Download elastic deformation

ELASTIC DEFORMATION
AND
POISSON'S RATIO
19
Background Unit: Elastic Deformation and Poisson's Ratio
Introduction
We're all aware that a very large force will produce a permanent, or irreversible,
distortion of even a large piece of a strong material. The permanent distortion may be a bending
which doesn't spring back when the force is removed or it may be fracture. Designers usually
work to assure that their machines and structures aren't loaded by forces large enough to produce
permanent distortion, but are loaded only within the range where the distortion "springs back"
when the force is removed. We usually think of the forces causing deformation to be externally
applied forces: forces such as the pressure force pressing on a piston's top within an engine or the
traffic loads on a bridge or the snow loads on the roof of a mountain cabin. However, forces can
be developed in many other ways: the load on a cable lowered into a deep oceanic canyon will be
due mainly to the weight of the cable itself. Of course, the loads on transmission lines are due to
the weight of the wire. Likewise most of the load on the bottom of the Golden Gate Bridge
towers is from the weight of the bridge itself, not the traffic on the bridge. Other types of loading
result from centrifugal forces (for example in a jet engine or steam turbine blade), from magnetic
forces (on the windings and structure of large electromagnets. including those in large motors
and generators), and from temperature differences within a piece of material causing unequal
expansion (for example a coffee cup or a nuclear reactor pressure vessel).
In this unit you will study how deformation is related to loading and how both are
measured.
Objectives
After studying this unit and completing the lab assignment you should be able to perform
the following tasks:
1. Write definitions for the following terms, including SI and English units when appropriate.
a. tensile force
b. elongation
c. normal elastic strain
d. normal stress
e. elastic modulus
2. Given original and loaded dimensions of a rectangular or cylindrical rod, be able to calculate
elongation and axial strain. Given strain and the loaded length, be able to calculate the
unloaded length; given strain and the unloaded length, be able to calculate the loaded length.
3. Given an applied normal force and the dimensions of the part, calculate the stress on planes
normal to the force. Also be able to work backward from the stress to the force or crosssectional area dimensions.
4. Use the definition of elastic modulus to solve problems involving simple elastic normal
strains and stresses.
5. Write the equation defining Poisson's Ratio. Define each term in the Poisson's Ratio
equation, indicate the sign of Poisson's Ratio and the maximum possible value for Poisson's
Ratio.
6. Apply Poisson’s Ratio and the elastic modulus to problems involving normal stress and
strain.
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7. Discuss the principles of the wire strain gage including:
a. a sketch of a simple strain gage
b. an explanation of why the strain gage works (i.e. what properties of the gage material are
responding to the strain)
8. With assistance of other students, obtain accurate measurements of the longitudinal and/or
transverse strain as a function of load in a properly instrumented metal part which is
uniaxially loaded in a universal testing machine.
9. Calculate Poisson’s ratio and the elastic modulus from a proper graph which you prepared
from a set of load-longitudinal elastic strain-transverse elastic strain data and the initial
specimen dimensions.
Elastic Deformation
Lo
Deformation which “springs back” when the force is removed is
called elastic deformation. Elastic deformation occurs as the
interatomic bonds stretch when a load is applied; the atoms retain
their original nearest neighbors and they “spring back” to their
original positions when the load is removed. Fig. 1 shows an
exaggerated view of the elastic deformation of a steel bar when
the bar is subjected to an axial tensile (i.e. elongation or tension)
force. We expect the tensile force to increase the length of the
bar a small amount. This increase in length is called the
elongation of the bar.
Lo + ΔL
F
elongation = L – Lo = ΔL
where:
Lo = original (unloaded) length
Figure 1
F
L = final (loaded) length
The deformation is usually "normalized" to the original length of the bar by dividing the
elongation by the original length. The normalized elastic deformation is called the elastic strain.
elastic strain =
= ε = epsilon
The units of elongation are length, so the units of strain are (length/length). Since the
units cancel, strain is dimensionless. It doesn't matter if the measurements are made in
inches/inch, meters/meter, or miles/mile, the strain will always be the same. You may find it
helpful to write use units for strain although it is not necessary and is usually not done.
