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 𝜃 [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 𝜃
*(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 𝜃 [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 𝜃
*(5 points)*

𝜃 = -15

2𝜃 = -30

R = = 350/2 = 175 MPa

Point C: = R = 175 MPa

Point D: = R + Rcos|2𝜃| = 175 + 175 cos30 = 326.55 MPa

= R – Rcos|2𝜃| = 175 – 175 cos30 = 23.44 MPa

-R sin (2𝜃) = 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 𝜃 [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 𝜃
*(5 points)*

𝜃 = -10

2𝜃 = -20

R = = 30/2 = 15 MPa

Point C: = R = 15 MPa

Point D: = R + Rcos|2𝜃| = 15 + 15 cos20 = 29.09 MPa

= R – Rcos|2𝜃| = 15 – 15 cos20 = 0.9 MPa

-R sin (2𝜃) = -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 𝜃 [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 𝜃
*(5 points)*

𝜃 = -65

2𝜃 = -130

R = = 2100/2 = 1050 MPa

Point C: = R = 1050 MPa

Point D: = R + Rcos|2𝜃| = 1050 + 1050 cos130 = 375.07 MPa

= R – Rcos|2𝜃| = 1050 – 1050 cos130 = 1724.9 MPa

-R sin (2𝜃) = -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 𝜃 [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:

- Identify the state of the stress for the element oriented at an angle 𝜃
*(5 points)*

𝜃 = -55

2𝜃 = -110

R = = -40/2 = -20 MPa

Point C: = R = -20 MPa

Point D: = R + Rcos|2𝜃| = -20 + (-20) cos110 = -13.2 MPa

= R – Rcos|2𝜃| = -20 – 6.8 = -26.8 MPa

-R sin (2𝜃) = 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)*