simple stress

Simple Stresses 

Simple stresses are expressed as the ratio of the applied force divided by the resisting 

area or 

σ = Force / Area. 

It is the expression of force per unit area to structural members that are subjected to 

external forces and/or induced forces. Stress is the lead to accurately describe and 

predict the elastic deformation of a body. 

Simple stress can be classified as normal stress, shear stress, and bearing stress. 

Normal stress develops when a force is applied perpendicular to the cross-sectional 

area of the material. If the force is going to pull the material, the stress is said to be 

tensile stress and compressive stress develops when the material is being 

compressed by two opposing forces. Shear stress is developed if the applied force is 

parallel to the resisting area. Example is the bolt that holds the tension rod in its 

anchor. Another condition of shearing is when we twist a bar along its longitudinal axis. 

This type of shearing is called torsion and covered in Chapter 3. Another type of simple 

stress is the bearing stress, it is the contact pressure between two bodies. 

Suspension bridges are good example of structures that carry these stresses. The 

weight of the vehicle is carried by the bridge deck and passes the force to the stringers 

(vertical cables), which in turn, supported by the main suspension cables. The 

suspension cables then transferred the force into bridge towers. 

Normal Stress 

Stress 

Stress is the expression of force applied to a unit area of surface. It is measured in psi 

(English unit) or in MPa (SI unit). Another unit of stress which is not commonly used is 

the dynes (cgs unit). Stress is the ratio of force over area. 

stress = force / area 

Simple Stresses 

There are three types of simple stress namely; normal stress, shearing stress, and 

bearing stress. 

Normal Stress 

The resisting area is perpendicular to the applied force, thus normal. There are two 

types of normal stresses; tensile stress and compressive stress. Tensile stress applied 

to bar tends the bar to elongate while compressive stress tend to shorten the bar. 

where P is the applied normal load in Newton and A is the area in mm

2

. The maximum 

stress in tension or compression occurs over a section normal to the load. 

SOLVED PROBLEMS IN NORMAL STRESS 

Problem 104 

A hollow steel tube with an inside diameter of 100 

mm must carry a tensile load of 400 kN. Determine 

the outside diameter of the tube if the stress is limited 

to 120 MN/m

2

. 

Solution 104 

Problem 105 

A homogeneous 800 kg bar AB is supported at either 

end by a cable as shown in Fig. P-105. Calculate the 

smallest area of each cable if the stress is not to exceed 

90 MPa in bronze and 120 MPa in steel. 

Figure P-105

Solution 105 

Problem 106 

The homogeneous bar shown in Fig. P-106 is 

supported by a smooth pin at C and a cable that runs 

from A to B around the smooth peg at D. Find the 

stress in the cable if its diameter is 0.6 inch and the 

bar weighs 6000 lb. 

Solution 106 

Problem 107 

A rod is composed of an aluminum section rigidly 

attached between steel and bronze sections, as shown 

in Fig. P-107. Axial loads are applied at the positions 

indicated. If P = 3000 lb and the cross sectional area 

of the rod is 0.5 in

2

, determine the stress in each 

section. 

Solution 107 

Problem 108 

An aluminum rod is rigidly attached between a steel rod and a bronze rod as shown in 

Fig. P-108. Axial loads are applied at the positions indicated. Find the maximum value of 

P that will not exceed a stress in steel of 140 MPa, in aluminum of 90 MPa, or in bronze 

of 100 MPa. 

Figure P-108

Solution 108 

Problem 109 

Determine the largest weight W that can be supported by two wires shown in Fig. P-

109. The stress in either wire is not to exceed 30 ksi. The cross-sectional areas of wires 

AB and AC are 0.4 in

2

 and 0.5 in

2

, respectively. 

Solution 109 

Problem 110 

A 12-inches square steel bearing plate lies between an 8-inches diameter 

wooden post and a concrete footing as shown in Fig. P-110. Determine 

the maximum value of the load P if the stress in wood is limited to 1800 

psi and that in concrete to 650 psi. 

Solution 110 

Problem 111 

For the truss shown in Fig. P-111, calculate the stresses in members CE, DE, and DF. 

The crosssectional area of each member is 1.8 in

2

. Indicate tension (T) or compression 

(C). 

Solution 111 

Problem 112 

Determine the crosssectional areas of members AG, BC, and CE for the truss shown in 

Fig. P-112 above. The stresses are not to exceed 20 ksi in tension and 14 ksi in 

compression. A reduced stress in compression is specified to reduce the danger of 

buckling. 

Solution 112 

Problem 113 

Find the stresses in members BC, BD, and CF for the truss shown in Fig. P-113. Indicate 

the tension or compression. The cross sectional area of each member is 1600 mm

2

. 

Solution 113 

Problem 114 

The homogeneous bar ABCD shown in Fig. P-114 is supported by a cable that runs from 

A to B around the smooth peg at E, a vertical cable at C, and a smooth inclined surface 

at D. Determine the mass of the heaviest bar that can be supported if the stress in each 

cable is limited to 100 MPa. The area of the cable AB is 250 mm

2

 and that of the cable 

at C is 300 mm

2

. 

Solution 114 

Shearing Stress 

Forces parallel to the area resisting the force cause shearing stress. It differs to tensile 

and compressive stresses, which are caused by forces perpendicular to the area on 

which they act. Shearing stress is also known as tangential stress. 

where V is the resultant shearing force which passes which passes through the centroid 

of the area A being sheared.  

SOLVED PROBLEMS IN SHEARING STRESS 

Problem 115 

What force is required to punch a 20-mm-diameter hole in a plate that is 25 mm thick? 

The shear strength is 350 MN/m

2

. 

Solution 115 

Problem 116 

As in Fig. 1-11c, a hole is to be punched out of a plate having a shearing strength of 40 

ksi. The compressive stress in the punch is limited to 50 ksi. (a) Compute the maximum 

thickness of plate in which a hole 2.5 inches in diameter can be punched. (b) If the 

plate is 0.25 inch thick, determine the diameter of the smallest hole that can be 

punched. 

Solution 116 

Problem 117 

Find the smallest diameter bolt that can be used in the clevis shown in Fig. 1-11b if P = 

400 kN. The shearing strength of the bolt is 300 MPa. 

Solution 117 

Problem 118 

A 200-mm-diameter pulley is prevented from rotating relative to 60-mm-diameter shaft 

by a 70-mm-long key, as shown in Fig. P-118. If a torque T = 2.2 kN·m is applied to the 

shaft, determine the width b if the 

allowable shearing stress in the key is 60 MPa. 

Solution 118 

Problem 119 

Compute the shearing stress in the pin at B for the member supported as 

shown in Fig. P-119. The pin diameter is 20 mm.  

Solution 119 

Problem 120 

The members of the structure in Fig. P-120 weigh 200 lb/ft. Determine the smallest 

diameter pin that can be used at A if the shearing stress is limited to 5000 psi. Assume 

single shear. 

Solution 120 

Problem 121  

Referring to Fig. P-121, compute the maximum force P that can be applied by the 

machine operator, if the shearing stress in the pin at B and the axial stress in the 

control rod at C are limited to 4000 psi and 5000 psi, respectively. The diameters are 

0.25 inch for the pin, and 0.5 inch for the control rod. Assume single shear for the pin 

at B. 

Solution 121 

Problem 122 

Two blocks of wood, width w and thickness t, are glued together along the joint inclined 

at the angle θ as shown in Fig. P-122. Using the free-body diagram concept in Fig. 1-4a, 

show that the shearing stress on the glued joint is τ = P sin 2θ/2A, where A is the cross-

sectional area. 

Solution 122 

Problem 123 

A rectangular piece of wood, 50 mm by 100 mm in cross section, is used as a 

compression block shown in Fig. P-123. Determine the axial force P that can be safely 

applied to the block if the compressive stress in wood is limited to 20 MN/m

2

 and the 

shearing stress parallel to the grain is limited to 5 MN/m

2

. The grain makes an angle of 

20° with the horizontal, as shown. (Hint: Use the results in Problem 122.) 

Solution 123 

Bearing Stress 

Bearing stress is the contact pressure between the separate bodies. It differs from 

compressive stress, as it is an internal stress caused by compressive forces.  

SOLVED PROBLEMS IN BEARING STRESS 

Problem 125 

In Fig. 1-12, assume that a 20-mm-diameter rivet joins the plates that are each 110 

mm wide. The allowable stresses are 120 MPa for bearing in the plate material and 60 

MPa for shearing of rivet. Determine (a) the minimum thickness of each plate; and (b) 

the largest average tensile stress in the plates. 

Solution 125 

Problem 126 

The lap joint shown in Fig. P-126 is fastened by four ¾-in.-diameter rivets. Calculate 

the maximum safe load P that can be applied if the shearing stress in the rivets is 

limited to 14 ksi and the bearing stress in the plates is limited to 18 ksi. Assume the 

applied load is uniformly distributed among the four rivets. 

Solution 126 

Problem 127 

In the clevis shown in Fig. 1-11b, find the minimum bolt diameter and the minimum 

thickness of each yoke that will support a load P = 14 kips without exceeding a shearing 

stress of 12 ksi and a bearing stress of 20 ksi. 

Solution 127 

Problem 128 

A W18 × 86 beam is riveted to a W24 × 117 girder by a connection similar to that in 

Fig. 1-13. The diameter of the rivets is 7/8 in., and the angles are each 4 × 31/2 × 3/8 

in. For each rivet, assume that the allowable stresses are τ = 15 ksi and σ

b

 = 32 ksi. 

Find the allowable 

load on the connection.  

Solution 128 

Note: Textbook is Strength of Materials 4th edition by Pytel and Singer 

Problem 129 

A 7/8-in.-diameter bolt, having a diameter at the root of the threads of 0.731 in., is 

used to fasten two timbers together as shown in Fig. P-129. The nut is tightened to 

cause a tensile stress of 18 ksi in the bolt. Compute the shearing stress in the head of 

the bolt and in the threads. Also, determine the outside diameter of the washers if their 

inside diameter is 9/8 in. and the bearing stress is limited to 800 psi. 

Solution 129 

Problem 130 

Figure P-130 shows a roof truss and the detail of the riveted connection at joint B. Using 

allowable stresses of τ = 70 MPa and σ

b

= 140 MPa, how many 19-mm diameter rivets 

are required to fasten member BC to the gusset plate? Member BE? What is the largest 

average tensile or compressive stress in BC and BE? 

Solution 130 

Problem 131 

Repeat Problem 130 if the rivet diameter is 22 mm and all other data remain 

unchanged. 

Solution 131 

Thin-Walled Pressure Vessels 

A tank or pipe carrying a fluid or gas under a pressure is subjected to tensile forces, 

which resist bursting, developed across longitudinal and transverse sections. 

TANGENTIAL STRESS 

(Circumferential Stress) 

Consider the tank shown being subjected to an internal pressure p. The length of the 

tank is L and the wall thickness is t. Isolating the right half of the tank: 

If there exist an external pressure p

o

 and an internal pressure p

i

, the formula may be 

expressed as: 

LONGITUDINAL STRESS, σ

L

Consider the free body diagram in the transverse section of the tank: 

The total force acting at the rear of the tank F must equal to the total longitudinal stress 

on the wall P

T

 = σ

L

 A

wall

. Since t is so small compared to D, the area of the wall is close 

to πDt 

If there exist an external pressure p

o

 and an internal pressure p

i

, the formula may be 

expressed as: 

It can be observed that the tangential stress is twice that of the longitudinal stress. 

