Principle of Statics in Structural Analysis | Conditions of Static Equilibrium |Sign Convention

Anonymous March 13, 2018
Statics is the study of structures that are under equilibrium conditions. In structural analysis , we determine the effect of forces or loads that act on a structural member. This effect is determined in the form of bending moment, shear force, deflection or displacement. 

So, the primary step of structural analysis is performed by various methods of analysis. The method is unique for structures that are in static equilibrium and different for those that does not follow static equilibrium. 

Principle of Statics in Structural Analysis | Conditions of Static Equilibrium |Sign Convention

This article is the primary introduction to the detailed study of the structural analysis method in civil engineering. You start learning what structural analysis is first, followed by statics and related static equilibrium conditions.

Principles of Statics in Structural Analysis

Equilibrium of the structure means - all the forces acting on the structure must balance, and the net torque must be equal to zero.
The principle of statics can be used to:
  • Analyze the Structures
  • Determine Internal acting on structures
  • Determine the External Forces acting on the structures

Structural Equilibrium

Equilibrium can be defined as a state of balance achieved by a structure or a state of rest occurred when all the forces are equal and opposite to each other.

In Structural analysis and structural design, we focus more on the magnitude, direction and the point of application of the forces as well as the resultant forces in order to ensure the state of equilibrium. 


Fig.1: (a)  The resultant force obtained after resolving must be provided with a reaction equal and opposite for maintaining the equilibrium (b) The moment at the cantilever support is the reaction formed that is equal and opposite to the force formed due to the point load at the end of the beam.


In more easy words, a structural element will be in equilibrium when it reacts with an equal and opposite force to that applied to it.

Newton's Third Law of Action and Equilibrium
As per Newton's third law of motion, for every action, there is an equal and opposite reaction along the same line of action as the original force. Every element part of a structural system will have reactions that will contribute to the equilibrium of the whole system.

Conditions of Equilibrium

Principles of a statics can be applied to a structure only if the structure is at rest. This condition is obtained when the sum of the applied loads, the support reaction is equal to zero and there is no resultant couple at any point in the structure. All the structural system must also be in equilibrium.

Then for the body to stay in static equilibrium the condition to be satisfied is:

ƩFx = 0   ;  ƩFy = 0   ;

For a rigid body to be in equilibrium, two conditions are:
  1. The vector sum of all the forces must be equal to zero. This will ensure translational equilibrium ∑Fx = 0, ∑Fy = 0,
  2. Next, the algebraic sum of all the moments of forces about any point must be equal to zero. This will ensure rotational equilibrium ∑M = 0
The above equations are called as equations of static equilibrium of a planar structure that is subjected to a system of forces.



Statically Determinate and Indeterminate Structure

A statically determinate structure is a structure whose internal and external forces can be determined by the equations of equilibrium (∑Fx = 0, ∑Fy = 0, ∑M = 0). Those structures that cannot be analysed by the equilibrium equations are called statically indeterminate structures. 

If the number of unknown reactions is greater than the number of equilibrium equations in a structure, then the structure is called an indeterminate structure.

In the case of indeterminate structures, we will see that, lots of unknown internal reactions exist that cannot be determined using static equilibrium. For this we need to use different compatibility equations and analysis method. This extra unknown forces are what we call as redundant and measured in the form of Degree of Redundancy. This condition of the structure is called indeterminacy of the structure. 


Sign Convention in Static Structural Analysis

The forces that are acting along the right are positive and the vertical forces acting upward forms positive.
The forces that are acting downwards and leftwards are considered negative. The moment acting in counter clock direction is positive and along the anti-clockwise direction forms negative.

Fig.3.Sign Convention of Forces in Structures


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