Copyrighted Materials 0p UyPress rm CHAPTER NINE Finite Element Analysis The finite element method offers a practical means of calculating the deformations of,and stresses and strains in,complex structures.A detailed description of the finite element method is beyond the scope of this book.Instead,we focus on those features specific to composite materials. Finite element analysis consists of the following major steps: 1.A mesh encompassing the structure is generated (Fig.9.1). 2.The stiffness matrix [k]of each element is determined. 3.The stiffness matrix [K]of the structure is determined by assembling the ele- ment stiffness matrices. 4.The loads applied to the structure are replaced by an equivalent force system such that the forces act at the nodal points. 5.The displacements of the nodal points d are calculated by k]d f. (9.1) where f is the force vector representing the equivalent applied nodal forces (Fig.9.1). 6.The vector d is subdivided into subvectors 6,each 6 representing the displace- ments of the nodal points of a particular element. 7.The displacements at a point inside the element are calculated by u=[W6, (9.2) where the vector u represents the displacements and [N]is the matrix of the shape vectors. 8.The strains at a point inside the elements are calculated by e=[B6, (9.3) where [B]is the strain-displacement matrix. 395CHAPTER NINE Finite Element Analysis The finite element method offers a practical means of calculating the deformations of, and stresses and strains in, complex structures. A detailed description of the finite element method is beyond the scope of this book. Instead, we focus on those features specific to composite materials. Finite element analysis consists of the following major steps: 1. A mesh encompassing the structure is generated (Fig. 9.1). 2. The stiffness matrix [k] of each element is determined. 3. The stiffness matrix [K] of the structure is determined by assembling the ele￾ment stiffness matrices. 4. The loads applied to the structure are replaced by an equivalent force system such that the forces act at the nodal points. 5. The displacements of the nodal points d are calculated by [K] d = f, (9.1) where f is the force vector representing the equivalent applied nodal forces (Fig. 9.1). 6. The vector d is subdivided into subvectors δ, each δ representing the displace￾ments of the nodal points of a particular element. 7. The displacements at a point inside the element are calculated by u = [N] δ, (9.2) where the vector u represents the displacements and [N] is the matrix of the shape vectors. 8. The strains at a point inside the elements are calculated by ε = [B] δ, (9.3) where [B] is the strain–displacement matrix. 395