ch.3 equilibrium.pdf

8/20/2019 Ch.3 Equilibrium.pdf

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Equilibrium

M. Berke Gür

Statics

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Lecture Objectives

• In this lecture, we will learn about drawing free-

body-diagrams (FBD)

• FBDs are the most important step in both static and

dynamic analysis

• Using FBDs, we will investigate the condition in

which the resultant of a system of forces acting on a

body is zero (i.e., static equilibrium condition)

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Representing Forces as Vectors

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Planar Supports & Connections

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Cable in tension

Pin supportWeld connection

Rocker support

Freely sliding surface Pin support


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Representing Forces as Vectors

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3-D Supports & Connections

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Thrust bearing

Journal bearing

Ball-&-socket


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Engineering Mechanics Analysis Methodology

• In performing a static or dynamic analysis of a

system, we must

1. Define the mechanical system we are analyzing

• Clearly identify all known and unknown quantities of the system

• The system should include all unknown quantities we are seeking

2. Isolate the system from its surroundings using a free-

body-diagram (FBD)

• Draw an external boundary around the system being analyzed and

remove all other bodies that are not a part of the system

The FBD is the most important single step in the solution

of mechanics problems

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3. Identify all external forces (contact & body) acting on the

system by marking them on the FBD

• Make sure to add a force to the FBD for every contacting or

attracting body that was removed

• Mark all information (magnitude, line of action, sense) readily

available on these external forces

• If the sense of the force vector is not known, make an arbitrary

assignment

4. Define & mark a coordinate system suitable for the

problem

• Make a clever choice for the moment centers to simplify

calculations (e.g., choose moment centers with as many unknown

forces passing through as possible)

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5. Also indicate pertinent dimensions

6. But, avoid cluttering the FBD with unnecessary and

unrelevant information

7. Identify and state the appropiate force and moment

equations governing the problem

8. Match the number of unknown quantities to the number

of independent equations

9. Carry out the solution and check your results

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Examples of FBDs

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Exercise: Completing FBDs

MCH2008 Eng. Mechanics

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Exercise: Constructing FBDs

MCH2008 Eng. Mechanics

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Equilibrium

• A body is said to be in complete equilibrium when

the resultant forces (R ) and couples (MO, any point

O) acting on the body is zero (necessary & sufficient

conditions)

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, , ,

0

0, 0, 0

l

l

x l y l z l

l l l

M M

M M M

, , ,

0

0, 0, 0

k

k

x k y k z k

k k k

R F

F F F

Equilibrium

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Two-Force & Three-Force Members

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For a two-force member, the forces

acting on the member are only along

the line joining the two ends of the

member

Two-Force & Three-Force Members

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Link AB is a two-force member

Link ABC is a three-force member


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• Alternative 1:

– Force balance in one direction and moment balances

about two points A and B

Alternative Formulations for Equilibrium

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0, 0, 0 x A B

F M M

M A = 0 suggests that R

must pass through A

M B = 0 suggests equilibrium

if AB is not to x axis

• Alternative 2:

– Moment balances about three points A, B, and C

Alternative Formulations for Equilibrium

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0, 0, 0 A B C

M M M

When equilibrium is expressed using

a set of dependent equations (more

equations than necessary), you mayend up with a trivial force or moment

balance equation of the form 0 = 0


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Equilibrium

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Example: Equilibrium in 2-D

• Determine the forcesC and T acting on the bridge-

truss

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• Ignoring the weights of the pulleys, determine the

tension T in the cable

• Find the the total force on the bearing of the pulley

C

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• Determine the tension P in the cable for lifting the

100 kg beam’s point B 3 m above A

• Determine the reaction at support A

• What is the angle made with the horizontal

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• Determine the magnitude T of the force in the cable

supporting the I-beam with a mass of 95 kg/m

• What is the reaction force at support A

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• Determine the forces exerted at the ball supports at

points A and B on the 200 kg beam

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• Determine the mass m that can be supported by the

200 N force applied at the handle

• Compute the radial force exerted on the shaft by

each bearing

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• Ignoring the weight of the frame shown below,

determine the tension in the cable CD

• Determine the reaction forces at loose-fitted ring B

•

Calculate the reaction forces at the ball-and-socket joint at point A

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Constraints

• A constraint is a restriction of motion

– Example: A roller is free to move horizontally

(no horizontal constraint)

A pin cannot move vertical or horizontal

(horizontally constrained)

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• The force and moment balance equations are

necessary and sufficient for equilibrium but may not

be adequate to determine all the unknown forces

Statical Determinacy

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• A mechanical system that has more constraints than

necessary to maintain an equilibrium is termed

statically indeterminate


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• Constraints that can be removed without disturbing

equilibrium are termed redundant constraints

• The number of redundant constraints is termed

degree of statical determinacy

• We will mostly deal with statically determinant

problems

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Degree of Statical Unknown External Number of IndependentEquilibrium EquationsDeterminacy Forces

Unknown External Number of IndependentEquilibrium EquationsForces


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Adequacy of Constraints

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Adequacy of Constraints

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Next Lecture

• Lecture topics

– Structures

• Reading assignment: Ch.5 in textbook

• Questions?

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ch.3 equilibrium.pdf

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