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Lo
The strains we considered above are
ΔL
often called normal strains because the force
causing them is acting perpendicular to a plane
of action. The normal strains are also found by
dividing the displacement parallel to the action
of the force, ΔL, by the distance between the
applied forces, Lo. Another basic type of strain
is possible: if the force causing the strain acts
parallel to a plane of action (shear) the strain is
a shear strain. Normal strains are usually
denoted by the Greek letter epsilon (ε), while
shear strains are denoted by the Greek letter
gamma (γ). Shear strain is calculated by
dividing the displacement parallel to the action
of the force, ΔL , by the perpendicular distance
F
between the applied shears, Lo. (Fig. 2)
For this lab we will concentrate on
Figure 2 Before and After Shearing
normal strains, although shear strains are very
important. You will learn about shears and shear
Front view of Cube, Edge Length = Lo
strains in E 45 lecture, and in statics, mechanics
of materials, structures, and machine design
courses.
Usually the forces acting on a body are normalized by dividing the force by the area of the
surface it's acting on to get a "force intensity". The force intensity is called stress. Normal stresses are
usually symbolized by the Greek letter sigma (σ) and shear stresses by the Greek letter tau (τ).
normal stress = σ
where:
shear stress = τ
F = applied force
Ao = area over which the force is applied
In the SI system, force has units of Newtons (N), so stress has units of Newtons/meter2, (N/m2).
A stress or pressure of 1 N/m2 is called 1 Pascal (Pa). Pascals are very small so it is quite common to find
data listed with units of MPa (MegaPascal = 106 x Pa) .In the English (British Engineering) system, force
has units of pounds-force (1bf) so stress has units of lbf/in2, usually abbreviated psi. (Remember, a poundforce is the force exerted by the acceleration of gravity, 32.2 ft/s2, on a mass of 1 pound.) Often lbf/in2 x
103 is abbreviated ksi. Robert Hooke noted that doubling the load on a spring will double its extension.
Hooke's Law is:
F=kX
where:
F = force (pounds-force, lbf, or Newtons, N)
X = extension (inches, in., or meters, m)
k = spring constant (lbf /in. or N/m)
Hooke’s Law describes linear elastic behavior. The extension (elongation) is directly proportional to the
applied force.
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F
Many engineering materials are linearly elastic (most metals and alloys, ceramic
materials, rigid plastics, but not soft plastics or rubbery materials). That means that the amount of
strain exhibited by the material is proportional to the amount of stress applied since strain is
related to extension and stress is related to applied force. The behavior of linearly elastic
materials follows Hooke’s Law. Each type of material has a characteristic proportionality
constant called the elastic modulus. For elastic deformation:
normal stress = elastic modulus . strain
σ=Ε.ε
Since strain is dimensionless, the elastic modulus must have the same units as stress, i.e.
N/m2 or psi. (The elastic modulus, E, is often called "Young's Modulus" and is also commonly
abbreviated as a "Y".) In an analogous way, the constant of proportionality between shear strain
and stress is G, the shear modulus, so:
τ=G.γ
Unlike most mechanical properties, the elastic constants (E and G) are independent of
details of composition, material processing, or heat treatment because they are related to the
interatomic bond strength. For example, virtually all aluminum alloys, regardless of how they
were processed have an elastic modulus of 6.9 x 1010 N/m2.
Plastic Deformation
After a material has reached it's elastic limit, or yielded, further straining will result in
permanent deformation. Plastic (or permanent) deformation will be examined further in the next
laboratory unit.
Stress-Strain Curves
Many tests are used to study the mechanical behavior of materials, but none is more
important than tension (tensile) or compression tests used to obtain stress- strain curves. The
whole idea behind the tests is to obtain a series of strain values as the stress on a test specimen is
increased. Most of the properties aren’t dependent on the specimen size. Experience shows that
the elastic modulus is the same whether we use a tensile test or a compression test to find it.
