16.21 Techniques of structural analysis and design

Home assignment 8
Solution Set

Question 3: Problem 7.42 from textbook

Follow the inverse steps of problem 5.3 to obtain:
PVD:

\(TraditionalForm\`∫\_0\%L\(∫\_A σ\ δϵ dA dx\)\  = \ ∫\_0\%L\( 
                q\_0\) \(\(δw\_0\)(x)\) dx\) for all admissible virtual displacements δ(Subscript[w, 0])

ϵ = (Subscript[u, 0])'  +  (1/2) ((((Subscript[w, 0])'))^2) -  
        z (Subscript[w, 0])''
σ = E ϵ
δϵ  =  δ(Subscript[u, 0])'  +  
          (Subscript[w, 0])' δ(Subscript[w, 0])' -  z δ(Subscript[w, 0])''

PVD:
\(TraditionalForm\`∫\_0\%L\((\(N(
                    x)\)\ δu\_0'\  + \ \(N(x)\) 
                  w\_0'\ δw\_0' - \ 
                M\ δw\_0'')\) dx\  = ∫\_0\%L\( q\_0\) \(\(δw\_0\)(x)\) dx\)

and from the definitions of N(x) and M(x):
\(TraditionalForm\`\(\(∫\_0\%L\( EA(
                    u\_0'\  + \ \(1\/2\) \((w\_0')\)\^2)\)\ \((δu\_0'\  + \ w\_0'\ δw\_0')\) + \ 
              EI\ w\_0''\ δw\_0''\)\()\)\) dx\  = ∫\_0\%L\( q\_0\) \(\(δw\_0\)(x)\) dx\)