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Calculation of Shear Force, Bending Moment, Slope and Deflection of a Beam

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Added on Β 2023-06-14

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ASSIGNMENT SOLTION
Answer 1)
Given Data
Young Modulud -Steel =210 GPa
Moment of Inertia-I xx = 420 x10 ^6 mm^4
a) The equation is given by :-
𝑑𝑑2𝑦𝑦
𝑑𝑑π‘₯π‘₯ 2 𝐸𝐸𝐸𝐸 = 𝑀𝑀........1)
The equation (1) shown above is Euler Bernoulli equation
All the notations given below
M = Bending moment
y = Deflection of beam
I = Area moment of inertia
E = Young modulus
b) Calculation for Shear force
The reactions at its support is calculated.
M A = 0: Total moments at point A is zero
- P 1*4 - P 2*8 + R B*10 = 0
M B = 0: Total moments at point B is zero
- R A*10 + P 1 *6 + P2*2 = 0
Calculation of Shear Force, Bending Moment, Slope and Deflection of a Beam_1
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Solve equations
H A = 0 (kN)
Calculate reaction of roller support at point B
R B = ( P 1*4 + P2*8) / 10 = ( 10*4 + 5*8) / 10 = 8.00 (kN)
Calculate reaction of pin support at point A
R A = ( P1*6 + P2*2) / 10 = ( 10*6 + 5*2) / 10 = 7.00 (kN)
First span of the beam: 0 ≀ x1 < 4
Shear force (Q)
Q(x 1) = + R A
Q 1(0) = + 7 = 7 (kN)
Q 1(4) = + 7 = 7 (kN)
Second span of the beam: 4 ≀ x2 < 8
Shear force (Q)
Q(x 2) = + R A - P1
Q 2(4) = + 7 - 10 = -3 (kN)
Q 2(8) = + 7 - 10 = -3 (kN)
Third span of the beam: 8 ≀ x3 < 10
Determine the equations for the shear force (Q)
Q(x 3) = + R A - P1 - P 2
Q 3(8) = + 7 - 10 - 5 = -8 (kN)
Q 3(10) = + 7 - 10 - 5 = -8 (kN)
Calculation of Shear Force, Bending Moment, Slope and Deflection of a Beam_2
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Graph shows that maximum shear force will be at location of P2 load of 8 KN .
c) Equation of bending moment
Bending moment (M)
M(x) = + R A*(x) - P1*(x - 4) - P 2*(x - 8)
M(x) = + 7*(x) - 10*(x - 4) - 5*(x - 8)
d) Bending moment diagram
First span of the beam: 0 ≀ x1 < 4
Calculation of Shear Force, Bending Moment, Slope and Deflection of a Beam_3

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