The three fundamental areas of engineering mechanics are statics, dynamics, and mechanics of materials. Statics and dynamics are devoted primarily to the study of external forces and motions associated with particles and rigid bodies (i.e., idealized objects in which any change of size or shape due to forces can be neglected). Mechanics of materials is the study of the internal effects caused by external loads acting on real bodies that deform (meaning objects that can stretch, bend, or twist). Why are the internal effects in an object important? The reason is that engineers are called upon to design and produce a variety of objects and structures, such as automobiles, airplanes, ships, pipelines, bridges, buildings, tunnels, retaining walls, motors, and machines—and these objects and structures are all subject to internal forces, moments, and torques that affect their properties and operation. Regardless of the application, a safe and successful design must address the following three mechanical concerns:
1. Strength: Is the object strong enough to withstand the loads that will be applied to it? Will it break or fracture? Will it continue to perform properly under repeated loadings?
2. Stiffness: Will the object deflect or deform so much that it cannot perform its intended function?
3. Stability: Will the object suddenly bend or buckle out of shape at some elevated load so that it can no longer continue to perform its function?
Addressing these concerns requires both an assessment of the intensity of the internal forces and deformations acting within the body and an understanding of the mechanical characteristics of the material used to make the object.
Bolted connection in a steel frame. The bolts must withstand the shear forces imposed on them by the members of the frame.
Normal Stress Under Axial Loading
In every subject area, there are certain fundamental concepts that assume paramount importance for a satisfactory comprehension of the subject matter. In mechanics of materials, such a concept is that of stress. In the simplest qualitative terms, stress is the intensity of internal force. Force is a vector quantity and, as such, has both magnitude and direction. Intensity implies an area over which the force is distributed. Therefore, stress can be defined as
Stress = Force / Area
To introduce the concept of a normal stress, consider a rectangular bar subjected to an axial force (Figure 1). An axial force is a load that is directed along the longitudinal axis of the member. Axial forces that tend to elongate a member are termed tension forces, and forces that tend to shorten a member are termed compression forces. The axial force P in Figure 1 is a tension force. To investigate internal effects, the bar is cut by a transverse plane, such as plane a–a of Figure 1, to expose a free-body diagram of the bottom half of the bar (Figure 2). Since this cutting plane is perpendicular to the longitudinal axis of the bar, the exposed surface is called a cross section.
Equilibrium of the lower portion of the bar is attained by a distribution of internal forces that develops on the exposed cross section. This distribution has a resultant internal force F that is normal to the exposed surface, is equal in magnitude to P, and has a line of action that is collinear with the line of action of P. The intensity of F acting in the material is referred to as stress.
The technique of cutting an object to expose the internal forces acting on a plane surface is often referred to as the method of sections. The cutting plane is called the section plane. To investigate internal effects, one might simply say something like “Cut a section through the bar” to imply the use of the method of sections. This technique will be used throughout the study of mechanics of materials to investigate the internal effects caused by external forces acting on a solid body.
P: The component of the resultant force that is perpendicular to the cross section, tending to elongate or shorten the bar, is called the normal force.
V: The component of the resultant force lying in the plane of the cross section, tending to shear (slide) one segment of the bar relative to the other segment, is called the shear force.
T: The component of the resultant couple that tends to twist (rotate) the bar is called the twisting moment or torque.
M: The component of the resultant couple that tends to bend the bar is called the bending moment.
