6. INTERNAL FORCES IN STRUCTURES

This chapter will start as a review of shear and moment diagrams which you have studied in both Statics and Strength of Materials, and will proceed with the analysis of statically determinate frames, arches and grids. By the end of this lecture, you should be able to draw the shear, moment and torsion (when applicable) diagrams for each member of a structure.

This chunk introduces the main topics of the chapter: shear and moment diagrams. These diagrams are graphical representations that show how internal forces, like shear forces and bending moments, behave within structural members under different loads. Understanding these diagrams helps us analyze how structures respond to external forces.

Think of a bridge as a long beam. When cars drive over it, they exert forces on the structure. The shear and moment diagrams help us visualize how these forces spread throughout the bridge, similar to how water flows through a pipe. Just as we need to ensure our pipes are suitable for the water pressure, we must ensure our bridge can handle the forces applied to it.

For instance, for flexural design, we will consider the section subjected to the highest moment, and make sure that the internal moment is equal and opposite to the external one.

In structural engineering, when designing beams, it is crucial to ensure that the internal bending moment (the internal force from the material that tries to resist bending) counters the external bending moments (the forces trying to bend the beam). This balance is necessary to prevent failure. The section of the beam that experiences the highest moment is particularly important, as it is where we must ensure sufficient strength.

Imagine pushing down on a ruler that is resting on the edge of a table. The more you push, the more the ruler wants to bend downwards. To keep it steady, you can imagine that the material of the ruler internally pushes back just as hard as you press down. In structural design, we're essentially ensuring that this 'pushing back' is strong enough to handle the pressure.

Before we (re)derive the Shear-Moment relations, let us arbitrarily define a sign convention. The sign convention adopted here is the one commonly used for design purposes: Load Positive along the beam’s local y-axis (assuming a right hand side convention), that is positive upward. Axial: tension positive. Flexure A positive moment is one which causes tension in the lower fibers and compression in the upper ones.

Sign conventions are essential in engineering because they establish a uniform way to represent the direction of forces and moments. According to this convention, any load acting upwards is positive, tension (a pulling force) is also positive, and a moment that causes the bottom part of the beam to stretch (tension) is considered positive as well.

Imagine pulling on a rubber band. When you pull it, you cause tension, which we define as 'positive.' Conversely, if you push down on the elastic, you would be compressing it, creating 'negative' tension in our sign convention. This consistency helps engineers communicate clearly about the forces at play.

Let us (re)derive the basic relations between load, shear and moment. Considering an infinitesimal length dx of a beam subjected to a positive load w(x). The infinitesimal section must also be in equilibrium.

This section discusses the mathematical relationships between load, shear, and moment in structures. By analyzing a small segment of a beam (denoted as dx), we can derive equations that link the external load applied to the beam with the internal shear and bending moments. This equilibrium condition ensures that all forces and moments acting on the segment balance out.

Imagine balancing a seesaw. When one side goes down (load applied), the other side (shear and moment) must respond by going up or maintaining balance, depending on where weight is placed. Similarly, in beams, every load creates reactions that need to be analyzed to maintain structural stability.

The slope of the shear curve at any point along the axis of a member is given by the load curve at that point. Similarly, the slope of the moment curve at any point along the axis of a member is given by the shear at that point.

These concepts add depth to our understanding of the relationships between different internal forces. The shear force at a given point in a beam reflects how much the load is applied there, and the moment at a point reflects how that shear force changes along the length of the beam. Mathematically, this relationship is captured through differential equations, allowing us to calculate shear and moment diagrams accurately.

Consider a car driving along a downhill slope. At any moment, the car's speed (akin to shear) depends on how steep the slope (the load) is. The faster the car goes, the sharper it cuts through the slope. The relationship between the car's speed and the slope mimics the relationship between shear and moment in structural analysis.

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