Stress and Strain

Abstract

When a body is subjected to external forces a system of internal forces is developed. It is important in engineering mechanics to determine the intensity of these forces on the various cross section portions of the body, as the resistances to applied forces depend on these intensities. This intensity is called stress and it is a measure of the resisting forces. Stress is determined by dividing the total applied load (force) F, by the total area of loaded cross section A. This is expressed as

$$ \sigma =F/A $$

Appendices

Chapter Summary

Engineers must first identify the external forces acting on a structural member, and then calculate the internal resistance, called stress, developed by the member. Based on this, and other related properties such as strain and the modulus of elasticity, the designer is able to select the material and design the size and shape of the member that will properly resist the applied forces.

The materials used vary in quality and physical and mechanical characteristics. Hence, it is important that a designer have good knowledge of the properties of the materials being used. In this chapter, we introduced the concepts of stress, strain, and deformation, and the relationship between stress and strain in different materials.

Stress is defined as the force per unit area for a specific material under loading. The unit of stress in the English system is psi, and in the metric system is Pa, or MPa. There are three normal stresses: tensile stress, compressive stress, and shear stress. These may be calculated using the stress formula:

$$ \sigma =F/A $$

Strain is defined as the total deformation, or stretching, per unit length of the member. The formula to calculate the strain is

$$ \varepsilon =\delta /L $$

Stress is directly proportional to the strain:

$$ \sigma =E\cdot \varepsilon $$

This relation is known as Hooke’s law and the proportionality constant is called the modulus of elasticity (E) of the material. The modulus of elasticity is a measure of a material’s resistance to deformation. The above linear equation is valid provided that the stress does not exceed the proportional limit of the material. The proportional limit is the maximum stress for which stress is proportional to strain.

Review Questions

• 1. What is stress?

• 2. What is strain?

• 3. What are the units of stress and strain?

• 4. What is a tensile load?

• 5. What is a compressive load?

• 6. What is shear?

• 7. What is the relation between stress and strain?

• 8. What is Hooke’s diagram, and where is it valid?

• 9. What is the modulus of elasticity, and what is meant by it in engineering problems?

• 10. What is the allowable stress?

• 11. What is the ultimate stress?

• 12. What is the relation between allowable stress and ultimate stress?

• 13. What is the proportional limit?

Problems

• 1. A tie rod 10.0 ft long and 2.2 in. in diameter is subjected to a tensile force of 3,000 lb. Find the stress, strain, and the total deformation. Use E = 2.8 × 10⁷ psi.

• 2. What is the tensile strain in a specimen of material if it is subjected to a tensile force of 2 × 10⁸ N/m²? Use E = 1.1 × 10¹¹ Pa.

• 3. A telephone wire 100 m long and 2.5 mm in diameter is subjected to a tensile force of 400 N. Find the stress, strain, and the modulus of elasticity. Assume that the final length is 100.30 m after stretching.

• 4. A 200–mm long metal alloy tube of 40 mm outer diameter and 30 mm inner diameter is subjected to a compressive load of 30, 000 N. Find the stress, strain, and final length. Use E = 9 × 10¹⁰ Pa.

• 5. Find the magnitude of the tensile force acting on a steel bar 1.50 in. in diameter if the strain in the bar is 0.0015. Use E = 3 × 10⁷ psi.

• 6. A rectangular steel bar ¾ in. × 7/8 in. and 20 in. long is subjected to a tensile load of 3,000 lb. Find the stress, strain, and the total deformation. Use E = 3 × 10⁷ psi.

• 7. A steel cable 50 ft long and 0.5 in. in diameter is subjected to a tensile force and stretches 0.37 in. Find the stress. Use E = 3 × 10⁷ psi.

• 8. A copper wire 5 m long and 2 mm in. diameter is subjected to a tensile load of 600 N, and stretches 12.5 mm. Calculate the modulus of elasticity of the wire.

• 9. A steel circular cross section of a column is subjected to a tensile load of 500,000 N.

  Find the size of the cross section of the column if the allowable stress is 1 × 10⁸ Pa.

• 10. A steel circular cross section with a diameter of 2 in. is subjected to an axial tensile force. Find the force if the resulting tensile strain is 0.0005.

Stephen P. Timoshenko (1878–1972)

Stepan Prokopovych Timoshenko was born on December 22, 1878 in the village of Shpotivka in the Ukrain. Timoshenko’s early life seems to have been a happy one in pleasant rural surroundings. He studied at a “realnaya” school from 1889 to 1896. Timoshenko continued his education towards a university degree at the St. Petersburg Institute of Engineering. After graduating in 1901, he stayed on teaching in this same institution from 1901 to 1903 and then worked at the St. Petersburg Polytechnic Institute under Viktor Kyrpychov 1903–1906. His restlessness and discontent with the educational system extant in Russia at that time motivated the young Timoshenko to venture out to explore, examine, and assimilate diverse pedagogical views and cultures in France, Germany, and England. In 1905 he was sent for 1 year to the University of Göttingen where he worked under Ludwig Prandtl.

In the fall of 1906 he was appointed to the Chair of Strengths of Materials at the Kyiv Polytechnic Institute. Thanks to his tormented spirit at this institute, Timoshenko took the plunge to writing his maiden Russian classic, Strength of Materials in 1908 (Part I) and 1910 (Part II). From 1907 to 1911 as a professor at the Polytechnic Institute he did research in the area of finite element methods of elastic calculations, and did excellent research work on buckling. He was elected dean of the Division of Structural Engineering in 1909.

In 1911 he was awarded the D. I. Zhuravski prize of St. Petersburg; he went there to work as a Professor in the Electro-technical Institute and the St. Petersburg Institute of the Railways (1911–1917). During that time he developed the theory of elasticity and the theory of beam deflection, and continued to study buckling.

In 1922 Timoshenko moved to the United States where he worked for the Westinghouse Electric Corporation from 1923 to 1927, after which he became a faculty professor at the University of Michigan where he created the first bachelor’s and doctoral programs in engineering mechanics. His textbooks have been published in 36 languages. His first textbooks and papers were written in Russian; later in his life, he published mostly in English.

The following 3 years 1935–1937, Timoshenko teamed up with Gere and Young for three more books: a condensed guide of strength of materials, elastic stability, and engineering mechanics. These unique texts explore intricate mathematical techniques to explain some subtle aspects underlying elasticity and stability to give new insight into the behavior of solids and structures for engineering design. From 1936 onward he was a professor at Stanford University.

The year 1953 saw the great Timoshenko epic, The History of Strength of Materials with a brief account of the history of the theory of elasticity and structural mechanics. Tracing the history all the way back to Archimedes, he carries the reader through the period of Leonardo da Vinci, Galileo, Hooke, Newton, Mariotte, Bernoulli, Euler, Lagrange, and Coloumb, reaching the end of the eighteenth century.

This missionary zeal of Timoshenko for writing books for improving teaching and for guiding practical engineers has played a key role in uplifting technical education worldwide, but more emphatically in the United States. In 1960 he moved to Wuppertal (Western Germany) to be with his daughter. He died in 1972 and his ashes are buried in Alta Mesa Memorial Park, Palo Alto, California.