An 8 m long simply-supported elastic beam of rectangular cross-section 100mm × 200 mm is subjected to a uniformly distributed load of 10 kN/m over its entire span. The maximum principal stress (in MPa, up to two decimal places) at a point located at the extreme compression edge of a cross-section and at 2 m from the support is ____________.Correct answer is '90'. Can you explain this answer?

Question

Answer

The given problem can be solved by analyzing the bending stress and the normal stress on the beam.

1. Calculate the Bending Stress:
The bending stress is given by the formula:
σ_b = (M * y) / I
where σ_b is the bending stress, M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia of the cross-section.

- Calculate the Bending Moment:
The bending moment can be calculated using the formula:
M = (w * L^2) / 8
where M is the bending moment, w is the uniformly distributed load, and L is the length of the beam.

Substituting the given values, we get:
M = (10 * 8^2) / 8 = 80 kNm = 80,000 Nm

- Calculate the Moment of Inertia:
The moment of inertia of a rectangular cross-section can be calculated using the formula:
I = (b * h^3) / 12
where I is the moment of inertia, b is the width of the cross-section, and h is the height of the cross-section.

Substituting the given values, we get:
I = (0.1 * 0.2^3) / 12 = 0.000133 m^4

- Calculate the Bending Stress:
Substituting the values of M, y, and I in the formula, we get:
σ_b = (80,000 * 0.1) / 0.000133 = 60,150,375 Pa = 60.15 MPa

2. Calculate the Normal Stress:
The normal stress is given by the formula:
σ_n = (P / A)
where σ_n is the normal stress, P is the force acting on the cross-section, and A is the cross-sectional area.

- Calculate the Force:
The force can be calculated using the formula:
P = (w * L)
where P is the force, w is the uniformly distributed load, and L is the length of the beam.

Substituting the given values, we get:
P = (10 * 8) = 80 kN = 80,000 N

- Calculate the Cross-sectional Area:
The cross-sectional area can be calculated using the formula:
A = (b * h)
where A is the cross-sectional area, b is the width of the cross-section, and h is the height of the cross-section.

Substituting the given values, we get:
A = (0.1 * 0.2) = 0.02 m^2

- Calculate the Normal Stress:
Substituting the values of P and A in the formula, we get:
σ_n = (80,000 / 0.02) = 4,000,000 Pa = 4 MPa

3. Calculate the Maximum Principal Stress:
The maximum principal stress can be calculated using the formula:
σ_max = (σ_n + σ_b) / 2

Substituting the values of σ_n and σ_b in the formula, we get:
σ_max = (4 + 60.15) / 2 = 32.075 MPa

Since the point of interest is located at the extreme compression edge, the maximum principal stress will be the compressive stress. Therefore, the maximum principal stress at this point is 32.075 MP

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