Chapter 4
Deflection and Stiffness
Omit 4.6, 4.9, 4.16, 4.17
 Rigid Body:
 A body is said to be rigid if it exhibits no change in size or shape under the influence
of forces or couples; the distance between any two points within the body remains
constant under the application of forces.
 All real bodies deform under load, either elastically or plastically. Classification of a
real body as a rigid is an idealization.
 Deflection analysis enters into design situations in many ways:
In a transmission, the gears must be supported by a rigid shaft. If the shaft bends too
much (too flexible) the teeth will not mesh properly, and the result will be excessive
impact, noise, wear, and early failure.
 Sometimes mechanical elements must be designed to have a particular force-deflection
characteristic:
The suspension system of an automobile, for example, must be designed within a very
narrow range to achieve an optimum vibration frequency for all conditions of vehicle
loading.
4.1 Spring Rates
 Elasticity is that property of a material that enables it to regain its original
configuration after having been deformed.
 Stiffness is the rigidity of an object — the extent to which it resists deformation in
response to an applied force.
 Flexibility is the ability of a body to distort. The complementary concept is stiffness.
The more flexible an object is the less stiff it is.
 A spring is a mechanical element that exerts a force when deformed; figures shown:
(a) A straight beam of length l simply supported at ends loaded by force F. The
deflection y is linearly related to the force, as long as the elastic limit of the material
is not exceeded, this beam can be described as a linear spring.
(b) A straight beam is supported on two cylinders; beam is shorter. A larger force is
required to deflect a short beam, it becomes stiffer. The force is not linearly related
to the deflection; the beam can be described as a nonlinear stiffening spring.
(c) An edge-view of a dish-shaped round disk. The force necessary to flatten the disk
increases at first and then decreases as the disk approaches a flat configuration. Any
mechanical
element having
such a
characteristic is
called a
nonlinear
softening
spring.
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 Consider that force and deflection are related F =F(y); then the spring rate is defined as
where y is measured in the direction of F and at the point of application of F.
 For linear spring:
Here k is called spring constant. Units lbf/in ; or N/m.
 Above Eqns. are quite general and apply equally well for torques and moments.
4.2 Tension, Compression, and Torsion
 Consider a uniform bar subjected to axial (or compressive) force F
where δ is the linear deformation.
This equation does not apply to a long bar loaded in compression
if there is a possibility of buckling.
The spring constant of an axially loaded bar is
 The angular deflection of a uniform solid or hollow round bar subjected to a twisting
moment T was given in Eq. (3–35), and is
where θ is the angular deformation expressed in radians.
Equation (4–5) can be rearranged to give the torsional spring
rate (kθ) as
Above equations apply only to circular cross sections. For noncircular cross section
refer to chapter 3.
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4.3 Deflection Due to Bending
 Beams deflect great deal more than axially loaded members.
 Bending of beams probably occurs more often than any other loading problem in
mechanical design.
 Shafts, axles, cranks, levers, springs, brackets, and wheels, as well as many other
elements, must often be treated as beams in the design and analysis of mechanical
structures and systems.
 Consider a beam segment of length L. After deformation, the length of the neutral
surface remains L. At other sections:
L     y  
  L  L    y        y
x 
m 

L
c


y

or
y

ρ
c
m
(strain varies linearly)
and
y
c
 x   m
; also
Mc

I
where σm denotes the maximum absolute value of the stress.
 So, the curvature of a beam subjected to a bending moment M
is given by
 m  E m  E
c


