## Step 1: **Equation of Elastic Curve (Bending Equation)** Given: - Simply supported beam - Length \( L = 12 \) m (6 m + 6 m) - Uniformly distributed load \( w = 12 \) kN/m - \( EI \) = constant **Bending Moment at a distance \( x \) from A:** For a simply supported beam with a UDL: 1. **Calculate Reactions:** - Total load = \( wL = 12 \times 12 = 144 \) kN - By symmetry, each support gets half: \( R_A = R_B = 72 \) kN 2. **Moment at a section \( x \) from A:** \[ M(x) = R_A x - w \frac{x^2}{2} \] Substitute values: \[ M(x) = 72x - 6x^2 \quad \text{(kNm)} \] 3. **Elastic curve differential equation:** \[ EI \frac{d^2y}{dx^2} = -M(x) \] \[ EI \frac{d^2y}{dx^2} = -(72x - 6x^2) \] Integrate twice to get the equation for the elastic curve: \[ EI \frac{dy}{dx} = -36x^2 + 2x^3 + C_1 \] \[ EI y = -12x^3 + \frac{1}{2}x^4 + C_1x + C_2 \] **Explanation:** This step sets up the governing equation for the deflection curve using the moment-curvature relationship and integrates it, including constants of integration. --- ## Step 2: **Boundary Conditions, Slope at A, and Maximum Deflection** ### **Boundary Conditions** - At \( x = \): \( y = \) (at A) - At \( x = L = 12 \): \( y = \) (at B) Apply at \( x = \): \[ = -12()^3 + \frac{1}{2}()^4 + C_1() + C_2 \implies C_2 = \] Apply at \( x = 12 \): \[ = -12(12)^3 + \frac{1}{2}(12)^4 + C_1(12) \] \[ = -12(1728) + .5 \times 20736 + 12C_1 \] \[ = -20736 + 10368 + 12C_1 \] \[ 12C_1 = 20736 - 10368 \] \[ 12C_1 = 10368 \implies C_1 = 864 \] So the **elastic curve equation** is: \[ EIy = -12x^3 + \frac{1}{2}x^4 + 864x \] ### **Slope at A (\( x = \))** \[ EI \frac{dy}{dx}\Big|_{x=} = -36()^2 + 2()^3 + 864 = 864 \] \[ \text{Slope at A: } \frac{dy}{dx}\Big|_{x=} = \frac{864}{EI} \] ### **Maximum Deflection (at Midspan, \( x = 6 \)):** \[ EIy_{max} = -12(6^3) + \frac{1}{2}(6^4) + 864(6) \] \[ = -12(216) + .5(1296) + 5184 \] \[ = -2592 + 648 + 5184 \] \[ = 324 \] \[ y_{max} = \frac{324}{EI} \] --- ## **Final Answers** 1. **Equation of the Elastic Curve:** \[ EI y = -12x^3 + \frac{1}{2}x^4 + 864x \] 2. **Slope at A:** \[ \left. \frac{dy}{dx} \right|_{x=} = \frac{864}{EI} \] 3. **Maximum Deflection (at midspan):** \[ y_{max} = \frac{324}{EI} \] **Summary:** - We derived the elastic curve by integrating the moment equation and applying boundary conditions. - Slope at A and maximum deflection were computed using the curve equation.