Normal Stress Under Axial Loading

1. The bar in Figure. has a constant width of 35 mm and a thickness of

10 mm. Determine the maximum average normal stress in the bar when it is subjected to the loading
shown. (σave = 85.7 MPa)

2. Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Knowing that
d1 = 50 mm and d2 =30 mm, find the average normal stress at the midsection of (a) rod AB, (b) rod BC.
(σAB = 35.7 MPa, σBC = 42.4 MPa)

3. The built-up shaft consists of a pipe AB and solid rod BC. The pipe has an inner diameter of 20 mm and
outer diameter of 28 mm. The rod has a diameter of 12 mm. Determine the average normal stress at points
D and E and represent the stress on a volume element located at each of these points.
(σD = 13.3 MPa (C), σE = 70.7 MPa (T))

Strength of Materials I · Concepts of Stresses · 1

4. Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Knowing that the
average normal stress must not exceed 175 MPa in rod AB and 150 MPa in rod BC, determine the smallest
allowable values of d1 and d2. (d1 = 22.6 mm, d2 = 15.96 mm)

5. Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Determine the
magnitude of the force P for which the tensile stress in rod AB has the same magnitude as the compressive
stress in rod BC. (P = 28.2 kips.)

6. Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. knowing that
P = 40 kips, determine the average normal stress at the midsection of
(a) rod AB. (σAB = 12.73 ksi.)
(b) rod BC. (σBC = -2.83 ksi.)

Strength of Materials I · Concepts of Stresses · 2

7. The cross-sectional area of bar ABCD is 600 mm2. Determine the maximum normal stress in the bar.
(58.3 MPa)

8. Axial loads are applied to the compound rod that is composed of an aluminum segment rigidly
connected between steel and bronze segments. What is the stress in each material given that P = 10 kN?
(σbr = 50 MPa (C), σal = 33.3 MPa (T), σst = 100 MPa (T))

9. Determine the largest internal normal force in the bar. (Nmax = 10 kN)

10. Determine the internal normal force at section A if the rod is subjected to the external uniformally
distributed loading along its length of 8 kN/m. (NA = 24 kN)

11. A 50-mm-wide steel bar has axial loads applied at points B, C, and D. If the normal stress magnitude in
the bar must not exceed 60 MPa, determine the minimum thickness that can be used for the bar.
(tmin = 21.7 mm)

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12. Two solid cylindrical rods (1) and (2) are joined together at flange B and loaded, as shown in Figure. The
diameter of rod (1) is d1 = 24 mm and the diameter of rod (2) is d2 = 42 mm. Determine the normal stresses
in rods (1) and (2). (σ1 = 176.8 MPa (T), σ2 = 144.4 MPa (C))

13. Two solid cylindrical rods (1) and (2) are joined together at flange B and loaded, as shown in Figure. If
the normal stress in each rod must be limited to 120 MPa, determine the minimum diameter required for
each rod. (d1 = 29.1 mm, d2 = 46.1 mm)

14. Two solid cylindrical rods (1) and (2) are joined together at flange B and loaded, as shown in Figure. If
the normal stress in each rod must be limited to 40 ksi, determine the minimum diameter required for
each rod. (d1 = 0.691 in., d2 = 1.545 in.)

Strength of Materials I · Concepts of Stresses · 4

15. Axial loads are applied with rigid bearing plates to the solid cylindrical rods shown in Figure. The
diameter of aluminum rod (1) is 2.00 in., the diameter of brass rod (2) is 1.50 in., and the diameter of steel
rod (3) is 3.00 in. Determine the normal stress in each of the three rods.
(σ1 = 5.09 ksi (C), σ2 = 7.92 ksi (T), σ3 = 3.68 ksi (C))

16. Two solid cylindrical rods support a load of P = 50 kN, as shown in Figure. If the normal stress in each
rod must be limited to 130 MPa, determine the minimum diameter required for each rod.
(d1 = 19.96 mm, d2 = 16.13 mm)

Strength of Materials I · Concepts of Stresses · 5

17. The 80-kg lamp is supported by two rods AB and BC as shown in Figure. If AB has a diameter of 10 mm
and BC has a diameter of 8 mm, determine the average normal stress in each rod.
(σBC = 7.86 MPa, σBA = 8.05 MPa)

18. Bar (1) in Figure. has a cross-sectional area of 0.75 in 2. If the stress in bar (1) must be limited to 30 ksi,
determine the maximum load P that may be supported by the structure. (Pmax = 13.50 kips.)