where ρ is the radius of curvature.
 From studies in Calculus the curvature of a plane curve is
given by:
where y is the deflection of the beam at any point x along its length.
 The slope of the beam at any point x is given by
 For many problems in bending, the slope is very small, and for these the denominator of
Eq. (4–9) can be taken as unity. Equation (4–8) can then be written
 Noting that Eqs. (3–3) and (3–4)
and successively differentiating Eq. (b) yields
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 It is convenient to display these relations in a group as follows:
Integration
The nomenclature and conventions are illustrated by the beam shown.
Example 4-1 (see textbook)
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4.4 Beam Deflection Methods
 Beams have intensities of loading that can be q = constant, variable intensity q(x), or
concentrated loads.
 There are many techniques employed to solve the integration problem for beam
deflection. Some of the popular methods include:
1. Integration of moment equation (example 4-1)
2. Superposition (section 4-5)
3. Moment-area method
4. Singularity functions (section 4-6) –Omitted.
5. Strain energy with Castigliano’s theorem (sections 4-7, 4-8)
6. Numerical integration.
4.5 Beam Deflections by Superposition
 Table A-9 provides some cases for results of beams subjected to simple loads and
boundary conditions.
 Superposition resolves the effect of combined loading on a structure by determining the
effect of each load separately and adding the results algebraically.
 In using the superposition principle, the followings are required:
1. Each effect is linearly related to the load that produces it.
2. A load does not create a condition that affects the results of another load.
3. The deformations resulting from any specific load are not large enough to
appreciably alter the geometric relations of the parts of the structure.
Example 4-2
Superposition:
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Example 4-3
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Example 4-4 (see textbook)
4.6 Beam Deflections by Singularity Functions (Omitted)
4.7 Strain Energy
 It is the potential energy stored in a body by virtue of an elastic deformation.
 The total work U done by the load as the rod undergoes a deformation x1 is thus
and is equal to the area under the load-deformation diagram between x = 0 and x = x1.
The work done by the load P as it is slowly applied to the rod must result in the
increase of some energy associated with the deformation of the rod. This energy is
referred to as the strain energy of the rod. We have, by definition,
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 In the case of a linear and elastic deformation, the portion of
the load-deformation diagram involved can be represented
by a straight line of equation P = kx (Fig. 11.4). Substituting
for P in Eq. (11.2), we have
OR
 If a member is deformed a distance y, and if the force deflection
relationship is linear, this energy is equal to the product of the
average force F and the deflection y, or, force F and the deflection
y, or,



This equation is general in the sense that the force F can also mean
torque, or moment, provided, of course, that consistent units are
used for k.
Strain Energy For tension and compression, we employ Eq. (4–4)
and obtain
where the first equation applies when all the terms are constant throughout the length,
and the more general integral equation allows for any of the terms to vary through the
length.
Strain energy for torsion, we employ (4–7) and get
Strain energy for direct shear, consider the element with one side fixed in Fig. 4–8a. The
force F places the element in pure shear, and the work done is
U = Fδ/2. Since the shear strain is γ = δ/l = τ/G = F/AG, we have
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
Strain energy stored in a beam or lever by bending may be obtained by referring to Fig.b.
Here AB is a section of the elastic curve of length ds having a radius
of curvature ρ. The strain energy stored in this element of the beam
is dU = (M/2) dθ.
Since ρ dθ = ds, we have
We can eliminate ρ by using Eq. (4–8), ρ = EI/M. Thus
For small deflections, ds = dx. Then, for the entire beam
Summarized to include both the integral and nonintegral form, the strain energy for
bending is

