Lecture 3 : Potential Energy Hans Welleman

Ir J.W. Welleman Work and Energy methods2 Potential Energy EpEp h F g =mg plane of reference mgh 2 mgh 1 h hh mgh 2 mgh 1 no kinetic energy (statics)

Ir J.W. Welleman Work and Energy methods3 Total Energy Sum of all energy in a system is constant Sum of all Potential energy = C Potential energy: –Loads (energy with respect to reference, reduces) –Strain energie (E v, increases)

Ir J.W. Welleman Work and Energy methods4 Stable Equilibrium Equilibrium x V pertubation Stationary Energy Function (hor. tangent)

Ir J.W. Welleman Work and Energy methods5 Potential Energy due to Loads Load Potential u F Situation 0Situation 1 F force spring characteristics u F displ. F Situation 2

Ir J.W. Welleman Work and Energy methods6 Stationary Energy Function at a Minimum Energy level Extreme = derivative with respect to a governing variable ( u ) must be zero Extreme is a minimum = 2nd deriv > 0 principle of minimum potential energy

Ir J.W. Welleman Work and Energy methods7 Application Approximate displacement field Demand Stationary Potential Energy: derivative(s) of V with respect to all governing variables a i must be zero.

Ir J.W. Welleman Work and Energy methods8 Example : Beam F l w(x)w(x) z, w x

Ir J.W. Welleman Work and Energy methods9 Solution

Ir J.W. Welleman Work and Energy methods10 Minimalise approximation

Ir J.W. Welleman Work and Energy methods11 Example : Rigid Block kkk 4a2a F a

Ir J.W. Welleman Work and Energy methods12 Displacement field u (assumption) k kk 4a2a F u1u1 u2u2 u3u3

Ir J.W. Welleman Work and Energy methods13 Result exact solution