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

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In summary, a simply supported beam is a type of structural element that is commonly used in construction and is supported at each end by a support or joint. When two equal point loads are placed on a simply supported beam, it creates a symmetrical load distribution and allows for more efficient weight distribution. The formula for calculating the reactions at the supports in this scenario is R = (W/2)(L/2). However, if the point loads are not equal, the reactions at the supports will also not be equal. The strength and stability of the beam can be affected by various factors such as material, dimensions, and external factors like weather and vibrations.
  • #1
SarahM1980
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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!
 
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  • #2
You can either use the equation w''(x) = M(x) / EI , or a dummy force method (which I personally prefer more).
 
  • #3


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
 

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

1. What is a simply supported beam?

A simply supported beam is a type of structural element that is supported at each end by a support or joint. It is commonly used in construction for bridges, floors, and roofs.

2. What is the significance of 2 equal point loads on a simply supported beam?

The two equal point loads on a simply supported beam create a symmetrical load distribution, with each load exerting the same amount of force on the beam. This allows for more efficient weight distribution and reduces the risk of bending or breaking the beam.

3. What is the formula for calculating the reactions at the supports of a simply supported beam with 2 equal point loads?

The formula for calculating the reactions at the supports of a simply supported beam with 2 equal point loads is: R = (W/2)(L/2), where R is the reaction force at each support, W is the magnitude of the point loads, and L is the length of the beam.

4. How do the reactions at the supports change if the point loads are not equal on a simply supported beam?

If the point loads are not equal on a simply supported beam, the reactions at the supports will also not be equal. The reaction force at each support will be proportional to the magnitude of the point loads and their distance from the support.

5. What factors can affect the strength and stability of a simply supported beam with 2 equal point loads?

The strength and stability of a simply supported beam with 2 equal point loads can be affected by factors such as the material and dimensions of the beam, the distance between the supports, and the magnitude and placement of the point loads. Other external factors such as weather and vibrations may also impact the beam's performance.

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