Summary Resultant of Force System - Engineering Mechanics - Civil Engineering (CE)

Resultant

The resultant of a system of forces is a single force which produces the same external effect on a rigid body as the original system of forces. Replacing a system by its resultant simplifies analysis of equilibrium, motion, and internal reactions.

Sign Conventions

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

When two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through the point of application. This is the parallelogram law.

For two forces F₁ and F₂ with the angle θ between their lines of action, the magnitude R of the resultant is given by

R² = F₁² + F₂² + 2 F₁ F₂ cos θ.

The direction (angle α measured from F₁ toward R) satisfies

tan α = (F₂ sin θ) / (F₁ + F₂ cos θ).

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

Resolution of a force means replacing a single force by two or more component forces which together are equivalent to the original force. It is most convenient to resolve a force into two orthogonal components (mutually perpendicular), typically along the x- and y-axes for planar problems.

Resolution of Coplanar Forces in Rectangular Coordinates

For a force F making angle θ with the positive x-axis, its rectangular components are

F_x = F cos θ,

F_y = F sin θ.

B. Resultant of Concurrent Coplanar Force Systems

For a system of coplanar forces concurrent at a point (or reduced to forces acting at a common point), resolve every force into x and y components and add algebraically to obtain the resultant components.

Note:

• If both R_x and R_y are positive, the resultant lies in the first quadrant.
• If both R_x and R_y are negative, the resultant lies in the third quadrant.
• If R_x is positive and R_y is negative, the resultant lies in the fourth quadrant.
• If R_x is negative and R_y is positive, the resultant lies in the second quadrant.

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

Moment

The moment (or torque) of a force about a point is the tendency of the force to produce rotation about that point. If a force F acts at point whose position vector relative to the chosen origin is r, then the moment M about the origin is given by the vector cross product

M = r × F.

The magnitude of the moment is M = F × d where d is the perpendicular distance (lever arm) from the point (or axis) to the line of action of the force. The direction of the moment is perpendicular to the plane of r and F and follows the right-hand rule.

Scalar Formulation

When using scalar sign convention for planar problems, take clockwise moments as positive and anti-clockwise moments as negative (or vice versa, but remain consistent). Varignon's theorem is useful to compute moments using resolved components.

Varignon's Theorem (Principle of Moments)

Varignon's theorem states that the moment of a force about any point equals the sum of the moments of its components about the same point. This allows moments to be computed using resolved components:

If F = F_xi + F_yj then M_O = xF_y - yF_x (for planar coordinates where appropriate).

Couple

A couple is a pair of equal, opposite, non-collinear forces whose lines of action do not coincide. A couple produces pure rotation only, with no resultant force. The moment of a couple equals one of the forces times the perpendicular distance (arm) between their lines of action.

A couple can be represented by a vector whose magnitude equals the moment of the couple and whose direction is perpendicular to the plane of the forces, determined by the right-hand rule.

Practical Examples : Force applied to a handle of a steering wheel

Differences between Moment and Couple

C. Resultant of Coplanar Non-concurrent Force Systems

For non-concurrent coplanar forces (forces not all meeting at a single point), the system can be reduced to an equivalent single resultant force acting at a specific point and a resultant couple (if required). The equivalent at a reference point O is obtained by summing forces to get R and summing moments about O to get the resultant moment M_O. The single resultant force R may be positioned so its line of action produces the same moment about O; otherwise a couple remains.

X-Y Intercepts of a Line of Action

Intercepts are the coordinates where a line of action meets the corresponding axes. For locating the point of action of a resultant in plane, intercepts are often used.

D. Resolution of a Force into a Force and Couple System

When shifting the line of action of a force from point A to a parallel line at point B, the original force can be represented at B plus a couple that accounts for the shift. The procedure is:

1. At point A, introduce two equal and opposite forces parallel to the given force; these do not change the net effect.
2. Combine one of these with the original force so that its line of action is shifted to point B.
3. The remaining pair (equal and opposite) produces a couple of magnitude F·d where d is the perpendicular distance between the lines of action.

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

For analysis, a force may be represented in different notations depending on convenience.

1. Scalar notation - magnitude and direction angle.
2. Vector notation - components along unit vectors (Cartesian basis) or position and direction form.

Vector Notation of Forces

Two-dimensional (coplanar) force: A force F in the xy-plane may be written as

F = F_x i + F_y j,

where F_x and F_y are scalar components along x and y, and i, j are the unit vectors.

Here F_x = F cos θ and F_y = F sin θ when θ is measured from x-axis.

Three-dimensional (non-coplanar) force: A force in space may be written as

F = F_x i + F_y j + F_z k.

Unit Vector A unit vector is a vector of magnitude one that specifies direction. Denote a unit vector by n. Any vector V may be written as V = V n where V is the scalar magnitude and n is the unit vector in the direction of V. The standard unit vectors along the rectangular coordinates are i, j, k.

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 60°, 45° and 120° respectively with x, y and z axes. Write the vector form of force.

Position Vector A position vector r locates a point in space relative to a reference origin. For point P with coordinates (x, y, z), the position vector r = x i + y j + z k. Position vector of B relative to A is rBA = rB - rA.

A. Resultant of Non-Coplanar Forces - by Vector Notation

1. Resultant of Non-Coplanar Concurrent Forces

In vector form, the resultant R of several concurrent forces F₁, F₂, ... , F_n is

R = Σ F_i = (Σ F_ix) i + (Σ F_iy) j + (Σ F_iz) k.

Thus compute components by summation and then form the resultant vector and its magnitude.

B. Moment and Couple - in Vector Notation

Moment

Vector form of moment about origin

M = r × F.

Varignon's Theorem - Vector Form

The moment of a resultant force about a point equals the sum of the moments of its components about the same point:

Couple - Vector Formulation

The moment of a couple produced by forces F and -F separated by vector d is

M = d × F.

Couple vectors are free vectors; their point of application is not significant.

Note: Cross product of vectors

2. Resultant of Non-Coplanar Non-Concurrent Forces For non-concurrent forces in space, reduce forces to a chosen reference point O by summing forces for resultant R and summing moments ri × Fi to obtain the net moment MO. If the resultant force R does not pass through O, its line of action may be located by using the relationship between resultant moment and position of R. Problems Q1. A table exerts the four forces shown on the floor surface. Reduce the force system to a force- couple system at point O. Determine the resultant of the following force and its location Q2. Replace the two forces acting on the post by a resultant force and couple moment at point O. Express the results in Cartesian vector form.

C. Equilibrium of Non-Coplanar Forces - by Vector Notation

For a body to be in static equilibrium in three dimensions, the following vector equations must hold simultaneously:

Σ F = 0 (sum of forces equals zero)

Σ M = 0 (sum of moments about any point equals zero)

1. Equilibrium of Non-Coplanar Concurrent Forces

When forces are concurrent, equilibrium reduces to the single vector equation

Σ F_i = 0.

The material above collects fundamental definitions, formulae and representative problems for the study of resultants, resolution of forces, moments, and couples in planar and spatial systems. Work through the practice questions, drawing free-body diagrams, resolving forces into components, and systematically applying the algebraic and vector summation procedures described.