Chapter 7 Trusses, Frames, and Machines - Drexel …cac542/L13.pdf · 1 MEM202 Engineering...
Transcript of Chapter 7 Trusses, Frames, and Machines - Drexel …cac542/L13.pdf · 1 MEM202 Engineering...
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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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7.2 Plane TrussesMethod of Joints
Reactions
( ) ( )( ) ( )( )( ) ( )( )
01000
0100048
020008100048
=+=
=−−=
=−−=
∑∑∑
xx
yB
yA
AF
AM
BM
lb 2500lb 500lb 1000
=
−=−=
y
y
x
BAA
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7.2 Plane TrussesMethod of Joints
⎩⎨⎧
=−+==−+=
∑∑
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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7.2 Plane TrussesMethod of Joints
? 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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7.2 Plane TrussesMore About Zero-Force Members
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)
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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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7.2 Plane TrussesMore About Zero-Force Members
Washington Crossing Bridge
Tension-only members