Since
strain = ε
the strain will be positive (greater than zero) for tensile testing and negative (less than zero) for
compression testing. And since
stress = σ = Εε
consistency requires that a tensile stress be considered positive and a compressive stress be
considered negative.
Stress-strain curves are created by measuring the length of change of an appropriate test
specimen as the load on the specimen is increased and plotting those data. The elongation values,
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stress
DL, divided by the original length, Lo, are the strains and the loads, F, divided by the original
cross-sectional area, Ao, are the stresses. The stress-strain curve is found by plotting the stress
values on the vertical axis (ordinate, usually the y-axis) and the strain values on the horizontal
axis (abscissa, usually the x axis).
Stress-strain curves for metals have the
general shape shown in Fig. 3. The linear portion of
the stress- strain curve indicates elastic, or
recoverable deformation. The non-linear portion
indicates plastic deformation. Plastic strain is defined
as permanent, non-recoverable deformation. In this
unit we will only investigate elastic behavior.
Recall that the elastic modulus is defined by
the relation:
σ = Εε
so by rearranging this equation we can write:
strain
Figure 3
This equation is valid only for the elastic portion of the curve, that is the linear region of
the stress-strain curve. We can easily calculate the elastic modulus of a material by calculating
the slope of the straight-line portion of the stress-strain curve.
Elastic Straining in Directions Normal to the Applied Stress
We are familiar with the contraction or thinning of a rubber band when it's stretched
elastically, or with the drawing down of bread dough or taffy when it's pulled, so there can be a
significant amount of deformation in a specimen perpendicular to the direction of loading.
Isotropic is a term meaning that the material's properties are the same in every direction. The
materials we are discussing are assumed to be isotropic; the elastic properties are independent of
direction. This is a reasonable assumption because most engineering materials are made up of
more or less randomly oriented grains so the isotropic case is the usual one encountered in
engineering practice. (The anisotropic case is much more complicated.)
When a piece of material is elastically loaded as shown in Fig. 4, the elastic strain parallel to the
direction of loading (the axial strain) is given by:
= εxx
elastic strain =
Here epsilon has been given the subscript “xx.” In the double subscript system for strains, the
first subscript indicates the direction in which the strain is occurring and the second subscript
indicates the direction of the stress causing the strain. Hence, exx means strain in the x direction
caused by a stress in the x direction.
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F
Y
Z
X
Lo
L
The definition of Poisson's Ratio is the
key concept when working with strains
in directions perpendicular to the applied
force (often called the transverse, lateral,
or diametral direction). Poisson's Ratio
relates the strain in the direction
perpendicular to the applied force (the
transverse strain) to the axial strain.
Poisson’s Ration = v (or μ)
Do
So in Fig. 4
Wo
D
W
Figure 4
For isotropic materials V will always be
positive since
loading type
F
________________sign__________________
εaxial
εtransverse
v
tensile
+
_
+
compressive
_
+
+
Also, for isotropic materials we will get the same result for v no matter in which direction
εtransverse is measured as long as it is measured perpendicular to εaxial.
Volume Change Associated with Elastic Straining
Consider the loaded dimensions of a round bar. If x represents the axial direction and d
the diametral direction:
therefore: loaded length = Lo + ΔL = Lo + Lo . εxx
= Lo(1 + εxx)
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And:
therefore: loaded diameter = Do + ΔD = Do . εdx
The original volume of the bar is, of course:
And the final volume of the bar will be:
Recalling that the original volume was:
We can write the fractional volume change as the result of elastic straining as:
so:
This expression can be greatly simplified by the following argument: with the exception
of rubber-like materials, the maximum elastic strain which an engineering material is likely to
encounter is about 0.01 because the yield strengths seldom exceed 1% of the elastic moduli of
the materials. Thus the terms in the above expression which involve (σx/E)3 or (σx /E)2 are only
10-4 to 10-2 as large as the terms involving (σx /E). So we can reasonably neglect the squared and
cubed terms, recognizing that this limits our accuracy to about 2 significant figures. The result is:
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The above result is also valid in general for isotropic materials since we assume there is
no directional dependence in the transverse strain as long as is it measured perpendicular to the
direction of the applied load.
