16.21 Techniques of structural analysis and design

Home assignment 5 
Handed out: 3/12/05
Due: 3/18/05
Note that the assignment has been shortened to 4 problems as you have one fewer day to work on it.

Solidworks practice exercise:
Create
a solid model for  the following Wing Box Section using Solidworks

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Question 1: Problem 5.1 from textbook: Use the principle of virtual displacements to determine the governing equations
and natural boundary conditions associated with the problem in Exercise 4.4.

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The kinematic assumption for the bar is:
(Subscript[u, 1])  =  u(x), ϵ = u'(x)
The constitutive law: σ = E ϵ
For the spring:
(Subscript[F, s]) =  k u(L)
The expression of internal virtual work is:
\(TraditionalForm\`δW\_int = \(∫\_v σ\ δϵ dx\ \  + \ \(F\_s\) \(δu(
                L)\)\  = ∫\_0\%L 
                A\ E\ u'\ δu' dx\  + \ \(F\_s\) \(δu(L)\)\)\)
The expression of external work is:
δ(Subscript[W, ext]) =  (∫_0^L) f(
                x) δu(x) dx  +  
          P δu(L)
The PVD reads: \(TraditionalForm\`\(\(δW\_int\)\(=\)\(δW\_ext\)\(\ \)\)\)∀ admissible δu
(∫_0^L) 
            A E u' δu' dx  +  (Subscript[F, s]) δu(
            L) = (∫_0^L) f(x) δu(
              x) dx  +  P δu(L), ∀ admissible δu
Integrating by parts the first term:
\(TraditionalForm\`∫\_0\%L 
            A\ E\ u'\ δu' dx\  = \ \(\(-\(∫\_0\%L\((A\ E\ u')\)'\ δu dx\)\)\  + \ ∫\_0\%L\((A\ E\ u'\ δu)\)' dx\  = 
\ \(\(\(-\(∫\_0\%L\((A\ E\ u')\)'\ δu dx\)\)\  + \ \((A\ E\ u'\ δu)\)\)\(\( | \_0\)\^L\)\)\)\)
applying the essential boundary conditions, i.e. δu(0) = 0
 \(TraditionalForm\`\(\(=\)\(\(-\(∫\_0\%L\((A\ E\ u')\)'\ δu dx\)\)\  + \ \((A\ E\ u'\ δu)\) \((L)\)\)\)\).
 Replacing everything  in PVD:
 \(TraditionalForm\`\(\(\(-\(∫\_0\%L\([\((A\ E\ u')\)'\  + \ 
                    f(x)\ ]\)\ δu dx\)\)\  + \ \((A\ E\ u'\ δu)\) \((L)\)\  + \ k\ \(u(L)\)\ \(δu(L)\)\  - \ 
          P\ \(δu(L)\)\)\(\ \)\(=\)\(\ \)\(0\)\(\ \)\)\) ∀ admissible δu, we obtain the conditions:
 (A E u')' + f(x) = o, 0<x<L
 essential boundary conditions: u(0) = 0
 natural boundary conditions:
 A E u'(L) + k u(L) - P = 0

Question 2: Problem 5.2 from textbook:Use the principle of virtual displacements to determine the governing equations
and natural boundary conditions associated with the problem in Exercise 4.5.

