16.21 Techniques of structural analysis and design

Home assignment 8
Solution Set

Question 1: Problem 7.43 from textbook

The unknown displacement field for this problem is the deflection of the beam's neutral axis \(TraditionalForm\`\((w\_0)\) . \ 
          We\ approximate\ this\ field\ using\ a\ Ritz\ approximation\ of\ the\ \(type : 
w\_0\ ~\ c\_i  ϕ\_i\)\)

The essential boundary conditions at the clamped end (x=0) of this cantilever beam require:
(Subscript[ϕ, i])(0)=0, (Subscript[ϕ, i])'(0) = 0
A good candidate family of functions (Subscript[ϕ, i])would be: (Subscript[ϕ, i]) =  (((x/L))^(i + 1)),  i = 1, n. Note (x^1) does not satisfy  the zero-slope boundary condition so it's been eliminated.

Clear["Global`*"]; 
    Off[General::spell, General::spell1];

1) approximation functions:

Ck[n_] :=  Table[ck[i], {i, n}]
ϕk[n_]  :=  
      Table[((((x/L)))^(i + 1)), {i, n}]
w[n_]  :=  Ck[n] . ϕk[n]
w[2]

2) approximate potential Π(Ck) for this problem

(Subscript[δ, L])[n_]  :=  
      w[n]  /.  x ->  L
Π[
        n_]  := (EI/2) (∫_0^L) ((((∂_(x, x))w[
                          n]))^2) \[DifferentialD]x  +  
          (k/2) (Subscript[δ, L])[n]^2 -  (∫_0^L) (Subscript[q, 0]) 
              w[n] \[DifferentialD]x // Expand

Π[2]

3) Minimize the potential to find the value of the coefficients that best approximate this problem

gradΠ[n_]  :=  
      Table[D[Π[n], ck[i]], {i, n}]

gradΠ[2]

solution[n_]  :=  
      Solve[gradΠ[n]  ==  0 ,  Ck[n]] // 
        Flatten

solution[1]

wapprox[n_]  :=  w[n]  /.  solution[n]

wapprox[1]