in the name of god Lecture by : LT “Sayed Dawod karimi” For NMAA and Kardan University Faculty of Engineering

Lesson #05 Strain 1. Lesson Objectives At the conclusion of this lesson, you should be able to do the following: Define the terms displacement, deformation, strain, modulus of elasticity, and Poisson’s Ratio. Apply Hooke's Law to calculate stress or strain in the elastic region. Apply Poisson's Ratio to calculate lateral strain and lateral deformation.

2. Orientation Why is this topic important? Deformation is an important measure of the performance of a structural element or an entire structure. Strain, a non-dimensional measure of the intensity of deformation, is used extensively in the characterization of materials. Because it is easily measured and can be mathematically related to stress, strain can be used to indirectly measure the stress in a member.

How does this topic relate to prior learning? In Lesson 15, we were introduced to normal stress in an axially loaded member. In Lesson 17, we will learn how stresses and strains are related to each other.

3. Reading Assignment Riley, Sturges, and Morris: Study Sections 4-4 and 4-5

4. Key Definitions Displacement – A movement of a point with respect to a reference axis system. In a body or system of bodies subjected to forces, displacements may be caused by the translation or rotation of a body as a whole, or they may result from a change in the dimensions or shape of a body. Deformation – A change in the dimensions or shape of a body. Normal (axial) deformation (d n ) is calculated as d n = L f - L o, where L f is the final length and L o is the original length. Strain – A measure of the intensity of deformation. Normal strain (e) is calculated as e = d n / L o, where d n is the normal (axial) deformation and L o is the original length. Strain is a non-dimensional quantity; however, normal strain is often expressed in terms of in/in or mm/mm. Tensile normal strain is considered to be positive (+), while compressive normal strain is considered to be negative (-). Poisson's Ratio – A material property representing the ratio of lateral to longitudinal strain. Poisson's Ratio (n) is calculated as n = -e lat /e long, where e lat is the lateral strain, measured perpendicular to the direction of the applied load, and e long is the longitudinal strain, measured parallel to the direction of the applied load. Modulus of Elasticity - A material property representing the slope of the normal stress-strain (s-e) curve in the elastic region. The modulus of elasticity is also called Young's Modulus. Hooke's Law - The linear relationship between normal stress (s) and normal strain (e). Hooke's Law is written mathematically as s = Ee, where E is the modulus of elasticity.

5. Lesson Notes Why are Deformations Important?

they did not have to allow for as much uncertainty in their design, and so they were able to use considerably less steel in the building. Less steel made the structure lighter and more economical than any previous skyscraper had been. However, the building was also much more flexible than it should have been. It was susceptible to much larger deformations than older, heavier buildings had been, and so even in light winds, the tower flexed and twisted far more than anyone had anticipated. Because window glass is itself very stiff, the windows were unable to flex along with the building frame, and they simply popped out of their mountings. It is important to note that the structural design was, in fact, safe. The stresses in the structural

members were within allowable limits. Nonetheless, the building failed to fulfill its intended function because of excessive deformations. A deformation is simply a change in the dimensions or shape of a body. Deformations can be caused by applied loads or by changes in temperature. Deformations are important to engineers for four principal reasons: Deformation is a measure of performance. As the case of the John Hancock tower illustrates, when structural elements deform excessively, an entire structure may fail to function as intended. The tendency to deform under load is one of several methods commonly used to characterize materials. Unlike stresses, deformations can be directly measured. And because there is a clear mathematical relationship between stress and deformation, measured deformations can be used to indirectly determine the stresses in a body. Thus deformations are very important to our understanding about how structures work. Deformations are particularly valuable for analyzing statically indeterminate structures. A statically indeterminate structure is one for which the equations of equilibrium alone are not sufficient to solve for all unknown reactions and internal forces. In such problems, deformations are used as the basis for formulating the additional equations necessary to analyze the structure. We will not analyze statically indeterminate structures in CE-301, but civil engineering majors will do so in a later course.

Deformation and Strain

where d h and d w are the changes in height and width dimensions, respectively. It is important to note that, while there are two different expressions for lateral strain, a given axially loaded member with a given applied load has just one value of lateral strain. The lateral deformations might be different in the two lateral directions—but the lateral strain is always the same in both directions. In all cases, the general expression for normal strain is the deformation divided by the original length. Because this calculation is always a length divided by a length, strain is actually a dimensionless quantity. However, as a standard convention, strain is usually expressed as inches per inch (in/in) or millimeters per millimeter (mm/mm), to emphasize how it is calculated.

It is important to recognize, however, that this equation is rarely applied in the form shown above. Because Poisson’s Ratio is a material property, it is usually obtained from reference tables like Table A-17 and A-18 in your textbook. The equation above is typically used to calculate lateral or longitudinal strain for a given material. Poisson’s Ratio is a scalar quantity, usually between 0.1 and 0.4 for most common materials. The minus sign in the equation above reflects that fact that longitudinal elongation (positive e long ) is always accompanied by a decrease in the lateral dimensions (negative e lat ), and longitudinal shortening (negative e long ) is always accompanied by an increase in the lateral dimensions (positive e lat ).

The Stress-Strain Curve and Hooke’s Law

At any given magnitude of load, this steel specimen experiences some normal stress, s, which is equal to the load divided by the cross-sectional area of the specimen. At the same time, this load causes the specimen to elongate. If we measure the elongation, we can calculate the corresponding longitudinal strain, e, as the measured deformation divided by the original length. As we increase the load applied to the test specimen, both the stress and strain will increase simultaneously as well. When the resulting relationship between stress and strain is plotted as a graph, the result is a stress-strain curve. Figure 4-28 in your textbook shows stress-strain curves for three different materials—steel, magnesium, and cast iron. For each of these examples, it is important to recognize that the stress- strain curve is a function solely of the material being tested. The size and shape of the test specimens have no effect on the curve. For a given material, the stress-strain curve will always be essentially the same. And note each of the three stress-strain curves has a characteristic shape and characteristic magnitudes of the key points on the curve—illustrating the important point that the stress-strain curve is used to characterize a material.

We will examine stress-strain curves in more detail next lesson. For now, it is only necessary to note two key characteristics of the example curves shown in Figure First, a stress-strain curve always starts at the origin. This is expected, because zero load results in zero deformation, and so zero stress should result in zero strain. Second, note that stress and strain tend to increase in direct proportion to each other within the lower portion of the curve, especially for steel and magnesium. This linear portion of the stress-strain is called the elastic region. The term elastic means that, if the load is removed from the material, it will return to its original shape with no permanent deformation. Since it is highly desirable that structural components and machine parts not experience permanent deformation under normal loading conditions, engineers almost always design such components to function safely within the elastic region.

The linear relationship between stress and strain in the elastic region of the stress-strain curve is one of the most important concepts underlying all of structural mechanics. This linear relationship is called Hooke’s Law, named after the great British scientist Robert Hook, and can be expressed mathematically as where E is the modulus of elasticity, the slope of the elastic portion of the stress-strain curve. The modulus of elasticity is a material property and is also sometimes called Young’s Modulus, after the British mathematician and scientist, Thomas Young. Because strain, e, is a dimensionless quantity, the units for the modulus of elasticity are the same as for stress—psi, ksi, N/m 2, Pa, MPa, etc. Textbook Example Problems 4-8 and 4-10 provide good examples of the calculation of stress, strain, modulus of elasticity, and Poisson’s ratio. In Example Problem 4-10, ignore the reference to the modulus of rigidity. We will work with this property in a future lesson.

In-Class Problem 17-1

End of lesson 1313 Question