19. Each of the four vertical links has an 8 X 36-mm uniform rectangular cross section and each of the four
pins has a 16-mm diameter. Determine the maximum value of the average normal stress in the links
connecting
(a) points B and D. (σBD = 101.6 MPa)
(b) points C and E. (σCE = -21.7 MPa)

Strength of Materials I · Concepts of Stresses · 6

 Direct Shear Stress
1. If the wood joint in Fig. has a width of 150 mm, determine the average shear stress developed along
shear planes a–a and b–b. For each plane, represent the state of stress on an element of the material.
(τa,ave = 200 kPa, τb,ave = 160 kPa)

2. When the force P reached 8 kN, the wooden specimen shown failed in shear along the surface indicated
by the dashed line. Determine the average shearing stress along that surface at the time of failure.
(τave = 5.93 MPa)

3. The wooden members A and B are to be joined by plywood splice plates that will be fully glued on the
surfaces in contact. As part of the design of the joint, and knowing that the clearance between the ends of
the members is to be 6 mm, determine the smallest allowable length L if the average shearing stress in the
glue is not to exceed 700 kPa. (L = 292 mm)

Strength of Materials I · Concepts of Stresses · 7

4. Two 6 in. wide wooden boards are to be joined by splice plates that will be fully glued on the contact
surfaces. The glue to be used can safely provide a shear strength of 120 psi. Determine the smallest
allowable length L that can be used for the splice plates for an applied load of P = 10,000 lb. Note that a
gap of 0.5 in. is required between boards (1) and (2). (Lmin = 14.39 in.)

5. In each case, determine the largest internal shear force resisted by the bolt. Include all necessary free-
body diagrams. (Vmax = 3 kN)

6. In each case, determine the largest internal shear force resisted by the bolt. Include all necessary free-
body diagrams. (Vmax = 5 kN)

7. For the clevis connection shown in Figure., determine the shear stress in the 22-mm diameter bolt for an
applied load of P = 90 kN. (τ = 118.4 MPa)

Strength of Materials I · Concepts of Stresses · 8

8. For the connection shown in Figure., determine the average shear stress in the 7/8-in. diameter bolts if
the load is P = 45 kips. (τ = 14.97 ksi.)

9. The five-bolt connection shown in Figure. must support an applied load of P = 300 kN. If the average
shear stress in the bolts must be limited to 225 MPa, determine the minimum bolt diameter that may be
used in the connection. (dbolt = 18.43 mm)

10. The three-bolt connection shown in Figure. must support an applied load of P = 40 kips. If the average
shear stress in the bolts must be limited to 24 ksi, determine the minimum bolt diameter that may be used
in the connection. (dbolt = 0.595 in.)

Strength of Materials I · Concepts of Stresses · 9

11. For the connection shown in Figure., the average shear stress in the 12-mm-diameter bolts must be
limited to 160 MPa. Determine the maximum load P that may be applied to the connection.
(Pmax = 108.6 kN)

12. Two wooden planks, each 22 mm thick and 160 mm wide, are joined by the glued mortise joint shown.
Knowing that the joint will fail when the average shearing stress in the glue reaches 820 kPa, determine the
smallest allowable length d of the cuts if the joint is to withstand an axial load of magnitude P = 7.6 kN.
(d = 60.2 mm.)

13. The inclined member in Figure. is subjected to a compressive force of 600 N. Determine the average
compressive stress along the smooth areas of contact defined by AB and BC, and the average shear stress
along the horizontal plane defined by DB.

