A cast iron block of 5Cm^2 cross section carries an axial compressive load of 50KN.Calculate the magnitude or normal and shear stresses on a plane whose normal is inclined at 30 deg.to the axis of the block,also determine the maximum shear stress in the block?
I'll give it a try.
We have 50kN on 5cm², or 0.0005 m².
So the axial stress
σx = 105 kPa
=100 MPa
σy=0
So draw the Mohr's circle (in the σ/τ plane) with between (0,0) and (100,0) as a diameter.
The centre is at (50,0).
To find the stresses at θ=30°
draw a diameter passing through the centre at 2*30=60° with the axis.
The stresses in that plane is shown by the extremities of the diameter, namely
σ=50±50cos(60°), and
τ=±50sin(60°).
Maximum shear stress is at θ=45° where
σx=σy=50, and
τxy=50
Hope that helps.
To calculate the normal and shear stresses on a plane inclined at 30 degrees to the axis of the block, we can first calculate the normal and shear forces acting on the inclined plane.
1. Calculate the normal force:
The normal force is the axial compressive load acting on the block. In this case, the normal force is 50 kN.
2. Calculate the shear force:
The shear force can be calculated by resolving the axial load into components parallel and perpendicular to the inclined plane.
The component perpendicular to the inclined plane is Fperpendicular = F * sin(30 degrees)
= 50 kN * sin(30 degrees)
= 25 kN
The shear force is equal to the force component perpendicular to the plane, so the shear force is 25 kN.
3. Calculate the cross-sectional area:
Given that the cross-sectional area of the block is 5 cm^2, we need to convert it to square meters.
1 cm^2 = 0.0001 m^2
Therefore, the cross-sectional area is 5 cm^2 * 0.0001 m^2/cm^2 = 0.0005 m^2.
4. Calculate the normal stress:
The normal stress is defined as the normal force divided by the cross-sectional area.
Normal stress = Normal force / Cross-sectional area = 50 kN / 0.0005 m^2
Normal stress = 100 MPa
5. Calculate the shear stress:
The shear stress on the inclined plane is given by:
Shear stress = Shear force / Cross-sectional area
Shear stress = 25 kN / 0.0005 m^2
Shear stress = 50,000 kPa
6. Calculate the maximum shear stress:
The maximum shear stress occurs when the inclined plane is at a 45-degree angle to the axial load direction. In this case, the maximum shear stress is equal to half the axial compressive load.
Maximum shear stress = 50 kN / 2
Maximum shear stress = 25 kN
So, the magnitude of the normal stress is 100 MPa, the magnitude of the shear stress on the inclined plane is 50,000 kPa, and the maximum shear stress in the block is 25 kN.
To calculate the magnitude of normal and shear stresses on a plane inclined at a certain angle to the axis of the block, you'll need to use the concepts of normal stresses and shear stresses in mechanics.
1. Let's start by calculating the normal stress:
The normal stress (σ) is the stress acting perpendicular to the cross-sectional area of the block. The formula to calculate the normal stress is:
σ = F / A,
where σ is the normal stress, F is the compressive load, and A is the cross-sectional area of the block.
Given:
- Compressive load (F) = 50 kN (50,000 N)
- Cross-sectional area (A) = 5 cm^2 (convert to m^2: 5 * 10^(-4) m^2)
Plug in the values into the formula:
σ = 50,000 N / (5 * 10^(-4) m^2)
Calculating that, we get:
Normal stress (σ) = 100,000,000 N/m^2 (or Pa)
2. Next, let's calculate the shear stress:
The shear stress (τ) is the stress acting parallel to the cross-sectional area of the block. When the plane is inclined at an angle to the axis, we use the equation:
τ = σ * sin(2θ),
where τ is the shear stress, σ is the normal stress, and θ is the angle between the normal to the plane and the axial axis of the block (given as 30 degrees in this case).
Plug in the values into the formula:
τ = 100,000,000 N/m^2 * sin(2 * 30°)
Calculating that, we get:
Shear stress (τ) = 50,000,000 N/m^2 (or Pa)
3. Finally, let's determine the maximum shear stress in the block:
The maximum shear stress occurs when the inclined plane is at a 45-degree angle to the axis. In this case, the shear stress (τ_max) can be determined by the following equation:
τ_max = σ / 2,
where τ_max is the maximum shear stress, and σ is the normal stress.
Plug in the value of the normal stress into the equation:
τ_max = 100,000,000 N/m^2 / 2
Calculating that, we get:
Maximum shear stress (τ_max) = 50,000,000 N/m^2 (or Pa).
So, to summarize:
- The magnitude of the normal stress on the inclined plane is 100,000,000 N/m^2 (or Pa).
- The magnitude of the shear stress on the inclined plane is 50,000,000 N/m^2 (or Pa).
- The maximum shear stress in the block is also 50,000,000 N/m^2 (or Pa).