Reactions and Supports
Equilibrium is the law that governs mechanics of materials! Usually, we refer to equilibrium as “the state of a body (or a system of bodies) at which the summation of all externally applied forces and moments are equal to zero”. But that is what we may call, also, static equilibrium. In this case, the body does not move at all, or, moves with a constant velocity.
More generally, since we said that Equilibrium is a law, we may also consider the state where the forces applied are related to the accelerations. Such case may be called dynamic equilibrium. Notice that the static equilibrium is a special case of dynamic equilibrium at which the accelerations are equal to zero.
In more sophisticated terms, we can say that Newton’s Second Law of motion is the law that governs our work in the field of mechanics of material. In such a spirit, we need to remember that the first thing we always need to satisfy, in this field of work, is Equilibrium.
What is a Support?
The word “support” may have different meaning in different contexts. As far as we are concerned, in mechanics of materials, a support is anything that prevents one or more form of motion of any part of the structure. The “prevention” does not have to be full stopping of motion (rigid support) rather, it may be also flexible support such as the ground under a building or the bed mattress under your body.
According to this definition, any part of the structure maybe considered as a support to the adjacent parts of the structure. Similarly, any structure carrying or holding another structure maybe considered a support to it.
However, in most of our discussions, we will be considering the supports to be rigid.
Types of Supports
At this point we will strict our discussion to 2-D problems. There are three main types of rigid supports that are important to 2-D problems; namely, roller support, pinned support, and fixed support.
Types of 2-D supports
The roller support is one that acts as if the structure is resting on a perfect cylinder that moves freely on a horizontal, vertical, or inclined surface. That cylinder is assumed not to create any kind of friction with the structure or the surface. However, the roller support in never allowed to move in a direction normal to the surface on which is rests. Thus, if we have a roller that is resting on a horizontal surface, it will move horizontally with no resistance, but, it is never allowed to move vertically. The roller support is assumed to create a reaction force normal to the surface in order to prevent the structure from moving in that direction.
Meanwhile, a pinned support is one that acts as if the structure attached to it is prevented from moving any linear motion. That is to say that the point at which the pinned support is connected can not move in the horizontal or vertical conditions, but, it may rotate freely around the pin. In order for the pinned support to perform its function of preventing linear motions, it may need to create up to two reaction forces (in two normal directions) to prevent the structure from moving. However, it can never create any moments around the point of fixation (the pin).
Finally, the fixed support prevents all kind of motion in the plane. At the point the structure is connected to a fixed support, the structure can not have any linear motion or rotation. Hence, the fixed support is assumed to create up to two forces to prevent the motion of the structure as well as a moment to prevent it from rotating.
Reactions and Free-Body Diagrams
A reaction is a force or a moment exerted by the support to prevent the point, at which the support is connected, from moving. The amount of force or moment is that needed to ensure equilibrium of the structure. When you need to calculate the reactions in a support, you pretend that the support is removed then replace each restricted motion with a force or a moment. That is what we call the free body diagram. When you have a free body diagram you can readily write down the equilibrium equations and solve them, if feasible, for the values of the reactions.
Point loads are NOT real! The contact between any two bodies can never occur, in reality, at a perfect point (zero area). However, when the load, applied to the structure, occupies a tiny area of the structure surface, we may assume that it is a point load. That is an extremely convenient assumption that allows us to analyze many cases of loading. For example, as you plate a bottle of water on a table, the bottle occupies an area of the table surface, however, if you want to calculate the reactions created in the table legs, assuming that the bottle occupies a single point on the surface of the table will simplify your calculations and provide you with accurate results.
Types of point loads
We have two types of point forces in 2-D problems. Point forces and point moments. Point forces may be perfectly aligned with the coordinate directions or not, thus, we usually resolve any inclined force into two forces parallel to the horizontal and vertical directions. Meanwhile, the point moment is always parallel to the third direction that is normal to the plane of the two dimensions we are concerned with.
Simply supported beam
A beam is supported by a roller at point A and a pin at point B. The beam length is L. A point vertical force is applied at point C which is a distance d from point A. Calculate the reactions at point A & B.
Simply supported beam with point load
First step in solving the problem will be drawing the free body diagram of the beam. To do that we assume that the supports are removed and replaced by forces and moments that correspond to the supports. Point A is supported on a roller which can produce one force normal to the surface (Horizontal surface), hence, we replace the roller with a vertical force Ay. The pinned support, on the other hand, creates two forces, hence, we replace it with Bx and By.
Now, we may write down the equations of equilibrium.
For the equilibrium in the x-direction, we add up all the horizontal forces
From this equation, we can easily declare that the horizontal reaction at point B is zero.
For the equilibrium in the y-direction, we add up all the vertical forces
Finally, we apply the equation of the moment equilibrium
When applying the moment equilibrium equation we have to:
1- Select a point at which the equation will be applied – Any point in the plane will will suffice, however, for convenience, we usually select a point that has some forces applied to in order to reduce the number of variables in the equations
2- Select the direction which will be considered the positive direction – It is advised to stick to the convention of the Cartesian coordinate system to create consistency in your work, but if you decided not to follow that convention, the results will still be correct as long as you are careful enough in setting the positive and negative signs in the equation.
Here we select the point A and the counter-clockwise direction for the moment equation.
From this equation, we may evaluate the reaction at point B to be
Now, we can use this result into the equation of vertical force equilibrium to get
From the above results, we can see that the vertical reaction force at points A&B depend on the distance d. If d becomes too small (almost zero) the reaction at point A will equal to the force Fy, while at be it becomes zero, and vise versa.
In other words, we may say that the closer the force to the support, the more the reaction force in the support becomes.