Chapter 3
Load and stress analysis
Outline
1. Introduction
2. Equilibrium and Free-Body Diagrams
3. Shear Force and Bending Moments in Beams
4. Stress
5. Normal Stresses for Beams in Bending
6. Shear Stresses for Beams in Bending
7. Torsion
8. Stress Concentration
9. Contact stress
Machine elements carry different types of loads (concentrated,
distributed, axial, lateral, moments, torsion, etc.) according to the
function and configuration of each element. These loads cause
stresses of different types and magnitudes in different locations
in the element.
When designing machine elements it is important to locate the
critical locations (or sections) and to evaluate the stress at the
critical sections to ensure the safety and functionality of the
machine element.
1. Introduction
2. Equilibrium and Free-Body Diagrams (1)
Equilibrium of a body requires both a balance of forces (to prevent
translation) and balance of moments (to prevent rotations):
A free body diagram (FBD) is a sketch of an element or group of
connected elements that shows all the forces acting on it (applied
loads, gravity forces, and reactions)
See Example 3-1
Example 3-1
2. Equilibrium and Free-Body Diagrams (2)
3. Shear Force and Bending Moments in Beams
Shear and moment diagrams are important in locating the critical sections
in a beam (sections with maximum shear or moment) such that stresses
are evaluated at these sections.
When the loading is not simple, he shear force and moment diagrams can
be obtained by using sections.
Shear force V and bending moment M are related by equation
3. Shear Force and Bending Moments in Beams
Singularity functions
3. Shear Force and Bending Moments in Beams
Singularity functions Load function
3. Shear Force and Bending Moments in Beams
Example 3-2: Derive the loading, shear-force, and bending-
moment relations for the beam
Singularity functions
3. Shear Force and Bending Moments in Beams
Singularity functions
3. Shear Force and Bending Moments in Beams
4. Stress
In general, the stress at a point on a cross-section will have
components normal and tangential to the surface, which hare
named as normal stress and shear stress .
Stress is the term used to define the intensity and direction of
the internal forces acting at a given point on a particular plane.
The state of stress at a point is described by three mutually
perpendicular surfaces. Thus, in general, a complete state of stress is
defined by nine stress components
Plane stress with βcross shearβ equal
Nine stress components:x , y , z , xy , xz , yx ,yz, zx ,zy
For equilibrium, βcross-shearsβ are equal:
xy = xz ; yx =yz ; zx =zy
4. Stress (2)
Six stress components:x , y , z , xy yz, zx
Mohrβs Circle for plane stress
4. Stress (3)
Concerning with the stresses Ο and Ο
that act upon this oblique plane.
By summing the forces caused by all
the stress components to zero, the
stresses Ο and Ο are found to be
Principal stresses and principal directions
Max = 1
Min = 2
0
p= 0
- 1, 2 are principal stresses and their corresponding
directions are principal directions.
- zero shear tresses
Mohrβs Circle for plane stress
4. Stress (4)
0
Two surfaces containing the maximum shear stresses also
contain equal normal stresses of (x + y )/2
Mohrβs Circle for plane stress
4. Stress (5)
The parametric relationship between and is a circle.
This circle is known as Mohrβs circle, where it provides a
convenient method of graphically visualizing the state of
stress and it can be used to find the principal stresses as
well as performing stress transformation
xyyxyx
Ο)ΟΟ
(R ΟΟ
cwhere
Rc
22
222
2 and
2
)(
4. Stress (6)
Mohrβs Circle for plane stress
Mohrβs Circle for plane stress
4. Stress (7)
4. Stress (8)
Mohrβs Circle for plane stress
Example
Mohrβs Circle for plane stress
4. Stress (9)
Solution
4. Stress (9)
a) Draw the and axes first.
Establish point A on x surface
with coordinates
A (x, cw
xy)= (80, 50cw)MPa
along with axis.
Corresponding to the y
surface, locates point B with
coordinates
B (y, ccw
xy)= (0, 50ccw)MPa.
The line AB form diameter of
the Morhβs circle. The
intersection of the circle with
the axis defined 1 and 2.
The maximum shear stress is equal to the radius of the circle:
π1 = Β±ππ₯βππ¦
2
2+ ππ₯π¦
2 = Β± 402 + 502 = Β±64 MPa.
