A cantilever beam, made of a material with a Young's modulus, E with a rectangular cross section with a moment of inertia I, is under a uniformly distributed load, q, across the length L of the cantilever at a point load P at the free end of the cantilever.
Visualization of Problem 3.1
What is the maximum deflection of the beam? Express your answer in terms of q,E,I,L,P.
To determine the maximum deflection of the cantilever beam, we can use the equation for deflection of a cantilever beam under a uniformly distributed load and a point load at the free end. The equation for the maximum deflection, delta_max, is given by:
delta_max = (5qL^4 - 10PL^3 + 2P^2L^2) / (384EI)
To understand how to arrive at this equation, let's break down the terms:
1. q: This represents the uniformly distributed load across the length L of the cantilever. It is exerted perpendicular to the beam and is measured in force per unit length.
2. E: The Young's modulus of the material determines the stiffness of the material. It relates stress to strain and is used to calculate the beam's bending deformation. It is measured in force per unit area.
3. I: The moment of inertia describes the cross-sectional shape of the beam. It is a measure of the beam's resistance to bending and is related to the distribution of material around the neutral axis. It is measured in length to the fourth power.
4. L: This is the length of the cantilever beam. It is the distance between the fixed support and the point of load application. It is measured in length.
5. P: The point load applied at the free end of the cantilever beam. It exerts a force perpendicular to the beam at a single point. It is measured in force.
Now, let's look at the formula:
- The numerator of the equation contains the terms (5qL^4 - 10PL^3 + 2P^2L^2).
- The term 5qL^4 represents the deformation caused by the uniform load across the cantilever beam.
- The term -10PL^3 represents the deformation caused by the point load P at the free end of the cantilever.
- The term 2P^2L^2 represents the deformation caused by the interaction between the uniform load and the point load.
- The denominator of the equation contains the terms 384EI, which combines the material properties and the moment of inertia of the cross-section.
By plugging in the given values of q, E, I, L, and P into the equation, you can calculate the maximum deflection, delta_max, of the cantilever beam.