A simply supported beam has 2 equal point loads (W/2)

[SarahM1980]

I have a simply supported beam 1.2m long
it has 2 = point loads (W/2) at equal distances (0.4m) along the beam
i have been asked to derive from first principles the expression for the mid span defelction.
I have got so far m= W/2x - W/2(x-L3) - W/2(x-L/6)
and i think the answer I'm looking for is something like 23WL 3 / 1296EI


can anyone help!

 

[radou]

You can either use the equation w''(x) = M(x) / EI , or a dummy force method (which I personally prefer more).

 

[ravenprp]

Sure, I can help with deriving the expression for the mid span deflection of a simply supported beam with two equal point loads. Let's start by defining some variables:

- L: Length of the beam (1.2m in this case)
- W: Total load applied on the beam (W/2 at each point)
- x: Distance from the left end of the beam to the point where we want to calculate the deflection
- L1: Distance from the left end of the beam to the first point load (0.4m in this case)
- L2: Distance between the two point loads (also 0.4m)

Now, let's consider the beam as two separate beams - one from the left end to the first point load and the other from the first point load to the right end. This can help us simplify the calculation.

For the first beam (from the left end to the first point load), the deflection can be calculated using the standard formula for a simply supported beam with a point load at the midspan:

δ1 = Wx^2(L1-x)/6EI

For the second beam (from the first point load to the right end), the deflection can be calculated using the standard formula for a simply supported beam with a point load at any distance from the support:

δ2 = W(x-L1)^2(L2-(x-L1))/6EI

Now, the total deflection at the midspan can be calculated by adding the deflections from both beams:

δ = δ1 + δ2

= Wx^2(L1-x)/6EI + W(x-L1)^2(L2-(x-L1))/6EI

= Wx^2(L1-x) + W(x-L1)^2(L2-(x-L1))/6EI

= Wx^2L1 - Wx^3 + Wx^2L2 - Wx^3 + Wx^3 - Wx^2L1 + Wx^3/6EI

= Wx^2(L1+L2) - Wx^3 + Wx^3/6EI

= Wx^2L - Wx^3 + Wx^3/6EI

= Wx^2L + Wx^3(1/6 - 1)/EI

= Wx^2L - Wx^3/6EI