Question:
For the elements in plane stress presented in the following pages:
- Identify Normal and Shear Stresses (5 points)
- Determine stresses acting on an element oriented at an angle ? [Using Equations]
(15 points)
- Show this stresses on a sketch of an element oriented at this angle (10 points)
- Determine the Principle stresses and Principle plane [Using Equations] (15 points)
- Determine average normal stress, maximum in-plane shear stress and maximum shear stress plane [Using Equations] (15 points)
- Draw the Mohr Circle diagram with the current state of stress in the Graph paper to scale (20 points)
- Identify the state of the stress for the element oriented at an angle ?(5 points)
- Determine Principle stresses, Maximum in-plane Shear Stress, Principle plane and Maximum shear stress plane from the diagram [show working] (15 points)
Answer:
The given element is shown above:
- Identify Normal and Shear Stresses (5 points)
The normal stress are:
= 350 MPa
= 112 MPa
And, the shear stress is:
= -120 MPa
- Determine stresses acting on an element oriented at an angle ? [Using Equations]
(15 points)
We know that,
= 350 MPa
= 112 MPa
= -120 MPa
= 15 clockwise = -15
=
=
= + (-120) (-0.5)
= – 60 = 274.09 MPa
=
=
= -59.5 -103.92 = -163.4 Mpa
=
= 350 + 112 – 274.09
= 187.7 Mpa
- Show this stresses on a sketch of an element oriented at this angle (10 points)
- Determine the Principle stresses and Principle plane [Using Equations] (15 points)
= 350 MPa
= 112 MPa
= -120 MPa
Principal Angles:
= = -22.61
= – 22.61 + 90 = 67.39
= = 231 -83.82 – 85.17 = 62.01 MPa
= = 231 + 83.82 + 85.17 = 399.9 MPa
- Determine average normal stress, maximum in-plane shear stress and maximum shear stress plane [Using Equations] (15 points)
Average Normal Stress: = = = 231 MPa
Max shearing stress is given by: = ± = ± 168.9 MPa
Maximum in-plane shear stress, = + = 231 + 168.9 = 399.9 MPa
- Draw the Mohr Circle diagram with the current state of stress in the Graph paper to scale (20 points)
The Mohr Diagram can be drawn as follows:
Now, substituting the values and drawing on the graph paper to scale, we can have the Mohr circle as drawn below:
- Identify the state of the stress for the element oriented at an angle ?(5 points)
? = -15
2? = -30
R = = 350/2 = 175 MPa
Point C: = R = 175 MPa
Point D: = R + Rcos|2?| = 175 + 175 cos30 = 326.55 MPa
= R – Rcos|2?| = 175 – 175 cos30 = 23.44 MPa
-R sin (2?) = 87.5 MPa
- Determine Principle stresses, Maximum in-plane Shear Stress, Principle plane and Maximum shear stress plane from the diagram [show working] (15 points)
Question 2
The given element is shown above:
- Identify Normal and Shear Stresses (5 points)
The normal stress are:
= 30 MPa
= -20 MPa
And, the shear stress is:
= 80 MPa
- Determine stresses acting on an element oriented at an angle ? [Using Equations]
(15 points)
We know that,
= 30 MPa
= -20 MPa
= 80 MPa
= 10 clockwise = -10
=
=
= (-0.34)
= 5 -27.36 = 1.14 MPa
=
=
= -8.5 + 75.2 = 66.7 Mpa
=
= 30 + (-20) – 1.14
= 8.86 Mpa
- Show this stresses on a sketch of an element oriented at this angle (10 points)
- Determine the Principle stresses and Principle plane [Using Equations] (15 points)
= 30 MPa
= -20 MPa
= 80 MPa
Principal Angles:
= = 36.32
= 36.32 + 90 = 126.32
= = 5 -7.45 – 76.35 = -78.8 MPa
= = 5 + 7.45 + 76.35 = 88.8 MPa
- Determine average normal stress, maximum in-plane shear stress and maximum shear stress plane [Using Equations] (15 points)
Average Normal Stress: = = = 10 MPa
Max shearing stress is given by: = ± = ± 167.6 MPa
Maximum in-plane shear stress, = + = 10 + 167.6 = 177.6 MPa
- Draw the Mohr Circle diagram with the current state of stress in the Graph paper to scale (20 points)
The Mohr Diagram can be drawn as follows:
Now, substituting the values and drawing on the graph paper to scale, we can have the Mohr circle as drawn below:
