Problem 11.1 (10 points) For the state of plane stress shown in the









Problem 11.1 (10 points) For the state of plane stress shown in the

Principal stress: 1. = 125 MPa 2. = 62.5 MPa
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213204 Problem 11.1 (10 points) For the state of plane stress shown in the

Problem 11.1 (10 points)

For the state of plane stress shown in the figure:

1. X and on

face Y. 2. shown in the figure and label this point as N .

ME 323: Mechanics of Materials Homework Set 11

Fall 2019 Due: Wednesday, November 20

Solution:

The give state of plane stress has the following stresses: Coordinates of point N and the normal and shear stresses on the inclined plane are as follows:

Shear Stress: ߬

Note: The rotation considered here is +૝૙࢕ǡ however a rotation of െ૞૙࢕is also valid (in this

Problem 11.2 (10 points)

For the loading conditions shown in cases (a) (b):

1. Determine the state of stress at points A and B

2. Represent the state of stress at points A and B in three-dimensional differential stress

elements. , determine:

3. The principal stresses and principal angles for the states of stress at A and B.

Note: Identify first which is the plane corresponding to the state of plane stress (namely, xy-plane, xz-plane or yz-plane) for each point and loading condition.

4. The maximum in-plane shear stresses at points A and B.

5. The absolute maximum shear stress at points A and B.

Case (a):

Solution: Case (a)

Making a cut at point H:

Internal resultant forces include only the torque.

POINT A

Stress distribution at point A:

୮ = polar moment of area There are no normal stresses acting on the point A, ߪ௫ൌͲǡߪ the xy plane, ߬ Three-dimensional differential stress element at A: Sinceǡߪ௭ൌͲǡ߬௬௭ൌ߬

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

POINT B

Stress distribution at point B:

୮ = polar moment of area There are no normal stresses acting on the point B, ߪ௫ൌͲǡߪ the xy plane, ߬ Three-dimensional differential stress element at B: Sinceǡߪ௭ൌͲǡ߬௬௫ൌ߬

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

Case (b):

Notice that there is no point B for this loading condition The element A will only experience hoop and axial stresses

Axial stress = ɐୟൌ୮୰

Hoop stress = ɐ୦ൌ୮୰

Three-dimensional differential stress element at A: Sinceǡߪ௭ൌͲǡ߬௬௭ൌ߬ Principal angle: =ߠ௣భൌͻͲι,ߠ

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

Problem 11.3 (10 points)

For the loading conditions shown in cases (c) ʹ (d):

1. Determine the state of stress at points A and B

2. Represent the state of stress at points A and B in three-dimensional differential stress elements.

3. The principal stresses and principal angles for the states of stress at A and B.

Note: identify first which is the plane corresponding to the state of plane stress (namely, xy- plane, xz-plane or yz-plane) for each point and loading condition.

4. The maximum in-plane shear stresses at points A and B.

5. The absolute maximum shear stress at points A and B.

Case (c):

FBD:

Making a cut at point H:

POINT A

Normal Stress Distribution due to axial loading:

Normal Stress Distribution due to bending:

Shear Stress Distribution due to transverse loading: Three-dimensional differential stress element at A:

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

POINT B

Normal Stress Distribution due to axial loading:

Normal Stress Distribution due to bending:

Shear Stress Distribution due to transverse loading: Three-dimensional differential stress element at B: Principal angle: =ߠ௣భൌͻͲι,ߠ

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

Case (d):

FBD: Making a cut at H and finding the internal resultant force, moment and torque we have:

POINT A

Normal Stress Distribution due to bending at A:

Shear Stress Distribution due to transverse loading at A: Shear stress distribution due to torsional loading at A: ୮ = polar moment of area Three-dimensional differential stress element at A:

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

POINT B

Normal Stress Distribution due to bending at B:

Shear Stress Distribution due to transverse loading at B: Stress distribution due to torsional loading at point B: ୮ = polar moment of area Three-dimensional differential stress element at A: stress.

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

Problem 11.4 (10 points)

Consider the elastic structure shown in the figure, where a force equal to 500 N i - 750 N j is applied at the end of the segment CH parallel to the z-axis.

1. Determine the internal resultants at cross section B (i.e., axial force, two shear forces,

torque, and two bending moments).

2. Show the stress distribution due to each internal resultant on the appropriate view of the

cross B (i.e., side view, front view or top view).

3. Determine the state of stress on points a and b on cross section B.

4. Represent the state of stress at points a and b in three-dimensional differential stress

elements.

5. Determine the principal stresses and the absolute maximum shear stress at point b.

FBD:

Moment balance about point B:

POINT ࢇ

Stress distribution due to torsional loading (ܠۻ ୮ = polar moment of area Stress distribution due to axial loading (ܠ۰ Stress distribution due to Shear force 1 (ܡ۰ Normal Stress Distribution due to bending moment 1 (ܡۻ Normal Stress Distribution due to bending moment 2 (ܢۻ State of stress at pointԢࢇԢ: ોܠ b Stress distribution due to torsional loading (ܠۻ ୮ = polar moment of area Normal Stress Distribution due to bending moment 1 (ܡۻ Normal Stress Distribution due to bending moment 2 (ܢۻ

State of stress at pointԢ࢈ᇱ:

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

Problem 11.1 (10 points)

For the state of plane stress shown in the figure:

1. X and on

face Y. 2. shown in the figure and label this point as N .