Biaxial Stresses
Z
1
Y
X
σy
σx
Figure 5
Consider the plate shown in Fig. 5 which has tensile stresses sx and sy in the x and y
directions respectively. Stress σx will tend to elongate the plate in the x direction and cause it to
contract in the y and z directions, while σy will tend to elongate it in the y direction and contract
it in the x and z directions. How can we analyze the combined effects of these stresses? Two
principles help with this problem: they will be stated here and you will have an opportunity to
apply them to a simple system in this exercise. (Hint: LEARN these principles. They are very
important and useful to engineering design. You will study in more detail in advanced courses in
mechanics of materials and design.)
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1. Boltzmann's Superposition Principle: Elastic stresses act independently so the strains in each
direction caused by each stress can be calculated independently. The total strain in any
direction can then be found by merely summing all the strains in that direction caused by all
the stresses.
2. The concept of principal axes and principal stresses: Regardless of the complexity of the
pattern of normal and shear forces acting on an object, at each point in the object there is an
orientation for an infinitesimal volume of material for which only compressive and tensile
stresses are acting on the surfaces of the volume. That is, at each point in the material an
orientation exists for an infinitesimal volume for which no shear stresses, but only normal
stresses, act on the volume. The axes normal to the infinitesimal volume which is oriented so
that no shear stresses are acting on it are the principal axes: the normal stresses acting on the
volume along the principal axes are the principal stresses.
The lack of shear stresses and the orthogonal orientation of the normal stresses acting on the
body in Fig. 5 mean that the principal axes in this case will coincide with the x, y, and z axes.
Let us now consider a unit volume located at a point such as “1” in Fig. 5, redrawn in Fig. 6:
Z
σy
σy
Y
σx
σx
X
Figure 6
εx = εxx + εxy
where:
and
so:
εy = εyx + εyy
where:
and
where:
and
so:
εz = εzx + εzy
so:
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Effective Stresses
Yielding is defined as permanent deformation. The yield strength is the stress at onset of
permanent deformation as measured in a tensile test. The yield strength determined in a tensile
test will be directly useful to predict the load at yielding in an actual part only if that part is
loaded in simple uniaxial tension or compression. If the part is loaded in complex ways by a
number of normal and shear forces, as is the usual case, we can use the uniaxial tensile yield
strength to predict yielding if we can determine the principal stresses at the most highly stressed
location within the part. If σ1, σ2, and σ3 are the principal stresses, the following formula can be
used to combine them to calculate a single effective stress, σ. Once we know s, we can compare
it with σys, the yield strength from a tension test. If sys is greater than s yielding will not occur,
but if sys is less than s yielding will occur.
Principle of Strain Gage Operation
The resistance of a wire of cross section A and length L made of a metal of resistivity r will be:
Direction of strain
If the wire is of rectangular cross section (width w and thickness t) then A = wt; if the wire is of
circular cross section with radius ro, then A = π(ro)2. The resistance will increase if the wire is
elastically stretched because L will increase and A will decrease. In addition to the resistance
increase caused by the dimensional change there will be some change in the resistivity of the
metal since the average space between the atoms will be slightly changed when it is elastically
stretched.
It is possible, therefore, to fasten a piece of thin wire or foil, to
the surface of a part that is to be stressed, measure the change in
resistance of the wire or foil as the part is stressed, and obtain an
indication of the magnitude of the strain in the metal in the region
directly under the wire. This is the principle of the strain gage. In
order to make sure we are measuring the strain in as localized a
manner as possible commercial strain gages are usually long wires
that are folded back and forth (as in Fig. 7) and mounted in a small
Figure 7
piece of polymer for easy attachment to the specimen.