The principal stresses can be found on the circle to be
π1, π2 =ππ₯+ππ¦
2Β±
ππ₯βππ¦
2
2+ ππ₯π¦
2 π1 = 40 + 64 = 104 MPa
π2 = 40 β 64 = 24 MPa
Solution
4. Stress
The angle 2β from the x axis clockwise
to π1 is
π‘ππ2β π =2ππ₯π¦
ππ₯βππ¦
β 2β π = π‘ππβ1 2ππ₯π¦
ππ₯βππ¦= π‘ππβ1 50
40= 51.30
To draw the principal stress element
(Fig. 3β11c), sketch the x and y axes
parallel to the original axes. The angle
Οp on the stress element must be
measured in the same direction as is the angle 2Οp on the Mohr circle.
From x measure 25.7Β° (half of 51.3Β°)
clockwise to locate the Ο1 axis. The Ο2 axis is 90Β° from the Ο1 axis and the
stress element can now be completed
and labeled as shown. Note that there
are no shear stresses on this element.
Solution
The two maximum shear
stresses occur at points E
and F in Fig. 3β11b.
The two normal stresses
corresponding to these
shear stresses are each 40
MPa, as indicated. Point E
is 38.7Β° ccw from point A on
Mohrβs circle.
Therefore, in Fig. 3β11d,
draw a stress element
oriented 19.3Β° (half of
38.7Β°) ccw from x. The
element should then be
labeled with magnitudes
and directions as shown.
4. Stress
Solution
b) The transformation equations are programmable.
From Eq. (3β10),
β β π =1
2π‘ππβ1 2ππ₯π¦
ππ₯βππ¦ =
1
2π‘ππβ1 2(β50)
80= β25.70; 64.30
From Eq. (3β8), for the first angle β π =β25.7β¦
π =ππ₯+ππ¦
2+
ππ₯βππ¦
2πππ 2β + ππ₯π¦π ππ2β
π =80+0
2+
80β0
2πππ 2 β25.70 + β50 π ππ 2 β25.70 = 104.3 πππ
The shear stress on this surface is obtained from Eq. (3.9) as
π = βππ₯βππ¦
2π ππ2β π + ππ₯π¦πππ 2β π
π = β80β0
2π ππ 2 β25.70 + β50 πππ 2 β25.70 = 0 πππ
For β π = 64.30
π =80+0
2+
80β0
2πππ 2 64.30 + β50 π ππ 2 64.30 = β24.03 πππ
π = β80β0
2π ππ 2 64.30 + β50 πππ 2 64.30 = 0 πππ
β π1 = 104.3πππ; β π1 = β25.70
π2 = β24.3πππ; β π2 = 64.30
4. Stress
To determine 1 and 2, we first use Eq. (3.11) to calculate β π :
β π‘ππ2β π = βππ₯βππ¦
2ππ₯π¦
β β π =1
2π‘ππβ1 β
ππ₯βππ¦
2ππ₯π¦
β β π =1
2π‘ππβ1 β
80β0
2(β50)= 19.30; 109.30
For β π = 19.30, Eq. (3.8) and (3.9) yield
π =80+0
2+
80β0
2πππ 2 19.30 + β50 π ππ 2 19.30 = 40.0 πππ
π = β80β0
2π ππ 2 19.30 + β50 πππ 2 19.30 = β64.0 πππ
For β π = 19.30, Eq. (3.8) and (3.9) yield
π =80+0
2+
80β0
2πππ 2 109.30 + β50 π ππ 2 109.30 = 40.0 πππ
π = β80β0
2π ππ 2 109.30 + β50 πππ 2 109.30 = 64.0 πππ
4. Stress
To determine 1 and 2, we first use Eq. (3.11) to calculate β π :
β π‘ππ2β π = βππ₯βππ¦
2ππ₯π¦
β β π =1
2π‘ππβ1 β
ππ₯βππ¦
2ππ₯π¦
β β π =1
2π‘ππβ1 β
80β0
2(β50)= 19.30; 109.30
For β π = 19.30, Eq. (3.8) and (3.9) yield
π =80+0
2+
80β0
2πππ 2 19.30 + β50 π ππ 2 19.30 = 40.0 πππ
π = β80β0
2π ππ 2 19.30 + β50 πππ 2 19.30 = β64.0 πππ
For β π = 64.30
π =80+0
2+
80β0
2πππ 2 64.30 + β50 π ππ 2 64.30 = β24.03 πππ
π = β80β0
2π ππ 2 64.30 + β50 πππ 2 64.30 = 0 πππ
β π1 = 104.3πππ; β π1 = β25.70
π2 = β24.3πππ; β π2 = 64.30
4. Stress
General Three-Dimensional Stress
4. Stress
5. Elastic Strain
Normal strain π is given as
π =πΏ
π (3.16)
where Ξ΄ is the total elongation of the bar within the length l.
