Describe St Venant's principle and its importance to the derived loading scenario
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
Explain what conditions would apply if plane strain can be assumed for analysis of the stress distribution
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.
Body force
Acts per unit volume (eg inertia)
Surface force
Acts per unit area (eg pressure)
Briefly explain the typical scenario that can be described as plane stress, detailing what additional info would be required to confirm that the assumption made for a square thin plate to be modelled as plane stress is valid.
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.
Determine what fundamental principles are satisfied if the biharmonic equation is satisfied
The conditions of compatibility and the equations of equilibrium are satisfied if the biharmonic equation is satisfied.
Outline the main features of St Venant's theory of the torsion of prismatic bars of arbitrary cross-section and show that the distribution of shear stresses in such a bar may be obtained by solving the following eqn subject to BCs:
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
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
Describe the mathematical princples behind the Finite Element Method for analysis of stresses and deformations in linear elastic solides
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
Describe the physical meaning of the Principle of Virtual Work and explain how it can be used to derive finite element equations
PVW - work done by "poking a node" or adding displacement
Used to derive RE eqns
Weak (variational) statement of equilibrium and BCs
