Theory of Elasticity Theory of elasticity governs response – Symmetric stress & strain components Governing equations – Equilibrium equations (3) – Strain-displacement equations (6) – Constitutive equations (6) Unknowns – Stress (6) – Strain (6) – Displacement (3)

Boundary & Initial Conditions Linear elastic material – Three partial differential equations in displacements – Second order in each coordinate – Second order in time On a free surface in each direction – Specify stress or displacement but not both Initial conditions for each direction specify – Displacement and velocity

Surface Forces Specify pressure (  zz ), shears (  zx,  zy ) or Specify displacements (u,v,w)

Rigid Body Motion – 2D

Rigid Body Displacements – 2D Strains vanish Integrating normal strains Integrating shear strain Hence ( U, V are constants) Displacement solution

Reactions with Excessive Constraints Extra constraint in horizontal direction will add excessive stress Vertical constraints and loads produce point load infinite stresses

Rigid Body Displacements – 3D All strains vanish In terms of displacements Integrating yields displacements Where And

Self-Equilibrating Forces Examples include: – Uniform pressure (submarine or bathysphere) – Thermal expansion BCs remove rigid body translations & rotations – Constrain six degrees of freedom (3 dofs at one point, 2 dofs at a second and 1 dof at a third)

Plate & Beam Dofs at Each Node Beam – 3 translations 3 rotations Plate – 3 translations 2 rotations

Nastran FE Code – Plate Elements All nodes for all elements types have six dofs – 3 for translation – 3 for rotation Flat plate models need dofs perpendicular to plane of model constrained (set to zero) Shells made of plate elements do not Solid elements need all three rotations at each node set to zero

Simply Supported Beam Example Fix six dofs – 5 translation and 1 rotation

Simply Supported Beam Example Fix six dofs – 5 translation and 1 rotation

Cantilever Beam Fix six dofs – 3 translation and 3 rotation

Internal Surfaces & Cracks Cracks Internal Surfaces

Hertz Contact - Gaps & Friction Hertz Contact Gap & Friction Elements

Transformations

Use of Symmetry Makes a large problem smaller Axisymmetry reduces a 3D problem to 2D Recall stress & strain symmetric Examples:

Periodic Boundary Conditions Stress & Strain are periodic Mean displacements can vary linearly with coordinates due to expansion and rotation For

Multi-Point Constraints - Tying where x i are specified degrees of freedom, c ij d j are known constants. or

Distant Boundary Conditions Build a sufficiently large model – At least 20 times length of largest dimension of interest Substructure a large coarse model – Use output from large model as input to a refined local model – Use super-elements or substructuring – Use infinite elements (when available)