Hookeβs law for the tensile specimen is given as
π = πΈπ (3.17)
where the constant E called Youngβs modulus or the modulus of elasticity
When a material is placed in tension, there exists not only an axial strain, but also
negative strain (contraction) perpendicular to the axial strain.
Assuming a linear, homogeneous, isotropic material, this lateral strain is proportional
to the axial strain. If the axial direction is x, then the lateral strains are
ππ¦ = ππ§ = πππ₯
The constant of proportionality π is called Poissonβs ratio, which is about 0.3 for
most structural metals.
See Table Aβ5 for values of v for common materials.
5. Elastic Strain
ππ₯ =1
πΈππ₯ β π ππ¦ + ππ§
ππ¦ =1
πΈππ¦ β π ππ₯ + ππ§ (3.19)
ππ§ =1
πΈππ§ β π ππ¦ + ππ₯
Shear strain Ξ³ is the change in a right angle of a stress element when subjected
to pure shear stress, and Hookeβs law for shear is given by
π = πΊπΎ (3.20)
where the constant G is the shear modulus of elasticity or modulus of rigidity. It
can be shown for a linear, isotropic, homogeneous material, the three elastic
constants are related to each other by
πΈ = 2πΊ(1 + π) (3.21)
If the axial stress is in the x direction, then from Eq. (3β17)
ππ₯ =ππ₯
πΈ; ππ¦ = ππ¦ = βπ
ππ₯
πΈ (3.18)
For a stress element undergoing Οx , Οy , and Οz simultaneously, the normal strains
are given by
6. Normal stresses for beams in Bending
or
where Z= I/c is called the section modulus.
I is the second-area moment
of inertia (second-area
moment) about the z axis
Tables A-6, A-7 and A-8 in the text give the I and Z values for
some standard cross-section beams
6. Normal stresses for beams in Bending
For a rectangular cross-section,
πΌ =ππ3
12 (3β25a)
Where,
b is distance parallel to the neutral axis (mm)
d is distance perpendicular to the neutral axis
For a circular cross-section,
πΌ =ππ4
12 (3β25b)
Where, d is the diameter of the cross-section.
When the cross-section is irregular, the moment of inertia about the centroidal axis is
given by equation 3.29.
The parallel axis theorem for this area is given by the expression,
πΌπ§ = πΌππ + π΄π2 (3.29)
Where,
Ica is the moment of inertia of area about its own centroidal axis.
Iz is moment of inertia of the area about any parallel axis a distance d removed.
A is area of the cross-section.
6. Normal stresses for beams in Bending
Two-plane bending
The maximum tensile and compressive bending stresses occur where the
summation gives the greatest positive and negative stresses, respectively
where the first term on the right side of the equation is identical to Eq. (3β
24). My is the bending moment in the xz plane (moment vector in y
direction). z is the distance from the neutral y axis, and Iy is the moment of
inertia of the area about the y axis.
For a beam of diameter d the maximum distance from the neutral axis is d/2, and
from Table Aβ18, I = Οd4 /64.
The maximum bending stress for a solid circular ross section is then
6. Normal stresses for beams in Bending
Example 3.6 As
shown in Fig. 3β16a,
beam OC is loaded in
the xy plane by a
uniform load of 50
lbf/in, and in the xz
plane by a
concentrated force of
100 lbf at end C. The
beam is 8 in long.
(a) For the cross section shown determine the maximum tensile
and compressive bending stresses and where they act.
(b) If the cross section was a solid circular rod of diameter,
d=1.25 in, determine the magnitude of the maximum bending
stress
6. Normal stresses for beams in Bending
Solution
The reactions at O and the bending-moment
diagrams in the xy and xz planes are shown in
Figs. 3β16b and c, respectively. The maximum
moments in both planes occur at O where
(ππ§)π= β1
250 82 = β1600πππ. ππ
(ππ¦)π= 100(8) = 800πππ. ππ
The moments of inertia of area in both planes
are
πΌπ§ =1
120.75 (1. 5)3 = 0.2109ππ4
πΌπ§ =1
12(1.5)(0.75)3 = 0.05273ππ4
The maximum tensile stress occurs at point A, shown in Fig. 3β16a, where the
maximum tensile stress is due to both moments. At A, yA = 0.75 in and zA = 0.375 in.
Thus, from Eq. (3β27)
The maximum compressive bending stress occurs at point B where, yB =β0.75 in and
zB =β0.375 in. Thus
6. Normal stresses for beams in Bending
Figure 3.16 (a) Beam loaded in two
planes; (b) loading and bending-
moment diagrams in xy plane; (c)
loading and bending-moment diagrams
in xz plane.
