 # Advanced Mechanics of Materials and Applied Elasticity: Analysis of Stress

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## Problems

### Sections 1.1 through 1.8

1.1

Two prismatic bars of a by b rectangular cross section are glued as shown in Fig. P1.1. The allowable normal and shearing stresses for the glued joint are 700 and 560 kPa, respectively. Assuming that the strength of the joint controls the design, what is the largest axial load P that may be applied? Use f = 40°, a = 50 mm, and b = 75 mm.

1.2

A prismatic steel bar of a = b = 50-mm square cross section is subjected to an axial tensile load P = 125 kN (Fig. P1.1). Calculate the normal and shearing stresses on all faces of an element oriented at (a) f = 70°, and (b) f = 45°.

1.3

A prismatic bar is under an axial load, producing a compressive stress of 75 MPa on a plane at an angle q = 30° (Fig. P1.3). Determine the normal and shearing stresses on all faces of an element at an angle of q = 50°.

1.4

A square prismatic bar of 1300-mm2 cross-sectional area is composed of two pieces of wood glued together along the x' plane, which makes an angle q with the axial direction (Fig. 1.6a). The normal and shearing stresses acting simultaneously on the joint are limited to 20 and 10 MPa, respectively, and on the bar itself, to 56 and 28 MPa, respectively. Determine the maximum allowable axial load that the bar can carry and the corresponding value of the angle q.

1.5

Calculate the maximum normal and shearing stresses in a circular bar of diameter d = 50 mm subjected to an axial compression load of P = 150 kN through rigid end plates at its ends.

1.6

A frame is formed by two metallic rectangular cross sectional parts soldered along their inclined planes as illustrated in Fig. P1.6. What is the permissible axial load P all that can be applied to the frame, without exceeding a normal stress of s all or a shearing stress of t all on the inclined plane? Given: a = 10 mm, b = 75 mm, t = 20 mm, q = 55°, s all = 25 MPa, and t all = 12 MPa. Assumption: Material strength in tension is 90 MPa.

1.7

Redo Prob. 1.6 for the case in which s all = 20 MPa, t all = 8 MPa, and q = 40°.

1.8

Determine the normal and shearing stresses on an inclined plane at an angle f through the bar subjected to an axial tensile force of P (Fig. P1.1). Given: a = 15 mm, b = 30 mm, f = 50°, P = 120 kN.

1.9

Redo Prob. 1.8, for an angle of f = 30° and P = –100 kN.

1.10

A cylindrical pipe of 160-mm outside diameter and 10-mm thickness, spirally welded at an angle of f = 40° with the axial (x) direction, is subjected to an axial compressive load of P = 150 kN through the rigid end plates (Fig. P1.10). Determine the normal s x' and shearing stresses t x'y' acting simultaneously in the plane of the weld.

1.11

The following describes the stress distribution in a body (in megapascals):

 s x = x 2 + 2y, s y = xy – y 2 z, t xy = –xy 2 + 1 t yz = 0, t xz = xz – 2x 2 y, s z = x 2 – z 2

Determine the body force distribution required for equilibrium and the magnitude of its resultant at the point x = –10 mm, y = 30 mm, z = 60 mm.

1.12

Given zero body forces, determine whether the following stress distribution can exist for a body in equilibrium:

 s x = –2c 1 xy, s y = c 2 z 2, s z = 0 t xy = c 1(c 2 – y 2) + c 3 xz, t xz = –c 3 y, t yz = 0

Here the c's are constants.

1.13

Determine whether the following stress fields are possible within an elastic structural member in equilibrium:

1. 2. The c's are constant, and it is assumed that the body forces are negligible.

1.14

For what body forces will the following stress field describe a state of equilibrium?

 s x = –2x 2 + 3y 2 – 5z, t xy = z + 4xy – 7 s y = –2y 2, t xz = –3x + y + 1 s z = 3x + y + 3z – 5, t yz = 0

### Sections 1.12 and 1.13

1.59

The state of stress at a point in an x, y, z coordinate system is

Determine the stresses and stress invariants relative to the x', y', z' coordinate system defined by rotating x, y through an angle of 30° counterclockwise about the z axis.

1.60

Redo Prob. 1.59 for the case in which the state of stress at a point in an x, y, z coordinate system is

1.61

The state of stress at a point relative to an x, y, z coordinate system is given by

Calculate the maximum shearing stress at the point.

1.62

At a point in a loaded member, the stresses relative to an x, y, z coordinate system are given by

Calculate the magnitude and direction of maximum principal stress.

1.63

For the stresses given in Prob. 1.59, calculate the maximum shearing stress.

