Chapter 2 Coplanar Concurrent Forces Introduction In the

Chapter 2 Coplanar Concurrent Forces

Introduction �In the first chapter we discussed about simple case of concurrent forces, where only two non parallel forces were considered. �However, in many cases, the lines of action of three or more forces may intersect at a single point, and can form a system of concurrent forces. �In this chapter we are going to discuss these type of coplanar concurrent forces.

Resultant of Concurrent Force (Algebraically) � Find ∑ Fx, ∑ Fy then determine the resultant force and the angle. (The sense in which each force acts is considered) � Ax = A cos α, Bx = B cos β, Cx = C cos γ � Ay = A sin α, By = B sin β, Cy = C sin γ � ∑ Fx = A cos α - B cos β - C cos γ � ∑ Fy = A sin α - B sin β - C sin γ � Resultant force, R = [(∑ Fx)2 + (∑ Fy)2]1/2 � Direction or angle, tanθ′ = ∑ Fy / ∑ Fx

Example- 16 �Find algebraically the resultant of the force system shown in Fig 20 (a)

Equilibrium �A system of forces where the resultant is zero is said to be in Equilibrium. (Defined by Newton’s First Law) �Now If, R= 0 then, �∑ Fx = 0 �∑ Fy = 0 �The above two equations can be used to find two unknowns.

Equilibrium

Problem 57 �R is the resultant of F, T and Q, shown in Fig. 33. F = 150 lb. , θ = 30º, R = 85 lb. Find Q and α. �Example 21 - Do it yourself

Free Body �Most bodies in equilibrium are at rest. �But a rigid body moving with constant speed in a straight path is also in equilibrium. �A rigid body may be any particular mass whose shape remains unchanged while it is being analyzed for the effect of forces. �Since no body is truly rigid, we mean by ‘rigid body’ is one whose deformation under force is negligible for the purposes of the problem. �A free body is a representation of an object, usually a rigid body , which shows all the forces acting on it. (See page 2223 of Analytic Mechanics by Virgil Moring Faires for details)

Example

Problem 72 �A 5000 lb sphere rests on a smooth plane inclined at an angle θ = 45 º with the horizontal and against a smooth vertical wall. What are reactions at the contact surface A and B, Fig. 39?

Equilibrium of Three Forces & Force Polygon

Equilibrium of Three Forces & Force Polygon Again from the equation of equilibrium, �∑Fx = F 2 cos 45º - F 1 cos 60º = 0 �∑Fy = F 2 sin 45º + F 1 sin 60º = 0

Problem: 91 �Two weights are suspended from a flexible cable as shown in Fig. 46. For θ = 120º, determine the internal forces in various parts of the cable and weight W.

Trusses ? �Trusses are mainly used to support Roofs and Bridges.

Trusses: Joint to joint method �A determinate structure is one wherein the internal forces in the various members of the structure may be obtained by the conditions of equilibrium. �Although truss is subjected to all manner of loading, for simplicity it is assumed that, the loads on truss are applied at pin joints �So, with a truss loaded in this manner, all the various members are two force members and the free body of each pin is a system of concurrent forces. �If all the external forces including the reactions at the supports are known, then the above principles can be used to determine the internal forces in each member. �To determine the member forces, it is assumed that the unknown forces are acting away from the pin, which means that the members are in tension. �Solve for each of the unknowns by using the conditions of equilibrium, i. e. , ∑ Fx = 0, ∑ Fy = 0.

Example 22 �A roof truss is constructed and loaded as shown in Fig. 27. At the pin N, the following internal forces have been found: NF = 16900 lb (tension), DN = 7000 lb (compression) and NH = 12565 lb (tension). For an external load at the pin of 6000, find the forces in the members NC and NB.

Closure �See Book Page: 32 from Analytic Mechanics by Virgil Moring Faires

Assignments From Book (Analytic Mechanics by Virgil Moring Faires): �Problem: 77 �Problem: 79 �Problem: 114 �Problem: 137 �Problem: 138

Assignments

Assignments