ME 323: Mechanics of Materials Homework Set 11

Fall 2019 Due: Wednesday, November 20

Solution:

The give state of plane stress has the following stresses: Coordinates of point N and the normal and shear stresses on the inclined plane are as follows:

Shear Stress: ߬

Note: The rotation considered here is +૝૙࢕ǡ however a rotation of െ૞૙࢕is also valid (in this

Problem 11.2 (10 points)

For the loading conditions shown in cases (a) (b):

1. Determine the state of stress at points A and B

2. Represent the state of stress at points A and B in three-dimensional differential stress

elements. , determine:

3. The principal stresses and principal angles for the states of stress at A and B.

Note: Identify first which is the plane corresponding to the state of plane stress (namely, xy-plane, xz-plane or yz-plane) for each point and loading condition.

4. The maximum in-plane shear stresses at points A and B.

5. The absolute maximum shear stress at points A and B.

Case (a):

Solution: Case (a)

Making a cut at point H:

Internal resultant forces include only the torque.

POINT A

Stress distribution at point A:

୮ = polar moment of area There are no normal stresses acting on the point A, ߪ௫ൌͲǡߪ the xy plane, ߬ Three-dimensional differential stress element at A: Sinceǡߪ௭ൌͲǡ߬௬௭ൌ߬

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

POINT B

Stress distribution at point B:

୮ = polar moment of area There are no normal stresses acting on the point B, ߪ௫ൌͲǡߪ the xy plane, ߬ Three-dimensional differential stress element at B: Sinceǡߪ௭ൌͲǡ߬௬௫ൌ߬

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

Case (b):

Notice that there is no point B for this loading condition The element A will only experience hoop and axial stresses

Axial stress = ɐୟൌ୮୰

Hoop stress = ɐ୦ൌ୮୰

Three-dimensional differential stress element at A: Sinceǡߪ௭ൌͲǡ߬௬௭ൌ߬ Principal angle: =ߠ௣భൌͻͲι,ߠ

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

Problem 11.3 (10 points)

For the loading conditions shown in cases (c) ʹ (d):

1. Determine the state of stress at points A and B

2. Represent the state of stress at points A and B in three-dimensional differential stress elements.

3. The principal stresses and principal angles for the states of stress at A and B.

Note: identify first which is the plane corresponding to the state of plane stress (namely, xy- plane, xz-plane or yz-plane) for each point and loading condition.

4. The maximum in-plane shear stresses at points A and B.

5. The absolute maximum shear stress at points A and B.

Case (c):

FBD:

Making a cut at point H:

POINT A

Normal Stress Distribution due to axial loading:

Normal Stress Distribution due to bending:

Shear Stress Distribution due to transverse loading: Three-dimensional differential stress element at A:

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

POINT B

Normal Stress Distribution due to axial loading:

Normal Stress Distribution due to bending:

Shear Stress Distribution due to transverse loading: Three-dimensional differential stress element at B: Principal angle: =ߠ௣భൌͻͲι,ߠ

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

Case (d):

FBD: Making a cut at H and finding the internal resultant force, moment and torque we have:

POINT A

Normal Stress Distribution due to bending at A:

Shear Stress Distribution due to transverse loading at A: Shear stress distribution due to torsional loading at A: ୮ = polar moment of area Three-dimensional differential stress element at A:

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

POINT B

Normal Stress Distribution due to bending at B:

Shear Stress Distribution due to transverse loading at B: Stress distribution due to torsional loading at point B: ୮ = polar moment of area Three-dimensional differential stress element at A: stress.

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬

Problem 11.4 (10 points)

Consider the elastic structure shown in the figure, where a force equal to 500 N i - 750 N j is applied at the end of the segment CH parallel to the z-axis.

1. Determine the internal resultants at cross section B (i.e., axial force, two shear forces,

torque, and two bending moments).

2. Show the stress distribution due to each internal resultant on the appropriate view of the

cross B (i.e., side view, front view or top view).

3. Determine the state of stress on points a and b on cross section B.

4. Represent the state of stress at points a and b in three-dimensional differential stress

elements.

5. Determine the principal stresses and the absolute maximum shear stress at point b.

FBD:

Moment balance about point B:

POINT ࢇ

Stress distribution due to torsional loading (ܠۻ ୮ = polar moment of area Stress distribution due to axial loading (ܠ۰ Stress distribution due to Shear force 1 (ܡ۰ Normal Stress Distribution due to bending moment 1 (ܡۻ Normal Stress Distribution due to bending moment 2 (ܢۻ State of stress at pointԢࢇԢ: ોܠ b Stress distribution due to torsional loading (ܠۻ ୮ = polar moment of area Normal Stress Distribution due to bending moment 1 (ܡۻ Normal Stress Distribution due to bending moment 2 (ܢۻ

State of stress at pointԢ࢈ᇱ:

Maximum in plane shear stresses: ߬

Absolute shear stress: ߬


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