WORLD FOR CIVIL

Example 2 - Concurrent, Coplanar Forces

In this relatively simple structure, we have a 500 lb. load supported by two cables, which in turn are attached to walls. Let's say that we would like to determine the forces (tensions) in each cable.

If we examine Diagram-1 for a moment we observe this problem may be classified as a problem involving Concurrent, Coplanar Forces. That is, the vectors representing the two support forces in Cable 1 and Cable 2, and the vector representing the load force will all intersect at one point (Point C, See Diagram 2). When the force vectors all intersect at one point, the forces are said to be Concurrent. Additionally, we note that this is a two-dimensional problem, that forces lie in the x-y plane only. When the problem involves forces in two dimensions only, the forces are said to be Coplanar.

(Notice in this problem, that since the two supporting members are cables, and cables can only be in tension, the directions the support forces act are easy to determine. In later problems this will not necessarily be the case, and will be discussed later.)

To "Solve" this problem, that is to determine the forces (tensions) in cable 1 and cable 2, we will now follow a very specific procedure or technique, as follows:

1. Draw a Free Body Diagram (FBD) of the structure or a portion of the structure. This Free Body Diagram should include a coordinate system and vectors representing all the external forces (which include support forces and load forces) acting on the structure. These forces should be labeled either with actual known values or symbols representing unknown forces. Diagram 2 is the Free Body Diagram of point C with all forces acting on point C shown and labeled.

2. Resolve (break) forces not in x or y direction into their x and y components. Notice that T1, and T2, the vectors representing the tensions in the cables are acting at angles with respect to the x-axis, that is, they are not simply in the x or y direction. Thus for the forces T1, T2, we must replace them with their horizontal and vertical components. In Diagram 3, the components of T1 and T2 are shown.

Since the components of T1 and T2 (T1 sin 53^o, T1 cos 53^o, T2 sin 30^o, T2 cos 30^o) are equivalent to T1 and T2, in the final diagram 1d, we remove T1 and T2 which are now represented by their components. Notice that we do not have to do this for the load force of 500 lb., since it is already acting in the y-direction only.

3. Apply the Equilibrium Conditions and solve for unknowns. In this step we will now apply the actual equilibrium equations. Since the problem is in two dimensions only (coplanar) we have the following two equilibrium conditions: The sum of the forces in the x direction, and the sum of the forces in the y direction must be zero. We now place our forces into these equations, remembering to put the correct sign with the force, that is if the force acts in the positive direction it is positive and if the force acts in the negative direction, it is negative in the equation.
or, T1 cos53^o - T2 cos30^o = 0

or, T1 sin53^o + T2 sin30^o - 500 lb= 0
Notice we have two equations and two unknowns (T1 and T2), and therefore can solve for the unknowns. There are several ways to solve these two 'simultaneous' equations. We could solve the first equation for T1 in terms of T2, (T1 = T2 cos 30^o/cos 53^o), and substitute the expression for T1 into the second equation [(T2 cos30^o/cos53^o)sin53^o + T2 sin30^o - 500 lb= 0], giving us only one equation and one unknown.

On solving the equations for T1 and T2 we obtain: T1 = 436 lb.; T2 = 302 lb. Thus, if the structure is to be in equilibrium, if the cables, acting at the angles given, are to support the 500 lb. load, then the forces in the cable must be as found above, 436 lb. and 302 lb., respectively. So when we go to purchase cables for our structure, we must be sure they will support loads at least equal to the tensions we found.