σ

t

 = 2 σ

L

SPHERICAL SHELL 

If a spherical tank of diameter D and thickness t contains gas under 

a pressure of p, the stress at the wall can be expressed as:  

SOLVED PROBLEMS IN  THIN WALLED PREASSURE VESSELS 

Problem 133 

A cylindrical steel pressure vessel 400 mm in diameter with a wall thickness of 20 mm, 

is subjected to an internal pressure of 4.5 MN/m

2

. (a) Calculate the tangential and 

longitudinal stresses in the steel. (b) To what value may the internal pressure be 

increased if the stress in the steel is limited to 120 MN/m

2

? (c) If the internal pressure 

were increased until the vessel burst, sketch the type of fracture that would occur. 

Solution 133 

Problem 134 

The wall thickness of a 4-ft-diameter spherical tank is 5/16 in. Calculate the allowable 

internal pressure if the stress is limited to 8000 psi. 

Solution 134 

Problem 135 

Calculate the minimum wall thickness for a cylindrical vessel that is to carry a gas at a 

pressure of 1400 psi. The diameter of the vessel is 2 ft, and the stress is limited to 12 

ksi.  

Solution 135 

Problem 136 

A cylindrical pressure vessel is fabricated from steel plating that has a thickness of 20 

mm. The diameter of the pressure vessel is 450 mm and its length is 2.0 m. Determine 

the maximum internal pressure that can be applied if the longitudinal stress is limited to 

140 MPa, and the circumferential stress is limited to 60 MPa. 

Solution 136 

Problem 137  

A water tank, 22 ft in diameter, is made from steel plates that are ½ in. thick. Find the 

maximum height to which the tank may be filled if the circumferential stress is limited 

to 6000 psi. The specific weight of water is 62.4 lb/ft

3

. 

Solution 137 

Problem 138 

The strength of longitudinal joint in Fig. 1-17 is 33 kips/ft, whereas for the girth is 16 

kips/ft. Calculate the maximum diameter of the cylinder tank if the internal pressure is 

150 psi. 

Solution 138 

Problem 139 

Find the limiting peripheral velocity of a rotating steel ring if the allowable stress is 20 

ksi and steel weighs 490 lb/ft

3

. At what revolutions per minute (rpm) will the stress 

reach 30 ksi if the mean radius is 10 in.? 

Solution 139 

Problem 140 

At what angular velocity will the stress of the rotating steel ring equal 150 MPa if its 

mean radius is 220 mm? The density of steel 7.85 Mg/m

3

. 

Solution 140 

Problem 141 

The tank shown in Fig. P-141 is fabricated from 1/8-in steel plate. Calculate the 

maximum longitudinal and circumferential stress caused by an internal pressure of 125 

psi. 

Solution 141 

Problem 142 

A pipe carrying steam at 3.5 MPa has an outside diameter of 450 mm and a wall 

thickness of 10 mm. A gasket is inserted between the flange at one end of the pipe and 

a flat plate used to cap the end. How many 40-mm-diameter bolts must be used to hold 

the cap on if the allowable stress in the bolts is 80 MPa, of which 55 MPa is the initial 

stress? What circumferential stress is developed in the pipe? Why is it necessary to 

tighten the bolt initially, and what will happen if the steam pressure should cause the 

stress in the bolts to be twice the value of the initial stress? 

Solution 142 

Strain 

Simple Strain 

Also known as unit deformation, strain is the ratio of the change in length caused by the 

applied force, to the original length. 

where δ is the deformation and L is the original length, thus ε is dimensionless. 

Stress-Strain Diagram 

Suppose that a metal specimen be placed in tension-compression testing machine. As 

the axial load is gradually increased in increments, the total elongation over the gage 

length is measured at each increment of the load and this is continued until failure of 

the specimen takes place. Knowing the original cross-sectional area and length of the 

specimen, the normal stress σ and the strain ε can be obtained. The graph of these 

quantities with the stress σ along the y-axis and the strain ε along the x-axis is called 

the stress-strain diagram. The stress-strain diagram differs in form for various 

materials. The diagram shown below is that for a medium carbon structural steel. 

Metallic engineering materials are classified as either ductile or brittle materials. A 

ductile material is one having relatively large tensile strains up to the point of rupture 

like structural steel and aluminum, whereas brittle materials has a relatively small strain 

up to the point of rupture like cast iron and concrete. An arbitrary strain of 0.05 

mm/mm is frequently taken as the dividing line between these two classes. 

PROPORTIONAL LIMIT (HOOKE'S LAW) 

From the origin O to the point called proportional limit, the stress-strain 

curve is a straight line. This linear relation between elongation and the 

axial force causing was first noticed by Sir Robert Hooke in 1678 and is 

called Hooke's Law that within the proportional limit, the stress is directly 

proportional to strain or 

The constant of proportionality k is called the Modulus of Elasticity E or Young's Modulus 

and is equal to the slope of the stress-strain diagram from O to P. Then 

ELASTIC LIMIT 

The elastic limit is the limit beyond which the material will no longer go back to its 

original shape when the load is removed, or it is the maximum stress that may e 

developed such that there is no permanent or residual deformation when the load is 

entirely removed. 

ELASTIC AND PLASTIC RANGES 

The region in stress-strain diagram from O to P is called the elastic range. The region 

from P to R is called the plastic range. 

YIELD POINT 

Yield point is the point at which the material will have an appreciable elongation or 

yielding without any increase in load. 

ULTIMATE STRENGTH 

The maximum ordinate in the stress-strain diagram is the ultimate strength or tensile 

strength. 

RAPTURE STRENGTH 

Rapture strength is the strength of the material at rupture. This is also known as the 

breaking strength. 

MODULUS OF RESILIENCE 

Modulus of resilience is the work done on a unit volume of material as the force is 

gradually increased from O to P, in Nm/m

3

. This may be calculated as the area under 

the stress-strain curve from the origin O to up to the elastic limit E (the shaded area in 

the figure). The resilience of the material is its ability to absorb energy without creating 

a permanent distortion. 

MODULUS OF TOUGHNESS 

Modulus of toughness is the work done on a unit volume of material as the force is 

gradually increased from O to R, in Nm/m

3

. This may be calculated as the area under 

the entire stress-strain curve (from O to R). The toughness of a material is its ability to 

absorb energy without causing it to break. 

WORKING STRESS, ALLOWABLE STRESS, AND FACTOR OF SAFETY 

Working stress is defined as the actual stress of a material under a given loading. The 

maximum safe stress that a material can carry is termed as the allowable stress. The 

allowable stress should be limited to values not exceeding the proportional limit. 

However, since proportional limit is difficult to determine accurately, the allowable tress 

is taken as either the yield point or ultimate strength divided by a factor of safety. The 

ratio of this strength (ultimate or yield strength) to allowable strength is called the 

factor of safety. 

AXIAL DEFORMATION 

In the linear portion of the stress-strain diagram, the tress is proportional to strain and 

is given by 

σ = Eε 

since σ = P / A and εe = δ / L, then P / A = E δ / L. Solving for δ,  

To use this formula, the load must be axial, the bar must have a uniform cross-sectional 

area, and the stress must not exceed the proportional limit. If however, the cross-

sectional area is not uniform, the axial deformation can be determined by considering a 

differential length and applying integration. 

If however, the cross-sectional area is not uniform, the axial deformation can be 

determined by considering a differential length and applying 

integration. 

where A = ty and y and t, if variable, must be expressed in terms of x.  

For a rod of unit mass ρ suspended vertically from one end, the total elongation due to 

its own weight is 

where ρ is in kg/m

3

, L is the length of the rod in mm, M is the total mass of the rod in 

kg, A is the cross-sectional area of the rod in mm

2

, and g = 9.81 m/s

2

. 

STIFFNESS, k 

Stiffness is the ratio of the steady force acting on an elastic body to the resulting 

displacement. It has the unit of N/mm. 

k = P / δ 

SOLVED PROBLEMS IN AXIAL DEFORMATION 

Problem 206 

A steel rod having a cross-sectional area of 300 mm

2

 and a length of 150 m is 

suspended vertically from one end. It supports a tensile load of 20 kN at the lower end. 

If the unit mass of steel is 7850 kg/m

3

 and E = 200 × 10

3

 MN/m

2

, find the total 

elongation of the rod. 

Solution 206 

Problem 207  

A steel wire 30 ft long, hanging vertically, supports a load of 500 lb. Neglecting the 

weight of the wire, determine the required diameter if the stress is not to exceed 20 ksi 

and the total elongation is not to exceed 0.20 in. Assume E = 29 × 10

6

 psi. 

Solution 207  

Problem 208 

A steel tire, 10 mm thick, 80 mm wide, and 1500.0 mm inside diameter, is heated and 

shrunk onto a steel wheel 1500.5 mm in diameter. If the coefficient of static friction is 

0.30, what torque is required to twist the tire relative to the wheel? Neglect the 

deformation of the wheel. Use E = 200 GPa. 

Solution 208 

Problem 209 

An aluminum bar having a cross-sectional area of 0.5 in

2

 carries the axial loads applied 

at the positions shown in Fig. P-209. Compute the total change in length of the bar if E 

= 10 × 10

6

 psi. Assume the bar is suitably braced to prevent lateral buckling. 

Solution 209 

Problem 210 

Solve Prob. 209 if the points of application of the 6000-lb and the 4000-lb forces are 

interchanged. 

Solution 210 

Problem 211 

A bronze bar is fastened between a steel bar and an aluminum bar as shown in Fig. P-

211. Axial loads are applied at the positions indicated. Find the largest value of P that 

will not exceed an overall deformation of 3.0 mm, or the following stresses: 140 MPa in 

the steel, 120 MPa in the bronze, and 80 MPa in the aluminum. Assume that the 

assembly is suitably braced to prevent buckling. Use E

st

 = 200 GPa, E

al

 = 70 GPa, and 

E

br

 = 83 GPa. 

Solution 211 

Problem 212  

The rigid bar ABC shown in Fig. P-212 is hinged at A and supported by a steel rod at B. 

Determine the largest load P that can be applied at C if the stress in the steel rod is 

limited to 30 ksi and the vertical movement of end C must not exceed 0.10 in. 

Solution 212 

Problem 213  

The rigid bar AB, attached to two vertical rods as shown in Fig. P-213, is horizontal 

before the load P is applied. Determine the vertical movement of P if its magnitude is 50 

kN. 

Solution 213  

Problem 214 

The rigid bars AB and CD shown in Fig. P-214 are supported by pins at A and C and the 

two rods. Determine the maximum force P that can be applied as shown if its vertical 

movement is limited to 5 mm. Neglect the weights of all members. 

Solution 214 

Problem 215 

A uniform concrete slab of total weight W is to be attached, as shown in Fig. P-215, to 

two rods whose lower ends are on the same level. Determine the ratio of the areas of 

the rods so that the slab will remain level. 

Solution 215 

Problem 216 

As shown in Fig. P-216, two aluminum rods AB and BC, hinged to rigid supports, are 

pinned together at B to carry a vertical load P = 6000 lb. If each rod has a 

crosssectional area of 0.60 in

2

 and E = 10 × 10

6

 psi, compute the elongation of each 

rod and the horizontal and vertical displacements of point B. Assume α = 30° and θ = 

30°. 

Solution 216 

Problem 217 

Solve Prob. 216 if rod AB is of steel, with E = 29 × 10

6

 psi. Assume α = 45° and θ = 

30°; all other data remain unchanged. 

