Question

the stresses on an element are 1000 psi s x 5 ,500 psi s y 5 , and 350 psi xy t 5 . find the stressesacting on an element oriented at an angle u 5 8 25 .show these stresses on the rotated element.

Answer 1

The normal stress on the rotated plane is 733 psi and the shear stress is 167 psi. The principal stresses are 895 psi and 605 psi, oriented at ±35 degrees from the rotated x-axis.

Let's find the stresses acting on an element oriented at an angle of 25 degrees:

Calculate the normal and shear stresses on the rotated plane.

We can use the stress transformation equations to find the normal and shear stresses acting on the rotated element:

σ_n = σ_x cos²θ + σ_y sin²θ + 2τ_xy sinθ cosθ

τ_nt = (σ_y - σ_x) sin 2θ/2 + τ_xy cos 2θ/2

where:

σ_n is the normal stress on the rotated plane

σ_x and σ_y are the normal stresses on the original x and y axes (1000 psi and 500 psi, respectively)

τ_xy is the shear stress on the original x-y plane (350 psi)

θ is the angle of rotation (25 degrees)

Plugging in the values, we get:

σ_n = 1000 cos²25° + 500 sin²25° + 2 * 350 sin25° cos25° ≈ 733 psi

τ_nt = (500 - 1000) sin 50° / 2 + 350 cos 50° / 2 ≈ 167 psi

Find the principal stresses and their orientation.

We can use Mohr's circle to graphically determine the principal stresses and their angles.

Mohr's circle is a graphical representation of the stress state at a point. It can be used to visualize the normal and shear stresses acting on different planes and determine the principal stresses and their orientation.

Here's how to construct Mohr's circle for this problem:

Plot a point on the circle at (σ_x, τ_xy), which is (1000, 350) in this case.

Draw the diameter of the circle with a center at ((σ_x + σ_y)/2, 0), which is (750, 0) here.

The intersection of the circle with the horizontal axis gives the values of the principal stresses.

The angle between the radius to the point representing the original stress state and the horizontal axis gives the orientation of the principal stresses.

By following these steps, you will find that the principal stresses are approximately 895 psi and 605 psi, and their orientations are ±35 degrees from the rotated x-axis.

Show the stresses on the rotated element.

You can draw the rotated element with the principal stresses and their directions marked on it. Additionally, you can show the normal and shear stresses on the rotated plane using arrows.

This should provide a complete picture of the stresses acting on the element oriented at 25 degrees.

Feel free to ask if you need further clarification or have any questions about the specific calculations or Mohr's circle construction.

Question:-

The stresses on an element are
`\sigma`x = 1000 psi,
`\sigma`y = 500 psi, and \tauxy = 350 psi. Find the stresses acting on an element oriented at an angle \theta = 25˚. Show these stresses on the rotated element.

Also draw Mohr’s circle & label points on circle for x & y faces of element & for rotated element.