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From beam theory:
u(x)  =  -z (Subscript[w, 0])' (x), ϵ =  -z (Subscript[w, 0])'' (x), (Subscript[σ, 11]) =  E (Subscript[ϵ, 11])
\(TraditionalForm\`δW\_int = \(∫\_v σ\ δϵ dx\ \  + \ \(F\_s\) \(\(δw\_0\)(
                L)\)\  = ∫\_0\%L\ \(∫\_\(A(x)\)\(z\^2\) dA\ E\ w\_0'' \((x)\)\ \ δw\_0'' \((x)\) dx\)\  + \ \(F\_s\) \(\(δw\_0\)(L)\)\)\ \)
\(TraditionalForm\`\(\(=\)\(\ \)\(∫\_0\%L\ \(I(
                x)\)\ E\ w\_0'' \((x)\)\ \ δw\_0'' \((x)\) dx\  + \ \(\(F\_s\) \(\(δw\_0\)(L)\)\(\ \)\)\)\)\)
Using resultant forces: M = EI (Subscript[w, 0])''
\(TraditionalForm\`\(\(=\)\(\ \)\(∫\_0\%L\( M(
                x)\)\ \ δw\_0'' \((x)\) dx\  + \ \(\(F\_s\) \(\(δw\_0\)(L)\)\(\ \)\)\)\)\)
The first term can be integrated by parts twice to eliminate the derivatives in the virtual deflections as follows (essential boundary conditions are used along the way, enforcing admissibility of δ(Subscript[w, 0]): δ(Subscript[w, 0])(
          0)  =  δ(Subscript[w, 0])' (0)  =  0):
\(TraditionalForm\`∫\_0\%L\( M(
              x)\)\ \ δw\_0'' \((x)\) dx = \(\(\(\(-\(∫\_0\%L 
                      M' \((x)\)\ \ δw\_0' \((x)\) dx\)\)\  + \ \(M(
                  x)\)\ δw\_0' \((x)\)\)\(\( | \_0\)\^L\)\)\  = 
\ \(\(\(∫\_0\%L 
                      M'' \((x)\)\ \ \(\(δw\_0\)(
                      x)\) dx\  + \ \(M(
                    L)\)\ δw\_0' \((L)\)\  - \ 
                M' \((x)\)\ \(\(δw\_0\)(
                    x)\)\)\(\( | \_0\)\^L\)\) = 
∫\_0\%L M'' \((x)\)\ \ \(\(δw\_0\)(
                    x)\) dx\  + \ \(M(
                  L)\)\ δw\_0' \((L)\)\  - \ 
              M' \((L)\)\ \(\(δw\_0\)(L)\)\)\)\)


δ(Subscript[W, ext ]) =  (∫_0^L) 
                (Subscript[q, 0]) δ(Subscript[w, 0])(x) dx

PVD then reads:
(∫_0^L) M'' (x)  δ(Subscript[w, 0])(
                x) dx  +  M(
              L) δ(Subscript[w, 0])' (L)  -  
          M' (L) δ(Subscript[w, 0])(L) +  (Subscript[F, s]) δ(Subscript[w, 0])(
              L)  =  (∫_0^L) (Subscript[q, 0]) δ(Subscript[w, 0])(
              x) dx,  ∀  
        admissible δ(Subscript[w, 0]) 
This leads to:

M''(x) = (Subscript[q, 0]), 0<x<L
essential boundary conditions: (Subscript[w, 0])(0)  =  (Subscript[w, 0])' (0)  =  0
natural boundary conditions:
M(L) = 0
M'(L) = (Subscript[F, s]) =  k (Subscript[w, 0])(L)

Question 3: Problem 5.3 from textbook (modified): Use the principle of virtual displacements to derive the equations governing the
Euler-Bernoulli beam theory with geometric nonlinearity. Use a distributed
transverse load of q = q(x) and assume that the displacement field is given by
u(x, y, z) = (Subscript[u, 0])(
          x)  -  z (Subscript[w, 0])' (x) ,
w(x,y,z) = (Subscript[w, 0])(x)
and the only non-zero strain is (nonlinear)
(((Subscript[ϵ, xx]) =  
                (Subscript[u, 0])' (x)  +  (1/2)[(Subscript[w, 0])' (x)])^2)- z (Subscript[w, 0])'' (x)  
Here (Subscript[u, 0]) and (Subscript[w, 0]) are the axial and transverse displacements of a point (x; y; 0)
in the beam. Your answer should be expressed in terms of the area-integrated
quantities [e.g., stress resultants of Eq. (5.9)].

(Subscript[σ, xx])= E (Subscript[ϵ, xx])= E ((( (Subscript[u, 0])' (x)  +  (1/2)[
                          (Subscript[w, 0])' (x)])^2)- z (Subscript[w, 0])'' (x) ) 
N(
          x)  =  (∫_(A(x)))σ dA  =  
          AE ((Subscript[u, 0])' (x)  +  (1/2)[(Subscript[w, 0])' (x)]^2)
M(x) = - \(TraditionalForm\`∫\_\(A(x)\)σ\ z dA\  = \ EI\ w\_0'' \((x)\)\), as derived before

The virtual strains are:
δ(Subscript[ϵ, xx]) =  δ(Subscript[u, 0])' (x)  +  
                  (Subscript[w, 0])' (x) δ(Subscript[w, 0])'(x)- z δ(Subscript[w, 0])'' (x)  

