Attention - reading the theoretical introduction may help in understanding this example.

We calculate the support reactions, and write down the moment function from the left side

\begin{aligned} \sum M_F=0\\ &R_{AY}\cdot 11-30\cdot 9-8\cdot 3\cdot 5.5+20=0\\ &R_{AY}=34.73kN\\ &\sum M_A=0\\ &30\cdot 2+5\cdot 3\cdot 5.5+20=R_{FY}\cdot 11=0\\ &R_{FY}=19.27kN\\ \end{aligned}

I write down the bending moment equation for the most distant interval:

\begin{aligned} &M_g\left(x\right)=R_{AY}x-30\left(x-2\right)-\frac{1}{2}\cdot 8\left(x-4\right)^2+\frac{1}{2}\cdot 8\left(x-7\right)^2+20\left(x-9\right)^0\\ \end{aligned}

I write down the deflection equation, substituting the previously calculated reaction values:

\begin{aligned} &EIw''=-M_{g1}=-34.73\cdot x+30\cdot\left(x-2\right)+4\cdot\left(x-4\right)^2-4\cdot\left(x-7\right)^2-20\cdot\left(x-9\right)^0\\ &EIw'=-34.73\cdot \frac{x^2}{2}+30\cdot\frac{\left(x-2\right)^2}{2}+4\cdot\frac{\left(x-4\right)^3}{3}-4\cdot\frac{\left(x-7\right)^3}{3}-20\cdot\left(x-9\right)+C\\ &EIw=-34.73\cdot\frac{x^3}{6}+30\cdot\frac{\left(x-2\right)^3}{6}+4\cdot\frac{\left(x-4\right)^4}{12}-4\cdot\frac{\left(x-7\right)^4}{12}-20\cdot\frac{\left(x-9\right)^2}{2}+C\cdot x+D\\ \end{aligned}

Using boundary conditions, I calculate the integration constants:

\begin{aligned} &w\left(x=0\right)=0\\ &D=0\\ &w\left(x=11\right)=0\\ &EIw=-34.73\frac{{11}^3}{6}+30\frac{\left(11-2\right)^3}{6}+4\frac{\left(11-4\right)^4}{12}-4\frac{\left(11-7\right)^4}{12}-20\frac{\left(11-9\right)^2}{2}+C\cdot11+D=0\\ \end{aligned}

Hence:

\begin{aligned} &C=307.66kNm^3\\ \end{aligned}

Since we need to calculate the displacement of point C, I write:

\begin{aligned} &w\left(x=5\right)=\frac{1}{EI}\left(-34.73\cdot\frac{4^3}{6}+30\cdot\frac{\left(4-2\right)^3}{6}+307.66\cdot4\right)=\frac{900.187}{EI}\\ \end{aligned}

The function can also be written on the right side. We encourage you to check this option, calculate the desired displacement, and compare the results.

Deflections

Strength of materials - Deflections - Example 12

Example 12