Chapter 7 Trusses, Frames, and Machines - Drexel …cac542/L13.pdf · 1 MEM202 Engineering Mechanics - Statics MEM Chapter 7 Trusses, Frames, and Machines

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MEM202 Engineering Mechanics - Statics MEM

Chapter 7Trusses, Frames, and Machines

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7.2 Plane TrussesBefore this chapter In this chapter

1F 2F1R 2R1F 2F

1R 2R

Determine the reactions, R1and R2, of a rigid body subjected to a pair of forces F1, and F2.

Determine the reactions, R1and R2, and the forces in nine rigid members that are joined together with six pin joints, subjected to a pair of forces F1, and F2.

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7.2 Plane Trusses

Idealized trusses1. Members are connected together at their ends only.2. Members are connected together by frictionless pins.3. Loads are applied only at the joints. (Thus, all

members are two-force members.)4. Weights of members are neglected.

An actual riveted truss joint, which transmits both forces and moments among connecting members

An idealized frictionless pin connection, which transmits forces among connecting members, but not moments. This assumption can be justified so long as the members are long.

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7.2 Plane TrussesA triangle is the building block of all plane trusses

A

B

C

DF H

G

E

12

345

67

8

911

10

1213

32 −= jmA

B

C

12

3

m = NUMBER OF MEMBERS; j = NUMBER OF JOINTSTotal unknowns: m (one for each member) + 3 (3 support reactions).

Each joint yields two equations (ΣFx = 0, ΣFy = 0)Is a truss always “stable” and “solvable” when m = 2j -3 is satisfied?

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7.2 Plane TrussesA triangle is the building block of all plane trusses

A

B

C

DF H

G

E

12

345

67

8

911

10

1213

32 −= jmA

B

C

12

3

m = NUMBER OF MEMBERS; j = NUMBER OF JOINTSTotal unknowns: m (one for each member) + 3 (3 support reactions).

Each joint yields two equations (ΣFx = 0, ΣFy = 0)Is a truss always “stable” and “solvable” when m = 2j -3 is satisfied?

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7.2 Plane TrussesRigidity and Solvability of A Truss

32 −= jm rjm −= 2

rjm −= 2

m: # of Membersj: # of Jointsr: # of Reactions

mjmj

==−==

1532159

mrjrmj

==−===

1523159

mjmj

≠=−==

1532149

mrjrmj

==−===

1424149

mjmj

==−==

93296

mrjrmj

==−===

92396

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7.2 Plane Trussesrjm −= 2

121323128

=>=−===

mrjrmj

141323148

=<=−===

mrjrmj

1617231610

=>=−===

mrjrmj

mrjrmj

==−===

1323138

Yet, it is unstable!

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7.2 Plane TrussesMethod of Joints

1. Draw a free-body diagram of the entire structure and determine the reactions (if r = 3).

2. Draw free-body diagrams for all members (assume tensile forces in all members) and all joints.

3. Set up the equilibrium equations for each joint and solve them one joint at a time, begin with those that have at most two unknowns.

4. Check the results at the last joint.

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Reactions

( ) ( )( ) ( )( )( ) ( )( )

01000

0100048

020008100048

=+=

=−−=

=−−=

∑∑∑

xx

yB

yA

AF

AM

BM

lb 2500lb 500lb 1000

=

−=−=

y

y

x

BAA

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⎩⎨⎧

=−+==−+=

∑∑

0500sin01000cos

:θθ

ACADy

ACABx

TTFTTF

A

⎩⎨⎧

=+===

∑∑

025000

:BCy

ABx

TFTF

B

⎩⎨⎧

=−−−==−−=

∑∑

02000sin0cos

:θθ

ACBCy

ACCDx

TTFTTF

C

⎩⎨⎧

===+=

∑∑

001000

:ADy

CDx

TFTF

D

( ) ( )( ) ( )BCABACAB

BCACCDAD

TTBTTATTCTTD

,,,,

⇒⇒⇒

AB

CD

ACT

C

AACT

ACT

ACT

ABT

A B

ABT ABT ABT

ADT

ADT CDT

CDTCD

CDT CDT

BCT

BCT

lb 2500lb 500

lb 1000

BCT

BCT

B

C

ADT

ADTD

A

lb 2000

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? 2 ? rjm −=

mrjrmj

==−===

52354

542454

=<=−===

mrjrmj

mrjrmj

==−===

52354

542454

=<=−===

mrjrmj

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7.2 Plane TrussesZero-Force Members

AB

C

D

E

P

AB

C

D

E

P

AB

C

D

E

P

C

CDT ACT

CET

x′y′

BBDT ABT

BCT

00 =⇒=∑ ′ CDy TF00 =⇒=∑ BCy TF

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7.2 Plane TrussesZero-Force Members

AB

C

D

E

AB

C

D

E

AB

C

D

E

Zero-force member:AB, AC, BC

Zero-force member:AB, AC

Zero-force member:BC, CD

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7.2 Plane TrussesMore About Zero-Force Members

Zero force members cannot simply be removed from the truss and discarded, as they are needed to guarantee stability of the truss.

00sinsin

0coscos==⇒

⎪⎭

⎪⎬⎫

−=⇒=+=

=⇒=+−=

∑∑

CDDECDDECDDEy

CDDECDDEx TTTTTTF

TTTTF

φφ

φφ

Yet equilibrium at joint C requires that TCD ≠ 0. Thus, joint D will continue to buckle outward and the truss is no longer stable.

If members BD and AD are removed, then a slight disturbance would cause joint D to buckle outward. The equilibrium at joint D (see the free-body diagram) requires that

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Zero-force members may become non-zero-force members, or vice versa, as the load moves from one location to another.Consider a “Warren” truss with vertical supports.(See http://www.geocities.com/Baja/8205/truss.htm for other types of bridge truss)

http://www.geocities.com/Baja/8205/truss.htm

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7.2 Plane TrussesOther Bridge Trusses

Warren Truss w/o Vertical Supports

Pratt Truss

Quadrangular Warren Trusses

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Washington Crossing Bridge

Tension-only members

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Cross members are slender tension-only members. Any compressive force will buckle the member, render it useless.

C C C C C CT T T T T T

Chapter 7 Trusses, Frames, and Machines - Drexel …cac542/L13.pdf · 1 MEM202 Engineering Mechanics - Statics MEM Chapter 7 Trusses, Frames, and Machines

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