Laminated composite plates are widely used in many different structural applications because of the low weight and high strength properties. Such structures, however, exhibit a complex behavior including anisotropy, bending-extension coupling and significant shear deformations. For this reason, analytical and numerical tools for the analysis of laminated composites are an active research topic. In this work, a new quadrilateral 4-node element for laminated plates is formulated within the framework of the First order shear deformation theory (FSDT). The element is based on a hybrid stress formulation derived from the one presented by de Miranda and Ubertini (2006) for single layer isotropic plates. In particular, the same choice of linked displacement interpolation and stress approximation, co-ordinate invariant and equilibrated within each element, is made for the out of plane resultant forces and moments, while the approximation proposed by Pian (1993) is used for the resultant membrane forces. The variational formulation is extended to include the in plane-bending coupling typical of laminated theories. The resulting element has 4 nodes, five degrees of freedom per node, and involves only compatible displacement functions. It is locking-free and readily implementable into existing finite element codes. The described features ensure stability, accuracy, little sensitivity to geometric distortions and computational efficiency. An accurate evaluation of the stress state in the laminate, and in particular of transverse shear stresses along the thickness, is crucial in order to predict critical phenomena such as delamination. The kinematic hypothesis introduced in the first order shear deformation theory yields inaccurate piecewise constant shear stresses which are discontinuous between the laminae. However, accurate transverse shear stress profiles can be evaluated in the post-processing through three dimensional equilibrium equations. From the FEM internal moments and membrane forces it is possible to evaluate the curvatures and the membrane strains through the laminate constitutive equation and, so, the in plane strain tensor. The constitutive equations of each lamina allow then to determine the in plane stresses, which can be introduced in the three dimensional equilibrium equations to evaluate the transverse shear stresses. The shear stresses so obtained should satisfy the boundary conditions on both external surfaces of the laminate. Moreover, the shear stress resultants along the plate thickness should be equal to the transverse shear forces evaluated via finite element procedure. It can be shown that both conditions are automatically satisfied if the internal moments and membrane forces satisfy the plate equilibrium equations. Indeed, accuracy of the stress profiles depends upon the derivatives of the internal moments and membrane forces, therefore a good approximation not only of the internal forces but also of their derivatives is important. According to this, a two steps procedure is proposed to recover transverse shear stresses. First, a Recovery by Compatibility in Patches procedure is implemented in order to improve the rate of convergence of the FEM internal forces. Then, using the recovered internal forces, the transverse shear stress profiles are determined as described above. The effectiveness of the proposed element and the transverse shear stress recovery has been verified on various test problems.

### A FSDT-based hybrid stress finite element for composite laminates.

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*DAGHIA, FEDERICA;DE MIRANDA, STEFANO;UBERTINI, FRANCESCO;VIOLA, ERASMO*

##### 2007

#### Abstract

Laminated composite plates are widely used in many different structural applications because of the low weight and high strength properties. Such structures, however, exhibit a complex behavior including anisotropy, bending-extension coupling and significant shear deformations. For this reason, analytical and numerical tools for the analysis of laminated composites are an active research topic. In this work, a new quadrilateral 4-node element for laminated plates is formulated within the framework of the First order shear deformation theory (FSDT). The element is based on a hybrid stress formulation derived from the one presented by de Miranda and Ubertini (2006) for single layer isotropic plates. In particular, the same choice of linked displacement interpolation and stress approximation, co-ordinate invariant and equilibrated within each element, is made for the out of plane resultant forces and moments, while the approximation proposed by Pian (1993) is used for the resultant membrane forces. The variational formulation is extended to include the in plane-bending coupling typical of laminated theories. The resulting element has 4 nodes, five degrees of freedom per node, and involves only compatible displacement functions. It is locking-free and readily implementable into existing finite element codes. The described features ensure stability, accuracy, little sensitivity to geometric distortions and computational efficiency. An accurate evaluation of the stress state in the laminate, and in particular of transverse shear stresses along the thickness, is crucial in order to predict critical phenomena such as delamination. The kinematic hypothesis introduced in the first order shear deformation theory yields inaccurate piecewise constant shear stresses which are discontinuous between the laminae. However, accurate transverse shear stress profiles can be evaluated in the post-processing through three dimensional equilibrium equations. From the FEM internal moments and membrane forces it is possible to evaluate the curvatures and the membrane strains through the laminate constitutive equation and, so, the in plane strain tensor. The constitutive equations of each lamina allow then to determine the in plane stresses, which can be introduced in the three dimensional equilibrium equations to evaluate the transverse shear stresses. The shear stresses so obtained should satisfy the boundary conditions on both external surfaces of the laminate. Moreover, the shear stress resultants along the plate thickness should be equal to the transverse shear forces evaluated via finite element procedure. It can be shown that both conditions are automatically satisfied if the internal moments and membrane forces satisfy the plate equilibrium equations. Indeed, accuracy of the stress profiles depends upon the derivatives of the internal moments and membrane forces, therefore a good approximation not only of the internal forces but also of their derivatives is important. According to this, a two steps procedure is proposed to recover transverse shear stresses. First, a Recovery by Compatibility in Patches procedure is implemented in order to improve the rate of convergence of the FEM internal forces. Then, using the recovered internal forces, the transverse shear stress profiles are determined as described above. The effectiveness of the proposed element and the transverse shear stress recovery has been verified on various test problems.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.