ENGR221 Lecture 10

8/8/2019 ENGR221 Lecture 10

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Equilibrium and TrussesEquilibrium and Trusses

ENGR 221

February 17, 2003


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Lecture Goals

Lecture Goals

6-4

Equilibrium in Three Dimensions 7-1 Introduction to Trusses

7-2 Plane Trusses

7

-3 Space Trusses

7-4 Frames and Machines


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EquilibriumEquilibrium ProblemProblemDetermine the reactions

at A and the force in bar

CD due to the loading.


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EquilibriumEquilibrium ProblemProblemDraw the free-body

diagram of the main

body.

1 o6 in.tan 26.56512 in.

U ! !

RAx

RAy

TCD

U


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EquilibriumEquilibrium ProblemProblemLook at equilibrium

o

x x

o

x

o

y y

o

y

cos 26.565 125 lb 0

cos 26.565 125 lb

sin 26.565 40 lb 60 lb 80 lb 0

sin 26.565 20 lb

F R T

R T

F R T

R T

! !!

! !

!


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EquilibriumEquilibrium

ProblemProblemTake the moment about A

RAx

RAy

TCD

U

o

A CD

o

CD

CD

0 cos 26.565 6 in. 40 lb 4 in.

60 lb 8 in. 80 lb 12 in.

cos 26.565 6 in. 320 lb-in

59.628 lb

M T

T

T

! !

!

!


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EquilibriumEquilibrium

ProblemProblemTake the moment about A

RAx

RAy

TCD

U

o

x

o

y

59.628 lb cos 26.565 125 lb

71.667 lb

59.628 lb sin 26.565 20 lb

6.667 lb

R

R

!

! !

!


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Equilibrium in 3Equilibrium in 3--DimensionsDimensionsIn two dimensions, the equations are solved

using the summation of forces in the x, y and z

directions and the moment equilibrium includes

moment components in the x, y and z directions.

x y z0 0 0 F F F ! ! !

x y z0 0 0M M M! ! !


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TrussesTrusses --DefinitionDefinitionTrusses are structures

composed entirely of two

force members . They

consists generally of

triangular sub-element and

are constructed and

supported so as to prevent

any motion.


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FramesFrames --DefinitionDefinition

Frames are structures that

always contain at least one

member acted on by forces

at three or more points.

Frames are constructed and

supported so as to prevent

any motion. Frame like

structures that are not fully

constrained are called

machines or mechanisms.


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TrussTrussPlanar Trusses - lie in a

single plane and all

applied loads must lie in

the same plane.


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TrussTrussSpace Trusses - are structures that are not

contained in a single plane and/or are loaded out

of the plane of the structure.


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TrussTrussThere are four main assumptions made in the

analysis of truss

Truss members are connected together at theirends only.

Truss are connected together by frictionless

pins.

The truss structure is loaded only at the joints.

The weights of the members may be neglected.

1

2

3

4


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Simple Truss

Simple Truss

The basic building block of a

truss is a triangle. Large truss

are constructed by attachingseveral triangles together A

new triangle can be added

truss by adding two members

and a joint. A trussconstructed in this fashion is

known as a simple truss.


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Simple Truss

Simple Truss

It has been observed that the analysis of truss

can be done by counting the number member

and joints on the truss to determine the truss isdeterminate, unstable or indeterminate.


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Simple Truss

Simple Truss

A truss is analysis by using m=2*j-3, where m is

number of members, j represents the number of

joints and 3 represents the external supportreactions.


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Simple Truss

Simple Truss

If m< 2j-3, then the truss is unstable and will

collapse under load.

If m> 2j-3, then the truss has more unknowns

than know equations and is an indeterminate

structure.

If m= 2j-3, ensures that a simple plane truss is

rigid and solvable, it is neither sufficient nor

necessary to ensure that a non-simple plane truss

is rigid and solvable.


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Simple Truss

Simple Truss-- IdentifyIdentify

Determine type of simple truss is it

determinate, indeterminate or unstable.


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Method of Joints

Method of Joints --TrussTruss

The truss is made up of single bars, which are

either in compression, tension or no-load. The

means of solving force inside

of the truss use equilibriumequations at a joint. This

method is known as the

method of joints.


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Method of Joints

Method of Joints --TrussTrussThe method of joints uses the summation of

forces at a joint to solve the force in the

members. It does not use the

moment equilibrium equationto solve the problem. In a two

dimensional set of equations,

In three dimensions,

x y0 0F F! !

z 0F !


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Method of Joints

Method of Joints ExampleExample

Using the method of

joints, determine the

force in each member ofthe truss.


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Method of Joints

Method of Joints ExampleExample

Draw the free body

diagram of the truss and

solve for the equations

x x

x

y y

y

0

0 lb

0 2000 lb 1000 lb

3000 lb

F C

C

F E C

E C

! !

!

! !

!


