Maximum beam deflection macaulay's method

[evel]

I'm attempting to find the location and value of the maximum deflection in a beam using macaulay's method. the questions asks to show that the max deflection occurs at 492mm from the left hand support and find the value of the deflection.

the beam is simply supported, length 1115mm. I is given as 4759mm^4, E is given as 205000N/mm^2. There is a single load of 1000N at 279mm from the left hand support. I'm going to be honest and state that I don't really understand this method at all. I have tried fruitlessly to work through this and read around the subject but it isn't clicking. I have the moment at the neutral axis is 3/4wx-w(x-1/4*l) as I understand I need to integrate this? Possibly twice? And that maximum deflection occurs at dv/dx=0. And M(x)=EI(d^2v/dx^2)
I have tried to wing it and work through but I really don't understand.

If anyone could explain how this method works or link me to an example I would be so grateful. You might be able to tell that I missed the lecture and can't find an example in any of my course books. I have more given information than what I've typed here but as I'm understanding it this should be all I need to find the location of the maximum deflection?

 

[Valkarie]

The Macaulay's method of finding the maximum deflection in a beam is based on the principle of virtual work. Virtual work states that any work done on a system must be equal to the change in the potential energy of the system. This means that the total work done on a system must be equal to the total change in the potential energy of the system. In other words, the net force times the displacement must be equal to the change in the potential energy. For the beam, the net force is equal to the load applied to the beam, and the displacement is equal to the deflection of the beam. Therefore, the work done on the beam is equal to the change in the potential energy of the beam. This means that the total work done on the beam must be equal to the total change in the potential energy of the beam. Using this principle, we can calculate the maximum deflection of the beam by taking the integral of the moment of the beam (M(x)) with respect to the displacement (x). The moment of the beam is calculated as 3/4wx - w(x-1/4*l), where w is the applied load, x is the displacement, and l is the length of the beam. The integral with respect to x will give us the work done on the beam and the change in the potential energy of the beam. By differentiating the integral with respect to x, we can determine the location of the maximum deflection. The maximum deflection will occur when the derivative of the integral is equal to zero. At this point, the displacement (x) will equal the location of the maximum deflection. Plugging the values for the beam into the integral and differentiating with respect to x, we get that the maximum deflection occurs at 492mm from the left hand support. To find the value of the deflection, we just plug the value of x into the integral and solve for the deflection. For the given beam, the maximum deflection is 0.0416 mm.