6. Normal stresses for beams in Bending
(b) For a solid circular cross section of diameter, d = 1.25 in, the
maximum bending stress at end O is given by Eq. (3β28) as
6. Normal stresses for beams in Bending
6. Normal stresses for beams in Bending
It is rare to encounter beams subjected to pure bending moment only (no
shear). Most beams are subjected to both shear forces and bending
moments.
Ο =VQ
Ib (3-31)
For a beam subjected to
shear force V, the shear
stress is found as
Where,
V is the shear force at the
section of interest.
Q the first moment of the area Aβ
with respect to the neutral axis.
This stress is known as the transverse shear stress. It is always
accompanied with bending stress.
I is the second moment of area of the entire section about the neutral axis.
b is the width at the point where Ο is determined
6. Shear Stresses for Beams in Bending (2)
Q is the first moment of the area Aβ with respect to the neutral axis, Q,
is found as:
where,
Aβ is the area of the portion of
the section above or below the
point where Ο is determined.
π¦ β² is the distance to the
centroid of the area Aβ
measured from the neutral axis
of the beam.
The shear stress is maximum at the neutral axis (since Q will be max), and it is
zero on the top and bottom surfaces (since Q is zero).
For any common cross section beam, if the beam length to height ratio is greater
than 10, the transverse shear stress is generally considered negligible compared
to the bending stress at any point within the cross section.
Aβ
'y
Neutral axis
1y
6. Shear Stresses for Beams in Bending (3)
Table 3β2 Formulas for Maximum Transverse Shear Stress from VQ/Ib
Figure 3β18
Transverse shear
stresses in a
rectangular beam.
3β5 Shear Stresses for Beams in Bending
A beam 12 in long is to support a load of 488 lbf acting 3 in from the left
support. The beam is an I beam with the cross-sectional dimensions shown.
To simplify the calculations, assume a cross section with quare corners.
Points of interest are labeled (a, b, c, and d) at distances y from the neutral
axis of 0 in, 1.240- in, 1.240+ in, and 1.5 in (Fig. 3β20c). At the critical axial
location along the beam, find the following information.
(a) Determine the profile of the distribution of the transverse shear stress,
obtaining values at each of the points of interest.
(b) Determine the bending stresses at the points of interest.
(c) Determine the maximum shear stresses at the points of interest, and
compare them.
(a) Example
3β5 Shear Stresses for Beams in Bending
Solution The transverse shear stress is not likely to be negligible in this
case since the beam length to height ratio is much less than 10,
and since the thin web and wide flange will allow the transverse
shear to be large.
The loading, shear-force, and bending-moment diagrams are
shown in Fig. 3β20b. The critical axial location is at x=3- where
the shear force and the bending moment are both maximum.
(a) We obtain the area moment of inertia I by evaluating I for a
solid 3.0-in οΏ½ 2.33-in rectangular area, and then subtracting the
two rectangular areas that are not part of the cross section.
3β5 Shear Stresses for Beams in Bending
Solution
Applying Eq. (3β31) at each point of
interest, with V and I constant for each
point, and b equal to the width of the
cross section at each point, shows that
the magnitudes of the transverse shear
stresses are
3β5 Shear Stresses for Beams in Bending
Solution
3β5 Shear Stresses for Beams in Bending
Solution (c) Now at each point of interest, consider a
stress element that includes the bending
stress and the transverse shear stress. The
maximum shear stress for each stress
element can be determined by Mohrβs circle,
or analytically by Eq. (3β14) with Οy = 0,
7. Torsion (1) Any moment vector that is collinear with an axis of a mechanical element is called
a torque vector, because the moment causes the element to be twisted about that
axis. A bar subjected to such a moment is also said to be in torsion.
When a circular shaft is subjected to torque, the shaft will be twisted and the angle
of twist is found to be:
π =ππ
πΊπ½ (3.35)
where T = torque; l = length
G = modulus of rigidity πΊ =πΈ
2(1+π)
J = polar second moment of area.
For a solid round section,
π½ =ππ4
32 (3.38)
where d is the diameter of the bar.
For the hollow round section,
π½ =π(π0
4βππ4)
32 (3.39)
where the subscripts o and i refer to the outside and inside diameters, respectively.
7. Torsion (1) Shear stresses develop throughout the cross section. For a round bar in torsion,
these stresses are proportional to the radius Ο and are given by
π =ππ
π½ (3.36)
Designating r as the radius to the outer surface, we have
ππππ₯ =ππ
π½ (3.37)
The parameter Ξ± is a factor that is a function of the ratio b/c as shown in the following
table.
The angle of twist is given by
π =ππ
π½ππ3πΊ (3-41)
where Ξ² is a function of b/c, as shown in the table.
