Equilibrium of Rigid Bodies | Engineering Mechanics - Civil Engineering (CE) PDF Download

A. Beams

Beams are the best examples of a system of coplanar non-concurrent systems. We will restrict our study only to finding reactions of such determinate beams

Type Of Supports 

1. Simple Support : If one end of a beam simply rests on a support, the support is known as simple support. The beam is free to slide and rotate at simple support and the reaction is perpendicular to the surface. Eg : Scale resting on two tables

2. Roller Support : The end which is supported on a frictionless roller to permit contraction and expansion due to temperature or lateral forces is known as roller support. The reaction in roller support is normal to the surface

3. Hinged/Pined Support : This support does not permit movement along any direction but is free to rotate. The reaction components are taken along horizontal and vertical directions. Eg – Hinges of doors and windows

4. Fixed Support : At fixed support, the beam is not free to slide or rotate along any direction. The reaction components include horizontal, vertical and moment.

Types of Loadings 

1. Concentrated load or Point Load: If a load acts at a point, either horizontal, vertical or inclined, it is called a concentrated load or point load. If the load acts at a very small area, it is taken as a point load. It is expressed in N or Kn.

2. Uniformly Distributed Load :This type of load is spread over acertain length with the intensity of load being constant. It is expressed in N/m or kN/m. The resultant load R=wL acts at a midpoint of udl.

3. Uniformly Varying Load: This type of load is spread over a certain length with the intensity of load varying linearly. The resultant load R=wL/2 acts at a distance of 1/3 from point of action of w N/m.

4. Pure Moment : A pure moment force or couple can also act on beams. The unit of moment is kNm.

B. Equilibrium of General Coplanar Systems

Apart from beams there are a number of coplanar systems whose reactions is of interest. Here is one question put forth for those who wish to solve it.

1. Calculate the force and moment reactions at the bolted base O of the overhead traffic-signal assembly. Each traffic signal has a mass of 36 kg, while the masses of members OC and AC are 50 kg and 55 kg, respectively. The mass center of member AC is at G.

Types of Forces

Forces that act on a body can be divided into two general categories—

• Applied forces (action)
• Reactive forces (or, simply reactions)

Reaction

Reaction is the opposing force that a support offers whenever it is acted upon by external or inherent forces.

Free Body Diagram

Free body diagram is a diagram in which a rigid body is isolated from the system and all active forces applied to the body and reactive forces as a result of mechanical contact are represented.

Examples

Steps for Drawing Free Body Diagram
1. A sketch of the body is drawn assuming that all supports (surfaces of contact, supporting cables, etc.) have been removed.
2. All applied forces (including weight) and support reactions are drawn and labeled on the sketch.
3. Apply the weight of the body to its center of gravity (if it is uniform, then apply it to the centroid). If the sense of a reaction is unknown, it should be assumed

Equilibrium of Rigid Bodies

A rigid body is said to be in equilibrium if the resultant of all external and reactive forces and moments acting on it is zero.

Lami’s Theorem

If three coplanar concurrent forces acting on a body keep it in equilibrium, then each force is proportional to the sine of the angle between the other two”

Note: Lami’s theorem is applicable only to 3 coplanar concurrent forces in equilibrium

Proof

By applying polygon law of forces, draw triangle OAB representing the system of forces shown with external angles as indicated

Applying Sine Law

General Equations of Equilibrium

1. The algebraic sum of all forces in a force system is zero.

2. The algebraic sum of all moments in a force system is zero.

Equations of Equilibrium For Coplanar Systems