A simply supported beam of length l carries a uniformly…

A simply supported beam of length $l$ carries a uniformly distributed load of total weight $P$. Find the equation of the elastic curve and the deflection at the midpoint.

Step 1: Setup and reactions

Place the origin at the left support, $x$ to the right, $0\le x\le l$. The load intensity is $q=P/l$; by symmetry each reaction carries half the load: \[R_A = R_B = \frac{P}{2} = \frac{ql}{2}.\]

Step 2: Bending moment

At a section a distance $x$ from the left support (sagging positive): \[M(x) = R_A x - \frac{qx^2}{2} = \frac{q}{2}\,x(l-x). \tag{1}\]

Step 3: Differential equation of the elastic curve

With $y(x)$ the downward deflection, $EI\,y''=-M(x)$: \[EI\,y'' = -\frac{q}{2}(lx-x^2). \tag{2}\]

Step 4: First integration

\[ EI\,y' = -\frac{q}{2}\!\left(\frac{lx^2}{2}-\frac{x^3}{3}\right) + C_1. \]

By symmetry the slope vanishes at the midpoint, $y'(l/2)=0$: \[0 = -\frac{q}{2}\!\left(\frac{l^3}{8}-\frac{l^3}{24}\right)+C_1 = -\frac{q}{2}\cdot\frac{l^3}{12}+C_1 \ \Rightarrow\ C_1=\frac{ql^3}{24}.\]

\[ EI\,y' = \frac{q}{24}\bigl(l^3-6lx^2+4x^3\bigr). \tag{3} \]

Step 5: Second integration

\[ EI\,y = \frac{q}{24}\bigl(l^3x-2lx^3+x^4\bigr) + C_2. \] $y(0)=0 \Rightarrow C_2=0$; check the right support: $EI\,y(l)=\dfrac{q}{24}(l^4-2l^4+l^4)=0. $\checkmark$$

Step 6: Equation of the elastic curve

$y(x) = \frac{q$.

(using $q=P/l$).

Step 7: Deflection at the midpoint

Setting $x=l/2$: \[EI\,y\!\left(\frac{l}{2}\right) = \frac{q}{24}\!\left(\frac{l^4}{2}-\frac{l^4}{4}+\frac{l^4}{16}\right) = \frac{q}{24}\cdot\frac{5l^4}{16} = \frac{5ql^4}{384},\]

since $\tfrac12-\tfrac14+\tfrac1{16}=\tfrac{8-4+1}{16}=\tfrac{5}{16}$. With $q=P/l$:

$\delta_{\max$.

Step 8: Verification

Differentiate the elastic curve twice: $y'=\dfrac{P}{24EIl}(l^3-6lx^2+4x^3)$, so

\[ y'' = \frac{P}{24EIl}(-12lx+12x^2) = -\frac{P}{2EIl}\,x(l-x) \ \Rightarrow\ -EI\,y'' = \frac{P}{2l}x(l-x) = M(x). \checkmark \]

And the slope at midspan recovers zero directly: $y'(l/2)=\dfrac{P}{24EIl}\!\left(l^3-\tfrac{3l^3}{2}+\tfrac{l^3}{2}\right)=0. $\checkmark$$