Need help with multi-segment deflection problem

[Strawberry]

Homework Statement

There is a uniform, horizontal beam 4.8m in length supported by a pin at x = 0 and a roller at x= 3.2m. There is a distributed weight of 12kN/m from x = 0 to x = 1.6m. There is a concentrated, vertical force of 2.4kN in the downward direction at x = 4.8.
Find the deflection and define the coefficients of integration for each segment

Homework Equations

EIY'' = double integral of Mdx
Y is deflection

The Attempt at a Solution

I calculated the moment for each segment, but I can't seem to figure out what the coefficients are. I think I'm supposed to use symmetry conditions, and I know the deflection is 0 at x = 0 and x = 3.2 where there are supports. I can't figure out past that though.

M1 = 13.2x - 6x^2
M2 = -6x + 15.36
M3 = 2.4x - 11.52

EIV1 = 2.2x^3 - .5x^5 + C1x + C2
C2 = 0

EIV2 = -x^3 + 7.68x^2 + C1x + C2

EIV3 = .4x^3 - 5.76x^2 + C1x + C2

I believe the moments I calculated are correct, but I'm not sure. Any help would be appreciated.

 

[Strawberry]

Oh, I think I figured it out. V' is 0 at x = 1.1, from there I can find the rest of the coefficients

 

[piercas]

I would suggest using the Moment-Area Method to solve this multi-segment deflection problem. This method involves finding the area under the moment diagram for each segment and using it to determine the deflection at that point. The coefficients of integration can be found by applying the boundary conditions (symmetry conditions in this case) and solving for them.

First, we can find the area under the moment diagram for each segment using the equations given in the problem.
For segment 1 (0 ≤ x ≤ 1.6):
Area = ∫(13.2x - 6x^2)dx = 13.2x^2/2 - 6x^3/3 = 6.6x^2 - 2x^3 + C1
For segment 2 (1.6 ≤ x ≤ 3.2):
Area = ∫(-6x + 15.36)dx = -3x^2 + 15.36x + C2
For segment 3 (3.2 ≤ x ≤ 4.8):
Area = ∫(2.4x - 11.52)dx = 1.2x^2 - 11.52x + C3

Next, we can apply the boundary conditions to solve for the coefficients of integration. Since the deflection is 0 at x = 0 and x = 3.2, we can set the deflection equations for segments 1 and 2 equal to 0 at those points. This gives us:
For segment 1 (x = 0):
0 = 6.6(0)^2 - 2(0)^3 + C1
C1 = 0
For segment 2 (x = 3.2):
0 = -3(3.2)^2 + 15.36(3.2) + C2
C2 = 30.72 - 30.72 = 0

Finally, we can solve for the deflection at x = 4.8 by setting the equation for segment 3 equal to the given concentrated force of 2.4kN:
2.4 = 1.2(4.8)^2 - 11.52(4.8) + C3
C3 = 27.648 - 55.296 + 2.4 = -25.248