## What is a graphical method?

The graphical method is a method of solving equations involving more than one variable using the Cartesian coordinate system. In civil engineering, the use of the graphical method is limited to the problems involving two variables. The graphical method has a great impact in solving the questions regarding engineering mechanics.

In engineering mechanics, graphical methods can be used in the following topics:

- Triangle and Polygon Laws of Forces
- Representation of Space diagram, Vector diagram, and Bow's notation
- Resultant of Forces
- Equilibrium of Coplanar Forces

## Triangle and Polygon laws of forces

The triangle law of forces states that if two forces act simultaneously on a particle, and if the forces are represented as the two sides of a triangle in terms of both magnitude and direction, then the resultant of the two forces may be represented by the third side in terms of both magnitude and direction, but in the opposite order. To represent this graphically, two arrows representing the magnitude and direction of two forces can be drawn in the plane created by X-Y coordinates, and another arrow representing the resultant force can be drawn in the reverse direction as that of the two forces. The equation for the same can be derived from the graph.

The polygon law of forces states that if there are several forces acting on a particle, then all the forces can be represented as sides of a polygon in order, then the resultant of all the forces should be represented by the closing side of the polygon in the opposite direction. To represent this graphically, all the forces can be drawn with the help of arrows in the plane created by X-Y coordinates in a well-defined order and the resultant can be denoted by another arrow in the opposite direction as that of other forces. The equation for the same can hence be derived from the graph.

## Representation of force, Space diagram, Vector diagram, and Bow's notation

The force can be represented using the graphical method. To represent the force in graphs, a straight line is used and an arrowhead is used to represent the direction of the force. The diagram that shows the forces in space is known as the space diagram. In the space diagram, all the forces acting in space are represented with the help of straight lines, and arrowheads are used to represent their directions. A vector diagram is a diagram drawn in the X-Y plane of a Cartesian coordinate system using a suitable scale to represent the forces in terms of both magnitude and directions. The resultant of forces is drawn as a closure line of the polygon formed by the system of forces, but in the opposite direction as that of other forces. Bow's notation is the method of noting the forces in space and vector diagram. The forces in the space diagram are represented using two capital letters AB, BC, CD, DE, and so on, whereas the forces in the vector diagram are represented using two small notations, ab, bc, cd, de, and so on.

## Resultant of forces

The resolution of any of the two forces in a Cartesian coordinate system can be obtained using the graphical method. To resolve the forces, the parallelogram of force method is used. Initially, the resultant of the two forces are found by using the parallelogram force method. Two force vectors are first extended to converge at a point and then the resultant vector is drawn using the triangle theorem of forces. A parallelogram is formed by drawing lines parallel to the initial force vectors. The magnitude of the resultant is obtained by measuring the length of the vector using scale and then converting it.

## Equilibrium of coplanar forces

The equilibrium of coplanar forces can be used to solve problems related to the equilibrium conditions of a system of forces. To study this method graphically, two theorems are used- Converse of the law of triangle of forces and Converse of the law of polygon of the forces. The converse of the law of triangle of forces states that if two forces acting on a point, represented by its magnitude and direction are considered as two sides of the triangle, and if the results of the same represented by its magnitude and direction are considered as the third side of the triangle in the opposite direction, then the forces should be in equilibrium. The converse of the law of polygon of forces states that if the number of forces represented by their magnitude and direction is considered as the sides of a polygon and if the results of the same represented by its magnitude are considered as the closing side of the polygon in opposite direction, then the forces should be in equilibrium.

To understand this, an example is shown below in the form of figure 1. The figure on the left shows the system of forces and the figure on right shows the system of vector forces. To obtain the values of forces P1 and P2, the following steps need to be followed:

- Select a suitable scale. For example for 10 N = 1 cm.
- Select any point u on the Cartesian coordinate and draw a line uv parallel to the horizontal force of 90N to any suitable scale.
- Through v, draw a line of 100 N to a suitable scale, and parallel to the ray of 100 N from the system of forces.
- To draw lines of P1 and P2, draw lines parallel from the force diagram to the vector diagram.
- Follow the procedure for all the forces and form the final diagram as shown on the right side of figure 1.
- Measure the distance of lines P1 and P2 using scale and convert the same to get the value of forces.

## Context and Applications

The graphical method is useful for the students undergoing the following courses:

- Bachelors in Technology (Civil Engineering)
- Masters in Technology (Civil Engineering)
- Masters in Technology (Structural Engineering)

## Practice Problems

1. In civil engineering, problems involving maximum how many variables can be solved using the graphical method?

- Two
- Three
- Four
- Five

Answer: Option a

Explanation: In civil engineering, problems involving a maximum of two variables can be solved using the graphical method.

2. Graphical method cannot be used in which of the following topics of engineering mechanics?

- Triangle and polygon law of forces
- Resultant of forces
- Analysis of sway frames
- Equilibrium of coplanar forces

Answer: Option c

Explanation: Graphical method cannot be used in Analysis of sway frames topic of engineering mechanics.

3. According to Bow's notation, which of the following statements is true?

- The forces in the space diagram are represented using two capital letters.
- The forces in the space diagram are represented using two small letters.
- The forces in the vector diagram are represented using two capital letters.
- The forces in the vector diagram are represented using two small letters.

Answer: Option a

Explanation: According to Bow's notation, the sentence "The forces in the space diagram are represented using two capital letters" is true.

4. Which of the following is a method of solving equations involving more than one variable using the Cartesian coordinate system?

- Algebraic method
- Statistical method
- Triangular method
- Graphical method

Answer: Option d

Explanation: Graphical method is a method of solving equations involving more than one variable using the Cartesian coordinate system.

5. While representing the force in a graphical method, what does an arrowhead indicate?

- The magnitude of the force
- The direction of the force
- The angle of force
- The dimension of force

Answer: Option b

Explanation: While representing the force in a graphical method, the arrowhead indicates the direction of the force.

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