I have a simple beam with a pin support at A on the left side of my beam, and a

Question

I have a simple beam with a pin support at A on the left side of my beam, and a roller connection 20 feet  away on the far right side of the beam labeled C, with the midspan point (10 feet) labeled B. There is a distributed load acting on the beam starting from point A and going to point B at the midspan. The distributed load is 2k/ft.

The original question was asking to find the moment equations and then the deflection and rotation equations from the moment. These were asked to be found by making two seperate cuts. The first cut being anywhere between the distributed load with the x direction being taken from point A to the right (I said the limit states were 0 <(equal to) x <(equal to) 10). And the second cut was asked to be made between point B and C with the x direction again being measured from point A to the right. (I said the limit states were 10 <(equal to) x <(equal to) 20). I DO NOT KNOW FOR SURE IF THE LIMIT STATES ARE CORRECT.

I already solved these equations, but I cannot figure out the boundary conditions because the distributed load is not uniform in the middle of the beam, so it is throwing me off.

The Boundary Conditions that I think are correct is for the first cut, I said that when x = 0, y = 0, but I still need one more boundary condition to solve the equations.

The Boundary Conditions that I think are correct for the second cut, I said that when x = 20, y = 0, but I still need one more boundary condition to solve the equations.

All in all, if you can tell me what the limit states for each cut are, and what the correct boundary conditions would be then that is what I am truly looking for!

Explanation / Answer

The four conditions that you need to solve this problem are 1) 1st portion - x=0,y=0 2) 2nd portion - x=20,y=0 3) y(x=10) in right section =y(x=10) in left section 4) slope(x=10) in right section= slope(x=10) in left section ........slope is nothing but dy/dx...so these are the four conditions for evaluating the four constants

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