Tag Archives: 3D Bending

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 xx and yy axes. Find the minimum diameter required for the shaft if the maximum normal stress in the shaft cannot exceed 250 MPa250~ \text{MPa}. All dimensions are in mm\text{mm}.

Problem 2

The beam ABCDABCD shown in the figure is simply supported at AA and DD, and has a circular cross-section with a diameter of 80 mm80~\text{mm}. The distributed force acts at an angle of 60∘60^\circ to the ZZ axis and has components only along the ZZ and XX axis. The 1.2 kN1.2~\text{kN} force acts parallel to the ZZ axis and the 1.5 kN1.5~\text{kN} force acts parallel the XX axis. AB=0.6 mAB = 0.6~ \text{m}, BC=0.4 mBC = 0.4~ \text{m} and CD=1 mCD = 1~ \text{m}.

(a) Draw shear force and bending moment diagrams for the XYXY and YZYZ 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 mm35~\text{mm} in diameter, such that section A−CA-C is parallel to the yy-axis and section B−EB-E is parallel to xx-axis.

An unknown moment TT parallel to yy is applied at point AA, where there is also a spherical (ball) hinge, which constraints translations along xx, yy, and zz axes. The plane hinge at CC is contained in the x−zx-z plane, which means that it constrains translations along the xx and zz axes. The force of 1000 N1000~\text{N} at DD goes in −z-z, the force of 500 N500~\text{N} at DD goes in −y-y and the force of 500 N500~\text{N} at EE goes in −x-x. Given these conditions and the coordinate system provided, find:

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

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Homework 1 in MEE 322: Structural Mechanics | Internal Reactions in 3D and 2D Bending

Problem 1

The structure shown in Fig. 1.1 is made with a steel bar that has a rectangular cross-section 5 cm tall and 10 cm wide.

Fig 1.1. Frame supporting point and distributed load

Given the geometry, loads and supports in the structure draw the bending moment diagrams for segments A-C and D-E of the structure. Then, use the diagrams to find the critical section and calculate the maximum bending stress in segment AC.

Problem 2

The torsion rod with variable cross-section shown in Fig. 2.1 is clamped at A and carries point torques at B (4 kN.m), C (8 kN.m) and D (unknown value T) with the senses indicated in the figure. It is known that the diameter of AB is 25 mm, the diameter of BC is 50 mm and the diameter of CD is 20 mm. Furthermore, 6LAB = 6LBC = 5LCD = 1200 mm. If the shear modulus of the material is 80 GPa, find:

a) The value of T that would make the angle of twist at point C with respect to A equal to zero.

b) The maximum value of T that can applied with the sense shown such that failure does not occur for an allowable shear stress of 1.1 GPa. Can the condition from part a be achieved without failure?

Fig. 2.1: Torsion rod with variable cross-section.

Problem 3

The cantilever beam shown in Fig. 3.1 has a rectangular cross-section with h = 120 mm and b = 80 mm. The 12 kN and 10 kN forces act parallel to the x and z axis, respectively, and pass through the centroid of the beam cross-section at the locations they act. The 10 kN force acts at the free end of the cantilever, whereas the 12 kN force acts 250 mm away from the free end. The cross-section ABCD is 750 mm away from the free end.

a) Determine the magnitude and location of the maximum tensile and compressive bending stress at the cross-section ABCD and indicate the neutral axis on it.

b) If an additional force, F, is applied on the beam parallel to the z axis at a point 500 mm away from the free end, what should be its magnitude and direction to make the bending stress at point B zero?

Fig. 3.1: Cantilever beam with loads along two axes.

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Homework 1 in MEE 322: Structural Mechanics

This is the complete solution manual to the problems in Homework 1 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 homeworkhelp@engineering-math.org 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 help@engineering-math.org

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