INTRODUCTION - Arizona State University
6. STRUCTURES IN EQUILIBRIUM
6.1 TRUSSES
Trusses - structures supported and loaded at their joints (places where the bars are pinned together), with bar weights neglected
- each bar is a two-force member, so the forces at the ends of a member
must be equal in magnitude, opposite in direction, and directed along the line
between the joints (the force is called the axial force in the member);
when the forces are directed away from each other, the member is in tension,
when the forces are directed toward each other, the member is in compression
- examples: structures used to support bridges and the roofs of houses
6.2 The method of joints - preferred approach when the axial forces in all members
are to be determined
- involves drawing free-body diagrams of the joints of a
truss one by one, and using the equilibrium equations to
determine the axial forces in the members.
Procedure:
1. Draw a free-body diagram of the entire truss (treat the truss as a single object) and
determine the reactions at its supports.
2. Choose a joint (isolate it by “cutting” members pinned at considering joint) and
draw its free-body diagram (suppose the directions of unknown axial forces as
tension forces for a member).
3. Use the equilibrium equations to determine the unknown forces (if the result/force
is positive, the member is in tension, and if negative – in compression); for 2-D
problems, there are only 2 independent equilibrium equations (the forces are
concurrent and summing the moments about a point does not result in an additional
independent equation).
4. Repeat 2. and 3. for other joints
Particular types of joints:
Truss joints with two collinear members and no load
The sum of forces must equal zero ( T1 = T2 (the axial forces are equal)
Truss joints with two noncollinear members and no load
From the sum of the forces for the joint ( T2 = 0 = T1 (the axial forces are zero)
Truss joints with three members, two of which are collinear, and no load
From the sum of the forces in the x-direction ( T3 = 0, and from the sum of the
forces in the y-direction ( T1 = T2 (the axial forces in the collinear members are
equal, and the axial force in the third member is zero)
6.3 The method of sections - preferred approach when the axial forces only in a
few members are to be determined
- involves drawing a single free-body diagram of a
section of the truss, and using the equilibrium
equations to determine the axial forces in specific
members
Procedure:
1. Draw a free-body diagram of the entire truss (treat the truss as a single object) and
determine the reactions at its supports.
2. Choose a section - cut several members (usually no more then 3) including member(s)
whose axial force(s) are to be determined, and draw a free-body diagram of a section
(suppose the directions of unknown axial forces as tension forces for a member).
3. Use the equilibrium equations to determine the unknown forces
(if the result is positive, the member is in tension, and if negative – in compression);
for 2-D problems, there are 3 independent equilibrium equations
(the forces are not usually concurrent and summing the moments about a point does
result in an additional independent equation).
6.4 Space trusses
Supports
6.5 Frames and machines
Analyzing the Entire Structure - objects subjected to a system of forces and
Analyzing the Members - objects subjected to a system of forces and moments
Two-force Members
Loads applied at joints
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