Chapter 8 Internal Loadings
Key Questions
At the end of this chapter you should be able to answer these questions.
- Identify the three basic internal loadings in two-dimensional systems.
- Articulate the standard sign convention for internal forces.
- Develop free-body diagrams that include internal loads.
- Use static equilibrium to calculate the unknown internal loadings at a single point on a rigid body.
- Articulate the mathematical relationship between load, shear, and moment in beams.
- Construct shear and bending moment diagrams using multiple approaches.
One of the fundamental assumptions we make in statics is that bodies are rigid, that is, they do not deform, bend, or change shape when forces and moments are applied. While we know that this assumption is not true for real materials, we are building the analytical tools necessary to analyze deformation. In this chapter you will learn to compute the internal loads, which are the forces and moments within a body which hold it together as it supports its own weight and any applied loads.
The chapter begins with a discussion of internal forces and moments and defines a new sign conventions especially for them. Discussion of how to find internal loads at a specific point within a rigid body follows. The chapter concludes with three techniques to find internal loads throughout a beam. Note that the words loads or loading as opposed to forces are used here when we are referring to both forces and bending moments.
In courses you take after statics, finding the internal loading is the first step you will take to determine stresses, strains, deformations and ultimately select the appropriate shape, size, and material necessary for structural elements to safely support the applied loads. So you are advised to pay attention, and master this topic.
Application: Deformation.
The controlling design parameter for most engineering systems is deformation. Thankfully, due to a property called elasticity, most materials will bend, stretch, and compress, long before they ultimately break. For example, when designing the floor in a new building, the floor is often limited to deflecting less than the length of the span in inches, divided by 360. Any more deformation than this would be considered disconcerting to the building residents and also start damaging surface materials (like drywall). For example, for a \(\ft{20}\) span, the deflection would need to be less than
To meet this deformation limit, we need to consider the magnitude and location of applied loads, the size and shape of the floor beams, and the material the floor beams are made from. As deflection is an internal property of the flooring materials, the first step is to determine the internal loadings that arise from the externally applied loads, which you will learn in this chapter.