Solution 217 

Problem 218 

A uniform slender rod of length L and cross sectional area A is rotating in a horizontal 

ertical axis through one end. If the unit mass of the 

al 

plane about a v

rod is ρ, and it is rotating at a constant angular velocity of ω rad/sec, show that the tot

elongation of the rod is ρω

2

L

3

/3E. 

Solution 218 

Problem 219 

A round bar of length L, which tapers uniformly from a diameter D at one end to a 

r d at the other, is suspended vertically from the large end. If w is the 

e 

smaller diamete

weight per unit volume, find the elongation of the rod caused by its own weight. Us

this result to determine the elongation of a cone suspended from its base. 

Solution 219 

SOLVED PROBLEMS IN STRAIN AND  AXIAL DEFORMATION 

Problem 203 

The following data were recorded during the tensile test of a 14-mm-diameter mild steel 

rod. The gage length was 50 mm. 

Plot the stress-strain diagram and determine the following mechanical properties: (a) 

proportional limits; (b) modulus of elasticity; (c) yield point; (d) ultimate strength; and 

(e) rupture strength. 

Solution 203 

Problem 204 

The following data were obtained during a tension test of an aluminum alloy. The initial 

diameter of the test specimen was 0.505 in. and the gage length was 

2.0 in. 

Plot the stress-strain diagram and determine the following mechanical properties: (a) 

proportional limit; (b) modulus of elasticity; (c) yield point; (d) yield strength at 0.2% 

offset; (e) ultimate strength; and (f) rupture strength. 

Solution 204 

Problem 205 

A uniform bar of length L, cross-sectional area A, and unit mass ρ is suspended 

vertically from one end. Show that its total elongation is δ = ρgL

2

 / 2E. If the total mass 

of the bar is M, show also that δ = MgL/2AE. 

Solution 205 

Shearing Deformation 

Shearing forces cause shearing deformation. An element subject to shear does not 

change in length but undergoes a change in shape.  

The change in angle at the corner of an original rectangular element is called the shear 

strain and is expressed as  

The ratio of the shear stress τ and the shear strain γ is called the modulus of elasticity 

in shear or modulus of rigidity and is denoted as G, in MPa.  

The relationship between the shearing deformation and the applied shearing force is  

where V is the shearing force acting over an area A

s

. 

Poisson's Ratio 

When a bar is subjected to a tensile loading there is an increase in length of the bar in 

the direction of the applied load, but there is also a decrease in a lateral dimension 

perpendicular to the load. The ratio of the sidewise deformation (or strain) to the 

longitudinal deformation (or strain) is called the Poisson's ratio and is denoted by ν. For 

most steel, it lies in the range of 0.25 to 0.3, and 0.20 for concrete.  

where ε

x

 is strain in the x-direction and ε

y

 and ε

z

 are the strains in the perpendicular 

direction. The negative sign indicates a decrease in the transverse dimension when ε

x

 is 

positive. 

BIAXIAL DEFORMATION 

If an element is subjected simultaneously by ensile stresses, σ

x

 and σ

y

, in the x and y 

directions, the strain in the x-direction is σ

x

 / E and the strain in the y direction is σ

y

 / E. 

Simultaneously, the stress in the y direction will produce a lateral contraction on the x

x

direction of the amount -ν ε

y

 or -ν σ

y

/E. The resulting strain in the x direction will be 

TRIAXIAL DEFORMATION 

If an element is subjected simultaneously by three mutually perpendicular normal 

stresses σ

x

, σ

y

, and σ

z

, which are accompanied by strains ε

x

, ε

y

, and ε

z

, respectively,  

Tensile stresses and elongation are taken as positive. Compressive stresses and 

contraction are taken as negative. 

Relationship Between E, G, and 

ν 

The relationship between modulus of elasticity E, shear modulus G and Poisson's ratio ν 

is:  

Bulk Modulus of Elasticity or Modulus of Volume Expansion, K 

The bulk modulus of elasticity K is a measure of a resistance of a material to change in 

volume without change in shape or form. It is given as  

where V is the volume and ΔV is change in volume. The ratio ΔV / V is called volumetric 

strain and can be expressed as  

Solved Problems in Shearing Deformation 

Problem 222 

A solid cylinder of diameter d carries an axial load P. Show that its change in diameter is 

4Pν / πEd. 

Solution 222 

Problem 223  

A rectangular steel block is 3 inches long in the x direction, 2 inches long in the y 

direction, and 4 inches long in the z direction. The block is subjected to a triaxial loading 

of three uniformly distributed forces as follows: 48 kips tension in the x direction, 60 

kips compression in the y direction, and 54 kips tension in the z direction. If ν = 0.30 

and E = 29 × 10

6

 psi, determine the single uniformly distributed load in the x direction 

that would produce the same deformation in the y direction as the original loading. 

Solution 223 

Problem 224 

For the block loaded triaxially as described in Prob. 223, find the uniformly distributed 

load that must be added in the x direction to produce no deformation in the z direction. 

Solution 224 

Problem 225 

A welded steel cylindrical drum made of a 10-mm plate has an internal diameter of 1.20 

m. Compute the change in diameter that would be caused by an internal pressure of 1.5 

MPa. Assume that Poisson's ratio is 0.30 and E = 200 GPa. 

Solution 225 

Problem 226 

A 2-in.-diameter steel tube with a wall thickness of 0.05 inch just fits in a rigid hole. 

Find the tangential stress if an axial compressive load of 3140 lb is applied. Assume ν = 

0.30 and neglect the possibility of buckling. 

Solution 226 

Problem 227  

A 150-mm-long bronze tube, closed at its ends, is 80 mm in diameter and has a wall 

thickness of 3 mm. It fits without clearance in an 80-mm hole in a rigid 

block. The tube is then subjected to an internal pressure of 4.00 MPa. Assuming ν = 1/3 

and E = 83 GPa, determine the tangential stress in the tube. 

Solution 227 

Problem 228 

A 6-in.-long bronze tube, with closed ends, is 3 in. in diameter with a wall thickness of 

0.10 in. With no internal pressure, the tube just fits between two rigid end walls. 

Calculate the longitudinal and tangential stresses for an internal pressure of 6000 psi. 

Assume ν = 1/3 and E = 12 × 10

6

 psi. 

Solution 228 

Statically Indeterminate Members 

When the reactive forces or the internal resisting forces over a cross section exceed the 

number of independent equations of equilibrium, the structure is called statically 

indeterminate. These cases require the use of additional relations that depend on the 

elastic deformations in the members. 

Solved Problems in Statically Indeterminate Members 

Problem 233 

A steel bar 50 mm in diameter and 2 m long is surrounded by a shell of a cast iron 5 

mm thick. Compute the load that will compress the combined bar a total of 0.8 mm in 

the length of 2 m. For steel, E = 200 GPa, and for cast iron, E = 100 GPa. 

Solution 233 

Problem 234 

A reinforced concrete column 200 mm in diameter is designed to carry an axial 

compressive load of 300 kN. Determine the required area of the reinforcing steel if the 

allowable stresses are 6 MPa and 120 MPa for the concrete and steel, respectively. Use 

E

co

 = 14 GPa and E

st

 = 200 GPa. 

Solution 234 

Problem 235 

A timber column, 8 in. × 8 in. in cross section, is reinforced on each side by a steel 

plate 8 in. wide and t in. thick. Determine the thickness t so that the column will 

support an axial load of 300 kips without exceeding a maximum timber stress of 1200 

psi or a maximum steel stress of 20 ksi. The moduli of elasticity are 1.5 × 10

6

 psi for 

timber, and 29 × 10

6

 psi for steel. 

Solution 235 

Problem 236 

A rigid block of mass M is supported by three symmetrically spaced rods as shown in fig 

P-236. Each copper rod has an area of 900 mm

2

; E = 120 GPa; and the allowable stress 

is 70 MPa. The steel rod has an area of 1200 mm

2

; E = 200 GPa; and the allowable 

stress is 140 MPa. Determine the largest mass M which can be supported. 

Figure P-236 and P-237

Solution 236 

Problem 237 

In Prob. 236, how should the lengths of the two identical copper rods be changed so 

that each material will be stressed to its allowable limit? 

Solution 237 

Problem 238 

The lower ends of the three bars in Fig. P-238 are at the same level before the uniform 

rigid block weighing 40 kips is attached. Each steel bar has a length of 3 ft, and area of 

1.0 in.

2

, and E = 29 × 10

6

 psi. For the bronze bar, the area is 1.5 in.

2

 and E = 12 × 10

6

psi. Determine (a) the length of the bronze bar so that the load on each steel bar is 

twice the load on the bronze bar, and (b) the length of the bronze that will make the 

steel stress twice the bronze stress.  

Solution 238 

Problem 239 

The rigid platform in Fig. P-239 has negligible mass and rests on two steel bars, each 

250.00 mm long. The center bar is aluminum and 249.90 mm long. Compute the stress 

in the aluminum bar after the center load P = 400 kN has been applied. For each steel 

bar, the area is 1200 mm

2

 and E = 200 GPa. For the aluminum bar, the area is 2400 

mm

2

 and E = 70 GPa. 

Solution 239 

Problem 240 

Three steel eye-bars, each 4 in. by 1 in. in section, are to be assembled by driving rigid 

7/8-in.-diameter drift pins through holes drilled in the ends of the bars. The center-line 

spacing between the holes is 30 ft in the two outer bars, but 0.045 in. shorter in the 

middle bar. Find the shearing stress developed in the drip pins. Neglect local 

deformation at the holes. 

Solution 240 

Problem 241 

As shown in Fig. P-241, three steel wires, each 0.05 in.

2

 in area, are used to lift a load 

W = 1500 lb. Their unstressed lengths are 74.98 ft, 74.99 ft, and 75.00 ft. (a) What 

stress exists in the longest wire? (b) Determine the stress in the shortest wire if W = 

500 lb. 

Solution 241 

Problem 242 

The assembly in Fig. P-242 consists of a light rigid bar AB, pinned at O, that is attached 

to the steel and aluminum rods. In the position shown, bar AB is horizontal and there is 

a gap, Δ = 5 mm, between the lower end of the steel rod and its pin support at C. 

Compute the stress in the aluminum rod when the lower end of the steel rod is attached 

to its support.  

Solution 242 

Problem 243 

A homogeneous rod of constant cross section is attached to unyielding supports. It 

carries an axial load P applied as shown in Fig. P-243. Prove that the reactions are given 

by R

1

 = Pb/L and R

2

 = Pa/L.  

Solution 243 

Problem 244 

A homogeneous bar with a cross sectional area of 500 mm

2

 is attached to rigid 

supports. It carries the axial loads P1 = 25 kN and P2 = 50 kN, applied as shown in Fig. 

P-244. Determine the stress in segment BC. (Hint: Use the results of Prob. 243, and 

compute the reactions caused by P

1

 and P

2

 acting separately. Then use the principle of 

superposition to compute the reactions when both loads are applied.)  

Solution 244 

Problem 245  

The composite bar in Fig. P-245 is firmly attached to unyielding supports. Compute the 

stress in each material caused by the application of the axial load P = 50 kips.  

Solution 245 

Problem 246 

Referring to the composite bar in Prob. 245, what maximum axial load P can be applied 

if the allowable stresses are 10 ksi for aluminum and 18 ksi for steel. 

Solution 246 

Problem 247 

The composite bar in Fig. P-247 is stress-free before the axial loads P1 and P2 are 

applied. Assuming that the walls are rigid, calculate the stress in each material if P

1

 = 

150 kN and P

2

 = 90 kN.  

