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ME 323: Mechanics of Materials Homework Set 4

Fall 2019 Due: Wednesday, September 18

Problem 4.1 (10 points)

A rigid bar ADE is suspended horizontally using three elastic rods AB, CD, and EF as shown in Fig. 4.1. The rods have a cross sectional area of 𝑨 = 𝟏𝟎𝟎 𝐦𝐦𝟐 and the ends are attached to pinned connections. The rods are made of a material with elastic modulus 𝑬 = 𝟐 × 𝟏𝟎𝟓 𝐌𝐏𝐚 and coefficient of thermal expansion 𝜶 = 𝟏𝟎 𝟓 /˚𝐂. If the rod AB is not heated and the rods CD and EF are heated by 𝚫𝑻 and 𝟐𝚫𝑻 respectively, (where 𝚫𝑻 = 𝟑𝟎 ˚𝐂), determine the reaction forces developed in each rod.

Use 𝑳 = 𝟐𝟎𝟎 𝐦𝐦 for calculations.

Fig. 4.1

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Problem 4.2 (10 points)

A stepped shaft ABCD is made of circular solid members 1 and 3 and a circular hollow member 2 as shown in Fig 4.2. The allowable shear stress of each member is listed in the table below.

Member Maximum Allowable Shear Stress (in MPa) 1 60 2 25 3 40

If member 2 has an inner diameter of 𝒅𝟐 = 𝟖 𝐦𝐦 and a thickness 𝒕 = 𝟐 𝐦𝐦,

a) Determine the maximum value of the torque 𝑻 that can be applied without failure of member 2 in the shaft assembly.

b) For the calculated value of the torque 𝑻, determine the minimum diameters 𝒅𝟏 and 𝒅𝟑 of members 1 and 3, respectively, for safe design.

Note: Use 𝑳 = 𝟐𝟎𝟎 𝐦𝐦 for all calculations, if necessary.

Fig. 4.2

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Problem 4.3 (10 points)

A stepped composite bar assembly ABC is made of members 1, 2, and 3, which are joined by rigid connectors B and C as shown in Fig. 4.3a. Members 1 and 3 are circular solid rods with diameters 𝑫 and 𝟏. 𝟓𝑫 respectively, while member 2 is a hollow rod which is tightly fit on member 3 with an outer diameter of 𝟐𝑫. Members 1, 2, and 3 are made of materials with shear modulus 𝟐𝑮, 𝑮, and 𝟐𝑮 respectively. Torques of 𝑻 and 𝟐𝑻 are applied at B and C, respectively.

a) Determine the angle of twist generated at the connector C. b) Sketch the shear stress and shear strain distribution at the section cut XX’. (Note that the

cut XX’ can be anywhere in between the locations B and C)

Fig. 4.3a

Fig. 4.3b: Section view at cut XX’

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Problem 4.4 (10 points)

The gear-shaft system shown in Fig. 4.4 is supported by frictionless bearings and transmits the torque to the fixed end A using the rigid gears at B and C. Shafts 1 and 3 are made of a material with shear modulus 𝑮𝟏 = 𝑮𝟑 = 𝟏𝟎𝟎 𝐆𝐏𝐚 and shaft 2 is made of a material with shear modulus 𝑮𝟐 = 𝟓𝟎 𝐆𝐏𝐚. The diameters of the shafts are 𝒅𝟏 = 𝟑𝟐 𝐦𝐦, 𝒅𝟐 = 𝟔𝟒 𝐦𝐦, and 𝒅𝟑 = 𝟏𝟔 𝐦𝐦. Calculate the angle of twist generated at the free ends D and E.

Use 𝑳 = 𝟏 𝐦, 𝒅𝒄 = 𝟏𝟓𝟎 𝐦𝐦, 𝒅𝒃 = 𝟗𝟎 𝐦𝐦, and 𝑻𝟎 = 𝟓𝟎𝟎 𝐍 ∙ 𝐦

Fig. 4.4