For a cantilever beam subjected to a parabolic distributed load, what is the order of the internal moment expression of the beam as a function of the location

Internal moment is of 2nd order

Explanation:

A cantilever beam has a free end with no forces acting on it, and a fixed end.

First of all, we need to analyze the type of loads that are applied to the beam. We are told that it is a parabolic distributed load.

I've attached an image of what this load looks like on a cantilever.

This load will be given by a function of; q(x) = (q_o*x³)/L³.

That means that at the fixed end, there will be a force acting in the x - direction and y - direction as well as an internal moment. These different forces will need to be considered for us to derive an equation for the internal bending moment.

The free end of the beam (where it says A) has no forces acting on it and as a result of that, there'll be no shear force nor moment acting on it. However, due to the other forces acting on beam there will be an unknown deflection as well as an unknown angle of deformation.

However, at the free end, there are no forces acting on it and as such the boundary conditions will be: V(0) = 0 & M(0) = 0.

Where;

V(0) is shear force at boundary condition

M(0) is the internal bending moment at boundary condition.

Now, when we are trying to find an expression for the internal bending moment here, we'll need just boundary conditions for the parabolic distributed load, the shear force, and the internal bending moment.

Looking at the expression of the load which is q(x) = (q_o*x³)/L³.

We can set up the first differential equation for deflection as;

EI(d⁴v/dx⁴) = q(x) = -(q_o*x³)/L³

Where;

E = Young's Modulus

I = beam moment of inertia

v = beam deflection

q = the parabolic distributed load

L = the length of the cantilever beam

x = an unknown distance from the fixed point of the cantilever beam

d means differentiate.

Now, we will integrate the first differential to obtain the shear force.

Thus:

EI(d³v/dx³) = V(x) = ∫-(q_o*x³)/L³

EI(d³v/dx³) = V(x) = -(q_o*x⁴)/L⁴

Integrating again will yield internal moment.

Thus;

EI(d²v/dx²) = M(x) = ∫-(q_o*x⁴)/L⁴

EI(d²v/dx²) = M(x) = -(q_o*x^(5))/L^(5) + C1

At boundary, x = 0 and thus, C1 = 0

Thus,

EI(d²v/dx²) = M(x) = -(q_o*x^(5))/L^(5)

This is of order 2 differential equation

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