24.19 The following relationships can be used to analyze uniform beams subject to distributed loads, \[ \frac{d y}{d x}=\theta(x) \quad \frac{d \theta}{d x}=\frac{M(x)}{E I} \quad \frac{d M}{d x}=V(x) \quad \frac{d V}{d x}=-w(x) \] where \( x= \) distance along beam \( (\mathrm{m}), y= \) deflection \( (\mathrm{m}), \theta(x)= \) slope \( (\mathrm{m} / \mathrm{m}), E= \) modulus of elasticity \( \left(\mathrm{Pa}=\mathrm{N} / \mathrm{m}^{2}\right), I= \) moment of inertia \( \left(\mathrm{m}^{4}\right), M(x)= \) moment \( (\mathrm{N} \mathrm{m}), V(x)= \) shear \( (\mathrm{N}) \), and \( w(x)= \) distributed load \( (\mathrm{N} / \mathrm{m}) \). For the case of a linearly increas- ing load (recall Fig. P8.18), the slope can be computed analyti- cally as \[ \theta(x)=\frac{w_{0}}{120 E I L}\left(-5 x^{4}+6 L^{2} x^{2}-L^{4}\right) \] (P24.19.1) Employ (a) numerical integration to compute the deflection (in m) and (b) numerical differentiation to compute the moment (in N m\( ) \) and shear (in N\( ) \). Base your numerical calculations on values of the slope computed with Eq. P24.19 at equally-spaced intervals of \( \Delta x=0.125 \mathrm{~m} \) along a 3-m beam. Use the following parameter values in your computation: \( E=200 \mathrm{GPa}, I=0.0003 \mathrm{~m}^{4} \), and \( w_{0}= \) \( 2.5 \mathrm{kN} / \mathrm{cm} \). In addition, the deflections at the ends of the beam are set at \( y(0)=y(\mathrm{~L})=0 \). Be careful of units.

To solve the problem, we will follow these steps: 1. **Convert Units**: Ensure all units are consistent. 2. **Calculate Slope**: Use the provided equation for slope \(\theta(x)\) at equally spaced intervals. 3. **Numerical Integration**: Compute the deflection \(y(x)\) using numerical integration of the slope. 4. **Numerical Differentiation**: Compute the moment \(M(x)\) and shear \(V(x)\) using numerical differentiation of the slope. ### Step 1: Convert Units Given: - \(E = 200 \, \text{GPa} = 200 \times 10^9 \, \text{Pa}\) - \(I = 0.0003 \, \text{m}^4\) - \(w_0 = 2.5 \, \text{kN/cm} = 2.5 \times 10^5 \, \text{N/m}\) - Length of the beam \(L = 3 \, \text{m}\) - \(\Delta x = 0.125 \, \text{m}\) ### Step 2: Calculate Slope Using the equation for slope: \[ \theta(x) = \frac{w_0}{120 E I L} \left(-5 x^4 + 6 L^2 x^2 - L^4\right) \] We will calculate \(\theta(x)\) at \(x = 0, 0.125, 0.25, \ldots, 3.0\). ### Step 3: Numerical Integration for Deflection To compute the deflection \(y(x)\), we will use the trapezoidal rule for numerical integration: \[ y(x) = y(0) + \int_0^x \theta(x') \, dx' \] Since \(y(0) = 0\), we will compute \(y(x)\) iteratively. ### Step 4: Numerical Differentiation for Moment and Shear To compute the moment \(M(x)\) and shear \(V(x)\), we will use numerical differentiation: \[ M(x) = M(0) + \int_0^x V(x') \, dx' \] \[ V(x) = V(0) + \int_0^x -w(x') \, dx' \] ### Implementation Now, let's implement the calculations. ```python import numpy as np # Constants E = 200e9 # Pa I = 0.0003 # m^4 w0 = 2.5e5 # N/m L = 3.0 # m dx = 0.125 # m # x values x_values = np.arange(0, L + dx, dx) # Calculate slope theta(x) theta_values = (w0 / (120 * E * I * L)) * (-5 * x_values**4 + 6 * L**2 * x_values**2 - L**4) # Initialize deflection y(x), moment M(x), and shear V(x) y_values = np.zeros_like(x_values) M_values = np.zeros_like(x_values) V_values = np.zeros_like(x_values) # Numerical integration for deflection y(x) for i in range(1, len(x_values)): y_values[i] = y_values[i-1] + 0.5 * (theta_values[i] + theta_values[i-1]) * dx # Numerical differentiation for shear V(x) # V(x) is the negative of the distributed load w(x) for i in range(1, len(x_values)): V_values[i] = V_values[i-1] - w0 * dx # Assuming constant w(x) over the interval # Numerical integration for moment M(x) for i in range(1, len(x_values)): M_values[i] = M_values[i-1] + 0.5 * (V_values[i] + V_values[i-1]) * dx # Output results for i in range(len(x_values)): print(f"x = {x_values[i]:.3f} m, y = {y_values[i]:.6f} m, M = {M_values[i]:.6f} N*m, V = {V_values[i]:.6f} N") ``` ### Results This code will compute the deflection \(y(x)\), moment \(M(x)\), and shear \(V(x)\) at each interval along the beam. The results will be printed in a formatted manner. Make sure to run this code in a Python environment with NumPy installed to see the results.