Strength of Materials I · Concepts of Stresses · 10

14. The bell crank, which is in equilibrium under the forces shown in the figure, is supported by a 20-mm-
diameter pin at D that is in double shear. Determine (a) the required diameter of the connecting rod AB,
given that its tensile working stress is 100 MPa; and (b) the shear stress in the pin.
[(a) 19.92 mm, (b) 84.3 MPa]

Bearing Stress
1. A 2.5-in.-wide by 0.125-in.-thick steel plate is connected to a support with a 0.75-in.-diameter pin. The
steel plate carries an axial load of 1.8 kips. Determine the bearing stress in the steel plate. (σb = 19.20 ksi)

2. An axial load P is supported by a short steel column, which has a cross-sectional area of 11,400 mm 2.
If the average normal stress in the steel column must not exceed 110 MPa, determine the minimum
required dimension “a” so that the bearing stress between the base plate and the concrete slab does not
exceed 8 MPa. (a = 396 mm)

Strength of Materials I · Concepts of Stresses · 11

3. A 12 -mm.-diameter steel rod AB is fitted to a round hole near end C of the wooden member CD. For
the loading shown, determine
(a) the maximum average normal stress in the wood b (σw = 3.97 MPa]
(b) the distance b for which the average shearing stress is 620 kPa on the surfaces indicated by the dashed
lines [b = 202 mm]
(c) the average bearing stress on the wood. [(σb)w = 20.8 MPa]

4. A 6-mm-diameter pin is used at connection C of the pedal shown. Knowing that P = 500 N, determine
(a) the average shearing stress in the pin (τave = 23 MPa)
(b) the nominal bearing stress in the pedal at C (σb = 24.1 MPa)
(c) the nominal bearing stress in each support bracket at C. (σb = 21.7 MPa)

5. The two wooden boards shown in Figure are connected by a 0.5-in. diameter bolt. Washers are installed
under the head of the bolt and under the nut. The washer dimensions are D = 2 in. and d = 5/8 in. The nut
is tightened to cause a tensile stress of 9,000 psi in the bolt. Determine the bearing stress between the
washer and the wood.

Strength of Materials I · Concepts of Stresses · 12

6. For the assembly and loading in fig. determine
(a) the average shearing stress in the pin at B (τB = 80.8 MPa)
(b) the average bearing stress at B in member BD (σb = 127.0 MPa)
(c) the average bearing stress at B in member ABC, knowing that this member has a 10 x 50-mm uniform
rectangular cross section. (σb = 203 MPa)

7. The bell crank shown in Figure is in equilibrium for the forces acting in rods (1) and (2). The bell crank is
supported by a 10-mm-diameter pin at B that acts in single shear. The thickness of the bell crank is 5 mm.
Assume a = 65 mm, b = 150 mm, F1 = 1,100 N, and θ = 50°. Determine the following:
(a) the shear stress in pin B (τpin = 11.58 MPa)
(b) the bearing stress in the bell crank at B (σb = 18.19 Mpa)

Strength of Materials I · Concepts of Stresses · 13

8. The bell-crank mechanism shown in Figure is in equilibrium for an applied load of P = 7 kN applied at A.
Assume a = 200 mm, b = 150 mm, and θ = 65°. Determine the minimum diameter d required for pin B for
each of the following conditions:
(a) The shear stress in the pin may not exceed 40 MPa. (dmin = 14.42 mm.)
(b) The bearing stress in the bell crank may not exceed 100 MPa. (dmin = 16.33 mm.)
(c) The bearing stress in the support bracket may not exceed 165 MPa. (dmin = 6.60 mm.)

Stresses on Inclined Sections

1. Two wooden members of uniform rectangular cross section are joined by the simple glued scarf splice
shown. Knowing that P = 11 kN, determine the normal and shearing stresses in the glued splice.
(σ = 489 kPa, τ = 489 kPa)

2. Two wooden members of uniform rectangular cross section are joined by the simple glued scarf splice
shown. Knowing that the maximum allowable shearing stress in the glued splice is 620 kPa, determine
(a) the largest load P that can be safely applied. (P = 13.95 kN)
(b) the corresponding tensile stress in the splice. (σ = 620 kPa)

Strength of Materials I · Concepts of Stresses · 14

3. The 1.4-kip load P is supported by two wooden members of uniform cross section that are joined by the
simple glued scarf splice shown. Determine the normal and shearing stresses in the glued splice.
(σ = 70.0 psi, τ = 40.4 psi)

4. A 120-mm-wide steel bar with a butt-welded joint, as shown, will be used to carry an axial tension load
of P = 180 kN. If the normal and shear stresses on the plane of the butt weld must be limited to 80 MPa
and 45 MPa, respectively, determine the minimum thickness required for the bar. (tmin = 15.23 mm)

5. A structural steel bar with a 25 mm × 75 mm rectangular cross section is subjected to an axial load of
150 kN. Determine the maximum normal and shear stresses in the bar. (σmax = 80 MPa, τmax = 40 MPa)

6. A steel rod of circular cross section will be used to carry an axial load of 92 kips. The maximum stresses
in the rod must be limited to 30 ksi in tension and 12 ksi in shear. Determine the required diameter for the
rod. (dmin = 2.21 in.)

