Solids

St Venant's principle states that the stress distribution is independent of the exact mode of loading except in the immediate vicinity of the load application

Conditions of plane strain apply when the loaded body is very long in the z direction and where there is no displacement in the z direction. The cross-sections are all under the same conditions. For such problems, γ_yz = 0, γ_xy = 0 and E_z = 0. Since σ_z = ν(σ_x + σ_y), the problem reduces to three unkowns, σ_x, σ_y, and Τ_xy.

Acts per unit volume (eg inertia)

Acts per unit area (eg pressure)

Plane stress refers to a thin body where the loads are applied in the plane of the body and act over the thickness. Information on the loads acting on the plate needs to be known to confirm that conditions of plane stress apply.

The conditions of compatibility and the equations of equilibrium are satisfied if the biharmonic equation is satisfied.

Provides exact solutions to the shear stress distributions and warping deformation for a variety of non-circular sections

• Semi inverse method
• Direct methods = find stress function that satisfies geometry and BCs
• Inverse methods = find geometry and BCs that satisfy assumed stress function
• Semi-inverse: make assumptions regarding deformations and stresses, then find coefficients for a 'special' stress function

Has direct analogy with other physical phenomena (eg soap film technique) hence the method can be easily visualised

• St Venant's torsional stress function
• Satisfies Poisson's eqn
• Is constant (usually zero) along boundaries
• Volume under the surface proportional to the applied couple
• Shear stresses are given by the spatial derivatives of this function

• Satisfies Poisson's eqn
• Is constant (usually zero) along boundaries
• Volume under the surface proportional to the applied couple
• Shear stresses are given by the spatial derivatives of this function

• Principles of FEM:
• Numerical method for the solution of PDEs (partial differential eqns) and related boundary value problems (BVP)
• Based on the "variational" formulation of the problem, i.e. its "weak form" with ENERGY being the "functional" of the displacement function
• Thus in solid mechanics it is defined as an ENERGY method
• The "trial" functions of this variational method are the interpolated FE displacements given by the "shape functions"
• Both interpolation (of displacements) and integration (of stresses) are performed within the finite elements
• Suitable for the analysis of complex geometries due to its "element-by-element" nature

• PVW - work done by "poking a node" or adding displacement
• Used to derive RE eqns
• Weak (variational) statement of equilibrium and BCs
• At the element level:
• W[external] - W[internal] = 0