The 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.


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

Stress terms

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.

Stress parameters for resistance

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]

Strain parameters for resistance

Stress–strain relations

Main article: Stress–strain curve

Basic static response of a specimen under tension

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 linear-elastic 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]

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

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

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.

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


  1. ^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 210. ISBN 978-0-07-352938-7.
  2. ^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 7. ISBN 978-0-07-352938-7.
  3. ^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 5. ISBN 978-0-07-352938-7.
  4. ^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. pp. 9–10. ISBN 978-0-07-352938-7.
  5. ^ Beer, Ferdinand Pierre; Johnston, Elwood Russell; Dewolf, John T (2009). Mechanics of Materials (5th ed.). p. 52. ISBN 978-0-07-352938-7.
  6. ^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 60. ISBN 978-0-07-352938-7.
  7. ^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. pp. 693–696. ISBN 978-0-07-352938-7.
  8. ^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 47. ISBN 978-0-07-352938-7.
  9. ^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 49. ISBN 978-0-07-352938-7.
  10. ^ R. C. Hibbeler (2009). Structural Analysis (7 ed.). Pearson Prentice Hall. p. 305. ISBN 978-0-13-602060-8.
  11. ^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. pp. 53–56. ISBN 978-0-07-352938-7.
  12. ^ Beer & Johnston (2006). Mechanics of Materials (5thv ed.). McGraw Hill. pp. 27–28. ISBN 978-0-07-352938-7.
  13. ^ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. p. 28. ISBN 978-0-07-352938-7.

Further reading

  • Fa-Hwa Cheng, Initials. (1997). Strength of material. Ohio: McGraw-Hill
  • 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 0-486-60755-0.
  • 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 0-691-11726-8.
  • 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 0-13-088578-9.
  • 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 Science Series, Eds. Clarke, D.R., Suresh, S., Ward, I.M.Babu Tom.K (1998)