Summary: Resultant of Force System | Engineering Mechanics - Civil Engineering (CE) PDF Download

Resultant

Resultant is a single force that will replace a system of forces and produces the same effect on the rigid body as that of the system of forces.

A. Resultant of Two Concurrent Forces - Parallelogram Law of Forces

Problems

Q1. Find the magnitude of the two forces, such that if they act at right angles, their resultant is √10 N. But if they Act at 60°, their resultant is √13 N.

Q2. The greatest and least resultants of two forces F1 and F2 are 17 N and 3 N respectively. Determine the angles between them when their resultant is √149 N

Q3. A screw eye is subjected to two forces F1 and F2 as shown in figure. Determine the magnitude and direction of the resultant force by parallelogram method

Q4. The two structural members, one of which is in tension and the other in compression, exert the indicated forces on joint O. Determine the magnitude of the resultant R of the two forces and the angle which R makes with the positive x-axis.

Resolution Of Forces 

The replacement of a single force by a several components which will be equivalent in action to the given force is called resolution of a force.
Forces can be resolved in any 2 directions. However, it is convenient to resolve them into the two orthogonal components (mutually perpendicular directions)

Resolution of Coplanar Forces in Rectangular Coordinates

Problems for Practice

Q1. Four forces act on bolt A as shown. Determine the resultant of the force on the bolt.

Q2. If the magnitude of the resultant force is to be 9 kN directed along the positive x axis, determine the magnitude of force T acting on the eyebolt and its angle.

Q3. Determine the resultant of the 3 forces acting on the bracket and its direction.

Q4. The forces 20 N, 30 N, 40 N, 50 N and 60 N are acting at one of the angular points of a regular hexagon, towards the other five angular points, taken in order. Find the magnitude and direction of the resultant force.

Q5. If Φ = 30 and the resultant force acting on the gusset plate is directed along the positive x axis, determine the magnitudes of F2 and the resultant force.

 Q6. Determine the resultant of the forces shown below

Q7. Determine the resultant of the forces acting on the ring shown in figure.

Q8. Find the resultant of the three concurrent forces as shown on figure.

Q9. Find the magnitude and direction of the resultant of the following forces.

i. 20 N inclined at 30° towards North of East.

ii. 25 N towards North.

iii. 30 N towards North West and

iv. 35 N inclined at 40° towards South of West.

 Moment 

The tendency of a force to rotate the body in the direction of its application a force about a point that is not on the line of action of the force is called Moment of force or simply moment.
Moment is also referred to as torque.

Scalar Formulation 

Moment is a vector quantity whose direction is perpendicular to the plane of the body. The right-hand rule is used to establish the sense of direction of moment. Throughout the text, clockwise moments are taken as positive while anti-clockwise as negative.

Varignon’s Theorem or Principle of Moments

The moment of a force about any point is equal to the sum of the moments of the components of the force about the same point.”

Couple 

The moment produced by two equal, opposite, and non-collinear forces is called a couple. The perpendicular distance between the lines of action of the two and opposite parallel forces is known as arm of the couple.

A couple can be represented by a vector with magnitude and direction equal to the moment of the couple.

Differences between Moment and Couple

C. Resultant of Coplanar Non-concurrent Force Systems

A system of several forces and couple moments acting on a body can be reduced to equivalent single resultant force acting at a point O and a resultant couple moment.

D. Resolution of Force Into Force and Couple System

Suppose we have to sift he force from point A to B. The procedure to be followed

1. Apply 2 equal and opposite force at point A parallel to force B of the same magnitude

2. If the points are separated by a distance d, the opposite forces F and –F form a couple retaining force F at point B in the same direction as A

Problems

Q1. Replace the force system acting on the beam by an equivalent force and couple at point B.

Q2. Reduce the following force system into
a) A single force
b) Resultant force and couple acting at point A.
c) Resultant force and couple acting at point B.
d) Resultant force and couple acting at point C.

Q3. Replace the force system by a resultant force and couple moment at point O.

Q4. Two coplanar forces P and Q are shown in figure. Assume all squares of the same size.
i) If P = 4 kN, find the magnitude and direction of Q if their resultant passes through E
ii) If Q = 110 kN, find the magnitude and direction of P if their resultant passes through F

Representation of Force 

There are two ways of representation of force. The method used depends on the type of problem being solved and the easiest approach to finding a solution.

1. Scalar Notation
2. Vector Notation

Vector Notation Of Forces 

1. Two Dimensional Force Systems (Coplanar Forces) It is also possible to represent the x and y components of a force in terms of Cartesian unit vectors i and j.

where the scalars Fx and Fy are the x and y scalar components of the vector F.

2. Three Dimensional Force Systems(Non-Coplanar Forces)

Problems

Q1. A force vector F = 700i + 1500j is applied to a bolt. Determine the magnitude of force and the angle it forms with the horizontal.

Q2. A force of 500 N forms angles 60⁰, 45⁰ and 120⁰ respectively with x, y and z axes. Write the vector form of force.

A. Resultant of Non Coplanar Forces -by Vector Notation 

1. Resultant of Non Coplanar Concurrent Forces

In vector notation, the scalar components of the resultant vector can be obtained by adding algebraically the sum of the corresponding scalar components of the force vectors.

where

B. Moment and Couple – in Vector Notation 

Moment 

Moment is a vector quantity whose direction is perpendicular to the plane of the body. The right-hand rule is used to establish the sense of direction of moment.
Vector Formulation

Varignon’s Theorem

Vector Formulation

Couple 

Vector Formulation

Couple vectors are free vectors, i.e., the point of application is not significant.

C. Equilibrium of Non Coplanar Forces – by Vector Notation

1. Equilibrium of Non-Coplanar Concurrent Forces 

In vector notation, the equation of equilibrium can be summarized as