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