Equations (4–22) and (4–23) are exact only when a beam is subject to pure bending.
Even when transverse shear is present, these equations continue to give quite good
results, except for very short beams.
Strain energy due to shear in bending is a complicated problem. An approximate
solution can be obtained by using Eq. (4–20) with a correction factor whose value
depends upon the shape of the cross section. If we use C for the correction factor and V
for the shear force, then the strain energy due to shear in bending is
Values of the factor C are listed in Table 4–1.
Example 4-8 (see textbook)
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4.8 Castigliano’s Theorem
 It is an unusual, powerful, and often surprisingly simple approach to deflection analysis.
 It is a unique way of analyzing deflections and is even useful for finding the reactions of
indeterminate structures.
 Castigliano’s theorem states that when forces act on elastic systems subject to small
displacements, the displacement corresponding to any force, in the direction of the force,
is equal to the partial derivative of the total strain energy with respect to that force.
 Mathematically, for linear displacement, the theorem of Castigliano is
where δi is the displacement of the point of application of the force Fi and in its direction.
 For rotational displacement Eq. (4–26) can be written as
where θi is the rotational displacement, in radians, of the beam where the moment Mi
exists and in its direction.
 As an example, apply Castigliano’s theorem using Eqns. (4–16) and (4–18) to get the
axial and torsional deflections. The results are
Compare Eqns. (a) and (b) with Eqns. (4–3) and (4–5).
Example 4-9 (see textbook)
 The relative contribution of transverse shear to beam deflection decreases as the length-toheight ratio of the beam increases, and is generally considered negligible for l/d > 10.
Note that the deflection equations for the beams in Table A–9 do not include the effects
of transverse shear.
 Castigliano’s theorem can be used to find the deflection at a point even though no force
or moment acts there. The procedure is:
1. Set up the equation for the total strain energy U by including the energy due to a
fictitious force or moment Q acting at the point whose deflection is to be found.
2. Find an expression for the desired deflection δ, in the direction of Q, by taking the
derivative of the total strain energy with respect to Q.
3. Since Q is a fictitious force, solve the expression obtained in step 2 by setting Q equal
to zero. Thus, the displacement at the point of application of the fictitious force Q is
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 In cases where integration is necessary to obtain the strain energy, it is more efficient to
obtain the deflection directly without explicitly finding the strain energy, by moving the
partial derivative inside the integral. For the example of the bending case,
This allows the derivative to be taken before integration, simplifying the mathematics.
 The expressions for the common cases in Eqs. (4–17), (4–19), and (4–23) are rewritten
as
i 
i 
U
1  F
F


 Fi
G A   Fi 
U
C  V 
V


 Fi
G A  Fi 
direct shear
shear in bending
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Example
Example
Example 4-10 (see textbook)
Example 4-11 (see textbook)
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Stiffness (k) is the rigidity of an object — the extent to which it resists deformation in response
to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff
it is.[2]
Young's modulus (tensile modulus or elastic modulus) is a measure of the stiffness of an
elastic material and is a quantity used to characterize materials. It is defined as the ratio of the
stress along an axis over the strain along that axis in the range of stress in which Hooke's law
holds
Deflection is the degree to which a structural element is displaced under a load. It may refer to an
angle or a distance.
In materials science, deformation is a change in the shape or size of an object due to an applied
force or a change in temperature. The first case can be a result of tensile forces, compressive
forces, shear, bending or torsion. In the second case, the most significant factor, which is
determined by the temperature, is the mobility of the structural defects such as grain boundaries,
point vacancies, line and screw dislocations, stacking faults and twins in both crystalline and noncrystalline solids. Deformation is often described as strain.
In materials science, ductility is a solid material's ability to deform under tensile stress;
Malleability, a similar property, is a material's ability to deform under compressive stress;
A material is brittle if, when subjected to stress, it breaks without significant deformation
(strain). Brittle materials absorb relatively little energy prior to fracture, even those of high
strength. Brittle materials include most ceramics and glasses (which do not deform plastically)
and some polymers, such as PMMA and polystyrene. Many steels become brittle at low
temperatures, depending on their composition and processing.
A fracture is the separation of an object or material into two, or more, pieces under the action of
stress.
In materials science, toughness is the ability of a material to absorb energy and plastically
deform without fracturing; Material toughness is defined as the amount of energy per volume that
a material can absorb before rupturing. It is also defined as the resistance to fracture of a material
when stressed.
Resilience is the ability of a material to absorb energy when it is deformed elastically, and release
that energy upon unloading. The modulus of resilience is defined as the maximum energy that
can be absorbed per unit volume without creating a permanent distortion.
The yield strength or yield point of a material is defined in engineering and materials science as
the stress at which a material begins to deform plastically. Prior to the yield point the material
will deform elastically and will return to its original shape when the applied stress is removed.
Ultimate tensile strength (UTS), often shortened to tensile strength (TS) or ultimate
strength,[1][2] is the maximum stress that a material can withstand while being stretched or pulled
before failing or breaking. Tensile strength is the opposite of compressive strength and the values
can be quite different.
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