Consider a simply supported beam of length 50h with a rectangular cross-section of depth ‘h’ and width 2h, the load carried at mid-point. Find the ratio of the maximum shear stress to the maximum bending stress in the beam.:

To solve this problem, we need to understand the formulas for shear stress and bending stress.

1. The formula for shear stress (τ) is: τ = V/A, where V is the shear force and A is the area.

2. The formula for bending stress (σ) is: σ = My/I, where M is the moment, y is the distance from the neutral axis to the point of interest, and I is the moment of inertia.

Given that the load is at the midpoint of the beam, the shear force (V) is maximum at the supports and zero at the midpoint. The maximum bending moment (M) is at the midpoint of the beam.

The cross-sectional area (A) of the beam is 2h * h = 2h^2.

The moment of inertia (I) for a rectangular section is (1/12) * base * height^3 = (1/12) * 2h * h^3 = (1/6)h^4.

The distance from the neutral axis to the point of interest (y) is h/2.

Substituting these values into the formulas gives:

τ_max = V_max / A = V_max / 2h^2

σ_max = M_max * y / I = M_max * (h/2) / (1/6)h^4 = 3M_max / h^3

The ratio of the maximum shear stress to the maximum bending stress is therefore:

τ_max / σ_max = (V_max / 2h^2) / (3M_max / h^3) = V_max * h / 6M_max

Without specific values for V_max and M_max, this is as far as we can go with the calculation.

To solve this problem, we need to understand the formulas for shear stress and bending stress.

1. The formula for shear stress (τ) is: τ = V/A, where V is the shear force and A is the area.

2. The formula for bending stress (σ) is: σ = My/I, where M is the moment, y is the distance from the neutral axis to the point of interest, and I is the moment of inertia.

Given that the load is at the midpoint of the beam, the maximum shear force (V) will be half of the load. The area (A) of the cross-section is 2h*h = 2h^2.

The maximum moment (M) in a simply supported beam with a point load in the middle is WL/4, where W is the load and L is the length of the beam. The distance (y) will be h/2, as the maximum stress occurs at the extreme fiber from the neutral axis. The moment of inertia (I) for a rectangular section is bh^3/12, where b is the width and h is the height. Substituting b=2h, we get I = h^4/3.

Now, we can find the ratio of the maximum shear stress to the maximum bending stress:

τ/σ = (V/A) / (My/I) = (VI) / (AMy)

Substituting the values we have:

τ/σ = [(W/2)(h^4/3)] / [(2h^2)(W50h/4)(h/2)] = 1/300

So, the ratio of the maximum shear stress to the maximum bending stress in the beam is 1:300.