Equilibrium Of a Rigid Body

Objectives To develop the equations of equilibrium for a rigid body. To introduce the concept of the free-body diagram for a rigid body. To show how to solve rigid body equilibrium problems using the equations of equilibrium.

Part A

Conditions for Rigid-Body Equilibrium

2D Supports

Support Reactions General Rule: If a support prevents the translation of a body in a given direction, then a force is developed on the body in that direction. Likewise, if rotation is prevented a couple moment is exerted on the body.

Rocker

Smooth Surface

Pinned or Hinged

Fixed

Drawing a Free-Body Diagram

Modeling

Modeling

Procedure for Drawing a Free-Body Diagram Select co-ordinate axes. Draw outlined shape isolated or cut “free” from its constraints and connections. Show all forces and moments acting on the body. Include applied loadings and reactions. Identify each loading and give dimensions. Label forces and moments with proper magnitudes and directions. Label unknowns.

Free Body Diagrams

Free Body Diagrams

Important Points No equilibrium problem should be solved without first drawing the appropriate F.B.D. If a support prevents translation in a direction, then it exerts a force on the body in that direction. If a support prevents rotation of the body then it exerts a moment on the body.

Important Points Couple moments are free vectors and can be placed anywhere on the body. Forces can be placed anywhere along their line of action. They are sliding vectors.

Part B

Beams, Trusses, Frames and Cables Structure: Equilibrium in two dimensions

Types of Structures A structure refers to a system of connected parts used to support a load. Important examples related to civil engineering include buildings, bridges, and towers; and in other branches of engineering, ship and aircraft frames,…etc.

Classification of Structures 1. Structural Elements: Some of the more common elements from which structures are composed are as follows: I. Tie Rods: Structural members subjected to a tensile force are often referred to as tie rods or bracing struts

II. Beams: Beams are usually straight horizontal members used primarily to carry vertical loads.

III. Columns: Members that are generally vertical and resist axial compressive loads are referred to as columns

2. Types of Structures: The combination of structural elements and the materials from which they are composed is referred to as a structural system. Each system is constructed of one or more of four basic types of structures. I. Trusses: When the span of a structure is required to be large and its depth is not an important criterion for design, a truss may be selected. Trusses consist of slender elements, usually arranged in triangular fashion. Planar trusses are composed of members that lie in the same plane and are frequently used for bridge and roof support.

Roof Truss

II. Cables and Arches: Two other forms of structures used to span long distances are the cable and the arch. Cables are usually flexible and carry their loads in tension. They are commonly used to support bridges, and building roofs

III. Frames: Frames are often used in buildings and are composed of beams and columns that are either pin or fixed connected.

2D Equilibrium Scalar Equations

Procedure for Analysis Free-Body Diagrams Equations of Equilibrium

Example Determine the reactions at the supports.

Equations of Equilibrium

Example Determine the reactions at the supports.

Example Determine the reactions at the supports.

For the beams shown in Fig.(1) and (2). Example 1: For the beams shown in Fig.(1) and (2). I. Determine the resultant force and specify where it acts on the beam measured from A. II. Determine the support reactions. Fig.(1)

Fig.(2)

Determine the support reactions for the structures shown Example 2: Determine the support reactions for the structures shown in Fig. (3) to (5). Fig.(3)

Fig.(4)

Fig.(5)