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Concrete In Australia : September 2011
Concrete in Australia Vol 37 No 3 49  ⎥⎦⎤ ⎢⎣⎡ − − − = l x l x x w x w x / 1 / 2 / ) ( 0 0 1 ) ( , b (2) where d(x) = [εr κ ]T is the section generalised strain vector and  T 3 2 1 q q q = q is the generalised nodal deformation vector (without rigid body modes). e equilibrium equations for the free body Ax shown in Figure 1c, gives:  Q b D ) ( ), ( , ) ( x x w x x θ = (3) ] ] - - - - - = l x l x x w l x l x x x w x / 1 / ) ( / ) ( / ) ( 1 ) ( ), ( , θ θ θ b (4) where  T 3 2 1Q Q Q = Q denotes the nodal force vector (in the system without rigid body modes) and  T ) ( ) ( ) ( x M x N x= D is the section internal force vector. Substituting the total secant constitutive law σx = Ee (εx -- εpx) in the section equilibrium equation, 6 q Q6 5 5 q Qq4 4 y A B Q2 q2 x 1 1 Q Qq z 3 3q Q yx B A l0 l q Q11 2 2 Qq q3 3 Q y x B A 1 Q x Q2 3 Q V(x) N(x) M (x) w(x) (x)=w'(x)=dw/ dx l  T d d ) ( - = A y A x x x σ σ D , yields: ) ( ) ( ) ( ) ( x x x x p s D d k D + = (5)  T d . d . ) (- = A E y A E x px e px e p ε ε D (6) where Ee is the secant modulus of material, εpx is the plastic component of strain, ) (x p D is the residual plastic force vector for the section and ks(x) is the secant stiffness matrix of the section and inverting the stiffness matrix of the section the section flexibility matrix  1 ) ( ) ( − = x x s sk f mm, is obtained. Substituting Eqs. (3) and (5) into (1), gives the following relationship: p q Q F q − = (7) [ ] x x x w x x x w x s l d ) ( ), ( , ) ( ) ( , 0 T θ b f b F∫ = (8) x x x x w x p s l p d ) ( ) ( ) ( , 0 T D f b q∫ = (9) where F is the flexibility matrix of the simply supported configuration (without rigid body modes) and p q is the nodal generalised plastic deformation vector. It is observed that incorporating the geometrical nonlinearity into the formulation necessitates the deformed shape of the element to be available. In the displacement-based formulation, the deformed shape of the element is obtained based on the nodal displacement values and adopted shape functions. In the force-based element, however, there is no displacement shape function to be used and accordingly a composite Simpson s integration scheme combined with piecewise interpolation of curvature is used in this paper to update the geometry of the deformed configuration (Figure 2). 3.0 NONLOCAL INTEGRAL FORMULATION e transverse deflection of RC beams in a framed structure is associated with significant compressive force as well as compressive softening of concrete, which can be associated with a lack of objectivity and with spurious mesh sensitivity. Similarly, compressive softening of concrete and spurious mesh sensitivity may occur in reinforced concrete columns subjected Figure 1. Outline of (a) two-node frame element AB, (b) simply supported configuration and DOFs, and (c) Ax free body diagram, after deformation. (a) (c) (b) A B x z y xSub-element w(x) 3 4 5 2 order Parabola nd (x)=w'(x)=dw/ dx ... 5 2n-1 Integration point 3 12 n+1 Figure 2. Outline of the composite Simpson integration scheme, element and sub-elements.