Solution 247 

Problem 248 

Solve Prob. 247 if the right wall yields 0.80 mm. 

Solution 248 

Problem 249 

There is a radial clearance of 0.05 mm when a steel tube is placed over an aluminum 

tube. The inside diameter of the aluminum tube is 120 mm, and the wall thickness of 

each tube is 2.5 mm. Compute the contact pressure and tangential stress in each tube 

when the aluminum tube is subjected to an internal pressure of 5.0 MPa. 

Solution 249 

Problem 250  

In the assembly of the bronze tube and steel bolt shown in Fig. P-250, the pitch of the 

bolt thread is p = 1/32 in.; the cross-sectional area of the bronze tube is 1.5 in.

2

 and of 

steel bolt is ¾ in.

2

 The nut is turned until there is a compressive stress of 4000 psi in 

the bronze tube. Find the stresses if the nut is given one additional turn. How many 

turns of the nut will reduce these stresses to zero? Use Ebr = 12 × 10

6

 psi and Est = 29 

× 10

6

 psi. 

Solution 250  

Problem 251 

The two vertical rods attached to the light rigid bar in Fig. P-251 are identical except for 

length. Before the load W was attached, the bar was horizontal and the rods were 

stress-free. Determine the load in each rod if W = 6600 lb. 

Solution 251 

Problem 252 

The light rigid bar ABCD shown in Fig. P-252 is pinned at B and connected to two 

vertical rods. Assuming that the bar was initially horizontal and the rods stress-free, 

determine the stress in each rod after the load after the load P = 20 kips is applied. 

Solution 252 

Problem 253 

As shown in Fig. P-253, a rigid beam with negligible weight is pinned at one end and 

attached to two vertical rods. The beam was initially horizontal before the load W = 50 

kips was applied. Find the vertical movement of W. 

Solution 253 

Problem 254 

As shown in Fig. P-254, a rigid bar with negligible mass is pinned at O and attached to 

two vertical rods. Assuming that the rods were initially tress-free, what maximum load P 

can be applied without exceeding stresses of 150 MPa in the steel rod and 70 MPa in the 

bronze rod. 

Solution 254 

Problem 255 

Shown in Fig. P-255 is a section through a balcony. The total uniform load of 600 kN is 

supported by three rods of the same area and material. Compute the load in each rod. 

Assume the floor to be rigid, but note that it does not necessarily remain horizontal. 

Solution 255 

Problem 256 

Three rods, each of area 250 mm2, jointly support a 7.5 kN load, as shown in Fig. P-

256. Assuming that there was no slack or stress in the rods before the load was applied, 

find the stress in each rod. Use E

st

 = 200 GPa and E

br

 = 83 GPa. 

Solution 256 

Problem 257 

Three bars AB, AC, and AD are pinned together as shown in Fig. P-257. Initially, the 

assembly is stressfree. Horizontal movement of the joint at A is prevented by a short 

horizontal strut AE. Calculate the stress in each bar and the force in the strut AE when 

the assembly is used to support the load W = 10 kips. For each steel bar, A = 0.3 in.

2

and E = 29 × 10

6

 psi. For the aluminum bar, A = 0.6 in.

2

 and E = 10 × 10

6

 psi. 

Solution 257 

Thermal Stress 

Temperature changes cause the body to expand or contract. The amount δ

T

, is given by 

where α is the coefficient of thermal expansion in m/m°C, L is the length in meter, and 

T

i

 and T

f

 are the initial and final temperatures, respectively in °C. 

For steel, α = 11.25 × 10

–6

 / °C. 

If temperature deformation is permitted to occur freely, no load or stress will be 

induced in the structure. In some cases where temperature deformation is not 

permitted, an internal stress is created. The internal stress created is termed as thermal 

stress. 

For a homogeneous rod mounted between unyielding supports as shown, the thermal 

stress is computed as:  

deformation due to temperature changes;  

deformation due to equivalent axial stress;  

where σ is the thermal stress in MPa and E is the modulus of elasticity of the rod in MPa. 

If the wall yields a distance of x as shown, the following calculations will be made:  

where σ represents the thermal stress. 

Take note that as the temperature rises above the normal, the rod will be in 

compression, and if the temperature drops below the normal, the rod is in tension. 

Solved Problems in Thermal Stress  

Problem 261 

A steel rod with a cross-sectional area of 0.25 in

2

 is stretched between two fixed points. 

The tensile load at 70°F is 1200 lb. What will be the stress at 0°F? At what temperature 

will the stress be zero? Assume α = 6.5 × 10

-6

 in / (in·°F) and E = 29 × 10

6

 psi. 

Solution 261 

Problem 262 

A steel rod is stretched between two rigid walls and carries a tensile load of 5000 N at 

20°C. If the allowable stress is not to exceed 130 MPa at -20°C, what is the minimum 

diameter of the rod? Assume α = 11.7 µm/(m·°C) and E = 200 GPa. 

Solution 262 

Problem 263 

Steel railroad reels 10 m long are laid with a clearance of 3 mm at a temperature of 

15°C. At what temperature will the rails just touch? What stress would be induced in the 

rails at that temperature if there were no initial clearance? Assume α = 11.7 µm/(m·°C) 

and E = 200 GPa. 

Solution 263 

Problem 264 

A steel rod 3 feet long with a cross-sectional area of 0.25 in.

2

 is stretched between two 

fixed points. The tensile force is 1200 lb at 40°F. Using E = 29 × 10

6

 psi and α = 6.5 × 

10

-6

 in./(in.·°F), calculate (a) the temperature at which the stress in the bar will be 10 

ksi; and (b) the temperature at which the stress will be 

zero. 

Solution 264 

Problem 265 

A bronze bar 3 m long with a cross sectional area of 320 mm

2

 is placed between two 

rigid walls as shown in Fig. P-265. At a temperature of -20°C, the gap Δ = 25 mm. Find 

the temperature at which the compressive stress in the bar will be 35 MPa. Use α = 

18.0 × 10

-6

 m/(m·°C) and E = 80 GPa. 

Solution 265 

Problem 266 

Calculate the increase in stress for each segment of the compound bar shown in Fig. P-

266 if the temperature increases by 100°F. Assume that the supports are unyielding 

and that the bar is suitably braced against buckling.  

Solution 266 

Problem 267 

At a temperature of 80°C, a steel tire 12 mm thick and 90 mm wide that is to be shrunk 

onto a locomotive driving wheel 2 m in diameter just fits over the wheel, which is at a 

temperature of 25°C. Determine the contact pressure between the tire and wheel after 

the assembly cools to 25°C. Neglect the deformation of the wheel caused by the 

pressure of the tire. Assume α = 11.7 µm/(m·°C) and E = 200 GPa. 

Solution 267 

Problem 268 

The rigid bar ABC in Fig. P-268 is pinned at B and attached to the two vertical rods. 

Initially, the bar is horizontal and the vertical rods are stress-free. Determine the stress 

in the aluminum rod if the temperature of the steel rod is decreased by 40°C. Neglect 

the weight of bar ABC.  

Solution 268 

Problem 269 

As shown in Fig. P-269, there is a gap between the aluminum bar and the rigid slab that 

is supported by two copper bars. At 10°C, Δ = 0.18 mm. Neglecting the mass of the 

slab, calculate the stress in each rod when the temperature in the assembly is increased 

to 95°C. For each copper bar, A= 500 mm

2

, E = 120 GPa, and α = 16.8 µm/(m·°C). For 

the aluminum bar, A = 400 mm

2

, E = 70 GPa, and α = 23.1 µm/(m·°C).  

Solution 269 

Problem 270 

A bronze sleeve is slipped over a steel bolt and held in place by a nut that is turned to 

produce an initial stress of 2000 psi in the bronze. For the steel bolt, A = 0.75 in

2

, E = 

29 × 10

6

 psi, and α = 6.5 × 10

–6

 in/(in·°F). For the bronze sleeve, A = 1.5 in

2

, E = 12 × 

10

6

 psi and α = 10.5 × 10

–6

 in/(in·°F). After a temperature rise of 100°F, find the final 

stress in each material. 

Solution 270 

Problem 271 

A rigid bar of negligible weight is supported as shown in Fig. P-271. If W = 80 kN, 

compute the temperature change that will cause the stress in the steel rod to be 55 

MPa. Assume the coefficients of linear expansion are 11.7 µm/(m·°C) for steel and 18.9 

µm / (m·°C) for bronze.  

Solution 271 

Problem 272 

For the assembly in Fig. 271, find the stress in each rod if the temperature rises 30°C 

after a load W = 120 kN is applied. 

Solution 272 

Problem 273 

The composite bar shown in Fig. P-273 is firmly attached to unyielding supports. An 

axial force P = 50 kips is applied at 60°F. Compute the stress in each material at 120°F. 

Assume α = 6.5 × 10

–6

 in/(in·°F) for steel and 12.8 × 10

–6

 in/(in·°F) for aluminum. 

Figure P-273 and P-274

Solution 273 

Problem 274 

At what temperature will the aluminum and steel segments in Prob. 273 have 

numerically equal stress? 

Solution 274 

Problem 275 

A rigid horizontal bar of negligible mass is connected to two rods as shown in Fig. P-

275. If the system is initially stress-free. Calculate the temperature change that will 

cause a tensile stress of 90 MPa in the brass rod. Assume that both rods are subjected 

to the change in temperature. 

Solution 275 

Problem 276 

Four steel bars jointly support a mass of 15 Mg as shown in Fig. P-276. Each bar has a 

cross-sectional area of 600 mm2. Find the load carried by each bar after a temperature 

rise of 50°C. Assume α = 11.7 µm/(m·°C) and E = 200 GPa. 

Solution 276 

Torsion 

Consider a bar to be rigidly attached at one end and twisted at the other end by a 

torque or twisting moment T equivalent to F × d, which is applied perpendicular to the 

axis of the bar, as shown in the figure. Such a bar is said to be in torsion. 

TORSIONAL SHEARING STRESS, τ 

For a solid or hollow circular shaft subject to a twisting moment T, the torsional 

shearing stress τ at a distance ρ from the center of the shaft is 

where J is the polar moment of inertia of the section and r is the outer radius. 

For solid cylindrical shaft:  

For hollow cylindrical shaft:  

ANGLE OF TWIST 

The angle θ through which the bar length L will twist is  

where T is the torque in N·mm, L is the length of shaft in mm, G is shear modulus in 

MPa, J is the polar moment of inertia in mm

4

, D and d are diameter in mm, and r is the 

radius in mm. 

POWER TRANSMITTED BY THE SHAFT 

A shaft rotating with a constant angular velocity ω (in radians per second) is being acted 

by a twisting moment T. The power transmitted by the shaft is  

where T is the torque in N·m, f is the number of revolutions per second, and P is the 

power in watts. 

Solved Problems in Torsion 

Problem 304 

A steel shaft 3 ft long that has a diameter of 4 in. is subjected to a torque of 15 kip·ft. 

Determine the maximum shearing stress and the angle of twist. Use G = 12 × 10

6

 psi. 

Solution 304 

Problem 305 

What is the minimum diameter of a solid steel shaft that will not twist through more 

than 3° in a 6-m length when subjected to a torque of 12 kN·m? What maximum 

shearing stress is developed? Use G = 83 GPa. 

Solution 305 

Problem 306  

A steel marine propeller shaft 14 in. in diameter and 18 ft long is used to transmit 5000 

hp at 189 rpm. If G = 12 × 10

6

 psi, determine the maximum shearing stress. 

