5c-EnergyAndWeakForm-2010s

# 5c-EnergyAndWeakForm-2010s - MAE M168/CEE M135C...

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Unformatted text preview: MAE M168/CEE M135C Introduction to Finite Element Methods MAE 168 Lecture 5c Energy and the Weak Form Summary and Review • Strong form: • Residual function: BCs d du EA q dx dx + = + ( ) d du R x EA q dx dx = + du • Weighted residual: • Integrating by parts + NBCs → the weak form: ( ) ( ) ( ) L L d du R x v x dx EA q v x dx dx dx = + ∫ ∫ [ ( ), ( )] (0) ( ) 0, for all ( ) L L du dv G u x v x EA qvdx P v P v L v x dx dx =--- = ∫ Review of Linear Elasticity • Consider an arbitrary three- dimensional elastic body. • Volume Vol , and surface area S . • Components of displacement: u, v, and w . • Loads: • Work done by loads stored in strain energy . b: a distributed body force per unit volume , P: a concentrated force , p: a distributed traction or force per unit surface area . Stress and Strain • Stress block: nine active stress components at any point in the body. • The six unknown components that are labeled are independent. • To each of the six independent stress omponents there corresponds an components there corresponds an unknown strain component....
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5c-EnergyAndWeakForm-2010s - MAE M168/CEE M135C...

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