Deflection of a beam by successive integration

[mazz1801]

Homework Statement

A beam is built in at its end supports A and B, and is subject to a mid-span point load F. Using the method of successive integration, determine an expression for the deflection of this beam. Calculate the maximum deflection. Ignore any axial load effects.

Homework Equations

EI^d4v/_dx⁴=w(x)
EI^d3v/_dx³=V(x)
EI^d2v/_dx²=M(x)

The Attempt at a Solution

My attempt at the solution is pretty useless as far as I can tell. Any examples I can find online only demonstrate with a UDL and I am having trouble translating the information to my midspan point load.

My aim is to construct an equation for the moment in the beam. Integrate this equation twice to obtain an equation for the deflection of the beam. Using the boundary conditions that are applicable for a fixed end I will solve for the constants of integration and whaa-la I should have an answer.

Not so simple when I apply my method...

When it comes to creating an equation for the moment in the beam I am getting that the moment is 0 so when I integrate twice to obtain an equation for deflection I am only left with two integration constants that also equal 0.

Knowledge dictates the deflection cannot be zero.

If there is some hero out there that can help me solve this problem I will be forever in debt to you. I have an exam in about 24 hours and this question is worth 10%

Thank you in advance!

 

[pongo38]

Start with V(x) for the left half of the beam. It should be a constant. Integrate it, and try to determine the arbitrary constant from the end conditions of the beam.

 

[SteamKing]

To amplify what pongo38 said, start by constructing a shear and moment diagram for your beam. Some of the reactions may be unknown, but by constructing the necessary integrals and applying the correct boundary conditions, these unknowns can be determined. Don't disregard the example with the UDL; it will help you structure your calculations.

 

[mazz1801]

Thank you so much guys, I am just about to head to the exam and have just figured out my problem! You ledgends!

 

[rezeew]

Dear student,

Thank you for reaching out for help with this problem. I understand that you are struggling to apply the method of successive integration to determine the deflection of a beam with a mid-span point load. This can be a challenging problem, but with a clear understanding of the process, you should be able to solve it successfully.

First, let's review the basic steps of the method of successive integration:

1. Determine the moment equation for the beam using the equation EId2v/dx2=M(x).

2. Integrate this equation twice to obtain an equation for the deflection, v(x).

3. Apply the boundary conditions to solve for the integration constants.

4. Use the resulting equation for deflection to determine the maximum deflection.

Now, let's apply these steps to the given problem. We know that the moment equation for a beam with a point load at the mid-span is given by:

M(x) = -F(x-a)

where F is the point load and a is the distance from the support A to the point load.

Integrating this equation twice with respect to x, we get:

v(x) = -F/24EIx^2(x-a)^2 + C1x + C2

Applying the boundary conditions, we can determine the values of the integration constants C1 and C2. Since the beam is built in at its end supports, the deflection at these points must be zero. This gives us the following equations:

v(0) = 0

v(L) = 0

where L is the length of the beam.

Substituting these values into our equation for deflection, we get:

C1 = 0

C2 = F/24EIa^2(L-a)^2

Therefore, the final equation for deflection is:

v(x) = -F/24EIx^2(x-a)^2 + F/24EIa^2(L-a)^2

To determine the maximum deflection, we can take the derivative of this equation with respect to x and set it equal to zero:

dv/dx = -F/12EIx(x-a)(2x-2a) = 0

Solving for x, we get:

x = a/2

Substituting this value back into our equation for deflection, we get:

vmax = F/384EIa^2(L-a)^2

I hope this