In this problem, you will construct the 2D Mohr's circle for two plane stress states defined in terms of the Cartesian components of the stress tensor in the x,y reference frame: [ σx,σy,τxy].
For each of these stress states, you must construct and use the Mohr's circle to obtain the numerical values of:
The normal and shear stresses (in units of MPa) acting on the faces of an an element oriented at 45∘ with respect to the x,y reference frame: these are the components [σx1, σy1, τx1y1] of the stress tensor in the reference frame (x1,y1) defined, as indicated in the figure below, with the coordinate axis x1 rotated by 45∘ CCW from the x-axis. Notice that in the figure we indicate the stress directions for positive values of the components.
The maximum and minimum principal stresses, σ1 and σ2, and the maximum shear stress, τmax, in units of MPa.
The angle ΘP (in degrees) that takes you from the x,y reference frame to the 1,2 principal frame. As shown in the figure below, take ΘP as the angle from the x-axis to the 1-axis, positive CCW, with −90∘≤ΘP ≤90∘.
For [σx=10MPa,σy=0MPa,τxy=0MPa], obtain:
σx1=
σy1=
τx1y1=
σ1=
σ2=
τmax=
ΘP=
For [σx=−11MPa,σy=1MPa,τxy=8MPa], obtain:
σx1=
σy1=
τx1y1=
σ1=
σ2=
τmax=
ΘP=
Oh boy, time to get creative with some math! Let's start with the stress state [σx=10MPa,σy=0MPa,τxy=0MPa]:
Using Mohr's circle, I'll doodle some numbers for you:
- σx1: Oh, look at you, Mr. Normal Stress! You're all alone here at 10MPa. You'll be the same when you're facing 45 degrees, so no changes for you!
- σy1: Well, well, well, you're a real party pooper, always staying at 0MPa! No changes for you either.
- τx1y1: Ah, the shear stress, you cheeky devil! You're also stuck at 0MPa because there's no rotation happening here.
Now, let's move on to the principal stresses and maximum shear stress:
- σ1: So, the maximum principal stress, eh? It's the sum of σx and σy, which is 10MPa + 0MPa, giving us a total of 10MPa. Congrats, you're the biggest stress in this party!
- σ2: Now, the minimum principal stress. It's the difference between σx and σy, so that's 10MPa - 0MPa, which is still 10MPa because zero is a strange number and doesn't really change things.
- τmax: Ah, the maximum shear stress, my favorite dance move! It's just half of the difference between σx and σy, so that's (10MPa - 0MPa)/2, which is 5MPa. Keep grooving, my friend!
Lastly, let's find the angle ΘP:
- ΘP: Oh, you want me to do angles too? Fine, be that way! The angle ΘP is gonna be easy peasy lemon squeezy for this stress state because there's no τxy. Since σy is 0MPa, we don't need any fancy trigonometry. ΘP is just 0 degrees. Yep, you heard it right, no rotation at all. Boring, I know.
Alright, it's time for the next stress state: [σx=−11MPa,σy=1MPa,τxy=8MPa]. Let's crunch some numbers:
- σx1: Poor σx, you're the same old -11MPa, no changes for you, my friend!
- σy1: Oh, you want to be different, huh? Well, you can finally shine at 1MPa when facing 45 degrees. Good for you!
- τx1y1: Hello, shear stress! You're a bit more exciting this time, moving up to 8MPa when facing 45 degrees.
Now, let's find the principal stresses and maximum shear stress:
- σ1: Here comes the big dog, σx, with -11MPa. I hope you're ready to add some positive vibes, because σy is 1MPa. So σ1 is gonna be -11MPa + 1MPa, giving us -10MPa. Negativity at its finest!
- σ2: The minimum principal stress is just the difference between σx and σy, so that's -11MPa - 1MPa, resulting in -12MPa. Keep those negative vibes flowing!
- τmax: It's time to boogie with the maximum shear stress! Just like before, it's half of the difference between σx and σy, so that's (-11MPa - 1MPa)/2, giving us -6MPa. Keep dancing to your own rhythm!
Lastly, let's find ΘP:
- ΘP: Brace yourself, it's time for some angle action! Since τxy is 8MPa this time, we need to use some trigonometry. The formula to find ΘP is 0.5 * atan((2 * τxy) / (σx - σy)). So, plug in the numbers, do the math, and we get ΘP = -45 degrees. Time to twist and turn, my friend!
There you have it, the Mohr's circle results for both stress states! Hope I managed to make it entertaining for you. Now go forth and conquer the world of stress analysis with a smile on your face!
To construct the 2D Mohr's circle for the given stress states, follow these steps:
Step 1: Calculate the center of the circle (C):
The center of the Mohr's circle is given by the average of the normal stresses of the stress states.
For the stress state [σx,σy,τxy], the center is:
C = ( (σx + σy) / 2 , 0 )
Step 2: Calculate the radius of the circle (R):
The radius of the Mohr's circle is given by half of the difference between the normal stresses of the stress states.
For the stress state [σx,σy,τxy], the radius is:
R = ( (σx - σy) / 2 )
Step 3: Calculate the coordinates of the points on the circle:
The coordinates (σx1,τx1y1) of the point on the circle corresponding to an angle of 45 degrees can be found by using the formula:
σx1 = Cx + R * cos(θ)
τx1y1 = Cx + R * sin(θ)
where θ is the angle in radians (45 degrees = π/4 radians).
