he field of strength of materials (also called mechanics of materials) typically refers to various
methods of calculating the stresses and strains in structural members, such as beams, columns,
and shafts. The methods employed to predict the response of a structure under loading and its
susceptibility to various failure modes takes into account the properties of the materials such as
its yield strength, ultimate strength, Young's modulus, and Poisson's ratio. In addition, the
mechanical element's macroscopic properties (geometric properties) such as its length, width,
thickness, boundary constraints and abrupt changes in geometry such as holes are considered.
The theory began with the consideration of the behavior of one and two dimensional members of
structures, whose states of stress can be approximated as two dimensional, and was then
generalized to three dimensions to develop a more complete theory of the elastic and plastic
behavior of materials. An important founding pioneer in mechanics of materials was Stephen
Timoshenko.
Definition[edit]
In the mechanics of materials, the strength of a material is its ability to withstand an applied load
without failure or plastic deformation. The field of strength of materials deals with forces and
deformations that result from their acting on a material. A load applied to a mechanical member
will induce internal forces within the member called stresses when those forces are expressed on
a unit basis. The stresses acting on the material cause deformation of the material in various
manners including breaking them completely. Deformation of the material is called strain when
those deformations too are placed on a unit basis.
The stresses and strains that develop within a mechanical member must be calculated in order to
assess the load capacity of that member. This requires a complete description of the geometry of
the member, its constraints, the loads applied to the member and the properties of the material of
which the member is composed. The applied loads may be axial (tensile or compressive), or
rotational (strength shear). With a complete description of the loading and the geometry of the
member, the state of stress and state of strain at any point within the member can be calculated.
Once the state of stress and strain within the member is known, the strength (load carrying
capacity) of that member, its deformations (stiffness qualities), and its stability (ability to maintain
its original configuration) can be calculated.
The calculated stresses may then be compared to some measure of the strength of the member
such as its material yield or ultimate strength. The calculated deflection of the member may be
compared to deflection criteria that are based on the member's use. The calculated buckling load
of the member may be compared to the applied load. The calculated stiffness and mass
distribution of the member may be used to calculate the member's dynamic response and then
compared to the acoustic environment in which it will be used.
Material strength refers to the point on the engineering stress–strain curve (yield stress) beyond
which the material experiences deformations that will not be completely reversed upon removal of
the loading and as a result, the member will have a permanent deflection. The ultimate strength
of the material refers to the maximum value of stress reached. The fracture strength is the stress
value at fracture (the last stress value recorded).
Types of loadings[edit]
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Transverse loadings – Forces applied perpendicular to the longitudinal axis of a member.
Transverse loading causes the member to bend and deflect from its original position, with
internal tensile and compressive strains accompanying the change in curvature of the
member.[1] Transverse loading also induces shear forces that cause shear deformation of the
material and increase the transverse deflection of the member.
Axial loading – The applied forces are collinear with the longitudinal axis of the member. The
forces cause the member to either stretch or shorten.[2]

Torsional loading – Twisting action caused by a pair of externally applied equal and
oppositely directed force couples acting on parallel planes or by a single external couple
applied to a member that has one end fixed against rotation.
Stress terms[edit]
A material being loaded in a) compression, b) tension, c) shear.
Uniaxial stress is expressed by
where F is the force [N] acting on an area A [m2].[3] The area can be the undeformed area or
the deformed area, depending on whether engineering stress or true stress is of interest.
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Compressive stress (or compression) is the stress state caused by an applied load that
acts to reduce the length of the material (compression member) along the axis of the
applied load, it is, in other words, a stress state that causes a squeezing of the material.
A simple case of compression is the uniaxial compression induced by the action of
opposite, pushing forces. Compressive strength for materials is generally higher than
their tensile strength. However, structures loaded in compression are subject to
additional failure modes, such as buckling, that are dependent on the member's
geometry.
Tensile stress is the stress state caused by an applied load that tends to elongate the
material along the axis of the applied load, in other words, the stress caused
by pulling the material. The strength of structures of equal cross-sectional area loaded in
tension is independent of shape of the cross-section. Materials loaded in tension are
susceptible to stress concentrations such as material defects or abrupt changes in
geometry. However, materials exhibiting ductile behaviour (most metals for example) can
tolerate some defects while brittle materials (such as ceramics) can fail well below their
ultimate material strength.
Shear stress is the stress state caused by the combined energy of a pair of opposing
forces acting along parallel lines of action through the material, in other words, the stress
caused by faces of the material sliding relative to one another. An example is cutting
paper with scissors[4] or stresses due to torsional loading.
Stress parameters for resistance[edit]
Material resistance can be expressed in several mechanical stress parameters. The
term material strength is used when referring to mechanical stress parameters. These
are physical quantities with dimension homogeneous to pressure and force per unit surface.
The traditional measure unit for strength are therefore MPa in the International System of
Units, and the psi between the United States customary units. Strength parameters include:
yield strength, tensile strength, fatigue strength, crack resistance, and other parameters.[citation
needed]