Solution 306 

Problem 307 

A solid steel shaft 5 m long is stressed at 80 MPa when twisted through 4°. Using G = 

83 GPa, compute the shaft diameter. What power can be transmitted by the shaft at 20 

Hz? 

Solution 307 

Problem 308 

A 2-in-diameter steel shaft rotates at 240 rpm. If the shearing stress is limited to 12 

ksi, determine the maximum horsepower that can be transmitted. 

Solution 308 

Problem 309 

A steel propeller shaft is to transmit 4.5 MW at 3 Hz without exceeding a shearing stress 

of 50 MPa or twisting through more than 1° in a length of 26 diameters. Compute the 

proper diameter if G = 83 GPa. 

Solution 309 

Problem 310 

Show that the hollow circular shaft whose inner diameter is half the outer diameter has 

a torsional strength equal to 15/16 of that of a solid shaft of the same outside diameter. 

Solution 310 

Problem 311 

An aluminum shaft with a constant diameter of 50 mm is loaded by torques applied to 

gears attached to it as shown in Fig. P-311. Using G = 28 GPa, determine the relative 

angle of twist of gear D relative to gear A. 

Solution 311 

Problem 312 

A flexible shaft consists of a 0.20-in-diameter steel wire encased in a stationary tube 

that fits closely enough to impose a frictional torque of 0.50 lb·in/in. Determine the 

maximum length of the shaft if the shearing stress is not to exceed 20 ksi. What will be 

the angular deformation of one end relative to the other end? G = 12 × 10

6

 psi. 

Solution 312 

Problem 313 

Determine the maximum torque that can be applied to a hollow circular steel shaft of 

100-mm outside diameter and an 80-mm inside diameter without exceeding a shearing 

stress of 60 MPa or a twist of 0.5 deg/m. Use G = 83 GPa. 

Solution 313 

Problem 314 

The steel shaft shown in Fig. P-314 rotates at 4 Hz with 35 kW taken off at A, 20 kW 

removed at B, and 55 kW applied at C. Using G = 83 GPa, find the maximum shearing 

stress and the angle of rotation of gear A relative to gear C. 

Solution 314 

Problem 315  

A 5-m steel shaft rotating at 2 Hz has 70 kW applied at a gear that is 2 m from the left 

end where 20 kW are removed. At the right end, 30 kW are removed and another 20 

kW leaves the shaft at 1.5 m from the right end. (a) Find the uniform shaft diameter so 

that the shearing stress will not exceed 60 MPa. (b) If a uniform shaft diameter of 100 

mm is specified, determine the angle by which one end of the shaft lags behind the 

other end. Use G = 83 GPa. 

Solution 315 

Problem 316 

A compound shaft consisting of a steel segment and an aluminum segment is acted 

upon by two torques as shown in Fig. P-316. Determine the maximum permissible value 

of T subject to the following conditions: τ

st

 = 83 MPa, τ

al

 = 55 MPa, and the angle of 

rotation of the free end is limited to 6°. For steel, G = 83 GPa and for aluminum, G = 

28 GPa. 

Solution 316 

Problem 317 

A hollow bronze shaft of 3 in. outer diameter and 2 in. inner diameter is slipped over a 

solid steel shaft 2 in. in diameter and of the same length as the hollow shaft. The two 

shafts are then fastened rigidly together at their ends. For bronze, G = 6 × 10

6

 psi, and 

for steel, G = 12 × 10

6

 psi. What torque can be applied to the composite shaft without 

exceeding a shearing stress of 8000 psi in the bronze or 12 ksi in the steel? 

Solution 317 

Problem 318 

A solid aluminum shaft 2 in. in diameter is subjected to two torques as shown in Fig. P-

318. Determine the maximum shearing stress in each segment and the angle of rotation 

of the free end. Use G = 4 × 10

6

 psi. 

Solution 318 

Problem 319 

The compound shaft shown in Fig. P-319 is attached to rigid supports. For the bronze 

segment AB, the diameter is 75 mm, τ ≤ 60 MPa, and G = 35 GPa. For the steel 

segment BC, the diameter is 50 mm, τ ≤ 80 MPa, and G = 83 GPa. If a = 2 m and b = 

1.5 m, compute the maximum torque T that can be applied. 

Solution 319 

Problem 320 

In Prob. 319, determine the ratio of lengths b/a so that each material will be stressed to 

its permissible limit. What torque T is required? 

Solution 320 

Problem 321 

A torque T is applied, as shown in Fig. P-321, to a solid shaft with built-in ends. Prove 

that the resisting torques at the walls are T

1

 = Tb/L and T

2

 = Ta/L. How would these 

values be changed if the shaft were hollow? 

Solution 321 

Problem 322 

A solid steel shaft is loaded as shown in Fig. P-322. Using G = 83 GPa, determine the 

required diameter of the shaft if the shearing stress is limited to 60 MPa and the angle 

of rotation at the free end is not to exceed 4 deg. 

Solution 322 

Problem 323 

A shaft composed of segments AC, CD, and DB is fastened to rigid supports and loaded 

as shown in Fig. P-323. For bronze, G = 35 GPa; aluminum, G = 28 GPa, and for steel, 

G = 83 GPa. Determine the maximum shearing stress developed in each segment. 

Solution 323 

Problem 324 

The compound shaft shown in Fig. P-324 is attached to rigid supports. For the bronze 

segment AB, the maximum shearing stress is limited to 8000 psi and for the steel 

segment BC, it is limited to 12 ksi. Determine the diameters of each segment so that 

each material will be simultaneously stressed to its permissible limit when a torque T = 

12 kip·ft is applied. For bronze, G = 6 × 10

6

 psi and for steel, G = 12 × 10

6

psi. 

Solution 324 

Problem 325 

The two steel shaft shown in Fig. P-325, each with one end built into a rigid support 

have flanges rigidly attached to their free ends. The shafts are to be bolted together at 

their flanges. However, initially there is a 6° mismatch in the location of the bolt holes 

as shown in the figure. Determine the maximum shearing stress in each shaft after the 

shafts are bolted together. Use G = 12 × 10

6

 psi and neglect deformations of the bolts 

and  

flanges. 

Solution 325 

Flanged Bolt Couplings 

In shaft connection called flanged bolt couplings (see figure above), the torque is 

transmitted by the shearing force P created in he bolts that is assumed to be uniformly 

distributed. For any number of bolts n, the torque capacity of the coupling is 

If a coupling has two concentric rows of bolts, the torque capacity 

is 

where the subscript 1 refer to bolts on the outer circle an 

subscript 2 refer to bolts on the inner circle. See figure. 

For rigid flanges, the shear deformations in the bolts are proportional to their radial 

distances from the shaft axis. The shearing strains are related by  

Using Hooke’s law for shear, G = τ / γ, we have 

If the bolts on the two circles have the same area, A

1

 = A

2

, and if the bolts are made of 

the same material, G

1

 = G

2

, the relation between P

1

 and P

2

 reduces to 

Solved Problems in Flanged Bolt Couplings  

Problem 326 

A flanged bolt coupling consists of ten 20-mmdiameter bolts spaced evenly around a 

bolt circle 400 mm in diameter. Determine the torque capacity of the coupling if the 

allowable shearing stress in the bolts is 40 MPa. 

Solution 326 

Problem 327  

A flanged bolt coupling consists of ten steel ½ -in.-diameter bolts spaced evenly around 

a bolt circle 14 in. in diameter. Determine the torque capacity of the coupling if the 

allowable shearing stress in the bolts is 6000 psi. 

Solution 327 

Problem 328 

A flanged bolt coupling consists of eight 10-mmdiameter steel 

bolts on a bolt circle 400 mm in diameter, and six 10-mm-

diameter steel bolts on a concentric bolt circle 300 mm in 

diameter, as shown in Fig. 3-7. What torque can be applied 

without exceeding a shearing stress of 60 MPa in the bolts? 

Solution 328 

Problem 329 

A torque of 700 lb-ft is to be carried by a flanged bolt coupling that consists of eight ½ -

in.-diameter steel bolts on a circle of diameter 12 in. and six ½ -in.-diameter steel bolts 

on a circle of diameter 9 in. Determine the shearing stress in the bolts. 

Solution 329 

Problem 330 

Determine the number of 10-mm-diameter steel bolts that must be used on the 400-

mm bolt circle of the coupling described in Prob. 328 to increase the torque capacity to 

14 kN·m 

Solution 330 

Problem 331 

A flanged bolt coupling consists of six ½ -in. steel bolts evenly spaced around a bolt 

circle 12 in. in diameter, and four ¾ -in. aluminum bolts on a concentric bolt circle 8 in. 

in diameter. What torque can be applied without exceeding 9000 psi in the steel or 

6000 psi in the aluminum? Assume G

st

 = 12 × 10

6

 psi and G

al

 = 4 × 10

6

 psi. 

Solution 331 

Problem 332 

In a rivet group subjected to a twisting couple T, show that the torsion formula τ = Tρ/J 

can be used to find the shearing stress t at the center of any rivet. Let J = ΣAρ

2

, where 

A is the area of a rivet at the radial distance ρ from the centroid of the rivet group. 

Solution 332 

Problem 333 

A plate is fastened to a fixed member by four 20-mm diameter rivets arranged as 

shown in Fig. P-333. Compute the maximum and minimum shearing stress developed. 

Solution 333 

Problem 334 

Six 7/8-in-diameter rivets fasten the plate in Fig. P-334 to the fixed member. Using the 

results of Prob. 332, determine the average shearing stress caused in each rivet by the 

14 kip loads. What additional loads P can be applied before the shearing stress in any 

rivet exceeds 8000 psi? 

Solution 334 

Problem 335 

The plate shown in Fig. P-335 is fastened to the fixed member by five 10-mm-diameter 

rivets. Compute the value of the loads P so that the average shearing stress in any rivet 

does not exceed 70 MPa. (Hint: Use the results of Prob. 332.) 

Solution 335 

Torsion of Thin-Walled Tubes 

The torque applied to thin-walled tubes is expressed as 

where T is the torque in N·mm, A is the area enclosed by the centerline of the tube (as 

shown in the stripefilled portion) in mm

2

, and q is the shear flow in N/mm. 

The average shearing stress across any thickness t is 

Thus, torque T ca also be expressed as 

Solved Problems in Torsion of Thin-Walled Tubes  

Problem 337 

A torque of 600 N·m is applied to the rectangular section shown in Fig. P-337. 

Determine the wall thickness t so as not to exceed a shear stress of 80 MPa. What is the 

shear stress in the short sides? Neglect stress concentration at the corners. 

Solution 337 

Problem 338 

A tube 0.10 in. thick has an elliptical shape shown in Fig. 

P-338. What torque will cause a shearing stress of 8000 

psi? 

Solution 338 

Problem 339 

A torque of 450 lb·ft is applied to the square section shown in Fi

P-339. Determine the smallest permissi

g.

ble dimension a if the 

shearing stress is limited to 6000 psi. 

Solution 339 

Problem 340 

A tube 2 mm thick has the shape shown in Fig. P-340. Find 

the shearing stress caused by a torque of 600 N·m. 

Solution 340 

Problem 341 

Derive the torsion formula τ = Tρ/J for a solid circular section by assuming the section is 

composed of a series of concentric thin circular tubes. Assume that the shearing stress 

at any point is proportional to its radial distance. 

