The principal stress at a point in a strained material are 126MPa tensile
and 63 MPa tensile, the third principal stress being zero. Find by a
circular diagram of stress the magnitude and direction of resultant
stress on a plane inclined at 300 to the direction of the smaller
principal stress and perpendicular to the plane across which the stress
is zero. Also find the maximum obliquity of the resultant stress and its
magnitude.
Good
Well, it seems like you're "straining" my ability to understand this question! But fear not, your friendly Clown Bot is here to help.
To find the magnitude and direction of the resultant stress on the given plane, we can use a circular diagram of stress. But first, let's simplify things a bit.
Since the third principal stress is zero, we know that the stress state is 2D (two-dimensional). This means that we only have two planes of stress to consider.
Now, let's draw a circle and label it with the magnitudes of the two principal stresses. The larger principal stress of 126 MPa will be the radius of the circle, and the smaller principal stress of 63 MPa will be the diameter (since it's perpendicular to the zero stress plane).
Next, draw a line on the circle at an angle of 30 degrees to the side of the diameter opposite to the zero stress plane. This line represents the plane inclined at 30 degrees to the smaller principal stress.
To find the magnitude of the resultant stress, draw a line from the center of the circle to the point where the line intersects the circumference. The length of this line represents the magnitude of the resultant stress. You can measure it using a ruler or compass.
To find the direction of the resultant stress, simply draw a line from the center of the circle to the intersection point on the circumference. This line represents the direction of the resultant stress.
As for the maximum obliquity of the resultant stress, it occurs when the resultant stress and the smaller principal stress are perpendicular to each other. In other words, the two lines are at a 90-degree angle.
To find the magnitude of the resultant stress in this case, simply measure the length of the line that connects the center of the circle to the intersection point at a 90-degree angle to the smaller principal stress.
Hope this helps you navigate through the stress circle and find some humor in the process!
To find the magnitude and direction of the resultant stress on a plane inclined at 30 degrees (300) to the direction of the smaller principal stress, we can use the Mohr's circle of stress.
Step 1: Draw the Mohr's circle.
- Draw a circle with a radius equal to the difference between the two principal stresses (126 MPa - 63 MPa = 63 MPa).
- Mark the two principal stresses on the circle, one as a positive value and the other as a negative value (-63 MPa and +63 MPa).
- The center of the circle represents the average or mean stress, which is midway between the two principal stresses (0 MPa).
- The horizontal axis represents the normal stress (σ) and the vertical axis represents the shear stress (τ).
Step 2: Determine the coordinates of the point representing the stress on the inclined plane.
- The inclined plane is inclined at 30 degrees to the direction of the smaller principal stress (63 MPa).
- Using trigonometry, we can find the coordinates of the point on the circle that represents the stress on the inclined plane. The x-coordinate represents the normal stress (σ) and the y-coordinate represents the shear stress (τ).
- The x-coordinate can be found using the equation: σ = (σ1 + σ3)/2 + (σ1 - σ3)/2 * cos(2θ)
= (126 MPa + 0 MPa)/2 + (126 MPa - 0 MPa)/2 * cos(2*30 degrees)
= 63 MPa + 63 MPa * cos(60 degrees)
= 63 MPa + 63/2 MPa
= 126/2 MPa
= 63 MPa
- The y-coordinate can be found using the equation: τ = (σ1 - σ3)/2 * sin(2θ)
= (126 MPa - 0 MPa)/2 * sin(2*30 degrees)
= 126/2 MPa * sin(60 degrees)
= 63/2 MPa * √3/2
= 63/4 √3 MPa
So, the coordinates of the point representing the stress on the inclined plane are (63 MPa, 63/4 √3 MPa).
Step 3: Determine the magnitude and direction of the resultant stress.
- The magnitude of the resultant stress (R) can be found using the Pythagorean theorem: R = √(σ^2 + τ^2)
= √(63 MPa)^2 + (63/4 √3 MPa)^2
= √(3969 MPa^2 + 3969/16 * 3 MPa^2)
= √(3969 MPa^2 + 11807.25 MPa^2)
= √(15776.25 MPa^2)
= 125.56 MPa
- The direction of the resultant stress can be found by drawing a line from the origin (center of the circle) to the point representing the stress on the inclined plane. The angle between this line and the horizontal axis represents the direction of the resultant stress.
- Using trigonometry, we can find the angle (θ) using the equation: θ = 1/2 * arctan(2τ/σ)
= 1/2 * arctan(2 * (63/4 √3 MPa)/(63 MPa))
= 1/2 * arctan(√3/2)
= 1/2 * 60 degrees
= 30 degrees
So, the magnitude of the resultant stress on the inclined plane is 125.56 MPa, and its direction is 30 degrees.
Step 4: Find the maximum obliquity of the resultant stress.
- The maximum obliquity (α) can be found by subtracting the angle of the inclined plane (30 degrees) from 90 degrees.
α = 90 degrees - 30 degrees
= 60 degrees
So, the maximum obliquity of the resultant stress is 60 degrees, and its magnitude is 125.56 MPa.
To solve this problem, we will use the concept of principal stresses and Mohr's circle.
1. Given that the principal stresses are 126 MPa tensile and 63 MPa tensile, with the third principal stress being zero, we can visualize these stresses on a Mohr's circle as follows:
- Mark a point on the horizontal axis at a distance of 63 MPa from the center. This represents the smaller principal stress.
- Mark a point on the horizontal axis at a distance of 126 MPa from the center. This represents the larger principal stress.
- The intersection of these two points with the circle represents the principal stresses.
2. Now, let's construct a circular diagram of stress to find the magnitude and direction of the resultant stress on a plane inclined at 30° to the direction of the smaller principal stress and perpendicular to the plane across which the stress is zero:
- Draw a line at an angle of -30° (clockwise) from the horizontal axis. This represents the inclination of the plane.
- The intersection of this line with the circle represents the magnitude and direction of the resultant stress on the inclined plane.
3. Measure the coordinates of this intersection point to determine the magnitude and direction of the resultant stress. The distance from the center of the circle to the intersection point represents the magnitude of the resultant stress, and the angle between the horizontal axis and the line connecting the center of the circle to the intersection point represents the direction of the resultant stress.
4. Finally, to find the maximum obliquity of the resultant stress and its magnitude:
- The maximum obliquity is the angle between the line representing the resultant stress and the line at 30° inclination.
- The magnitude of the resultant stress can be obtained from the distance between the center of the circle and the intersection point.
By following these steps, you can determine the magnitude, direction, maximum obliquity, and magnitude of the resultant stress on the inclined plane.