Yield strength is the lowest stress that produces a permanent deformation in a material. In
some materials, like aluminium alloys, the point of yielding is difficult to identify, thus it is
usually defined as the stress required to cause 0.2% plastic strain. This is called a 0.2%
proof stress.[5]
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Compressive strength is a limit state of compressive stress that leads to failure in a
material in the manner of ductile failure (infinite theoretical yield) or brittle failure (rupture
as the result of crack propagation, or sliding along a weak plane – see shear strength).
Tensile strength or ultimate tensile strength is a limit state of tensile stress that leads to
tensile failure in the manner of ductile failure (yield as the first stage of that failure, some
hardening in the second stage and breakage after a possible "neck" formation) or brittle
failure (sudden breaking in two or more pieces at a low-stress state). The tensile strength
can be quoted as either true stress or engineering stress, but engineering stress is the
most commonly used.
Fatigue strength is a more complex measure of the strength of a material that considers
several loading episodes in the service period of an object,[6] and is usually more difficult
to assess than the static strength measures. Fatigue strength is quoted here as a
simple range (
). In the case of cyclic loading it can be appropriately expressed as
an amplitude usually at zero mean stress, along with the number of cycles to failure
under that condition of stress.

Impact strength is the capability of the material to withstand a suddenly applied load and
is expressed in terms of energy. Often measured with the Izod impact strength
test or Charpy impact test, both of which measure the impact energy required to fracture
a sample. Volume, modulus of elasticity, distribution of forces, and yield strength affect
the impact strength of a material. In order for a material or object to have a high impact
strength, the stresses must be distributed evenly throughout the object. It also must have
a large volume with a low modulus of elasticity and a high material yield strength.[7]
Strain parameters for resistance[edit]
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Deformation of the material is the change in geometry created when stress is applied ( as
a result of applied forces, gravitational fields, accelerations, thermal expansion, etc.).
Deformation is expressed by the displacement field of the material.[8]
Strain or reduced deformation is a mathematical term that expresses the trend of the
deformation change among the material field. Strain is the deformation per unit
length.[9] In the case of uniaxial loading the displacement of a specimen (for example a
bar element) lead to a calculation of strain expressed as the quotient of the displacement
and the original length of the specimen. For 3D displacement fields it is expressed as
derivatives of displacement functions in terms of a second order tensor (with 6
independent elements).
Deflection is a term to describe the magnitude to which a structural element is displaced
when subject to an applied load.[10]
Stress–strain relations[edit]
Main article: Stress–strain curve
Basic static response of a specimen under tension

Elasticity is the ability of a material to return to its previous shape after stress is released.
In many materials, the relation between applied stress is directly proportional to the
resulting strain (up to a certain limit), and a graph representing those two quantities is a
straight line.
The slope of this line is known as Young's modulus, or the "modulus of elasticity." The
modulus of elasticity can be used to determine the stress–strain relationship in the linearelastic portion of the stress–strain curve. The linear-elastic region is either below the yield
point, or if a yield point is not easily identified on the stress–strain plot it is defined to be
between 0 and 0.2% strain, and is defined as the region of strain in which no yielding
(permanent deformation) occurs.[11]

