Strength of Materials: Internal Forces, Simple Stress, and Analysis
1.1 Strength of Materials
The study of material resistance extends the concepts of forces from mechanics, but there’s a clear distinction between the two. Mechanics primarily focuses on the relationships between forces acting on a solid object. Statics analyzes solids at rest, while dynamics studies solids in motion, although dynamic equilibrium can be established by incorporating inertial forces. In contrast, strength of materials investigates the relationship between externally applied loads and their internal effects on a solid. Unlike mechanics, which often assumes ideally rigid bodies, strength of materials considers deformations, however small, as they are crucial for understanding material behavior. The properties of materials used in constructing a structure or machine significantly influence their selection and design, as they must meet specific strength and rigidity requirements. The differences between rigid body mechanics and strength of materials can be illustrated through the following examples: Determining the force (Fig 1-1) needed at the end of a lever to lift a given weight is a straightforward statics problem. The sum of moments about the fulcrum determines the value of P. This static solution assumes the lever is sufficiently rigid and strong to function properly. However, strength of materials provides a more comprehensive solution. It involves analyzing the bar itself to ensure it won’t break or deform excessively under the load.
1.2 Internal Forces Analysis
Consider a solid of any shape subjected to multiple forces, as depicted in Figure 1-2. In mechanics, the goal is to determine the resultant force to ascertain whether the solid is in equilibrium. If the resultant force is zero, static equilibrium is achieved, a common requirement for structures. If the resultant force is non-zero, introducing inertial forces into the system allows for dynamic equilibrium analysis. Strength of materials delves into the internal distribution of stresses caused by the applied external forces. To achieve this, a common approach is to conceptually cut the solid along a section and examine the forces required at this section to maintain equilibrium for each of the separated parts. Generally, the internal forces can be represented by a resultant force and a resultant moment, which are further decomposed into components normal and tangential to the section, as shown in Figure 1-3. The origin of the coordinate system is always placed at the centroid, which serves as the reference point for the section. If the X-axis is perpendicular to the section, it’s referred to as the surface or face X. The orientation of the Z and Y axes within the section plane is typically chosen to align with the principal axes of inertia of the section. The notation used in Figure 1-3 identifies both the action of exploration and the direction of the force and moment components. The first subscript indicates the face on which the component acts, while the second subscript specifies the direction of the component. For instance, Pxy represents the force acting on face X in the Y direction.
Components of Internal Effects on the Section:
- Pxx Axial Force: This component represents the pulling (or pushing) force on the section. Pulling is a tensile force that tends to elongate the solid, while pushing is a compressive force that tends to shorten it. It’s usually denoted by P.
- Pxy, Pxz Shear: These components represent the resistance to the relative sliding motion of one portion of the solid along the section plane with respect to the other portion. The total shear force is typically denoted by V, and its components, Vy and Vz, determine its direction.
- Mxx Torsional Moment: This component measures the resistance of the solid to twisting and is usually represented by T.
- Mxy, Mxz Bending Moment: These components measure the solid’s resistance to bending or warping about the Y or Z axis, respectively. They are commonly denoted by My and Mz.
1.3 Simple Stress
The force per unit area that a material withstands is often called stress and is mathematically expressed as: σ = P/A (1.1) where σ represents stress, P is the applied load, and A is the cross-sectional area. It’s important to note that the maximum tensile or compressive stress occurs in a section perpendicular to the load, as illustrated in Figure 1-4. However, even a seemingly simple expression like (1.1) requires careful interpretation. Dividing the load by the sectional area doesn’t provide the stress value at every point within the section but rather the average stress. A more precise determination of stress involves dividing the differential load dP by the differential area dA over which it acts. A situation where the stress is constant throughout the section is called a simple stress state. A uniform stress distribution can only exist if the resultant of the applied forces passes through the centroid of the considered section.
Problem No. 01
An aluminum tube is rigidly clamped between a brass rod and a steel rod. As shown in the figure, axial loads are applied at the indicated positions. Determine the stress in each material.
Solution
To calculate the stress in each section, we must first determine the axial load in each one. The appropriate free-body diagrams are shown in the figures below, which allow us to determine the axial load in each section as: Pb = 20 kN (compression), Pal = 5 kN (compression), Ps = 10 kN (tension).