Solution 341 

Helical Springs 

When close-coiled helical spring, composed of a wire of round rod of diameter d wound 

into a helix of mean radius R with n number of turns, is subjected to an axial load P 

produces the following stresses and elongation: 

The maximum shearing stress is the sum of the direct shearing stress τ

1

 = P/A and the 

torsional shearing stress τ

2

 = Tr/J, with T = PR. 

This formula neglects the curvature of the spring. This is used for light spring where the 

ratio d/4R is small. 

For heavy springs and considering the curvature of the spring, a more precise formula is 

given by: (A.M.Wahl Formula) 

where m is called the spring index and (4m – 1) / (4m – 4) is the Wahl Factor. 

The elongation of the bar is 

Notice that the deformation δ is directly proportional to the applied load P. The ratio of P 

to δ is called the spring constant k and is equal to 

SPRINGS IN SERIES 

For two or more springs with spring laid in series, the resulting spring constant k is 

given by 

where k

1

, k

2

,… are the spring constants for different springs. 

SPRINGS IN PARALLEL 

Solved Problems in Helical Springs  

Problem 343 

Determine the maximum shearing stress and elongation in a helical steel spring 

composed of 20 turns of 20-mm-diameter wire on a mean radius of 90 mm when the 

spring is supporting a load of 1.5 kN. Use Eq. (3-10) and G = 83 GPa. 

Solution 343 

Problem 344 

Determine the maximum shearing stress and elongation in a bronze helical spring 

composed of 20 turns of 1.0-in.-diameter wire on a mean radius of 4 in. when the 

spring is supporting a load of 500 lb. Use Eq. (3-10) and G = 6 × 10

6

 psi. 

Solution 344 

Problem 345 

A helical spring is fabricated by wrapping wire ¾ in. in diameter around a forming 

cylinder 8 in. in diameter. Compute the number of turns required to permit an 

elongation of 4 in. without exceeding a shearing stress of 18 ksi. Use Eq. (3-9) and G = 

12 × 106 psi. 

Solution 345 

Problem 346 

Compute the maximum shearing stress developed in a phosphor bronze spring having 

mean diameter of 200 mm and consisting of 24 turns of 200-mm-diameter wire when 

the spring is stretched 100 mm. Use Eq. (3-10) and G = 42 GPa. 

Solution 346 

Problem 347 

Two steel springs arranged in series as shown in Fig. P-347 supports a load P. The 

upper spring has 12 turns of 25-mm-diameter wire on a mean radius of 100 mm. The 

lower spring consists of 10 turns of 20-mmdiameter wire on a mean radius of 75 mm. If 

the maximum shearing stress in either spring must not exceed 200 MPa, compute the 

maximum value of P and the total elongation of the assembly. Use Eq. (3-10) and G = 

83 GPa. Compute the equivalent spring constant by dividing the load by the total 

elongation. 

Solution 347 

Problem 348 

A rigid bar, pinned at O, is supported by two identical springs as shown in Fig. P-348. 

Each spring consists of 20 turns of ¾-in-diameter wire having a mean diameter of 6 in. 

Determine the maximum load W that may be supported if the shearing stress in the 

springs is limited to 20 ksi. Use Eq. (3-9). 

Solution 348 

Problem 349 

A rigid bar, hinged at one end, is supported by two identical springs as shown in Fig. P-

349. Each spring consists of 20 turns of 10-mm wire having a mean diameter of 150 

mm. Compute the maximum shearing stress in the springs, using Eq. (3-9). Neglect the 

mass of the rigid bar. 

Solution 349 

Problem 350 

As shown in Fig. P-350, a homogeneous 50-kg rigid block is suspended by the three 

springs whose lower ends were originally at the same level. Each steel spring has 24 

turns of 10-mm-diameter on a mean diameter of 100 mm, and G = 83 GPa. The bronze 

spring has 48 turns of 20-mm-diameter wire on a mean diameter of 150 mm, and G = 

42 GPa. Compute the maximum shearing stress in each spring using Eq. (3-9). 

Solution 350 

Shear & Moment in Beams 

DEFINITION OF A BEAM 

A beam is a bar subject to forces or couples that lie in a plane containing the 

longitudinal of the bar. According to determinacy, a beam may be determinate or 

indeterminate. 

STATICALLY DETERMINATE BEAMS 

Statically determinate beams are those beams in which the reactions of the supports 

may be determined by the use of the equations of static equilibrium. The beams shown 

below are examples of statically determinate beams. 

STATICALLY INDETERMINATE BEAMS 

If the number of reactions exerted upon a beam exceeds the number of equations in 

static equilibrium, the beam is said to be statically indeterminate. In order to solve the 

reactions of the beam, the static equations must be supplemented by equations based 

upon the elastic deformations of the beam. 

The degree of indeterminacy is taken as the difference between the umber of reactions 

to the number of equations in static equilibrium that can be applied. In the case of the 

propped beam shown, there are three reactions R

1

, R

2

, and M and only two equations 

(∑M = 0 and sum;F

v

 = 0) can be applied, thus the beam is indeterminate to the first 

degree (3 – 2 = 1). 

TYPES OF LOADING 

Loads applied to the beam may consist of a concentrated load (load applied at a point), 

uniform load, uniformly varying load, or an applied couple or moment. These loads are 

shown in the following figures. 

Shear and Moment Diagrams 

Consider a simple beam shown of length L that 

carries a uniform load of w (N/m) throughout its 

length and is held in equilibrium by reactions R

1

and R

2

. Assume that the beam is cut at point 

distance of x from he left support and the portion of 

the beam to the right of C be removed. The portion 

removed must then be replaced by vertical 

shearing force V together with a couple M to hold

the left portion of the bar in equilibrium under the 

action of R

1

 and wx. The couple M is called the resisting moment or moment and the 

force V is called the resisting shear or shear. The sign of V and M are taken to be 

positive if they have the senses ind

C a 

icated above. 

Solved Problems in Shear and Moment Diagrams  

INSTRUCTION 

Write shear and moment equations for the beams in the following problems. In each 

problem, let x be the distance measured from left end of the beam. Also, draw shear 

and moment diagrams, specifying values at all change of loading positions and at points 

of zero shear. Neglect the mass of the beam in each problem. 

Problem 403 

Beam loaded as shown in Fig. P-403. 

Solution 403 

Problem 404 

Beam loaded as shown in Fig. P-404. 

Solution 404 

Problem 405 

Beam loaded as shown in Fig. P-405. 

Solution 405 

Problem 406 

Beam loaded as shown in Fig. P-406. 

Solution 406 

Problem 407 

Beam loaded as shown in Fig. P-407. 

Solution 407 

Problem 408 

Beam loaded as shown in Fig. P-408. 

Solution 408 

Problem 409 

Cantilever beam loaded as shown in Fig. P-409. 

Solution 409 

Problem 410 

Cantilever beam carrying the uniformly varying load shown in Fig. P-410. 

Solution 410 

Problem 411 

Cantilever beam carrying a distributed load with intensity varying from wo at the free 

end to zero at the wall, as shown in Fig. P-411. 

Solution 411 

Problem 412 

Beam loaded as shown in Fig. P-412. 

Solution 412 

Problem 413 

Beam loaded as shown in Fig. P-413. 

Solution 413 

Problem 414 

Cantilever beam carrying the load shown in Fig. P-414. 

Solution 414 

Problem 415 

Cantilever beam loaded as shown in Fig. P-415. 

Solution 415 

Problem 416 

Beam carrying uniformly varying load shown in Fig. P-416. 

Solution 416 

Problem 417 

Beam carrying the triangular loading shown in Fig. P- 417. 

Solution 417 

Problem 418 

Cantilever beam loaded as shown in Fig. P-418. 

Solution 418 

Problem 419 

Beam loaded as shown in Fig. P-419. 

Solution 419 

Problem 420 

A total distributed load of 30 kips supported by a uniformly distributed reaction as 

shown in Fig. P-420. 

Solution 420 

Problem 421  

Write the shear and moment equations as functions of the angle θ for the built-in arch 

shown in Fig. P-421. 

Solution 421  

Problem 422 

Write the shear and moment equations for the semicircular arch as shown in Fig. P-422 

if (a) the load P is vertical as shown, and (b) the load is applied horizontally to the left 

at the top of the arch. 

Solution 422 

Relationship between Load, Shear, and Moment 

The vertical shear at C in the figure shown in previous section  is taken as 

where R

1

 = R

2

 = wL/2 

If we differentiate M with respect to x: 

thus, 

Thus, the rate of change of the bending moment with respect to x is equal to the 

shearing force, or the slope of the moment diagram at the given point is the 

shear at that point. 

Differentiate V with respect to x gives 

Thus, the rate of change of the shearing force with respect to x is equal to the load or 

the slope of the shear diagram at a given point equals the load at that point. 

PROPERTIES OF SHEAR AND MOMENT DIAGRAMS 

The following are some important properties of shear and moment diagrams: 

1.  The area of the shear diagram to the left or to the right of the section is equal to 

the moment at that section.  

2.  The slope of the moment diagram at a given point is the shear at that point.  

3.  The slope of the shear diagram at a given point equals the load at that point.  

4.  The maximum moment occurs at the point of zero shears. 

This is in reference to property number 2, that when the 

shear (also the slope of the moment diagram) is zero, the 

tangent drawn to the moment diagram is horizontal.  

5.  When the shear diagram is increasing, the moment diagram is concave upward.  

6.  When the shear diagram is decreasing, the moment diagram is concave 

downward.  

SIGN CONVENTIONS 

The customary sign conventions for shearing force and bending moment are 

represented by the figures below. A force that tends to bend the beam downward is said 

to produce a positive bending moment. A force that tends to shear the left portion of 

the beam upward with respect to the right portion is said to produce a positive shearing 

force. 

An easier way of determining the sign of the bending moment at any section is that 

upward forces always cause positive bending moments regardless of whether they act 

to the left or to the right of the exploratory section. 

Solved Problems in Relationship between Load, Shear, and Moment

INSTRUCTION 

Without writing shear and moment equations, draw the shear and moment diagrams for 

the beams specified in the following problems. Give numerical values at all change of 

loading positions and at all points of zero shear. (Note to instructor: Problems 403 to 

420 may also be assigned for solution by semi graphical method describes in this 

article.) 

Problem 425 

Beam loaded as shown in Fig. P-425. 

Solution 425 

Problem 426 

Cantilever beam acted upon by a uniformly distributed load and a couple as shown in 

Fig. P-426. 

Solution 426 

Problem 427 

Beam loaded as shown in Fig. P-427. 

Solution 427 

Problem 428 

Beam loaded as shown in Fig. P-428. 

Solution 428 

Problem 429 

Beam loaded as shown in Fig. P-429. 

Solution 429 

Problem 430 

Beam loaded as shown in P-430. 

Solution 430 

Problem 431 

Beam loaded as shown in Fig. P-431. 

Solution 431 

Problem 432 

Beam loaded as shown in Fig. P-432. 

Solution 432 

Problem 433 

Overhang beam loaded by a force and a couple as shown in Fig. P-433. 

Solution 433 

Problem 434 

Beam loaded as shown in Fig. P-434. 

Solution 434 

Problem 435 

Beam loaded and supported as shown in Fig. P-435. 

Solution 435 

Problem 436 

A distributed load is supported by two distributedreactions as shown in Fig. P-436. 

Solution 436 

Problem 437 

Cantilever beam loaded as shown in Fig. P-437 

Solution 437 

Problem 438 

The beam loaded as shown in Fig. P-438 consists of two segments joined by a 

frictionless hinge at which the bending moment is zero. 