Step 4: Calculate the principal stresses and maximum shear stress:
The principal stresses are given by:
σ1 = Cx + R
σ2 = Cx - R
The maximum shear stress is given by:
τmax = R
Step 5: Calculate the angle ΘP:
The angle ΘP can be found by using the formula:
ΘP = 0.5 * atan( (2 * τxy) / (σx - σy) ) in degrees.
Now let's calculate the values for each stress state:
Stress State: [σx=10MPa, σy=0MPa, τxy=0MPa]
Step 1: Calculate the center (C):
C = ( (10 + 0) / 2 , 0 ) = ( 5 , 0 )
Step 2: Calculate the radius (R):
R = ( 10 - 0 ) / 2 = 5
Step 3: Calculate the coordinates of the point on the circle corresponding to θ = 45 degrees:
σx1 = 5 + 5 * cos(π/4) ≈ 9.142 MPa
τx1y1 = 0 + 5 * sin(π/4) ≈ 3.536 MPa
Step 4: Calculate the principal stresses and maximum shear stress:
σ1 = 5 + 5 = 10 MPa
σ2 = 5 - 5 = 0 MPa
τmax = 5 MPa
Step 5: Calculate the angle ΘP:
ΘP = 0.5 * atan( (2 * 0) / (10 - 0) ) = 0 degrees
Therefore, for the stress state [σx=10MPa, σy=0MPa, τxy=0MPa], we have:
σx1 ≈ 9.142 MPa
σy1 = 0
τx1y1 ≈ 3.536 MPa
σ1 = 10 MPa
σ2 = 0 MPa
τmax = 5 MPa
ΘP = 0 degrees
Now let's calculate the values for the second stress state:
Stress State: [σx=−11MPa, σy=1MPa, τxy=8MPa]
Step 1: Calculate the center (C):
C = ( (-11 + 1) / 2 , 0 ) = ( -5 , 0 )
Step 2: Calculate the radius (R):
R = ( -11 - 1 ) / 2 = -6
Step 3: Calculate the coordinates of the point on the circle corresponding to θ = 45 degrees:
σx1 = -5 + (-6) * cos(π/4) ≈ -8.071 MPa
τx1y1 = 0 + (-6) * sin(π/4) ≈ -4.243 MPa
Step 4: Calculate the principal stresses and maximum shear stress:
σ1 = -5 + (-6) = -11 MPa
σ2 = -5 - (-6) = 1 MPa
τmax = -6 MPa
Step 5: Calculate the angle ΘP:
ΘP = 0.5 * atan( (2 * 8) / (-11 - 1) ) ≈ -49.35 degrees
Therefore, for the stress state [σx=−11MPa, σy=1MPa, τxy=8MPa], we have:
σx1 ≈ -8.071 MPa
σy1 = 0
τx1y1 ≈ -4.243 MPa
σ1 = -11 MPa
σ2 = 1 MPa
τmax = -6 MPa
ΘP ≈ -49.35 degrees
To construct Mohr's circle for a given stress state, you need to follow these steps:
1. Determine the values of σx, σy, and τxy for the given stress state.
2. Plot the stress state on a Cartesian coordinate system, where σx is the x-axis and τxy is the y-axis. Mark the point representing the stress state.
3. Draw a circle with the stress state point as the center. The radius of the circle is equal to the average of σx and σy.
4. Construct the tangents to the circle from the x-axis and y-axis. These tangents intersect at the center of the circle, which represents the average stress value.
5. Draw the line connecting the center of the circle and the stress state point. This is the mean stress line.
6. Determine the coordinates of the point where the mean stress line intersects the circle. These coordinates represent the principal stresses, σ1 and σ2.
7. Calculate the maximum shear stress, τmax, which is half the difference between σ1 and σ2.
8. Calculate the angle ΘP in degrees, which represents the rotation from the x-axis to the principal axis.
Using these steps, we can calculate the required values for the given stress states:
For [σx=10MPa,σy=0MPa,τxy=0MPa]:
1. σx = 10 MPa, σy = 0 MPa, τxy = 0 MPa.
2. Plot (10, 0) on the Cartesian coordinate system.
3. Draw a circle with a radius of (10 + 0) / 2 = 5 MPa.
4. Construct tangents from the x-axis and y-axis, which intersect at the center of the circle.
5. Draw a line connecting the center to the stress state point.
6. Determine the coordinates of the intersection between the mean stress line and the circle. Let's call it (σ1', τ1').
7. Calculate the maximum shear stress, τmax = (σ1' - τ1') / 2.
8. Calculate the angle ΘP using the formula ΘP = (1/2) * atan(2 * τxy / (σx - σy)) in degrees.
For [σx=-11MPa,σy=1MPa,τxy=8MPa]:
1. σx = -11 MPa, σy = 1 MPa, τxy = 8 MPa.
2. Plot (-11, 8) on the Cartesian coordinate system.
3. Draw a circle with a radius of (-11 + 1) / 2 = -5 MPa.
4. Construct tangents from the x-axis and y-axis, which intersect at the center of the circle.
5. Draw a line connecting the center to the stress state point.
6. Determine the coordinates of the intersection between the mean stress line and the circle. Let's call it (σ1', τ1').
7. Calculate the maximum shear stress, τmax = (σ1' - τ1') / 2.
8. Calculate the angle ΘP using the formula ΘP = (1/2) * atan(2 * τxy / (σx - σy)) in degrees.
By following these steps, you can calculate the values of σx1, σy1, τx1y1, σ1, σ2, τmax, and ΘP for both stress states.