The purpose of this instructable is to go through a fourth-order differential equation so that you may find the internal moment of a beam. For this instructable the only boundary conditions are a distributed load, a free end of the beam, and a side of the beam that is fixed. Upon going through the differential equation (which will be specified later) you will find an equation for what the value of the internal bending moment of the beam is. This process has many real world engineering applications that are implemented when looking at the necessary designs so that safety is ensured.
Step 1: Problem Definition
The first step is to analyze what type of loads are applied to the given beam. In the above picture, there is a distributed load that has a given function of w(x)=(w0*x^3)/L^3, a free end of the beam with no forces acting on it, and a fixed end of the beam that makes it a canitlever. That means on the fixed end of the beam (at point B) there is a force in x-direction, y-direction, and an internal moment. You will take all of these different forces into account to formulate an equation for the internal bending moment of the beam.
Step 2: State the Boundary Conditions
The free end of the beam (where it says A) has no forces acting on it. Because of that there is no shear force and no moment acting on it. However, because of the other forces acting on beam there is an unknown deflection and an unknown angle of deformation.
For the free end of the beam the boundary conditions are: V(0)=0 and M(0)=0. The capital V refers to shear force and the capital M refers to the internal bending moment.
When finding the internal bending moment of the beam you will only need boundary conditions for the distributed load, shear force, and the internal bending moment. Note: if you were to go through the full differential integration you would need boundary conditions for angle of deflection and beam deflection but for this example those will not be used.
Step 3: Set-up the Differential Equation
The first picture is the equation that you start with to find beam deflection. This equation is equal to the equation for the distributed load (which is given for this problem and shown in a previous picture) so you plug the given equation in for w(x).
In the picture there are several variables that I will define for you: E is Young's Modulus, I is the moment of inertia for the beam, v is for beam deflection, w is the distributed load, L is the length of the beam, x is some arbitrary distance along the beam, and d stands for differentiate.
Step 4: First Integration to Obtain V(x)
In this step you will integrate the first equation to get the second equation (which is on the bottom of the first picture). Upon integrating for the first time you will have an equation for shear force that is a function of x. However, upon integrating a constant of integration will appear. This is an unknown value that prohibits you from actually using the equation. To get rid of the unknown constant integration you have to plug in the boundary conditions that were established earlier (V(0)=0).
In the second picture the boundary condition is implemented and we see that when you solve for the unknown constant, c1, you find out that it is equal to zero. Now that we know c1 is zero we have a fully defined function for shear force and you are ready to integrate again.
Step 5: Second Integration to Obtain M(x)
In this step you will integrate the final equation for shear force, V(x), to obtain an equation for the internal bending moment, M(x). This is illustrated in the first picture. Like the previous step, you will end up with an unknown constant of integration, c2. You will implement the boundary condition M(0)=0 to solve for c2.
When you implement the boundary condition M(0)=0 you will see that c2 is also equal to zero (as shown in the second picture).
Step 6: Defining the Full Equation for the Internal Bending Moment
After finding out that c2 is equal to zero, the full equation for the internal bending moment as a function of x is fully defined in the accompanying picture. The way this equation works is that you can plug in any point along the beam in for the variable x and a value for the internal bending moment can be computed.