Solved: Calculating beam deflections using superposition

Chapter 4, Problem 4-18

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Calculating beam deflections using superposition is quite convenient provided you have a comprehensive table to refer to. Because of space limitations, this book provides a table that covers a great deal of applications, but not all possibilities. Take for example, Prob. 4-19, which follows this problem. Problem 4-19 is not directly solvable from Table A-9, but with the addition of the results of this problem, it is. For the beam shown, using statics and double integration, show that

\(R_1\) = \(\frac{w a}{2 l}(2 l-a) \quad R_2=\frac{w a^2}{2 l} \quad V_{A B}=\frac{w}{2 l}\left[2 l(a-x)-a^2\right] \quad V_{B C}=-\frac{w a^2}{2 l}\)

\(M_{A B}\) = \(\frac{w x}{2 l}\left(2 a l-a^2-l x\right) \quad M_{B C}=\frac{w a^2}{2 l}(l-x)\) 

\(y_{A B}\) = \(\frac{w x}{24 E I l}\left[2 a x^2(2 l-a)-l x^3-a^2(2 l-a)^2\right] \quad y_{B C}=y_{A B}+\frac{w}{24 E I}(x-a)^4\)

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