Problem 11.1 (10 points) For the state of plane stress shown in the
Principal stress: 1. = 125 MPa 2. = 62.5 MPa
HW solution corrected V
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1/2. = 16.2 ! 106 N/m2 = 16.2 MPa. 8.7 Suppose that a wing component on an aircraft is fabricated (b) Determine the fatigue strength at 5 × 105 cycles.
Assignment solutions
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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: Sinceǡߪ௭ൌͲǡ߬௬௭ൌ߬Maximum 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: Sinceǡߪ௭ൌͲǡ߬௬௫ൌ߬Maximum 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 stressesAxial stress = ɐୟൌ୮୰
Hoop stress = ɐ୦ൌ୮୰
Three-dimensional differential stress element at A: Sinceǡߪ௭ൌͲǡ߬௬௭ൌ߬ Principal angle: =ߠభൌͻͲι,ߠMaximum 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: =ߠభൌͻͲι,ߠMaximum 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 (ܢۻState 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