Plasticity or plastic deformation is the opposite of elastic deformation and is defined as
unrecoverable strain. Plastic deformation is retained after the release of the applied
stress. Most materials in the linear-elastic category are usually capable of plastic
deformation. Brittle materials, like ceramics, do not experience any plastic deformation
and will fracture under relatively low strain, while ductile materials such as metallics, lead,
or polymers will plastically deform much more before a fracture initiation.
Consider the difference between a carrot and chewed bubble gum. The carrot will stretch
very little before breaking. The chewed bubble gum, on the other hand, will plastically deform
enormously before finally breaking.
Design terms[edit]
Ultimate strength is an attribute related to a material, rather than just a specific specimen
made of the material, and as such it is quoted as the force per unit of cross section area
(N/m2). The ultimate strength is the maximum stress that a material can withstand before it
breaks or weakens.[12] For example, the ultimate tensile strength (UTS) of AISI 1018 Steel is
440 MPa. In Imperial units, the unit of stress is given as lbf/in² or pounds-force per square
inch. This unit is often abbreviated as psi. One thousand psi is abbreviated ksi.
A factor of safety is a design criteria that an engineered component or structure must
achieve.
, where FS: the factor of safety, R: The applied stress, and UTS: ultimate
stress (psi or N/m2)[13]
Margin of Safety is also sometimes used to as design criteria. It is defined MS = Failure
Load/(Factor of Safety × Predicted Load) − 1.
For example, to achieve a factor of safety of 4, the allowable stress in an AISI 1018 steel
component can be calculated to be
= 440/4 = 110 MPa, or
= 110×106 N/m2. Such
allowable stresses are also known as "design stresses" or "working stresses."
Design stresses that have been determined from the ultimate or yield point values of the
materials give safe and reliable results only for the case of static loading. Many machine
parts fail when subjected to a non-steady and continuously varying loads even though the
developed stresses are below the yield point. Such failures are called fatigue failure. The
failure is by a fracture that appears to be brittle with little or no visible evidence of yielding.
However, when the stress is kept below "fatigue stress" or "endurance limit stress", the part
will endure indefinitely. A purely reversing or cyclic stress is one that alternates between
equal positive and negative peak stresses during each cycle of operation. In a purely cyclic
stress, the average stress is zero. When a part is subjected to a cyclic stress, also known as
stress range (Sr), it has been observed that the failure of the part occurs after a number of
stress reversals (N) even if the magnitude of the stress range is below the material's yield
strength. Generally, higher the range stress, the fewer the number of reversals needed for
failure.
Failure theories[edit]
Main article: Material failure theory
There are four failure theories: maximum shear stress theory, maximum normal stress theory,
maximum strain energy theory, and maximum distortion energy theory. Out of these four
theories of failure, the maximum normal stress theory is only applicable for brittle materials,
and the remaining three theories are applicable for ductile materials. Of the latter three, the
distortion energy theory provides most accurate results in a majority of the stress conditions.
The strain energy theory needs the value of Poisson's ratio of the part material, which is often
not readily available. The maximum shear stress theory is conservative. For simple
unidirectional normal stresses all theories are equivalent, which means all theories will give
the same result.

