Question 15.2: For the beam and loading shown determine (a) the equation of...

STRATEGY: Determine the elastic curve directly from the load distribution using Eq. (15.8), applying the appropriate boundary conditions. Use this equation to find the desired slope and deflection.
MODELING and ANALYSIS:
Differential Equation of the Elastic Curve. From Eq. (15.8),

\frac{d^4y}{dx^4}=-\frac{w(x)}{EI} \quad \quad \quad \pmb{(15.8)} \\ EI\frac{d^4y}{dx^4}=-w(x)=-w_0\sin\frac{\pi x}{L} \quad \quad \quad \pmb{(1)}

Integrate Eq. (1) twice:

EI\frac{d^3y}{dx^3}=V=+w_0\frac{L}{\pi} \cos\frac{\pi x}{L}+C_1 \quad \quad \quad \pmb{(2)} \\ EI\frac{d^2y}{dx^2}=M=+w_0\frac{L^2}{\pi^2}\sin\frac{\pi x}{L}+C_1x +c_2 \quad \quad \quad \pmb{(3)}

Boundary Conditions:      Refer to Fig. 1.
[x = 0, M = 0]:                             From Eq. (3),                              C_2 = 0
[x = L, M = 0]:                             Again using Eq. (3),

0=w_0\frac{L^2}{\pi^2} \sin \pi+C_1L \quad \quad C_1=0

Thus,

EI\frac{d^2y}{dx^2}=+w_0\frac{L^2}{\pi^2}\sin\frac{\pi x}{L} \quad \quad \quad \pmb{(4)}

Integrate Eq. (4) twice:

EI\frac{dy}{dx}=EI \ \theta=-w_0\frac{L^3}{\pi^3}\cos\frac{\pi x}{L}+C_3 \quad \quad \quad \pmb{(5)} \\ EI \ y=-w_0\frac{L^4}{\pi^4}\sin\frac{\pi x}{L}+C_3x+C_4 \quad \quad \quad \pmb{(6)}

Boundary Conditions:    Refer to Fig. 1.

[x = 0, y = 0]:                             Using Eq. (6),                            C_4 = 0
[x = L, y = 0]:                             Again using Eq. (6),                C_3 = 0

a. Equation of Elastic Curve.            EIy=-w_0\frac{L^4}{\pi ^4}\sin\frac{\pi x}{L}

b. Slope at End A.    Refer to Fig. 2. For x = 0,

EI \ \theta_A=-w_0\frac{L^3}{\pi ^3}\cos0 \quad \quad \quad \theta_A=\frac{w_0 L^3}{\pi^3 EI} \measuredangle

c. Maximum Deflection.     Referring to Fig. 2, for x=\frac{1}{2}L,

ELy_{\text{max}}=-w_0\frac{L^4}{\pi^4}\sin\frac{\pi}{2} \quad \quad \quad y_{\text{max}}=\frac{w_0 L^4}{\pi ^4 EI}\downarrow

REFLECT and THINK: As a check, we observe that the directions of the slope at end A and the maximum deflection are consistent with the deflected shape anticipated for this loading (Fig. 1).