It is necessary to assure that the strain gage is bonded to the part so that it will strain with
the part and it is also necessary to electrically insulate the strain gage from the part. Neither of
these problems present much difficulty as long as the temperature is not above a few hundred
degrees Fahrenheit. We need to have a calibration factor for the gage, the gage factor, to convert
the electrical output to
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strain. The gage factor is defined as the fractional change in resistance per unit strain:
where:
R(0) = resistance of the gage at zero strain
R(ε) = resistance of the gage at
Notice the gage factor is dimensionless (the units are ohm/ohm per inch/inch). For metals with a
Poisson's ratio of 0.3. the gage factor would be 1.6 if the only factor contributing to the change in
resistance of the gage was the dimensional change of the wire. Actual gage factors of metals
commonly used in strain gages range from 2.0 to 3.5. with the most strain gages having gage
factors of roughly 2.
Measurement of Resistance Changes and Strain
The simplest way of making a highly accurate measurement of the resistance change in the strain
gage as it is elongated is to use a Wheatstone bridge circuit (Fig. 8).
A
At Balance:
I4
I1
IG = 0
R4
IG
R1
I4 = I3
I3 = I2
B
G
and
R2
R3
I2
I3
C
V
Figure 8
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The Wheatstone bridge is composed of a voltage source, four resistances arranged in two
parallel paths, each path having two resistances, and a galvanometer connected between the two
nodes which are not connected to the voltage source. In Fig. 8 the voltage source is connected to
nodes “B” and “D” while the galvanometer is connected to nodes “A” and “C.” The
galvanometer is just an extremely sensitive ammeter; when the galvanometer shows no
deflection then IG, the current through the galvanometer is zero and the voltages at points A and
C (relative to any reference, such as ground) must be the same.
Rl is a strain gage, with R2, R3, and R4 being internal resistances within the strain indicator.
Furthermore, R3 and R4 are variable resistors. With the specimen to which the gage is bonded in
an unstressed state, the bridge can be "balanced" by adjusting R3 so that no current flows through
the galvanometer. When the bridge is balanced we must necessarily have the conditions:
VBA = VBC (voltage drop from B to A equals that from B to C)
VAD = VCD (voltage drop from A to D equals that from C to D)
Now we can use Ohm’s Law which says that the voltage drop V across a resistance R is
equal to the product of the resistance and the current I flowing through the resistance. We can
also note from Fig. 8 that as long as the bridge is balanced IG = 0; current I1 will flow through R1
and R4 while current I2 will flow through R2 and R3. So we can use the voltage relationships
above and Ohm Is Law to write:
I1Rl = I2R2
I1R4 = I2R3
By taking the ratio of the two equations above we can write the condition for balance in the
Wheatstone bridge as:
or
If the part to which the strain gage Rl is bonded is strained, R1 will experience the same
strain and the resistance of Rl will increase throwing the bridge out of balance. When the bridge
is out of balance the result is current flow through the galvanometer. From above, the bridge can
be rebalanced by increasing R4. In practice, the strain gage Rl is connected to a Wheatstone
bridge which is in an instrument called a strain indicator. The strain indicator is calibrated to read
strain directly in micro-strain (strain x 10-6) rather than in resistance units (ohms).
A separate adjustment on the strain indicator compensates for the gage factor and must be set to
correspond to the gage factor of the particular strain gage being used. Some strain indicators
indicate zero strain in the initial unstressed condition; successive readings with these instruments
give strain readings directly. Other strain indicators give a non-zero initial reading which must be
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subtracted from each of the subsequent readings to get the actual strain values relative to the
initial unstrained condition.
Temperature and Transverse Strain Effects
If we cement a strain gage to a test piece at room temperature, then warm the test piece
(but without stressing it) would you expect the strain gage to indicate a strain in the material?
