Equlibrium Concept, Action / Reaction

As Sir Isaac Newton discovered centuries ago, a force, (if unbalanced by an equal reaction acting on the opposite direction) will create an acceleration on the body, as in F=ma, where F is the force in newtons, m is the mass in kg and a is acceleration in m/s2.

Same is true for turning effects. A net moment or torsion acting on a body, if unbalanced by an equal and opposite moment or torsion reaction, will create a rotational acceleration.

But if you look out of your window right now, all the buildings seem to be standing still. This means, the net (resultant / sum) forces and turning effects acting on all these structures must be zero. Although we definitely know that there are so many individual forces and turnning effects are acting on them, at any given time, we know the bottom line. The bottom line is zero. The buildings are not moving, no matter how many thousands of forces or moments are acting on them. So overall it means, equal and opposite reactions are created from the ground, to hold the building in place and not moving.

So, when we determine the loads acting on a structure, and the reactions that hold that structure in place, by using these forces and reactions, we can design our overall structure and the our individual structural members.   

In order to determine the unknown reactions that support a structure, a Free Body Diagram (FBD) of the structure is drawn, as you saw before, in the table and glass example. A FBD shows all forces and moments as vectors in proper directions along with a simplified sketch of the structure itself. So, a free body diagram is the graphical representation of all forces and moments that act on a rigid body.

Figure 1f shows a FBD. Look at the upper portion. The beam is fixed to a wall, while a compressive force acts on its end, and also another point force acts on it laterally. The bottom part of figure 1f is called the Free Body Diagram. It is statically equivalent to the upper figure. The external forces that act on the right side of the beam, produce reactions at the wall. All vertical forces, horizontal forces cancel each other. Also, you might have noticed that a moment is also drawn at the left side. This is because the vertical external force and the vertical reaction force have distance between each other, which would produce a moment, since moment = force x distance. So the wall also produces a reaction moment, to resist this moment effect. These reactions are produced, because, the beam wall connection here is capable of producing it in the first place. This may not have been the case though.

What if the left side of the beam was not firmly inserted to the wall, but connected to the wall by a hinge that could turn freely? Then this beam would rotate. Because although the horizontal and vertical forces acting on the right would be cancelled out, the hinge would not be able to resist the moment, and that net moment would cause the beam to rotate, as in the figure below.

FBD of Beam with hinge. Image Copyright: A. Tuter

So, as seen here, if the beam was supported by a hinge at the wall, it would still create horizontal and vertical reactions, as in the case of fixed beam, so it would not travel sideways or vertically, but, unlike the fixed beam, this time it would not be able to resist the turning effect, and would rotate.

This logic is the fundamental concept in structural engineering. In other words, all analysed structures, must satisfy equilibrium, which means, all forces in all x y z directions, and moments and torsions along all x y z axes must be equal to zero. If not, a structure would accelerate linearly or rotationally, due to Newton’s Law, as seen in the latest example, when the beam did not remain static, but rotated.

So the static bodies that are not moving fall into the realm of statics. The movement and acceleration of bodies fall into the subject of structural dynamics, such as earthquake analysis, which we will not cover in this post series, except introducing it later.

Let’s look further into our beam:

Uniformly loaded beam with fixed end support. The bottom figure shows fibers, and parts in tension and compression. Middle has no tension or compression and therefore is the neutral axis.

Figure 1g is very similar to Figure 1f, except that this time instead of a point load at the end, a uniform load is acting along the length of the beam. But the effect is similar as far as the reactions produced, except, this time the magnitudes of the reactions would be different at the wall. The bottom figure shows the bent shape of the beam. Do not be confused. You would obtain a very similar bent shape from figure 1f too – we just didn’t show it. So the only difference in Figure 1g from Figure 1f would be the magnitude of reactions and the exact bent shape.

We also purposefully showed the beam fibers here. Now as you see, the upper fibers of the beam is in tension, and the bottom fibers of the beam is in compression (shown with letters T and C). Take a look again to Figure 1c, and then, the fibers of the beam in Figure 1g. Do you see similarity? As you can see, the fibers of this beam are trying to slide against each other, which is similar to the situation in Fig.1c. For example, take several rulers in your hand and put them on top of each other. Then try to bend the whole assembly. Each ruler would tend to slide away from each other, as you bend more. Now you see nothing in this beam is sliding against each other, and that is only because the material of the beam is strong enough to resist this sliding, which means shear stress is produced along these fibers just like in Figure 1c.

For example, if you glued those rulers to each other with a strong glue, and do the bending again, upto a certain point, no ruler would slide against each other, but the shear stress for the glue would keep building up. As you bend more, after a certain stress, the glue would not hold and the rulers would now start sliding.

Concrete for example, is strong in compression but weak in tension and shear, and the reinforcing steel that is placed in concrete not only resists tension but also shear stresses just like the situation described here.    

In the next post, we will introduce the term “Types of supports”

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