PH 001 Equilibrium

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These requirements are both necessary and sufficient conditions

for equilibrium

When a body is in equilibrium, the resultant of all forces acting on

it is zero.

Thus the resultant force FR and the resultant couple MR are both

zero.

R

FR = ∑ F = 0

MR = ∑ M = 0

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.

• If rotation is prevented, a couple moment is exerted on the body.

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Support for rigid bodies subjected to two dimensional force systems

Flexible cable, belt, chain, rope (Weightless)

Smooth Surfaces

Rough Surfaces

Freely sliding guide

Pin Connections Pin free to turn

Pin not free to turn

Roller Support

Built in support of fixed support

Gravitational attraction

Spring actions

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To construct free-body diagram for a rigid body or any group of bodies considered

as a single system, the following steps should be performed:

Draw Outlined Shape:

Imagine the body to be isolated or cut "free" from its constraints and connections

and draw (sketch) its outlined shape.

Show All Forces and Couple Moments.

Identify all the known and unknown external forces and couple moments that act

on the body. Those generally are due to

(I) Applied loading. (2) reactions occurring at the supports or at points of contact

with other bodies, and (3) the weight of the body.

Identify Each Loading and Give Dimensions.

The forces and couple moments that are known should be labeled with their proper

magnitudes and directions.

Establish an xy coordinate system so that these unknowns. Ax, Ay can be identified.

Finally, indicate the dimensions of the body necessary for calculating.

Procedure for Drawing a Free - Body Diagram

Examples of Free Body Diagrams

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Determine the horizontal and vertical components of reaction on the beam caused

by the pin at B and the rocker at A as shown below. Neglect the weight of the

beam.

Free Body diagram:

Problem:1

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Support for rigid bodies subjected to three dimensional force systems

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The uniform 7-m steel shaft has a mass of 200 kg and is supported by a ball and

socket joint at A in the horizontal floor. The ball end B rests against the smooth

vertical wall as shown. Compute the forces exerted by the walls and the floor on

the ends of the shaft.

Free Body diagram:

Problem: 2

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Vertical position of B

7 = 2� + 6� + ℎ�

h = 3 m

Weight W = mg = 200(9.81)

= 1962 N

A (2,6,0) B(0,0,3) G(1,3,1.5)

The welded tubular frame is secured to

the horizontal x-y plane by a ball and

socket joint at A and receives support

from the loose-fitting ring at B. Under the

action of the 2-kN load, rotation about a

line from A to B is prevented by the cable

CD, and the frame is stable in the position

shown. Neglect the weight of the frame

compared with the applied load and

determine the tension T in the cable, the

reaction at the ring, and the reaction

components at A.

Problem: 3

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A(0,0,0) B(0,4.5,6) C(-3,0,6) D (-1,2.5,0) F (2.5,0,6)

Taking moment about A

0)2()()(21

=×+++×++× jrkjirkir zyxzxAB TTTBB

Equating the coefficients of k and j to zero

0125.25.4

055.25.4

=−+

=+−−−

zz

xyx

TB

TTB

On solving

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Determine the components of reaction that the ball-and-socket joint at A,

the smooth journal bearing at B , and the roller support at C exert on the

rod assembly.

The reactive forces of the supports will prevent the assembly from rotating

about each coordinate axis, and so the journal bearing at B only exerts

reactive forces on the member.

No couple moments are required.

Free-Body Diagram

Problem: 4

Equations of Equilibrium:

0)900()4.04.0()()2.16.0()(8.0 =×+−−×+−++× kjikjikijCzx

FBB

Taking moment about A

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The boom is used to support the 75-lb

flowerpot. Determine the tension

developed in wires AB and AC .

Free-Body Diagram

Problem: 5

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Constraints and Statical Determinacy

Redundant Constraints

When a body has redundant supports i.e., more supports than are necessary to hold

it in equilibrium, It becomes statically indeterminate.

Statically indeterminate means that there will be more unknown loadings on the

body than equations of equilibrium available for their solution.

Improper Constraints

Lines of action of the reactive forces are concurrent at A

Lines of action of the reactive forces intersect a common axis

the loading P will rotate the

member about the AB axis

The applied loading P will cause

the beam to rotate slightly

about A

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Improper constraining leads to instability when the reactive

forces are parallel