Solution 438 

Problem 439 

A beam supported on three reactions as shown in Fig. P-439 consists of two segments 

joined by frictionless hinge at which the bending moment is zero. 

Solution 439 

Problem 440 

A frame ABCD, with rigid corners at B and C, supports the concentrated load as shown 

in Fig. P-440. (Draw shear and moment diagrams for each of the three parts of the 

frame.) 

Solution 440 

Problem 441 

A beam ABCD is supported by a roller at A and a hinge at D. It is subjected to the loads 

441, which act at the ends of the vertical members 

shown in Fig. P-

BE and CF. These vertical members are rigidly attached to the beam at B and C. (Draw 

shear and moment diagrams for the beam ABCD only.) 

Solution 441 

Problem 442 

Beam carrying the uniformly varying load shown in Fig. P-442. 

Solution 442 

Problem 443 

Beam carrying the triangular loads shown in Fig. P-443. 

Solution 443 

Problem 444 

Beam loaded as shown in Fig. P-444. 

Solution 444 

Problem 445 

Beam carrying the loads shown in Fig. P-445. 

Solution 445 

Problem 446 

Beam loaded and supported as shown in Fig. P-446. 

Solution 446 

Finding the Load & Moment Diagrams with Given Shear 
Diagram 

INSTRUCTION 

In the following problems, draw moment and load diagrams corresponding to the given 

shear diagrams. Specify values at all change of load positions and at all points of zero 

shear. 

Problem 447 

Shear diagram as shown in Fig. P-447. 

Solution 447 

Problem 448 

Shear diagram as shown in Fig. P-448. 

Solution 448 

Problem 449 

Shear diagram as shown in Fig. P-449. 

Solution 449 

Problem 450 

Shear diagram as shown in Fig. P-450. 

Solution 450 

Problem 451 

Shear diagram as shown in Fig. P-451. 

Solution 451 

Moving Loads 

From the previous section, we see that the maximum moment occurs at a point of zero 

shears. For beams loaded with concentrated loads, the point of zero shears usually 

occurs under a concentrated load and so the maximum moment. 

Beams and girders such as in a bridge or an overhead crane are subject to moving 

concentrated loads, which are at fixed distance with each other. The problem here is to 

determine the moment under each load when each load is in a position to cause a 

maximum moment. The largest value of these moments governs the design of the 

beam. 

SINGLE MOVING LOAD 

For a single moving load, the maximum moment occurs when the load is at the midspan 

and the maximum shear occurs when the load is very near the support (usually 

assumed to lie over the support). 

TWO MOVING LOADS 

For two moving loads, the maximum shear occurs at the reaction when the larger load 

is over that support. The maximum moment is given by 

where P

s

 is the smaller load, P

b

 is the bigger load, and P is the total load (P = P

s

 + P

b

). 

THREE OR MORE MOVING LOADS 

In general, the bending moment under a particular load is a maximum when the center 

of the beam is midway between that load and the resultant of all the loads then on the 

span. With this rule, we compute the maximum moment under each load, and use the 

biggest of the moments for the design. Usually, the biggest of these moments occurs 

under the biggest load. 

The maximum shear occurs at the reaction where the resultant load is nearest. Usually, 

it happens if the biggest load is over that support and as many a possible of the 

remaining loads are still on the span. 

The maximum shear occurs at the reaction where the resultant load is nearest. Usually, 

it happens if the biggest load is over that support and as many a possible of the 

remaining loads are still on the span. In determining the largest moment and shear, it is 

sometimes necessary to check the condition when the bigger loads are on the span and 

the rest of the smaller loads are outside. 

Solved Problems in Moving Loads  

Problem 453 

A truck with axle loads of 40 kN and 60 kN on a wheel base of 5 m rolls across a 10-m 

span. Compute the maximum bending moment and the maximum shearing force. 

Solution 453 

Problem 454 

Repeat Prob. 453 using axle loads of 30 kN and 50 kN on a wheel base of 4 m crossing 

an 8-m span. 

Solution 454 

Problem 455 

A tractor weighing 3000 lb, with a wheel base of 9 ft, carries 1800 lb of its load on the 

rear wheels. Compute the maximum moment and maximum shear when crossing a 14 

ft-span. 

Solution 455 

Problem 456 

Three wheel loads roll as a unit across a 44-ft span. The loads are P

1

 = 4000 lb and P

2

 = 

8000 lb separated by 9 ft, and P

3

 = 6000 lb at 18 ft from P

2

. Determine the maximum 

moment and maximum shear in the simply supported span. 

Solution 456 

Problem 457 

A truck and trailer combination crossing a 12-m span has axle loads of 10, 20, and 30 

kN separated respectively by distances of 3 and 5 m. Compute the maximum moment 

and maximum shear developed in the span. 

Solution 457 

Stresses in Beams 

Forces and couples acting on the beam cause bending (flexural stresses) and shearing 

stresses on any cross section of the beam and deflection perpendicular to the 

longitudinal axis of the beam. If couples are applied to the ends of the beam and no 

forces act on it, the bending is said to be pure bending. If forces produce the bending, 

the bending is called ordinary bending. 

ASSUMPTIONS 

In using the following formulas for flexural and shearing stresses, it is assumed that a 

plane section of the beam normal to its longitudinal axis prior to loading remains plane 

after the forces and couples have been applied, and that the beam is initially straight 

and of uniform cross section and that the moduli of elasticity in tension and 

compression are equal. 

Flexure Formula 

Stresses caused by the bending moment are known as flexural or bending stresses. 

Consider a beam to be loaded as shown. 

Consider a fiber at a distance y from the neutral axis, because of the beam’s curvature, 

as the effect of bending moment, the fiber is stretched by an amount of cd. Since the 

curvature of the beam is very small, bcd and Oba are considered as similar triangles. 

The strain on this fiber is 

By Hooke’s law, ε = σ / E, then 

which means that the stress is proportional to the distance y from the neutral axis. 

Considering a differential area dA at a distance y from N.A., the force acting over the 

area is 

The resultant of all the elemental moment about N.A. must be equal to the bending 

moment on the section. 

but 

then 

substituting ρ = Ey / f

b

then 

and 

The bending stress due to beams curvature is 

The beam curvature is: 

where ρ is the radius of curvature of the beam in mm (in), M is the bending moment in 

N·mm (lb·in), f

b

 is the flexural stress in MPa (psi), I is the centroidal moment of inertia 

in mm

4

 (in

4

), and c is the distance from the neutral axis to the outermost fiber in mm 

(in). 

SECTION MODULUS 

In the formula 

the ratio I/c is called the section modulus and is usually denoted by S with units of mm

3

(in

3

). The maximum bending stress may then be written as 

This form is convenient because the values of S are available in handbooks for a wide 

range of standard structural shapes. 

Solved Problems in Flexure Formula  

Problem 503 

A cantilever beam, 50 mm wide by 150 mm high and 6 m long, carries a load that 

varies uniformly from zero at the free end to 1000 N/m at the wall. (a) Compute the 

magnitude and location of the maximum flexural stress. (b) Determine the type and 

magnitude of the stress in a fiber 20 mm from the top of the beam at a section 2 m 

from the free end. 

Solution 503 

Problem 504 

A simply supported beam, 2 in wide by 4 in high and 12 ft long is subjected to a 

concentrated load of 2000 lb at a point 3 ft from one of the supports. Determine the 

maximum fiber stress and the stress in a fiber located 0.5 in from the top of the beam 

at midspan. 

Solution 504 

Problem 505 

A high strength steel band saw, 20 mm wide by 0.80 mm thick, runs over pulleys 600 

mm in diameter. What maximum flexural stress is developed? What minimum diameter 

pulleys can be used without exceeding a flexural stress of 400 MPa? Assume E = 200 

GPa. 

Solution 505 

Problem 506 

A flat steel bar, 1 inch wide by ¼ inch thick and 40 inches long, is bent by couples 

applied at the ends so that the midpoint deflection is 1.0 inch. Compute the stress in 

the bar and the magnitude of the couples. Use E = 29 × 10

6

 psi. 

Solution 506 

Problem 507 

In a laboratory test of a beam loaded by end couples, the fibers at layer AB in Fig. P-

507 are found to increase 60 × 10

–3

 mm whereas those at CD decrease 100 × 10

–3

 mm 

in the 200-mm-gage length. Using E = 70 GPa, determine the flexural stress in the top 

and bottom fibers. 

Solution 507 

Problem 508 

Determine the minimum height h of the beam shown in Fig. P-508 if the flexural stress 

is not to exceed 20 MPa. 

Solution 508 

Problem 509 

A section used in aircraft is constructed of tubes connected by thin webs as shown in 

Fig. P-509. Each tube has a cross-sectional area of 0.20 in2. If the average stress in the 

tubes is no to exceed 10 ksi, determine the total uniformly distributed load that can be 

supported in a simple span 12 ft long. Neglect the effect of the webs. 

Solution 509 

Problem 510 

A 50-mm diameter bar is used as a simply supported beam 3 m long. Determine the 

largest uniformly distributed load that can be applied over the right two-thirds of the 

beam if the flexural stress is limited to 50 MPa. 

Solution 510 

Problem 511 

A simply supported rectangular beam, 2 in wide by 4 in deep, carries a uniformly 

distributed load of 80 lb/ft over its entire length. What is the maximum length of the 

beam if the flexural stress is limited to 3000 psi? 

Solution 511 

Problem 512 

The circular bar 1 inch in diameter shown in Fig. P-512 is bent into a semicircle with a 

mean radius of 2 ft. If P = 400 lb and F = 200 lb, compute the maximum flexural stress 

developed in section a-a. Neglect the deformation of the bar. 

Solution 512 

Problem 513 

A rectangular steel beam, 2 in wide by 3 in deep, is loaded as shown in Fig. P-513. 

Determine the magnitude and the location of the maximum flexural stress. 

Solution 513 

Problem 514 

The right-angled frame shown in Fig. P-514 carries a uniformly distributed loading 

equivalent to 200 N for each horizontal projected meter of the frame; that is, the total 

load is 1000 N. Compute the maximum flexural stress at section a-a if the cross-section 

is 50 mm square. 

Solution 514 

Problem 515 

Repeat Prob. 524 to find the maximum flexural stress at section b-b. 

Solution 515 

Problem 516 

A timber beam AB, 6 in wide by 10 in deep and 10 ft long, is supported by a guy wire 

AC in the position shown in Fig. P-516. The beam carries a load, including its own 

weight, of 500 lb for each foot of its length. Compute the maximum flexural stress at 

the middle of the beam. 

Solution 516 

Problem 517 

A rectangular steel bar, 15 mm wide by 30 mm high and 6 m long, is simply supported 

at its ends. If the density of steel is 7850 kg/m

3

, determine the maximum bending 

stress caused by the weight of the bar. 

Solution 517 

Problem 518 

A cantilever beam 4 m long is composed of two C200 × 28 channels riveted back to 

back. What uniformly distributed load can be carried, in addition to the weight of the 

beam, without exceeding a flexural stress of 120 MPa if (a) the webs are vertical and 

(b) the webs are horizontal? Refer to Appendix B of text book for channel properties. 

Solution 518 

Problem 519 

A 30-ft beam, simply supported at 6 ft from either end carries a uniformly distributed 

load of intensity w

o

 over its entire length. The beam is made by welding two S18 × 70 

(see appendix B of text book) sections along their flanges to form the section shown in 

Fig. P-519. Calculate the maximum value of wo if the flexural stress is limited to 20 ksi. 