You probably would answer something like, 'Well, let's see...if the gage metal has a lower
coefficient of thermal expansion than the test piece, then the test piece would expand more than
the gage. So the test piece would be pulling on the gage, and the gage would indicate a tensile
strain in the test piece even though no stress is applied." Handbooks on experimental stress
analysis give lots of details about various ways of compensating for temperature. It's sufficient to
note here that the strain gage manufacturers have used some tricky metallurgy to produce their
gages with a whole range of thermal expansion coefficients. If we need a gage to put on an
aluminum part, we buy the gages with a thermal expansion coefficient to match aluminum's;
likewise, if we are measuring strains in a forged steel connecting rod, we buy a gage with the
same thermal expansion coefficient as the steel.
In this laboratory you are exploring elastic strains perpendicular to the applied stress.
What about the same kind of strains in strain gages? That is, how sensitive are strain gages to
the stresses perpendicular to the direction of the gage windings? Usually, the "transverse
sensitivity" of gages can be neglected, although in exacting cases a correction must be made.
Young's Modulus and Poisson's Ratio
The elastic behavior of engineering materials can normally be characterized by the following
four engineering elastic constants:
units
Elastic (Young’s) Modulus,
psi (N / m2)
Poisson’s Ratio,
dimensionless
Bulk Modulus,
psi (N / m2)
Shear Modulus,
psi (N / m2)
In this experiment you will measure the elastic modulus and Poisson’s ratio of a common
engineering material, quenched and tempered 4140 steel. This steel is commonly used for
hardened steel parts such as heavy duty shafts, heavy crankshafts, steering knuckles, etc. The
heat treatment which was given to the specimen was: heated to between 1500°F and 1550°F for
1 hour, quenched in oil with stirring, then reheated (tempered or drawn) to 530°F for 1 hour.
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Measurement of the elastic modulus and Poisson's ratio allows calculation of the other
two engineering elastic constants by one of these formulas:
= Shear Modulus (or Modulus of Rigidity)
= Bulk Modulus
You won't usually find the above values tabulated; handbook authors frequently tabulate only E
and v values because they assume you know about the above relationships for G and B.
The elastic constants of engineering materials are of critical importance in design. In
general, the elastic constants of crystalline materials (metals, ceramics, some polymers) are not
affected significantly by thermal or mechanical treatments. The elastic modulus and Poisson’s
ratio are the same regardless of whether the axial stress is tensile or compressive. Elastic
behavior is discussed in more detail in your text.
Bibliography
1. American Society for Testing and Materials, ASTM Standard E 111, “Determination of
Young's Modulus at Room Temperature.”
2. ASTM Standard E132, “Determination of Poisson's Ratio at Room Temperature.”
3. ASTM Standard E251, “Performance Characteristics of Bonded Resistance Strain Gages.”
4. H.K.P. Neubert, Strain Gages, Kinds and Uses. St. Martin's Press, New York, 1967.
5. C.C. Perry and H.R. Lissner, The Strain Gage Primer. 2nd ed., McGraw-Hill, New York,
1962.
6. James W. Daly and William F. Riley, Experimental Stress Analysis, 2nd ed., McGraw-Hill,
New York, 1978.
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Laboratory Unit: Elastic Strain and Poisson's Ratio
Introduction
In this unit you will learn to use electrical strain gages to monitor the transverse and longitudinal
strains of a steel specimen under tensile loads and from these data determine the Poisson’s ratio
of the steel.
Tensile Test
Materials and Equipment
1. Tensile test specimen with two strain gages attached. One strain gage is aligned parallel to
the specimen's loading axis and the other is perpendicular to the loading axis.
2. Strain indicator (Wheatstone bridge).
3. Switch and balance unit.
4. Micrometer and scale.
Testing Procedure
1. Check the dimensions of the test specimen using a micrometer and scale and calculate the
cross-sectional area.
2. The specimen will be installed in the universal testing machine and should be connected to a
switch and balance unit and strain indicator assembly. Follow operating instructions on the
inside cover to the strain indicator, except for the balancing instruction. Instead of balancing
the bridge in the unstressed state using the balance knob on the strain indicator, switch the
channel selector on the switch and balance unit to one of the wired channels and balance the
bridge using the balance knob for that specific channel. Switch the selector to the second
wired channel and balance the bridge with the balance knob for that channel.
