Beam Deflection

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When force is applied on a long thick piece of a body (called a beam which is capable of holding the load to restrict the bending) there is deflection in it, which is called beam deflection.

When we placed a uniform load of W (force/length units), there will be Shear V, Moment M and deflection δ. We can calculate the deflection using the modulus of elasticity E and moment of inertia I.

Condition 1: Pinned-pinned beam with uniform load:

V = W(L/2 – x)

M = Wx(L-x)/2

δ = Wx(L^3 – 2Lx^2 + x^3)/24EI

Condition 2: Free-fixed beam with uniform load:

V = -Wx

M = -Wx^2/2

δ = W(x^4 – 4L^3x + 3L^4)/24EI

Example: A free-fixed beam of length 45 cm is put uniform load of 180 gm having x as 35 cm. Calculate the moment. 

Solution: Given that,
W = 180 gm = 0.18 kg
L = 45 cm = 0.45 m
x = 35 cm = 0.35 m
Now, we can use the equation for moment,
          M = -Wx^2/2
          = -0.18(0.35)^2/2
          = -0.011 kg m^2

Example: A pinned-pinned beam of length 350 cm is put uniform load of 75 gm having x as 15 cm. Calculate the shear. 

Solution: Given that,
W = 75 gm = 0.075 kg
L = 350 cm = 3.5 m
x = 15 cm = 0.15 m
Now, we can use the equation for shear
V = W(L/2 – x)
= 0.075(3.5/2 – 0.15)
= 0.12 kg m.

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