using a variant of Bernoulli’s elastic curve equation by substituting the effective flexural rigidity of the system. The temporal y displacements are expressed as Tracking the nodal x position relies on a back looking approach. The temporal x displacements of the previous node are used for information to track the following node. Once the x and y nodal positions are known, the rotation referenced from the un-deformed shape, for the beam, for each of the nodes, can be calculated using Once the nodes are tracked, with their respective angles, the nodal normal forces can be calculated through recalling Equation 8. The normal nodal forces become These nodal forces are used to initiate the looped convergence process of determine the actual nodal normal forces that stiffen the system when accounting for the presence of the frictional interactions. These normal forces will require an adjustment when an out-of-plane compressive force is present in the system. This is taken into account by linearly decomposing the compressive forces. Once these are decomposed these are added directly to the nodal normal forces in Equation 9 and the nodal normal forces are then expressed as It is assumed that the compression remains constant when the composite experiences bending. For calculating the decreased displacements from the nodal normal forces, the nodal normal forces can be averaged since all the nodes are assumed to move simultaneously along their respective interface. This is done by Using the average nodal normal force, the averaged frictional stress is found by Once Gs has been calculated, the displacements of the composite can be calculated with the inclusion of the frictional stiffness. Gs can be used to represent an artificial shear strength at the layer interfaces. The shear stress for a solid beam with the same geometry as the composite must first be calculated. Timoshenko and Goodier [5] express this as τxy must be calculated for each of the interface locations, with respect to the solid beam neutral axis and then summed. If the artificial shear stress is less than the shear stress for the solid beam, then the load step chosen was too large and a smaller load step is necessary. Convergence is achieved once the ratio of the artificial shear stress and the shear stress of the beam approach 1. Using the sum of τxy and interfacial stiffness of the entire composite can be calculated using ( )( ) eff eff n vn yn E I F x L x 6 3 2 − = δ ( ) 1 2 1 2 2 − + = − xn i y x xn d d δ δ δ δ ( ) x y d d δ δ ϕ − = −1 tan ( )( ) n vn Nn F F ϕ cos = 1− = + n F F F c Nn Tn ( )( )( ) ( ) ( ) ( ) ( )1 1 1 3 1 1 2 2 1 − − − − − − + + − − − − = ni T i n ni T n ni n Tni Tn x x F x x F F F x x F η η η A F G s s µ = − = 2 2 2 4 nl xynl y t I F τ (6) (7) (8) (9) (10) (11) (12) (13) L 292
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