1.64

At a specified point in a member, the state of stress with respect to a Cartesian coordinate system is given by

Calculate the magnitude and direction of the maximum principal stress.

1.65

At a point in a loaded structure, the stresses relative to an x, y, z coordinate system are given by

Determine by expanding the characteristic stress determinant: (a) the principal stresses; (b) the direction cosines of the maximum principal stress.

1.66

The stresses (in megapascals) with respect to an x, y, z coordinate system are described by

 s x = x 2 + y, s z = –x + 6y + z s y = y 2 – 5, t xy = t xz = t yz = 0

At point (3, 1, 5), determine (a) the stress components with respect to x', y', z' if

and (b) the stress components with respect to x", y", z" if , , and n 3 = 1. Show that the quantities given by Eq. (1.34) are invariant under the transformations (a) and (b).

1.67

Determine the stresses with respect to the x', y', z' axes in the element of Prob. 1.64 if

1.68

For the case of plane stress, verify that Eq. (1.33) reduces to Eq. (1.20).

1.69

Obtain the principal stresses and the related direction cosines for the following cases:

1. 2. ### Sections 1.14 through 1.17

 1.7 The stress at a point in a machine component relative to an x, y, z coordinate system is given by Referring to the parallelepiped shown in Fig. P1.70, calculate the normal stress s and the shear stress t at point Q for the surface parallel to the following planes: (a) CEBG, (b) ABEF, (c) AEG. [Hint: The position vectors of points G, E, A and any point on plane AEG are, respectively, r g = 3 i , r e = 4 j , r a = 2 k , r = x i + y j + z k . The equation of the plane is given by Equation P1.70 from which The direction cosines are then 1.71 Re-solve Prob. 1.70 for the case in which the dimensions of the parallelepiped are as shown in Fig. P1.71. 1.72 The state of stress at a point in a member relative to an x, y, z coordinate system is Determine the normal stress s and the shearing stress t on the surface intersecting the point and parallel to the plane: 2x + y – 3z = 9. 1.73 For the stresses given in Prob. 1.62, calculate the normal stress s and the shearing stress t on a plane whose outward normal is oriented at angles 35°, 60°, and 73.6° with the x, y, and z axes, respectively. 1.74 At a point in a loaded body, the stresses relative to an x, y, z coordinate system are Determine the normal stress s and the shearing stress t on a plane whose outward normal is oriented at angles of 40°, 75°, and 54° with the x, y, and z axes, respectively. 1.75 Determine the magnitude and direction of the maximum shearing stress for the cases given in Prob. 1.69. 1.76 The stresses at a point in a loaded machine bracket with respect to the x, y, z axes are given as Determine (a) the octahedral stresses; (b) the maximum shearing stresses. 1.77 The state of stress at a point in a member relative to an x, y, z coordinate system is given by Calculate (a) the principal stresses by expansion of the characteristic stress determinant; (b) the octahedral stresses and the maximum shearing stress. 1.78 Given the principal stresses s 1, s 2, and s 3 at a point in an elastic solid, prove that the maximum shearing stress at the point always exceeds the octahedral shearing stress. 1.79 Determine the value of the octahedral stresses of Prob. 1.64. 1.8 By using Eq. (1.38b), verify that the planes of maximum shearing stress in three dimensions bisect the planes of maximum and minimum principal stresses. Also find the normal stresses associated with the shearing plane by applying Eq. (1.37). 1.81 A point in a structural member is under three-dimensional stress with s x = 100 MPa, s y = 20 MPa, t xy = 60 MPa, and s z , as shown in Fig. P1.81. Calculate (a) the absolute maximum shear stress for s z = 30 MPa; (b) the absolute maximum shear stress for s z = –30 MPa. 1.82 Consider a point in a loaded body subjected to the stress field represented in Fig. P1.82. Determine, using only Mohr's circle, the principal stresses and their orientation with respect to the original system. 1.83 Re-solve Prob. 1.82 for the case of a point in a loaded body subjected to the following nonzero stress components: s x = 80 MPa, s z = –60 MPa, and t xy = 40 MPa. 1.84 The state of stress at a point in a loaded structure is represented in Fig. P1.84. Determine (a) the principal stresses; (b) the octahedral stresses and maximum shearing stress. 1.85 Find the normal and shearing stresses on an oblique plane defined by , and . The principal stresses are s 1 = 35 MPa, s 2 = –14 MPa, and s 3 = –28 MPa. If this plane is on the boundary of a structural member, what should be the values of surface forces px , py , and px on the plane? 1.86 Redo Prob. 1.85 for an octahedral plane, s 1 = 40 MPa, s 2 = 15 MPa, and s 3 = 25 MPa.