- Identify the state of the stress for the element oriented at an angle ?(5 points)
? = -10
2? = -20
R = = 30/2 = 15 MPa
Point C: = R = 15 MPa
Point D: = R + Rcos|2?| = 15 + 15 cos20 = 29.09 MPa
= R – Rcos|2?| = 15 – 15 cos20 = 0.9 MPa
-R sin (2?) = -5.13 MPa
- Determine Principle stresses, Maximum in-plane Shear Stress, Principle plane and Maximum shear stress plane from the diagram [show working] (15 points)
Question 3
The given element is shown above:
- Identify Normal and Shear Stresses (5 points)
The normal stress are:
= 2100 MPa
= 300 MPa
And, the shear stress is:
= -560 MPa
- Determine stresses acting on an element oriented at an angle ? [Using Equations]
(15 points)
We know that,
= 2100 MPa
= 300 MPa
= -560 MPa
= 65 clockwise = -65
=
=
= + 428.9 = 1050.4 MPa
=
=
= 689.4 +359.96= 1049.4 Mpa
=
= 2100 + 300 – 1050.4
= 1349.6 Mpa
- Show this stresses on a sketch of an element oriented at this angle (10 points)
- Determine the Principle stresses and Principle plane [Using Equations] (15 points)
= 2100 MPa
= 300 MPa
= -560 MPa
Principal Angles:
= = -15.94
= – 15.94 + 90 = 74.06
= = 1200 -773.06 – 295.09 = 139.8 MPa
= = 1200 + 764.24 + 295.75 = 2259.9 MPa
- Determine average normal stress, maximum in-plane shear stress and maximum shear stress plane [Using Equations] (15 points)
Average Normal Stress: = = = 1200 MPa
Max shearing stress is given by: = ± = ± 2128.19 MPa
Maximum in-plane shear stress, = + = 1200 + 2128.19 = 3328.19 MPa
- Draw the Mohr Circle diagram with the current state of stress in the Graph paper to scale (20 points)
The Mohr Diagram can be drawn as follows:
Now, substituting the values and drawing on the graph paper to scale, we can have the Mohr circle as drawn below:
- Identify the state of the stress for the element oriented at an angle ?(5 points)
? = -65
2? = -130
R = = 2100/2 = 1050 MPa
Point C: = R = 1050 MPa
Point D: = R + Rcos|2?| = 1050 + 1050 cos130 = 375.07 MPa
= R – Rcos|2?| = 1050 – 1050 cos130 = 1724.9 MPa
-R sin (2?) = -428.9 MPa
- Determine Principle stresses, Maximum in-plane Shear Stress, Principle plane and Maximum shear stress plane from the diagram [show working] (15 points)
Question No 4
The given element is shown above:
- Identify Normal and Shear Stresses (5 points)
The normal stress are:
= -40 MPa
= 80 MPa
And, the shear stress is:
= 0 MPa
- Determine stresses acting on an element oriented at an angle ? [Using Equations]
(15 points)
We know that,
= -40 MPa
= 80 MPa
= 0 MPa
= 55 clockwise = -55
=
=
= = 40.5 MPa
=
=
= 56.38 Mpa
=
= -40 + 80 – 40.5
= -0.5 Mpa
- Show this stresses on a sketch of an element oriented at this angle (10 points)
- Determine the Principle stresses and Principle plane [Using Equations] (15 points)
= -40 MPa
= 80 MPa
= 0 MPa
Principal Angles:
= = 0
= 0 + 90 = 90
= = 20 +60 = 80 MPa
= = 20 -60 = -40 MPa
- Determine average normal stress, maximum in-plane shear stress and maximum shear stress plane [Using Equations] (15 points)
Average Normal Stress: = = = 20 MPa
Max shearing stress is given by: = ± = ± 60 MPa
Maximum in-plane shear stress, = + = 60+20 = 80 MPa
- Draw the Mohr Circle diagram with the current state of stress in the Graph paper to scale (20 points)
The Mohr Diagram can be drawn as follows:
Now, substituting the values and drawing on the graph paper to scale, we can have the Mohr circle as drawn below:
- Identify the state of the stress for the element oriented at an angle ?(5 points)
? = -55
2? = -110
R = = -40/2 = -20 MPa
Point C: = R = -20 MPa
Point D: = R + Rcos|2?| = -20 + (-20) cos110 = -13.2 MPa
= R – Rcos|2?| = -20 – 6.8 = -26.8 MPa
-R sin (2?) = 18.79 MPa
Determine Principle stresses, Maximum in-plane Shear Stress, Principle plane and Maximum shear stress plane from the diagram [show working] (15 points)
