From
Case No. 7 of
Summary of Beam Loadings, the deflection equations are
$EI \, y = \dfrac{Pbx}{6L}(L^2 - x^2 - b^2) \text{ for } 0 \lt x \lt a$
$EI \, y = \dfrac{Pb}{6L} \left[ \dfrac{L}{b}(x - a)^3 + (L^2 - b^2)x - x^3 \right] \text{ for } a \lt x \lt L$
The point under the load $P$ is generally located at $x = a$ and at this point, both equations above will become
$EI \, y = \dfrac{Pab}{6L}(L^2 - a^2 - b^2)$
Deflection under the 500 N load
EIδ = EIδ due to 500 N load + EIδ due to 800 N load
$EI \, \delta = \dfrac{500(2)(3)}{6(5)}(5^2 - 2^2 - 3^2) + \dfrac{800(1)(2)}{6(5)}(5^2 - 2^2 - 1^2)$
$EI \, \delta = 1200 + 1066.67$
$EI \, \delta = 2266.67 \, \text{ N}\cdot\text{m}^3$ answer
Deflection under the 800 N load
EIδ = EIδ due to 500 N load + EIδ due to 800 N load
$EI \, \delta = \dfrac{500(3)}{6(5)} \left[ \dfrac{5}{3}(4 - 2)^3 + (5^2 - 3^2)(4) - 4^3 \right] + \dfrac{800(1)(4)}{6(5)}(5^2 - 4^2 - 1^2)$
$EI \, \delta = 666.67 + 853.33$
$EI \, \delta = 1520 \, \text{ N}\cdot\text{m}^3$ answer
How come the value of x in
How come the value of x in finding the deflection under the 500N load is the same? shouldn't x=4 for the EIy due to 800N load?
$x$ is the location where…
In reply to How come the value of x in by Hi (not verified)
$x$ is the location where your are solving for the deflection. You are referring to the location of the load. The location of the load is not necessarily the location of the deflection you wish to calculate.
$x = 2$ is the location of the 500 N force to cause a deflection at $x = 2$ also.
$x = 4$ is the location of the 800 N force, while $x = 2$ is the location of the deflection caused by the force. Notice that the location of the force is not the same as the location of the deflection.
Sir what if we want to find…
In reply to $x$ is the location where… by Jhun Vert
Sir what if we want to find EI at midpoint? Do we just substitute x=2.5 in the equation?
Yes.
In reply to Sir what if we want to find… by boybantoten
Yes. But there is another way, a better way I suppose, to calculate midspan deflections. Check it from here: https://mathalino.com/node/955.
Use $x = 2.5$ from here and compare it to the results you've got from the Midspan Deflection link.