Question

Problem 3- Shear Stress (35 pts)

Consider the cross section shown here. Determine the max-
imum shear stress in the beam, Tmax, and the shear stress at
point A (TA) at the top of the beam's web. Use V = 12 kN.
200 mm
20 mm
B
200 mm
20 mm
300 mm
20 mm

Answer 1

To determine the maximum shear stress in the beam and the shear stress at point A, we can use the formula for shear stress:

τ = VQ / Ib

where τ is the shear stress, V is the shear force, Q is the first moment of area of the section about the neutral axis, I is the second moment of area of the section about the neutral axis, and b is the width of the section.

First, we need to calculate the values of Q and I for the given cross-section. The neutral axis is located at the centroid of the cross-section, which is at a distance of 110 mm from the top and bottom edges. Therefore, the first moment of area about the neutral axis is:

Q = (200 mm * 20 mm * 20 mm * 100 mm) + (300 mm * 20 mm * 150 mm * 90 mm) = 126,000,000 mm^3

The second moment of area about the neutral axis is:

I = (1/12) * (20 mm * 200 mm^3) + (20 mm * 300 mm^3) + (20 mm * 200 mm * (110 mm - 20 mm)^2) + (20 mm * 20 mm * (290 mm - 110 mm)^2) = 67,200,000 mm^4

Now we can calculate the maximum shear stress and the shear stress at point A.

At the maximum shear stress location, the shear force is acting along the web of the beam, which has a height of 200 mm and a width of 20 mm. Therefore, the width b is 20 mm.

Maximum shear stress:

τ_max = VQ / Ib = (12 kN) * (126,000,000 mm^3) / (67,200,000 mm^4 * 20 mm) = 13.33 MPa

The maximum shear stress in the beam is 13.33 MPa.

To find the shear stress at point A, we need to calculate the distance from the neutral axis to point A. This distance is equal to the distance from the top of the beam to the neutral axis minus half the height of the web.

Distance from neutral axis to top of beam: 110 mm
Height of web: 200 mm / 2 = 100 mm
Distance from neutral axis to point A: 110 mm - 100 mm = 10 mm

Now we can calculate the shear stress at point A:

τ_A = VQ / Ib = (12 kN) * (20 mm * 300 mm * 10 mm) / (67,200,000 mm^4 * 20 mm) = 0.0225 MPa

The shear stress at point A is 0.0225 MPa.

Answer 2

Step-by-step explanation:

To calculate the maximum shear stress (Tmax) and the shear stress at point A (TA) in the given beam, we can use the formula for shear stress:

Shear Stress (τ) = V / A

where V is the shear force and A is the cross-sectional area subjected to the shear force.

Given data:

Shear force (V) = 12 kN = 12,000 N

Dimensions of the cross-section:

Width of the web (20 mm)

Height of the web (200 mm)

Height of the flange (20 mm)

Length of the flange (300 mm)

Step 1: Calculate the cross-sectional area of the web (A_web):

A_web = Width * Height

A_web = 20 mm * 200 mm

A_web = 4000 mm²

Step 2: Calculate the cross-sectional area of one flange (A_flange):

A_flange = Width * Length

A_flange = 20 mm * 300 mm

A_flange = 6000 mm²

Step 3: Calculate the total cross-sectional area subjected to the shear force (A_total):

A_total = A_web + 2 * A_flange (since there are two flanges)

A_total = 4000 mm² + 2 * 6000 mm²

A_total = 16000 mm²

Step 4: Calculate the maximum shear stress (Tmax):

Tmax = V / A_total

Tmax = 12,000 N / 16000 mm²

Tmax = 0.75 N/mm²

Step 5: Calculate the shear stress at point A (TA) at the top of the beam's web:

TA = V / A_web

TA = 12,000 N / 4000 mm²

TA = 3 N/mm²

So, the maximum shear stress in the beam (Tmax) is 0.75 N/mm², and the shear stress at point A (TA) at the top of the beam's web is 3 N/mm².