Be sure to include the weight of the beam. 

Solution 519 

Problem 520 

A beam with an S310 × 74 section (see Appendix B of textbook) is used as a simply 

supported beam 6 m long. Find the maximum uniformly distributed load that can be 

applied over the entire length of the beam, in addition to the weight of the beam, if the 

flexural stress is not to exceed 120 MPa. 

Solution 520 

Problem 521 

A beam made by bolting two C10 × 30 channels back to back, is simply supported at its 

ends. The beam supports a central concentrated load of 12 kips and a uniformly 

distributed load of 1200 lb/ft, including the weight of the beam. Compute the maximum 

length of the beam if the flexural stress is not to exceed 20 ksi. 

Solution 521 

Problem 522 

A box beam is composed of four planks, each 2 inches by 8 inches, securely spiked 

together to form the section shown in Fig. P-522. Show that I

NA

 = 981.3 in

4

. If w

o

 = 300 

lb/ft, find P to cause a maximum flexural stress of 1400 psi. 

Solution 522 

Problem 523 

Solve Prob. 522 if w

o

 = 600 lb/ft. 

Solution 523 

Problem 524 

A beam with an S380 &times 74 section carries a total uniformly distributed load of 3W 

and a concentrated load W, as shown in Fig. P-524. Determine W if the 

flexural stress is limited to 120 MPa. 

Solution 524 

Problem 525 

A square timber beam used as a railroad tie is supported by a uniformly distributed 

loads and carries two uniformly distributed loads each totaling 48 kN as shown in Fig. P-

525. Determine the size of the section if the maximum stress is limited to 8 MPa. 

Solution 525 

Problem 526 

A wood beam 6 in wide by 12 in deep is loaded as shown in Fig. P-526. If the maximum 

flexural stress is 1200 psi, find the maximum values of w

o

 and P which can be applied 

simultaneously? 

Solution 526 

Problem 527 

In Prob. 526, if the load on the overhang is 600 lb/ft and the overhang is x ft long, find 

the maximum values of P and x that can be used simultaneously. 

Solution 527 

Economic Sections

From the flexure formula f

b

 = My / I, it can be seen that the bending stress at the 

neutral axis, where y = 0, is zero and increases linearly outwards. This means that for a 

rectangular or circular section a large portion of the cross section near the middle 

section is understressed. 

For steel beams or composite beams, instead of adopting the rectangular shape, the 

area may be arranged so as to give more area on the outer fiber and maintaining the 

same overall depth, and saving a lot of weight. 

When using a wide flange or I-beam section for long beams, the compression flanges 

tend to buckle horizontally sidewise. This buckling is a column effect, which may be 

prevented by providing lateral support such as a floor system so that the full allowable 

stresses may be used, otherwise the stress should be reduced. The reduction of stresses 

for these beams will be discussed in steel design. In selecting a structural section to be 

used as a beam, the resisting moment must be equal or greater than the applied 

bending moment. Note: ( f

b

 )

max

 = M/S. 

The equation above indicates that the required section modulus of the beam must be 

equal or greater than the ratio of bending moment to the maximum allowable stress. A 

check that includes the weight of the selected beam is necessary to complete the 

calculation. In checking, the beams resisting moment must be equal or greater than the 

sum of the live-load moment caused by the applied loads and the dead-load moment 

caused by dead weight of the beam. 

Dividing both sides of the above equation by ( f

b

 )

max

, we obtain the checking equation 

Assume that the beams in the following problems are properly braced against lateral 

deflection. Be sure to include the weight of the beam itself. 

Solved Problems in Economic Sections  

Problem 529 

A 10-m beam simply supported at the ends carries a uniformly distributed load of 16 

kN/m over its entire length. What is the lightest W shape beam that will not exceed a 

flexural stress of 120 MPa? What is the actual maximum stress in the beam selected? 

Solution 529 

Problem 530 

Repeat Prob. 529 if the distributed load is 12 kN/m and the length of the beam is 8 m. 

Solution 530 

Problem 531 

A 15-ft beam simply supported at the ends carries a concentrated load of 9000 lb at 

midspan. Select the lightest S section that can be employed using an allowable stress of 

18 ksi. What is the actual maximum stress in the beam selected? 

Solution 531 

Problem 532 

A beam simply supported at the ends of a 25-ft span carries a uniformly distributed load 

of 1000 lb/ft over its entire length. Select the lightest S section that can be used if the 

allowable stress is 20 ksi. What is the actual maximum stress in the beam selected? 

Solution 532 

Problem 533 

A beam simply supported on a 36-ft span carries a uniformly distributed load of 2000 

lb/ft over the middle 18 ft. Using an allowable stress of 20 ksi, determine the lightest 

suitable W shape beam. What is the actual maximum stress in the selected beam? 

Solution 533 

Problem 534 

Repeat Prob. 533 if the uniformly distributed load is changed to 5000 lb/ft. 

Solution 534 

Problem 535 

A simply supported beam 24 ft long carries a uniformly distributed load of 2000 lb/ft 

over its entire length and a concentrated load of 12 kips at 8 ft from left end. If the 

allowable stress is 18 ksi, select the lightest suitable W shape. What is the actual 

maximum stress in the selected beam? 

Solution 535 

Problem 536 

A simply supported beam 10 m long carries a uniformly distributed load of 20 kN/m 

over its entire length and a concentrated load of 40 kN at midspan. If the allowable 

stress is 120 MPa, determine the lightest W shape beam that can be used. 

Solution 536 

Floor Framing 

In floor framing, the subfloor is supported by light beams called floor joists or simply 

joists which in turn supported by heavier beams called girders then girders pass the 

load to columns. Typically, joist act as simply supported beam carrying a uniform load 

of magnitude p over an area of sL, 

where 

p = floor load per unit area 

L = length (or span) of joist 

s = center to center spacing of joists and 

w

o

 = sp = intensity of distributed load in joist. 

Solved Problems in Floor Framing  

Problem 538 

Floor joists 50 mm wide by 200 mm high, simply supported on a 4-m span, carry a floor 

loaded at 5 kN/m

2

. Compute the center-line spacing between joists to develop a 

bending stress of 8 MPa. What safe floor load could be carried on a center-line spacing 

of 0.40 m? 

Solution 538 

Problem 539 

Timbers 12 inches by 12 inches, spaced 3 feet apart on centers, are driven into the 

ground and act as cantilever beams to back-up the sheet piling of a coffer dam. What is 

the maximum safe height of water behind the dam if water weighs = 62.5 lb/ft

3

 and ( f

b

)

max

 = 1200 psi? 

Solution 539 

Problem 540 

Timbers 8 inches wide by 12 inches deep and 15 feet long, supported at top and 

bottom, back up a dam restraining water 9 feet deep. Water weighs 62.5 lb/ft

3

. (a) 

Compute the center-line spacing of the timbers to cause f

b

 = 1000 psi. (b) Will this 

spacing be safe if the maximum f

b

, ( f

b

 )

max

 = 1600 psi, and the water reaches its 

maximum depth of 15 ft? 

Solution 540 

Problem 541 

The 18-ft long floor beams in a building are simply supported at their ends and carry a 

floor load of 0.6 lb/in

2

. If the beams have W10 × 30 sections, determine the center-line 

spacing using an allowable flexural stress of 18 ksi. 

Solution 541 

Problem 542 

Select the lightest W shape sections that can be used for the beams and girders in 

Illustrative Problem 537 of text book if the allowable flexural stress is 120 MPa. Neglect 

the weights of the members. 

Solution 542 

Problem 543 

A portion of the floor plan of a building is shown in Fig. P-543. The total loading 

(including live and dead loads) in each bay is as shown. Select the lightest suitable W if 

the allowable flexural stress is 120 MPa. 

Solution 543 

Unsymmetrical Beams 

Flexural Stress varies directly linearly with distance from the neutral axis. Thus for a 

symmetrical section such as wide flange, the compressive and tensile stresses will be 

the same. This will be desirable if the material is both equally strong in tension and 

compression. However, there are materials, such as cast iron, which are strong in 

compression than in tension. It is therefore desirable to use a beam with unsymmetrical 

cross section giving more area in the compression part making the stronger fiber 

located at a greater distance from the neutral axis than the weaker fiber. Some of these 

sections are shown below. 

The proportioning of these sections is such that the ratio of the distance of the neutral 

axis from the outermost fibers in tension and in compression is the same as the ratio of 

the allowable stresses in tension and in compression. Thus, the allowable stresses are 

reached simultaneously. 

In this section, the following notation will be use: 

f

bt

 = flexure stress of fiber in tension 

f

bc

 = flexure stress of fiber in compression 

N.A. = neutral axis 

y

t

 = distance of fiber in tension from N.A. 

y

c

 = distance of fiber in compression from N.A. 

M

r

 = resisting moment 

M

c

 = resisting moment in compression 

M

t

 = resisting moment in tension 

Solved Problems in Unsymmetrical Beams  

Problem 548 

The inverted T section of a 4-m simply supported beam has the properties shown in Fig. 

P-548. The beam carries a uniformly distributed load of intensity w

o

 over its entire 

length. Determine wo if f

bt

 ≤ 40 MPa and f

bc

 ≤ 80 MPa. 

Solution 548 

Problem 549 

A beam with cross-section shown in Fig. P-549 is loaded in such a way that the 

maximum moments are +1.0P lb·ft and -1.5P lb·ft, where P is the applied load in 

pounds. Determine the maximum safe value of P if the working stresses are 4 ksi in 

tension and 10 ksi in compression. 

Solution 549 

Problem 550 

Resolve Prob. 549 if the maximum moments are +2.5P lb·ft and -5.0P lb·ft. 

Solution 550 

Problem 551 

Find the maximum tensile and compressive flexure stresses for the cantilever beam 

shown in Fig. P-551. 

Solution 551 

Problem 552 

A cantilever beam carries the force and couple shown in Fig. P-552. Determine the 

maximum tensile and compressive bending stresses developed in the beam. 

Solution 552 

Problem 553 

Determine the maximum tensile and compressive bending stresses developed in the 

beam as shown in Fig. P-553. 

Solution 553 

Problem 554 

Determine the maximum tensile and compressive stresses developed in the 

overhanging beam shown in Fig. P-554. The cross-section is an inverted T with the 

given properties. 

Solution 554 

Problem 555 

A beam carries a concentrated load W and a total uniformly distributed load of 4W as 

shown in Fig. P-555. What safe value of W can be applied if f

bc

 ≤ 100 MPa and f

bt

 ≤ 60 

MPa? Can a greater load be applied if the section is inverted? Explain. 

Solution 555 

Problem 556 

A T beam supports the three concentrated loads shown in Fig. P-556. Prove that the NA 

is 3.5 in. above the bottom and that I

NA

 = 97.0 in

4

. Then use these values to determine 

the maximum value of P so that f

bt

 ≤ 4 ksi and f

bc

 ≤ 10 ksi. 

Solution 556 

Problem 557 

A cast-iron beam 10 m long and supported as shown in Fig. P-557 carries a uniformly 

distributed load of intensity wo (including its own weight). The allowable stresses are f

bt

≤ 20 MPa and f

bc

 ≤ 80 MPa. Determine the maximum safe value of wo if x = 1.0 m. 

Solution 557 

Problem 558 

In Prob. 557, find the values of x and w

o

 so that w

o

 is a maximum. 

Solution 558