16.21 Techniques of structural analysis and design

Home assignment 6
Solution set

Question 1: Problem 5.11 from textbook

Clear["Global`*"]
M1  =  -((3 (Subscript[F, 0]) L)/4) - (((Subscript[q, 0]) (L^2))/8) + ((Subscript[F, 0]) + (((Subscript[q, 0]) L)/2)) x - (((Subscript[q, 0]) (x^2))/2);
M2  =  -(Subscript[F, 0]) (((3 L)/4) - 
              x);
Mcbar  =  -Pcbar ((L/2) - x);
Mebar  =  -Pebar (L - x);

wc  =  
      (1/Pcbar) Integrate[
          M1 (Mcbar/EI), {x, 0, (L/2)}]
we  =  (1/Pebar) (Integrate[M1 (Mebar/EI), {x, 0, (L/2)}]  +  
              Integrate[M2  (Mebar/EI), {x, (L/2), (3/4) L}]) // 
        Simplify

wc - (we/2) // Simplify

Integrate[M1 (Mcbar/EI), {x, 0, L/2}]

Question 2: Problem 5.12 from textbook

\(δ\_B = \ \(1\/P\&_\) \((∫\_0\%a\( F\_0\) 
                x\ \(2\/3\) \(P\&_\) 
                x\ \[DifferentialD]x\  + ∫\_a\%\(2  a\)\(F\_0\) 
                a\ \(1\/3\) \(P\&_\) \((3\ a\  - 
                    x)\)\ \[DifferentialD]x\  + \ ∫\_\(2\ a\)\%\(3  a\)\(F\_0\) \((3\ a\  - x)\)\ \(1\/3\) \(P\&_\) \((3\ a\  - 
                    x)\)\ \[DifferentialD]x\ )\)\)

Question 3: Problem 5.14 from textbook

(Subscript[M, 1]) =  -(Subscript[q, 0]) ((L^2)/2) +  (Subscript[F, 0]) (b/(a + b)) 
            x  -  (Subscript[q, 0]) ((L^2)/2) 
            (x/(a + b));
(Subscript[M, 2]) =  (Subscript[M, 1]) -  (Subscript[F, 0]) (x - a);
\(θ\_B = \ \(1\/\(\(M\&~\)\ EI\)\) \((∫\_0\%a\( 
                      M\_1\) \(\(\(M\)\(\ \)\)\&~\) \((a + b\  - \ x)\)\/\(a + b\)\ \[DifferentialD]x\  + ∫\_a\%\(a + b\)\(M\_2\) \(\(\(M\)\(\ \)\)\&~\) \((a + b\  - \ x)\)\/\(a + b\)\ \[DifferentialD]x\ )\) // 
        Simplify\)

values  =  {a ->  8,  b ->  12,  (Subscript[F, 0]) ->  40,  
        (Subscript[q, 0]) ->  0.5,  L ->  16}
(Subscript[θ, B]) /.  values