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Method ofMethod of

JointsJoints ExampleExample

Solve the moment about C

C

y

0 2000 lb 24 ft 1000 lb 12 ft 6 ft

10000 lb

C 3000 lb 10000 lb 7000 lb

M E

E

! !

!

! !


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Method ofMethod of


Look at joint A

y AD

AD AD

x AD AB AB

AB AB

40 2000 lb

5

2500 lb 2500 lb C

3 30 2500 lb

5 5

1500 lb 1500 lb T

F F

F F

F F F F

F F

! !

! !

! ! !

! !


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Method ofMethod of


Look at joint D

y

x

4 4 4 40 2500 lb

5 5 5 5

2500 lb 2500 lb T

3 30

5 5

3 32500 lb 2500 lb

5 5

3000 lb 3000 lb

F F F F

F F

F F F F

F

F F

! ! !

! !

! !

!

! !


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Method ofMethod of


Look at joint B

y

x

4 4

0 1000 lb5 5

4 42500 lb 1000 lb

5 5

3750 lb 3750 lb

3 305 5

3 32500 lb 1500 lb 3750 lb

5 5

5250 lb 5250 lb T

F F F

F

F F

F F F F F

F

F F

! !

!

! !

! !

!

! !


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Method ofMethod of


Look at joint E

y

x

4 4

0 10000 lb5 5

4 43750 lb 10000 lb

5 5

8750 lb 8750 lb

3 305 5

3 33750 lb 3000 lb

5 5

8750 lb 8750 lb

F F F

F

F F

F F F F

F

F F

! !

!

! !

! !

!

! !


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Method ofMethod of


Look at joint C to check

the solution

y CE

x CE CB x

40 7000 lb

5

48750 lb 7000 lb 0 OK!

5

305

38750 lb 5250 lb 0 0

5

F F

F F F C

! !

! !

! !

! !


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Method of Joints

Method of Joints Class ProblemClass Problem

Determine the forces BC,

DF and GE. Using themethod of Joints.


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Method of

Sections

Method of

Sections --TrussTruss

The method of joints is most effective when

the forces in all the members of a truss are to

be determined. If however, the force is only

one or a few members are needed, then the

method of sections is more efficient.


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Method of

Sections

Method of

Sections --TrussTruss

If we were interested in the

force of member CE. We

can use a cutting line or

section to breakup the truss

and solve by taking the

moment about B.


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Method of

Sections

Method of

Sections ExampleExample

Determine the forces in members FH, GH and GI

of the roof truss.


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Method of

Sections

Method of


Draw a free body diagram and solve for the

reactions.

RAx

RAy

L

x x

x

y

y

0

0 k

0

20 k

F R

R

F

L R

! !

!

!

!


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Method ofMethod of

Sections


Solve for the

moment at A.

RAx

RAy

L

A

Ay

6 kN 5 m 6 kN 10 m 6 kN 15 m

1 kN 20 m 1 kN 25 m 30 m

7.5 kN

12.5 kN

M

L

L

R

!

!

!


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Method of

Sections

Method of


Solve for the member GI. Take a cut between the

third and fourth section and draw the free-bodydiagram.

HI HI

HI

1 o

8 m 10 m8 m

15 m 10 m 15 m

5.333 m

8 mtan 28.1

15 m

ll

l

E

! !

!

! !


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Method ofMethod of

Sections


The free-body diagram of

the cut on the right side.

H GI

GI GI

1 k 5 m 7.5 k 10 m 5.333 m

13.13 k 13.13 k T

M F

F F

!

! !


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Method of

Sections

Method of


Use the line of action of the forces and take the moment

about G it will remove the FGI and FGH and shift FFH to the

perpendicular of G.


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Method ofMethod of

Sections


Take the moment at G

G

o

FH

FH FH

1 kN 5 m 1 kN 10 m 7.5 kN 15 m

cos 28.1 8 m

13.82 kN 13.82 kN C

M

F

F F

!

! !


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Method of

Sections

Method of


Use the line of action of the forces and take the moment

about L it will remove the FGI and FFH and shift FGH to

point G.

1 o5 mtan 133.25.333 m

F ! !


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Method ofMethod of

Sections


Take the moment at L

o

L GH

GH GH

1 k 5 m 1 k 10 m cos 43.2 15 m

1.372 k 1.372 k

M F

F F

!

! !


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Method ofSectionsMethod ofSections ClassClass

ProblemProblem

Determine the forces in members CD and CE using method

of sections.


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H

omework (Due 2/24/03)H

omework (Due 2/24/03)Problems:

6-34, 6-37, 6-38, 6-40, 6-45, 6-63


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TrussTruss Bonus ProblemBonus ProblemDetermine whether the

members are unstable,

determinate orindeterminate.


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TrussTruss Bonus ProblemBonus ProblemDetermine the loads in

each of the members.


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ENGR221 Lecture 10

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