tailieunhanh - Introduction to Elasticity Part 10

Tham khảo tài liệu 'introduction to elasticity part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 1 Tx -2000 1 Ty 0 1 Tz 0 2 Tz 0 3 Tz 0 4 Tx 2000 4 Ty 1000 4 Tz 0 5 Tz 0 Material Usage Summary Material steel Number 6 Length Mass Total mass The vertical displacement of node 3 the DOF 2 value is the same value obtained by the closed-form methods of Module 5. Figure 6 shows the velvet graphical output for the truss deflections greatly magnified . Figure 6 The 6-element truss in its original and deformed shape. General Stress Analysis The element stiffness matrix could be formed exactly for truss elements but this is not the case for general stress analysis situations. The relation between nodal forces and displacements are not known in advance for general two- or three-dimensional stress analysis problems and an approximate relation must be used in order to write out an element stiffness matrix. In the usual displacement formulation of the finite element method the governing equations are combined so as to have only displacements appearing as unknowns this can be done by using the Hookean constitutive equations to replace the stresses in the equilibrium equations by the strains and then using the kinematic equations to replace the strains by the displacements. This gives LTa LTDe LT DLu 0 3 9 Of course it is often impossible to solve these equations in closed form for the irregular boundary conditions encountered in practical problems. However the equations are amenable to discretization and solution by numerical techniques such as finite differences or finite elements. Finite element methods are one of several approximate numerical techniques available for the solution of engineering boundary value problems. Problems in the mechanics of materials often lead to equations of this type and finite element methods have a number of advantages in handling them. The method is particularly well suited to problems with irregular geometries and boundary conditions and it can be implemented in general computer codes that can be used for many .