16.21 - Techniques of Structural Analysis and Design
Spring 2005

Midterm Test

Student Name:

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b) Use the Unit Dummy Load Method, which is a special case of the Principle of Virtual Forces. The principle imposes the principle of compatibility. Equilibrium and constitutive behavior needs to be brought, as the principle does not enforce them. It's particularly useful when the actual (equilibrated) stress field can be computed independently, as in statically-determined structures like this problem. 


a) 

δP \(TraditionalForm\`\(w\_0\)(
          L)\  = \ ∫\_0\%\(L\_1\)\(\(M\_1\ δM\_1\)\/\(\(E\_1\) I\_1\)\) dx\  + ∫\_0\%\(L\_2\)\(\(M\_2\ δM\_2\)\/\(\(E\_2\) I\_2\)\) \(\(d\)\(x\)\(\ \)\)\), ∀ admissible δP

In this expression, δP is a virtual force applied at the tip of the cantilever beam where the value of the deflection is sought, δ(Subscript[M, 1]) and δ(Subscript[M, 2]) are the moment distributions produced by the virtual force δP, (Subscript[M, 1]) and (Subscript[M, 2]) are the actual (equilibrated) moment distributions. The success of the method relies on the ability of computing these moment distributions easily, as in this case:


(Subscript[M, 1]) =  P L
δ(Subscript[M, 1]) =  δP L

(Subscript[M, 2]) =  P x, where x is measured from the free end in the horizontal beam
δ(Subscript[M, 2]) =  δP x

canceling δP in the expression above since it must hold for all admissible δP:

\(TraditionalForm\`\(w\_0\)(
          L)\  = \ \(∫\_0\%\(L\_1\)\(\(P\ L\ \ L\)\/\(\(E\_1\) 
                      
                      I\_1\)\) dx\  + ∫\_0\%\(L\_2\)\(\(P\ x\ x\)\/\(\(E\_2\) 
                      I\_2\)\) dx = \ \(\(\(P\ L\^3\)\/EI\) \((1\  + \ 1\/3)\)\  = \ \(4\/3\) PL\^3\/EI\)\)\)

For the values of the problem:

EI  =  200 (10^9) 3 (10^(-6))
L  =  3
P  =  1000
w  =  (4/3) P ((L^3)/EI) // N