 σ δϵ=E {((( (Subscript[u, 0])'  +  (1/2)[(Subscript[w, 0])'])^2)) (δ(Subscript[u, 0])'  +  (Subscript[w, 0])' δ(Subscript[w, 0])') -
z [(Subscript[w, 0])'' (δ(Subscript[u, 0])' +  (Subscript[w, 0])'δ(Subscript[w, 0])') + δ(Subscript[w, 0])''((( (Subscript[u, 0])'  +  (1/2)[(Subscript[w, 0])'])^2))] +
(z^2) 
                  (Subscript[w, 0])'' δ(Subscript[w, 0])''}
 δ(Subscript[W, int])= \(∫\_V σ\ δϵ dV\) =  \(∫\_0\%L\(∫\_A σ\ δϵ dA dx\)\) =(∫_0^L)E{((( (Subscript[u, 0])'  +  (1/2)[
                                    (Subscript[w, 0])'])^2)) (δ(Subscript[u, 0])'  +  (Subscript[w, 0])' δ(Subscript[w, 0])')(∫_A)dA  -
(∫_A) 
                            z dA [  ...  ]  + \(∫\_A\( 
                                z\^2\) dA\ w\_0''\ δw\_0''\)} dx =
(∫_0^L)E{ A ((( (Subscript[u, 0])'  +  (1/2)[
                                    (Subscript[w, 0])'])^2)) (δ(Subscript[u, 0])'  +  (Subscript[w, 0])' δ(Subscript[w, 0])') - 0 [  ...  ]  + I (Subscript[w, 0])'' δ(Subscript[w, 0])'' }  \(\(d\)\(x\)\(
\)\)
 Using the definition of the stress resultants above:
 \(TraditionalForm\`δW\_int = ∫\_0\%L\((N\ δu\_0'\  - \ N\ w\_0'\ δw\_0'\  + \ 
                M\ δw\_0'')\) \(\(d\)\(x\)\(\ \)\)\)
 
The bending moment term can be integrated by parts twice as before (no specific boundary conditions assumed):
\(TraditionalForm\`∫\_0\%L\( M(
              x)\)\ \ δw\_0'' \((x)\) dx = \ \(\(∫\_0\%L M'' \((x)\)\ \ \(\(δw\_0\)(
                  x)\) dx\  + \ \(M(
                x)\)\ δw\_0' \((x)\)\)\(\( | \_0\)\^L\)\(\ \)\(\(-\ 
              M'\) \((x)\)\ \(\(δw\_0\)(
              x)\)\)\(\( | \_0\)\^L\)\(\ \ \)\)\)

The "N" terms must be integrated by parts as well to get rid of derivatives in the virtual displacements:
\(TraditionalForm\`∫\_0\%L 
            N\ δu\_0' dx\  = \ \(\(\(-\(∫\_0\%L 
                    N'\ \(δu\_0\) dx\)\)\  + \ \(N(
                x)\) \(\(δu\_0\)(
                x)\)\)\(\ \)\(\( | \_0\)\^L\)\(\ \)\)\)
\(TraditionalForm\`∫\_0\%L\((\(-\ 
                  N\)\ w\_0'\ δw\_0'\ )\) dx\  = \ \(\(∫\_0\%L\ \((N\ w\_0'\ )\)' δw\_0\ dx\ \  - \ \ \(N(x)\)\ \(\(w\_0\)(x)\)'\ \(\(δw\_0\)(
                x)\)\)\(\( | \_0\)\^L\)\(\ \ \ \)\)\)

Putting it all together:
\(TraditionalForm\`δW\_int = ∫\_0\%L\([M''\  + \((N\ w\_0'\ )\)'\ ]\)\ \(\(δw\_0\)(
                x)\) dx - ∫\_0\%L 
                N'\ \(δu\_0\) dx\ \  + \ \(\(boundary\)\(\ \)\(terms\)\(\ \)\)\)

For the external work, assume generic distributed bending  (Subscript[q, 0])(x) and axial (Subscript[p, 0])(x) forces:
δ(Subscript[W, ext]) =  (∫_0^L) (Subscript[q, 0])(
                x) δ(Subscript[w, 0])(
                x) dx  +  (∫_0^L) p(
                x) δ(Subscript[u, 0])(x) dx

Invoking PVD: δ(Subscript[W, int]) =  δ(Subscript[W, ext]),  ∀  
        admissible δ(Subscript[w, 0]),  δ(Subscript[u, 0])
leads to:

M'' + (N (Subscript[w, 0])' )'  =  (Subscript[q, 0]),  0 < x < L
N'= p, 0<x<L

+boundary conditions derived from above as usual

Question 4: Problem 5.5 from textbook

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