Department of Mechanical Engineering

ENGR 0135

Chapter 6-2

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Equilibrium equations in 2D

Two-force membersStatically determinate problemsFrames and MachinesStatically indeterminate problems

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Equilibrium in two dimension (xy-plane)

Basic equations– Two force balance equations + 1 moment balance equation– The moment balance is about any point in the plane

Alternative set of equations– Maximum number of independent equations = 3– Alternatively,

» 1 force balance + 2 moment balance equations» 3 moment balance equations

0kC

0jiR

==

=+=

∑∑∑

z

yx

M

FF 000 === ∑∑∑ zyx MFF

Scalar equationsVector equations

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Two-Force Members

= frequently used in 2D (and 3D) structures= requirements:

– The ends of the member are pins– Member weight is neglected – The external forces act at the ends– The equilibrium of the member requires these force to be equal,

opposite, and collinear passes the ends of the member

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Two-force membersThe shape of the member determines the internal force

Straight member Only axial (normal) internal

force

Curved members May have normal, lateral

(shearing) and moment

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Statically determinate structure

The support reactions can be determined from the equilibrium equations

Number of unknowns < 3

Find the reaction (support) forces

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2D problem - example Given:

– Geometry– Forces– Mass of W = 100 kg

Assumption:– Member weight is neglected

Questions:– Reactions at the supports A

(pin) and B (roller)

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Solving the problem Construct the FBD

– Remove the supports– Replace them with reaction

forces– Identify the unknowns: Bx, Ax,

Ay

– Replace the weight with a vertical downward force

Utilize the force and moment balance equations

Solve the unknowns

000 === ∑∑∑ zyx MFF

Roller

Smooth pin

This structure is called a TRUSS (consists only of 2-force members)

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Many ways on how to solve the 2D determinate problems

Given– Structure geometry– Weight of the ladder AB (W) = 250 lb– Notice: point D = intersection of the lines of

action of reaction forces at A and C

Questions:– Determine the support reactions

» Contact at C normal to the surface» Cable tension along the cable, in

tension» Contact at A normal to surface

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Many ways on how to solve the 2D determinate problems

Method 1– Use 2 force balance equations and 1

moment equation about A

Method 2– Use 3 moment equations about D, C, and

A (why about these points?)

Method 3– Use moment equation about D and two

force balance equations

000 === ∑∑∑ zyx MFF

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Problems involving curved 2-force members

Pins

FBD

Four unknonws:Ax, Ay, Bx, By

Note: Member AC is a 2-force member.The reaction force must pass both ends.

3 unknowns: A, Bx, By

C

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Problems involving distributed load

First, find the resultant force and its directionand location of the distributed load

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Problems involving pulleys

Pulleys change the direction of the cable tension

It is assumed to be frictionless It does not change the

magnitude

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Problems involving pulleys

Given:– Weight of the person– Frictionless pulleys

Questions:– The force must be exerted on the rope

to support himself

In the text:– Uses two FBDs– Two force balance equations (both in

vertical direction)

WTT =+ 21

02 21 =−TTWT =13

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Alternative solution

W

TT

T

WT =3

They are coming from the same cable the same magnitude