each of them is ignored: (14) (18)

6 + = Thus, after incline cracks appear, the shear rigidity can be change as: c = η = η + ρs D s c (2) (21) 2.6 Constitutive model The constitutive material model for the proposed beam model is totally based on the Stevens model which is based closely on the MCFT (Frank et al., 1986). The analytical results show good agreement with eperimental data with this model used for the 2D-plane stress analysis (Navarro and Miguel, 27). The details of the model are available in the literature (Stevens et al., 1991). The concrete and reinforcement contribution are first set up separately and then added. Hence, σ c c c ρ s s σ c c c ρ s s = + τ c τ c τ cτ c c c c c c dσ ρ D d s s c c c dσ ρ D = + d s s c c c dτ d c c c c c c (22) (23) cknowledgements This work is part of the projects financially supported by the Ministry of Science and Technology (MST) of China (Grant No. 27BF23B2-6) and National Natural Science Foundation of China (NSFC) (Grant No ),The authors gratefully acknowledge the financial support of MST and NSFC. References BUMNM T, RüSCH H.,197. Schubversuche mit indirekter Krafteinleitung. Deutscher usschuss für Stahlbeton, Heft 21, Berlin 197. FRNK J. VCCHIO, MICHL P. COLLINS., The Modified Compression-Field Theory for Reinforced Concrete lements Subjected to Shear. CI JOURNL, Proceedings V. 83, No.2, Mar-pr.1986:219~231 HVLIGR P.R. and RDDY J.N., higher order beam finite element for bending and vibration problems. J.Sound Vib 126,(39~326).1988 I. MCHB,. TOUNSI, M.. BNTT,.. DD BDI, 28. Deformation of short composite beam using refined theories. J. Math. nal. ppl.346 (28)468~479 J. NVRRO GRGPRI, P. MIGUL SOS, M.. FRNáNDZ PRD, FILIP C. FILIPPOU, 27. 3D numerical model for reinforced and prestressed concrete elements subjected to combined aial, bending, shear and torsion loading. ngineering Structures 29(27):344~3419

7 LVINSON M..,198. n accurate, simple theory of the statics and dynamics of elastic plates. Mech. Res. Commun,198,7:343~35 RRDDY J.N., simple higher order theory for laminated composite plate, SN J. ppl. Mech.51(1984)745~752 STVNS NJ, UZUMRI SM, COLLINS MP, WILL GT.,1991. Constitutive model for reinforced concrete finite element analysis. CI Structural Journal 1991:49~59 TIMOSHNKO S. P., On the correction for shear of the differential equation for transverse vibration of prismatic bars. Phil. Mag. 41,744~746 (1921)