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


Maximum Shear Stress Theory – This theory postulates that failure will occur if the
magnitude of the maximum shear stress in the part exceeds the shear strength of the
material determined from uniaxial testing.
Maximum Normal Stress Theory – This theory postulates that failure will occur if the
maximum normal stress in the part exceeds the ultimate tensile stress of the material as
determined from uniaxial testing. This theory deals with brittle materials only. The
maximum tensile stress should be less than or equal to ultimate tensile stress divided by
factor of safety. The magnitude of the maximum compressive stress should be less than
ultimate compressive stress divided by factor of safety.
Maximum Strain Energy Theory – This theory postulates that failure will occur when the
strain energy per unit volume due to the applied stresses in a part equals the strain
energy per unit volume at the yield point in uniaxial testing.
Maximum Distortion Energy Theory – This theory is also known as shear energy theory
or von Mises-Hencky theory. This theory postulates that failure will occur when the
distortion energy per unit volume due to the applied stresses in a part equals the
distortion energy per unit volume at the yield point in uniaxial testing. The total elastic
energy due to strain can be divided into two parts: one part causes change in volume,
and the other part causes change in shape. Distortion energy is the amount of energy
that is needed to change the shape.
Fracture mechanics was established by Alan Arnold Griffith and George Rankine Irwin.
This important theory is also known as numeric conversion of toughness of material in
the case of crack existence.
A material's strength is dependent on its microstructure. The engineering processes to which
a material is subjected can alter this microstructure. The variety of strengthening
mechanisms that alter the strength of a material includes work hardening, solid solution
strengthening, precipitation hardening, and grain boundary strengthening and can be
quantitatively and qualitatively explained. Strengthening mechanisms are accompanied by
the caveat that some other mechanical properties of the material may degenerate in an
attempt to make the material stronger. For example, in grain boundary strengthening,
although yield strength is maximized with decreasing grain size, ultimately, very small grain
sizes make the material brittle. In general, the yield strength of a material is an adequate
indicator of the material's mechanical strength. Considered in tandem with the fact that the
yield strength is the parameter that predicts plastic deformation in the material, one can make
informed decisions on how to increase the strength of a material depending its
microstructural properties and the desired end effect. Strength is expressed in terms of the
limiting values of the compressive stress, tensile stress, and shear stresses that would cause
failure. The effects of dynamic loading are probably the most important practical
consideration of the strength of materials, especially the problem of fatigue. Repeated loading
often initiates brittle cracks, which grow until failure occurs. The cracks always start at stress
concentrations, especially changes in cross-section of the product, near holes and corners at
nominal stress levels far lower than those quoted for the strength of the material.
See also[edit]
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Creep (deformation) – Tendency of a solid material to move slowly or deform permanently
under mechanical stress
Deformation mechanism map – Microscopic processes responsible for changes in a
material's structure, shape and volume
Dynamics – Branch of physics studying forces and their effect on motion
Fatigue (material) – Initiation and propagation of cracks in a material due to cyclic loading
Forensic engineering – Investigation of failures associated with legal intervention
Fracture mechanics – Field of mechanics that studies the propagation of cracks in
materials
Fracture toughness – Stress intensity factor at which a crack's propagation increases
drastically
List of materials properties § Mechanical properties
Material selection – selection process of a material for a particular application
Molecular diffusion – Thermal motion of liquid or gas particles at temperatures above
absolute zero
Specific strength – Ratio of strength to mass for a material
Statics – Branch of mechanics concerned with balance of forces in nonmoving systems
Universal testing machine – Type of equipment for determining tensile or compressive
strength of a material
References[edit]
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 210. ISBN 978-007-352938-7.
^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 7. ISBN 978-007-352938-7.
^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 5. ISBN 978-007-352938-7.
^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. pp. 9–
10. ISBN 978-0-07-352938-7.
^ Beer, Ferdinand Pierre; Johnston, Elwood Russell; Dewolf, John T (2009). Mechanics of
Materials (5th ed.). p. 52. ISBN 978-0-07-352938-7.
^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 60. ISBN 978-007-352938-7.
^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. pp. 693–
696. ISBN 978-0-07-352938-7.
^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 47. ISBN 978-007-352938-7.
^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 49. ISBN 978-007-352938-7.
^ R. C. Hibbeler (2009). Structural Analysis (7 ed.). Pearson Prentice Hall. p. 305. ISBN 9780-13-602060-8.
^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. pp. 53–
56. ISBN 978-0-07-352938-7.
^ Beer & Johnston (2006). Mechanics of Materials (5thv ed.). McGraw Hill. pp. 27–
28. ISBN 978-0-07-352938-7.
^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 28. ISBN 978-007-352938-7.
Further reading[edit]
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Fa-Hwa Cheng, Initials. (1997). Strength of material. Ohio: McGraw-Hill
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Mechanics of Materials, E.J. Hearn
Alfirević, Ivo. Strength of Materials I. Tehnička knjiga, 1995. ISBN 953-172-010-X.
Alfirević, Ivo. Strength of Materials II. Tehnička knjiga, 1999. ISBN 953-6168-85-5.
Ashby, M.F. Materials Selection in Design. Pergamon, 1992.
Beer, F.P., E.R. Johnston, et al. Mechanics of Materials, 3rd edition. McGraw-Hill,
2001. ISBN 0-07-248673-2
Cottrell, A.H. Mechanical Properties of Matter. Wiley, New York, 1964.
Den Hartog, Jacob P. Strength of Materials. Dover Publications, Inc., 1961, ISBN 0-48660755-0.
Drucker, D.C. Introduction to Mechanics of Deformable Solids. McGraw-Hill, 1967.
Gordon, J.E. The New Science of Strong Materials. Princeton, 1984.
Groover, Mikell P. Fundamentals of Modern Manufacturing, 2nd edition. John Wiley &
Sons,Inc., 2002. ISBN 0-471-40051-3.
Hashemi, Javad and William F. Smith. Foundations of Materials Science and
Engineering, 4th edition. McGraw-Hill, 2006. ISBN 0-07-125690-3.
Hibbeler, R.C. Statics and Mechanics of Materials, SI Edition. Prentice-Hall,
2004. ISBN 0-13-129011-8.
Lebedev, Leonid P. and Michael J. Cloud. Approximating Perfection: A Mathematician's
Journey into the World of Mechanics. Princeton University Press, 2004. ISBN 0-69111726-8.
Chapter 10 – Strength of Elastomers, A.N. Gent, W.V. Mars, In: James E. Mark, Burak
Erman and Mike Roland, Editor(s), The Science and Technology of Rubber (Fourth
Edition), Academic Press, Boston, 2013, Pages 473–516, ISBN 9780123945846,
10.1016/B978-0-12-394584-6.00010-8
Mott, Robert L. Applied Strength of Materials, 4th edition. Prentice-Hall, 2002. ISBN 0-13088578-9.
Popov, Egor P. Engineering Mechanics of Solids. Prentice Hall, Englewood Cliffs, N. J.,
1990. ISBN 0-13-279258-3.
Ramamrutham, S. Strength of Materials.
Shames, I.H. and F.A. Cozzarelli. Elastic and inelastic stress analysis. Prentice-Hall,
1991. ISBN 1-56032-686-7.
Timoshenko S. Strength of Materials, 3rd edition. Krieger Publishing Company,
1976, ISBN 0-88275-420-3.
Timoshenko, S.P. and D.H. Young. Elements of Strength of Materials, 5th edition. (MKS
System)
Davidge, R.W., Mechanical Behavior of Ceramics, Cambridge Solid State Science Series,
(1979)
Lawn, B.R., Fracture of Brittle Solids, Cambridge Solid State Science Series, 2nd Edn.
(1993)
Green, D., An Introduction to the Mechanical Properties of Ceramics, Cambridge Solid
State Scienc