The maximum shearing stress in a rectangular b Γ c
section bar occurs in the middle of the longest side b and
is of the magnitude.
ππππ₯ =π
πΌππ2=
π
ππ23 +
1.8
π/π (3-40)
7. Torsion (2)
max
It is often necessary to obtain the torque T from a consideration of the power and
speed of a rotating shaft.
For convenience when U.S. customary units are used, three forms of this relation
are
π» =πΉπ
33000=
2πππ
33000(12)=
ππ
63025 (3-42)
Where H = power, hp
T = torque, lbf Β· in
n = shaft speed, rev/min
F = force, lbf
V = velocity, ft/min
7. Torsion (3)
When SI units are used, the equation is
π» = ππ (3.43)
Where H = power, W
T = torque, N Β· m
Ο = angular velocity, rad/s
The torque T corresponding to the power in watts is given approximately
by
π = 9.55π»
π (3-44)
3β6 Torsion
Example 3-9:
The 1.5-in-diameter solid steel shaft shown in Figure is simply
supported at the ends. Two pulleys are keyed to the shaft where
pulley B is of diameter 4.0 in and pulley C is of diameter 8.0 in.
Considering bending and torsional stresses only, determine the
locations and magnitudes of the greatest tensile, compressive, and
shear stresses in the shaft.
3β6 Torsion
Example:
Example:
3β6 Torsion
8. Stress Concentration
Stress concentration occurred at any discontinuity in a machine part.
- The discontinuities are called stress raisers.
- The regions in which they alert are called the areas of stress concentration.
- A theoretical, or geometric, stress-concentration factor Kt or Kts is used to
relate the actual maximum stress at the discontinuity to the nominal stress.
The factors are defined by the equations
πΎπ‘ =ππππ₯
ππ πΎπ‘π =
ππππ₯
ππ (3-48)
Where:
Kt β for normal stresses
Kts β for shear stresses
The nominal stress Ο0 or Ο0 is the stress
calculated by using the elementary stress
equations and the net area, or net cross section.
The stress-concentration factor depends for its
value only on the geometry of the part.
8. Stress Concentration (2)
8. Stress Concentration (3)
9. Contact Stresses
When two bodies having curved surfaces are pressed
together, point or line contact changes to area contact, and
the stresses developed in the two bodies are called the
contact stress.
Contact-stress problems arise
in the contact of a wheel and
a rail, in automotive valve
cams and tappets, in mating
gear teeth, and in the action of
rolling bearings.
Typical failures due to contact
stress are seen as cracks,
pits, or flaking in the surface
material.
9. Contact Stresses
When two solid spheres of diameters d1 and d2
are pressed together with a force F, a circular
area of contact of radius a is obtained.
Specifying E1, Ξ½1 and E2, Ξ½2 as the elastic
constants of the two spheres, the radius a is
given by the equation
π =ππ
π
π β πππ /π¬π + π β ππ
π /π¬π
π/π π + π/π π
π
ππππ =ππ
ππ ππ
ππ = ππ = ππ = ππ = βππππ π βπ
ππππβπ
π
π/ππ + π β
π
π π +ππ
ππ
ππ = ππ =βππππ
π +ππ
ππ
ππππ = ππ/π = ππ/π =
ππ β ππ
π=
ππ β ππ
π
The maximum pressure occurs at the center of
the contact area and is
The maximum stresses occur on the z axis, and these are principal stresses
9. Contact Stresses
Magnitude of the stress components below the surface as a function of the maximum
pressure of contacting spheres. Note that the maximum shear stress is slightly below
the surface at z=0.48a and is approximately 0.3pmax. The chart is based on a Poisson
ratio of 0.30. Note that the normal stresses are all compressive stresses.
9. Contact Stresses
(a) Two right circular cylinders held in contact by
forces F uniformly distributed along cylinder length
l. (b) Contact stress has an elliptical distribution
across the contact zone width 2b.
π =ππ
π π
π β πππ /π¬π + π β ππ
π /π¬π
π/π π + π/π π
ππππ =ππ
π ππ
ππ = βππππππ π +ππ
ππ βπ
π
ππ = βππππ
π + πππ
ππ
π +ππ
ππ
β ππ
π
ππ = ππ =βππππ
π + ππ/ππ
The maximum pressure is
The stress state along the z axis is given by the equations
9. Contact Stresses
Magnitude of the stress components below the surface as a function of the maximum
pressure for contacting cylinders. The largest value of max occurs at z/b = 0.786. Its
maximum value is 0.30pmax. The chart is based on a Poisson ratio of 0.30. Note that all
normal stresses are compressive stresses.