Determine the reactions, R1 and R2, of a rigid body subjected to a pair of forces F1, and F2.
In this chapter
R2
R1
F1
F2
R2
Determine the reactions, R1 and 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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MEM202 Engineering Mechanics - Statics
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7.2 Plane Trusses Idealized trusses 1. 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. 3
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MEM202 Engineering Mechanics - Statics
7.2 Plane Trusses A triangle is the building block of all plane trusses 1 A
B
B
3
4 3 5
1
2 C
A
2 C
D
F 8
7
9
6
m = 2j−3
12 11 13
10
H
G
E
m = NUMBER OF MEMBERS; j = NUMBER OF JOINTS Total 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? 4
MEM
MEM202 Engineering Mechanics - Statics
7.2 Plane Trusses A triangle is the building block of all plane trusses 1 A
B
B
3
4 3 5
1
2 C
A
2 C
D
F 8
7
9
6
m = 2j−3
12 11 13
10
H
G
E
m = NUMBER OF MEMBERS; j = NUMBER OF JOINTS Total 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? 5
MEM202 Engineering Mechanics - Statics
MEM
7.2 Plane Trusses Rigidity and Solvability of A Truss
m=2j−r m: # of Members j: # of Joints r: # of Reactions
m = 2j−3
m=2j−r
j = 9 m = 15 2 j − 3 = 15 = m
j = 9 m = 15 r = 3 2 j − r = 15 = m
j = 9 m = 14 2 j − 3 = 15 ≠ m
j = 9 m = 14 r = 4 2 j − r = 14 = m
j=6 m=9 2j−3= 9 = m
j=6 m=9 r=3 2j−r =9=m
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MEM202 Engineering Mechanics - Statics
MEM
7.2 Plane Trusses m=2j−r j = 8 m = 12 r = 3 2 j − r = 13 > m = 12 j = 8 m = 14 r = 3 2 j − r = 13 < m = 14 j = 10 m = 16 r = 3 2 j − r = 17 > m = 16
j = 8 m = 13 r = 3 2 j − r = 13 = m
Yet, it is unstable!
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MEM202 Engineering Mechanics - Statics
MEM
7.2 Plane Trusses Method 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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MEM202 Engineering Mechanics - Statics
MEM
7.2 Plane Trusses Method of Joints Reactions
∑ M = (8)B − (4)(1000) − (8)(2000) = 0 ∑ M = −(8)A − (4)(1000) = 0 ∑ F = A + 1000 = 0 A
y
B
x
y
x
Ax = −1000 lb Ay = −500 lb B y = 2500 lb 9
MEM202 Engineering Mechanics - Statics
MEM
7.2 Plane Trusses Method of Joints ⎧∑ F A: ⎨ 2000 lb ⎩∑ F D
D
TCD
C TCD
TAD TCD TCD
TAC
TAD
C
D
C
A
B
TAD
TAC
A
TAD
TAC
A
1000 lb A TAB TAB 500 lb
TBC TAB TAB
y
= TAD + TAC sin θ − 500 = 0
⎧∑ Fx = −TCD − TAC cos θ = 0 C:⎨ ⎩∑ Fy = −TBC − TAC sin θ − 2000 = 0 ⎧∑ Fx = TCD + 1000 = 0 D:⎨ ⎩∑ Fy = TAD = 0
