tensile and Compressive stress and strain
The simplest elastic behavior to understand is the stretching of a bar, rod, or wire when its ends are pulled (Fig. 11.12a). Figure 11.14 shows an object that initially has uniform cross-sectional area A and length l0. We then apply forces of equal magnitude F# but opposite directions at the ends (this ensures that the object has no tendency to move left or right). We say that the object is in tension. We’ve already talked a lot about tension in ropes and strings; it’s the same concept here. The subscript # is a reminder that the forces act perpendicular to the cross section.
We define the tensile stress at the cross section as the ratio of the force F to the cross-sectional area A:
This is a scalar quantity because F# is the magnitude of the force. The SI unit of stress is the pascal (abbreviated Pa and named for the 17th-century French scientist and philosopher Blaise Pascal). Equation (11.8) shows that 1 pascal equals 1 newton per square meter (N>m2 ):
In the British system the most common unit of stress is the pound per square inch (lb/in.2 or psi). The conversion factors are
The units of stress are the same as those of pressure, which we will encounter often in later chapters.
Under tension the object in Fig. 11.14 stretches to a length l = l0 + ?l. The elongation ?l does not occur only at the ends; every part of the object stretches in the same proportion. The tensile strain of the object equals the fractional change in length, which is the ratio of the elongation ?l to the original length l0:
Tensile strain is stretch per unit length. It is a ratio of two lengths, always measured in the same units, and so is a pure (dimensionless) number with no units.
Experiment shows that for a sufficiently small tensile stress, stress and strain are proportional, as in Eq. (11.7). The corresponding elastic modulus is called Young’s modulus, denoted by Y:
Since strain is a pure number, the units of Young’s modulus are the same as those of stress: force per unit area. Table 11.1 lists some typical values. (This table also gives values of two other elastic moduli that we will discuss later in this chapter.) A material with a large value of Y is relatively unstretchable; a large stress is required for a given strain. For example, the value of Y for cast steel (2 * 1011 Pa) is much larger than that for a tendon (1.2 * 109 Pa).
When the forces on the ends of a bar are pushes rather than pulls (Fig. 11.15), the bar is in compression and the stress is a compressive stress. The compressive strain of an object in compression is defined in the same way as the tensile strain, but ?l has the opposite direction. Hooke’s law and Eq. (11.10) are valid for compression as well as tension if the compressive stress is not too great. For many materials, Young’s modulus has the same value for both tensile and compressive stresses. Composite materials such as concrete and stone are an exception; they can withstand compressive stresses but fail under comparable tensile stresses. Stone was the primary building material used by ancient civilizations such as the Babylonians, Assyrians, and Romans, so their structures had to be designed to avoid tensile stresses. Hence they used arches in doorways and bridges, where the weight of the overlying material compresses the stones of the arch together and does not place them under tension.
In many situations, bodies can experience both tensile and compressive stresses at the same time. For example, a horizontal beam supported at each end sags under its own weight. As a result, the top of the beam is under compression while the bottom of the beam is under tension (Fig. 11.16a). To minimize the stress and hence the bending strain, the top and bottom of the beam are given a large cross-sectional area. There is neither compression nor tension along the centerline of the beam, so this part can have a small cross section; this helps keep the weight of the beam to a minimum and further helps reduce the stress. The result is an I-beam of the familiar shape used in building construction (Fig. 11.16b)