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Homework 4 in MEE 322: Structural Mechanics | Two-Dimensional Stress Analysis

Problem 1

The state of stress at a point is given in the figure. Find τ and τxy directly using force equilibrium. Do NOT use the stress transformation equations.

Problem 2

The state of stress at a point is given in the figure. Find σ and τxy using stress transformation equations.

Problem 3

The state of stress at a point is given in the figure. Find the principal stresses, principal directions and the maximum shear stress using

(a) Eigenvalue problem approach

(b) Stress transformation equations

Problem 4

The shaft shown in the figure has a gear at B with a force of 2098 N in -y and 6456 N in +z applied at its tip. The force along z produces a torque that drives the component attached at C, which produces an equal and opposite torque to that produced at the gear as well as forces on the shaft of 6000 N along +y and +z. The bearing at A can be considered a spherical hinge, whereas the bearing at D can be considered a planar hinge in the y-z plane.

(b) Draw bending moment and torsion diagrams for the shaft and show diagrams of the cross-section of the shaft where the critical points occur, i.e., the locations where the maximum normal stresses due to bending and maximum shear stress due to torsion coincide. Indicate the internal reactions (bending and torsion moments) in this diagram, as well as the locations of the critical points.

(b) If the diameter of the shaft is 33 mm, find the stresses at the critical point and use them to find the maximum shear stress at that location as well as the maximum and minimum principal stresses. Note: the bending normal stress can be taken as σx, while the torsion shear stress can be assumed to be τxy for the effects of this calculation, all other stresses can be assumed to be zero.

(c) It is known that the material of the shaft is such that it will fail if the maximum shear stress reaches 300 MPa. Is the shaft safe? If so, calculate the factor of safety.

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Homework 4 in MEE 322: Structural Mechanics

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Homework 2 in MEE 322: Structural Mechanics | Normal and Shear Stress Under Combined Loading

Problem 1

The shaft with a circular cross-section is supported by two bearings at O and C. The bearings do not exert any moments or axial force on the shaft, and they act to constrain motion along the x and y axes. Find the minimum diameter required for the shaft if the maximum normal stress in the shaft cannot exceed 250~ \text{MPa}. All dimensions are in \text{mm}.

Problem 2

The beam ABCD shown in the figure is simply supported at A and D, and has a circular cross-section with a diameter of 80~\text{mm}. The distributed force acts at an angle of 60^\circ to the Z axis and has components only along the Z and X axis. The 1.2~\text{kN} force acts parallel to the Z axis and the 1.5~\text{kN} force acts parallel the X axis. AB = 0.6~ \text{m}, BC = 0.4~ \text{m} and CD = 1~ \text{m}.

(a) Draw shear force and bending moment diagrams for the XY and YZ planes.

(b) Determine the maximum tensile and compressive bending stress in the beam and show their locations on the cross-section of the beam.

Problem 3

The following steel structure will be made with a round bar 35~\text{mm} in diameter, such that section A-C is parallel to the y-axis and section B-E is parallel to x-axis.

An unknown moment T parallel to y is applied at point A, where there is also a spherical (ball) hinge, which constraints translations along x, y, and z axes. The plane hinge at C is contained in the x-z plane, which means that it constrains translations along the x and z axes. The force of 1000~\text{N} at D goes in -z, the force of 500~\text{N} at D goes in -y and the force of 500~\text{N} at E goes in -x. Given these conditions and the coordinate system provided, find:

(a) Diagrams of axial force, bending moment, and torsion moments for sections A-C and B-E. Use the given coordinate system to label the planes where you are making your internal reaction diagrams (x-y, x-z or z-y) and draw the corresponding axes.

(b) Calculate the maximum normal stress due to bending in this structure. Show in a diagram the point(s) in the cross-section of the structure where this stress occurs, and include the bending moments in each axis, the resultant moment, and the neutral axis. Use the given coordinate system to show the orientation of your diagram.

(c) Calculate the maximum shear stress due to torsion in this structure. Show in a diagram the point(s) in the cross-section of the structure where this stress occurs, including the torsion moment. Use the given coordinate system to show the orientation of your diagram.

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Homework 2 in MEE 322: Structural Mechanics

This is the complete solution manual to the problems in Homework 2 of ME322: Structural Mechanics. The 3 problems included in the manual are written above. The PDF document will be sent to your email address within 24 hours from the time of purchase. The email will be coming from [email protected] Do not forget to check on your spam folder. If you have not received your file in 24 hours, kindly send us an email at [email protected]

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