# A 160-lb force P is applied at point A of a structural member. Replace P with (a) an equivalent force-couple system at C, (b) an equivalent system consisting of a vertical force at B and a second force at D.

Question-AnswerCategory: Engineering MechanicsA 160-lb force P is applied at point A of a structural member. Replace P with (a) an equivalent force-couple system at C, (b) an equivalent system consisting of a vertical force at B and a second force at D.

A 160-lb force P is applied at point A of a structural member. Replace P with (a) an equivalent force-couple system at C, (b) an equivalent system consisting of a vertical force at B and a second force at D. Step: 1

(a)
Draw the figure to represent the force acting on the structural member. Step: 2

Draw the equivalent force-couple system at C. Step: 3

Let and are the equivalent force and couple system at C.
The magnitude of the equivalent force at C is: Substitute in the above equation and obtain the magnitude of the equivalent force at C. And the direction of the equivalent force is same as that of the force at A.
Therefore, the equivalent force at C is Step: 4

Determine the equivalent couple at C. Therefore, the equivalent couple is Step: 5

(b)
Draw the equivalent force system consisting of a vertical force at B and a force at D. Step: 6

Assume to be the vertical force at B and to be the inclined force at D.
The force acts at an angle from positive x – axis
Take the moments about D as follows Substitute in the above equation and determine the vertical force as below Therefore, the vertical force at B is .

Step: 7

From equilibrium condition, sum of the forces in y – direction is as follows Substitute and in the above equation  …… (1)
From equilibrium condition, sum of the forces in x – direction is as follows: Substitute in the equation  …… (2)

Step: 8

Square and add the equations (1) and (2) to obtain the force    Step: 9

Consider the equation (1) obtain the value of force  Substitute . Therefore, the force at D is 