3. Have your instructor check your set-up before proceeding.
4. Load the sample in a series of steps to the maximum allowable load (you determine this –
make sure you select a load that is below the yield strength of the material). Select at least 25
evenly spaced load intervals so that the intervals will be "nice round numbers" that are easy
to identify on the UTM load dial. In this experiment the loading must actually be stopped at
each desired load increment to allow both the transverse and longitudinal strain to be read.
Record the load, transverse strain and longitudinal strain at each step.
5. After reaching the maximum load, record a series of data at about 5 load values while
unloading. Again, choose the load values so that they will be easily identifiable on the UTM
dial. Be sure to record the final strain readings at zero load.
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Cantilever Beam
Materials and Equipment
1. Cantilever beam specimen with strain gages attached, top and bottom.
2. Strain recording device.
3. Micrometer and scale.
Testing Procedure
1. Check the dimensions of the specimen using an appropriate measuring device. Determine the
distance to the applied load(s), and the locations of the strain gages.
2. Attach the strain gages to the strain recording device. Make sure the device is zeroed
properly so that each strain gage reads zero when there is no load applied.
3. Apply the load(s) to the beam and record the strains at the locations indicated.
4. Remove the load(s) from the beam and record the strain values.
The stress at any point on the cantilever beam can be calculated by using the following
formula (you will derive and study this in detail in Mechanics of Materials):
c
c
where: M = moment of the force = F . d (N m, lbf in. or lbf ft)
(F = force, d = distance)
Figure 9
C = distance from the neutral axis at the centroid
of the beam (Fig. 9) (m, in., or ft)
h
I = moment of inertia =
(m4, in4, or ft4)
b
Figure 10
(b and h determined as in Fig. 10)
At any point along the beam there will be a linear stress distribution from the top of the
beam to the bottom. The stress will be zero at the centroid, maximum in tension at the top, and
maximum in compression at the bottom. This is a normal stress, and since this deformation is
elastic the strains can be estimated by relating these stresses to strain with the elastic modulus,
σ = εE.
The moment of force, M, varies linearly from the point of the applied load to the support.
In the case of one load applied at the end of the beam, M will be zero at the loaded end of the
beam and maximum at the support, where M = F . L, the applied force times the beam length.
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The maximum deflection in a cantilever beam (at the end of the beam) with one load
applied at the end of the beam (Fig. 11) can be determined by evaluating the following formula
(from Mechanics of Materials):
P
L
where:
P = load (N or lbf)
y
L = length (m, in., or ft)
Figure 11
2
E = modulus of elasticity (N/m , or psi)
I = moment of inertia (m4, in4, or ft4)
If the loading is more complicated, i.e. there is more than one applied load, the law of
superposition can be used to determine the maximum deflection; the resultant deflection from a
combination of loads is the sum of the deflection from each individual load. If there is a load
applied at the end of the beam and a load applied somewhere between the support and the end,
the maximum deflection is the sum of the deflections from each individual load. The formula
above is used to determine the maximum deflection when the load is applied at the end of the
beam, and the following formula can be used to determine the maximum deflection if the load is
applied between the support and the end of the beam (Fig. 12).
where:
P = load (N or lbf)
x
x = distance from support to
applied load (m, in., or ft)
L-x
y
L = length of the beam (m, in., or ft)
Figure 12
2
E = modulus of elasticity (N/m , or psi)
I = moment of inertia (m4, in4, or ft4)
By applying superposition, the maximum deflection at the end of a cantilever beam of length, L,
with two loads applied, P1 applied at the end of the beam, and P2 applied at a distance b from the
support (between the support and the end of the beam) (Fig. 13) is:
P2
P1
x
L-x
where E is the modulus of elasticity and I is the moment of inertia.
ytotal
Figure 13
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