Hi,
To calculate the deflection of a simply supported tapered beam under a uniform transverse load in MATLAB, you can follow a numerical approach such as the finite difference method or finite element method (FEM). Here, I'll outline a simple approach using the Euler-Bernoulli beam theory and numerical integration.
Assumptions:
- The beam is linearly tapered (i.e., the cross-sectional area changes linearly along its length).
- The load is uniformly distributed across the beam.
- The material is homogeneous and isotropic.
Beam Parameters:
- Length ( L )
- Modulus of Elasticity ( E )
- Moment of Inertia at each end ( I1 ) and ( I2 )
- Uniform load ( w )
MATLAB Program:
% Define beam parameters
L = 10; % Length of the beam (m)
E = 210e9; % Modulus of Elasticity (Pa)
I1 = 1e-4; % Moment of Inertia at left end (m^4)
I2 = 5e-4; % Moment of Inertia at right end (m^4)
w = 1000; % Uniform load (N/m)
% Number of segments for numerical integration
n = 1000;
dx = L / n;
% Initialize deflection array
x = linspace(0, L, n+1);
deflection = zeros(size(x));
% Calculate the moment of inertia at each point
I = I1 + (I2 - I1) * (x / L);
% Numerical integration to calculate deflection
for i = 2:n+1
% Calculate the bending moment at each point
M = w * (L * x(i) - x(i)^2 / 2) / 2;
% Calculate the curvature
curvature = M ./ (E * I(i));
% Integrate curvature to find slope and deflection
deflection(i) = deflection(i-1) + curvature * dx;
end
% Plot the deflection curve
figure;
plot(x, deflection, 'b', 'LineWidth', 2);
xlabel('Length along the beam (m)');
ylabel('Deflection (m)');
title('Deflection of a Simply Supported Tapered Beam');
grid on;
Explanation:
1. Beam Parameters: Define the length, modulus of elasticity, moments of inertia at both ends, and the uniform load.
2. Discretization: Divide the beam into small segments to perform numerical integration.
3. Moment of Inertia Calculation: Calculate the moment of inertia at each point along the beam using linear interpolation.
4. Numerical Integration: Compute the deflection using the curvature derived from the bending moment and integrate it over the beam's length.
5. Plotting: Visualize the deflection curve along the beam.
This script provides a basic numerical approach to estimate the deflection of a tapered beam under a uniform load. For more accurate results, especially for complex beam shapes or load conditions, consider using more sophisticated methods like the finite element method (FEM) with built-in MATLAB functions or toolboxes.
Thank you.