7. An axial load P is applied to the rectangular bar shown in Figure. The cross-sectional area of the bar is
400 mm2. Determine the normal stress perpendicular to plane AB and the shear stress parallel to plane AB
if the bar is subjected to an axial load of P = 70 kN. (σn = 117.4 MPa, τt = 82.2 MPa)

Strength of Materials I · Concepts of Stresses · 15

8. A compression load of P = 80 kips is applied to a 4 in. by 4 in. square post, as shown in Figure. Determine
the normal stress perpendicular to plane AB and the shear stress parallel to plane AB.
(σn = 1.645 ksi, τt = 2.35 ksi)

9. A 90 mm wide bar will be used to carry an axial tension load of 280 kN. The normal and shear stresses
on plane AB must be limited to 150 MPa and 100 MPa, respectively. Determine the minimum thickness t
required for the bar. (tmin = 15.32 mm)

Allowable Stress Design

1. Two forces are applied to the bracket BCD as shown.
(a) Knowing that the control rod AB is to be made of a steel having an ultimate normal stress of 600 MPa,
determine the diameter of the rod for which the factor of safety with respect to failure will be 3.3.
(dAB = 16.74 mm)
(b) The pin at C is to be made of a steel having an ultimate shearing stress of 350 MPa. Determine the
diameter of the pin C for which the factor of safety with respect to shear will also be 3.3. (dC = 22 mm)
(c) Determine the required thickness of the bracket supports at C knowing that the allowable bearing stress
of the steel used is 300 MPa. (t = 6 mm)

Strength of Materials I · Concepts of Stresses · 16

2. The rigid beam BCD is attached by bolts to a control rod at B, to a hydraulic cylinder at C, and to a fixed
support at D. The diameters of the bolts used are: dB = dD = 3/8 in., dC = 1/2 in. Each bolt acts in double
shear and is made from a steel for which the ultimate shearing stress is τU = 40 ksi. The control rod AB has
a diameter dA = 7/16 in. and is made of a steel for which the ultimate tensile stress is σU = 60 ksi. If the
minimum factor of safety is to be 3.0 for the entire unit, determine the largest upward force which may be
applied by the hydraulic cylinder at C. (τA = 6790 psi, τC = 7640 psi, σA = 2290 psi, (σb)c = 6000 psi,
τB = 171.4 psi)

3. In the steel structure shown, a 6-mm-diameter pin is used at C and 10-mm-diameter pins are used at B
and D. The ultimate shearing stress is 150 MPa at all connections, and the ultimate normal stress is 400
MPa in link BD. Knowing that a factor of safety of 3.0 is desired, determine the largest load P that can be
applied at A. Note that link BD is not reinforced around the pin holes. (P = 1.683 kN)

Strength of Materials I · Concepts of Stresses · 17

4. The rigid bar AB shown in Figure. is supported by a steel rod AC having a diameter of 20 mm and an
aluminum block having a cross sectional area of 1800 mm2. The 18-mm-diameter pins at A and C are
subjected to single shear. If the failure stress for the steel and aluminum is (st)fail = 680 MPa and
(al)fail = 70 MPa, respectively, and the failure shear stress for each pin is fail = 900 MPa, determine the
largest load P that can be applied to the bar. Apply a factor of safety of F.S. = 2. (P = 168 kN)

5. The joint is fastened together using two bolts. Determine the required diameter of the bolts if the failure
shear stress for the bolts is fail = 350 MPa. Use a factor of safety for shear of F.S. = 2.5. (d = 13.5 mm)

Strength of Materials I · Concepts of Stresses · 18