PLANE AND SPACE TRUSSES
A truss may be defined as a plane structure composed of a number of
members joined together at their ends by smooth pins so as to form a rigid
framework. Each member in a truss is a two-force member and is
subjected only to direct axial forces (tension or compression).
Common trusses may be classified according to their formation as simple,
compound and complex.
SIMPLE TRUSS
A simple truss is formed by a basic triangle; each new joint is connected to
the basic triangle by two new bars.
A rigid plane truss can always be formed by beginning with three bars
pinned together at their ends in the form of a triangle.
A number of typical trusses are shown below.
Typical Bridge Trusses
Typical Roof Trusses
METHOD OF JOINTS
In this method, a truss is analyzed by considering the equilibrium of each
pin successively, starting with a joint at which only two forces are unknown.
Steps:
1. Draw a free-body diagram of the entire truss, and use this diagram to
determine the reactions at the supports.
2. Locate a joint connecting only two members, and draw the free-body
diagram of its pin. Use this free-body diagram to determine the unknown
force in each of the two members. A positive answer means that the
member is in tension, a negative answer that the member is in
compression.
3. Next, locate a joint where the forces in only two of the connected
members are still unknown. Draw the free-body diagram of the pin and
use it as indicated above to determine the two unknown forces.
4. Repeat this procedure until the forces in all the members of the truss have
been found.
5. Note that the choice of the first joint is not unique. Once you have
determined the reactions at the supports of the truss, you can choose
either of two joints as a starting point for your analysis.
Sample Problem #1:
Using the method of joints, determine the force in each member of the
truss shown. State whether each member is in tension or compression.
METHOD OF SECTIONS
The method of joints is most effective when the forces in all the members
of a truss are to be determined. If, however, the force in only one member
or the forces in a very few members are desired, another method, the
method of sections, is more efficient.
Sample Problem #2:
A Warren bridge truss is loaded as shown. Determine the force in members
EG, FG, and FH.
MAXWELL DIAGRAM
The Cremona diagram, also known as the Cremona-Maxwell method, is a
graphical method used in the analysis of trusses to determine the forces in
members.
Two steps are involved in the graphic method of stress analysis of trusses:
1. The external reaction components must be determined by either
the algebraic or graphic methods.
2. The internal stresses in all members can be obtained from the stress
diagram, which is the superposition of all the individual force
polygons for the concurrent-force systems acting on each joint.
Each space between two adjacent external forces is labeled with a
letter in consecutive order, by starting with the letter “a” and then
proceeding in the clockwise direction.
Each triangle is labeled with a numeral by starting with the number “1”
and proceeding from left to right.
The two numbers, or one letter and one number, on opposite sides of
each member are used to represent the magnitude and direction of the
stress in the member.
Illustration:
COMPOUND TRUSS
A compound truss is formed from two or more simple trusses connected
together as one rigid framework either by three links neither parallel nor
concurrent, or by a link and a hinge.
COMPLEX TRUSS
A complex truss is a type of truss where you can't identify it either as simple
or compound. The arrangement of the members and joints form a
complicated framework in such sense that the method of joints and
sections don't help in analyzing the structure.
METHOD OF SUBSTITUTE MEMBERS:
Procedure for analysis:
1) Determine the reactions at the supports of truss.
2) Imagine how to analyze the truss by the method of joint by
removing a member from a joint with three members and replace
it by an imaginary member elsewhere in the truss.
3) Determine the forces (S'i) in all members due to the external loads
using joint method.
4) Remove the external loading and place equal but opposite
collinear unit load on the truss at the two joints from which the
member was removed.
5) Determine the force (si) in each member due to the unit load.
6) If the effects of the above two loadings are combined, the force in
the i th member of the truss will be Si = S’i + Xsi.
7) Use the equation to find the force in all members by substituting the
value of x obtained in the previous step.
Problem
Determine the force in all the members of the complex truss. State if the
members are in tension or compression.
SPACE TRUSSES
When several straight members are joined together at their extremities to
form a three-dimensional configuration, the structure obtained is called a
space truss.
The most elementary rigid space truss
consists of six members joined at their
extremities to form the edges of a
tetrahedron ABCD.
By adding three members at a time to this basic configuration, such as AE,
BE, and CE, attaching them to three existing joints, and connecting them
at a new joint, we can obtain a larger rigid structure which is defined as a
simple space truss.
Although the members of a space truss are actually joined together by
means of bolted or welded connections, it is assumed that each joint
consists of a ball-and-socket connection. Thus, no couple will be applied
to the members of the truss, and each member can be treated as a twoforce member. The conditions of equilibrium for each joint will be
expressed by the three equations ∑Fx = 0, ∑Fy = 0, and ∑Fz = 0. However,
to avoid the necessity of solving simultaneous equations, care should be
taken to select joints in such an order that no selected joint will involve
more than three unknown forces.
Problem:
Determine the force in each member of
the space truss and state if the members
are in tension or compression. The truss is
supported by ball-and-socket joints at A,
B, C, and D.