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3 Methods for Truss Analysis
Before discussing the
various methods of truss
analysis, it would be
appropriate to have a brief
introduction.
A structure that is composed
of a number of bars pin
connected at their ends to
form a stable framework is
called a truss. It is generally
assumed that loads and
reactions are applied to the truss only at the joints. A truss would
typically be composed of triangular elements with the bars on the
upper chord under compression and those along the lower chord
under tension. Trusses are extensively used for bridges, long span
roofs, electric tower, and space structures.
Trusses are statically determinate when the entire bar forces can
be determined from the equations of statics alone. Otherwise the
truss is statically indeterminate. A truss may be statically (externally)
determinate or indeterminate with respect to the reactions (more
than 3 or 6 reactions in 2D or 3D problems respectively).
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2. Sign convention
For truss
analysis, it is
assumed that:
Bars are
pin-connected.
Joints are frictionless hinges.
Loads are applied at the joints only.
Stress in each member is constant along its length.
The objective of truss analysis is to determine the reactions and
member forces. The methods used for carrying out the analysis with
the equations of equilibrium and by considering only parts of the
structure through analyzing its free body diagram to solve the
unknowns.
1. Method of Joints for Truss Analysis
We start by assuming that all members are in tension reaction. A
tension member experiences pull forces at both ends of the bar and
usually denoted by positive (+ve) sign. When a member is
experiencing a push force at both ends, then the bar is said to be in
compression mode and designated as negative (-ve) sign.
In the joints method, a virtual cut is made around a joint and the cut
portion is isolated as a Free Body Diagram (FBD). Using the
equilibrium equations of ∑ Fx = 0 and ∑ Fy = 0, the unknown
member forces can be solved. It is assumed that all members are
joined together in the form of an ideal pin, and that all forces are in
tension (+ve reactions).
An imaginary section may be completely passed around a joint in a
truss. The joint has become a free body in equilibrium under the
forces applied to it. The equations ∑ H = 0 and ∑ V = 0 may be
applied to the joint to determine the unknown forces in members
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3. RECOMMENDED
meeting there. It is evident that no more than two unknowns can be
determined at a joint with these two equations.
A simple truss model supported by pinned and roller support
at its end. Each triangle has the same length, L and it is
equilateral where degree of angle, θ is 60° on every angle.
The support reactions, Ra and Rc can be determined by
taking a point of moment either at point A or point C,
whereas Ha = 0 (no other horizontal force).
Here are some simple guidelines for this method:
Firstly draw the Free Body Diagram (FBD),1.
Solve the reactions of the given structure,2.
Select a joint with a minimum number of unknown (not more
than 2) and analyze it with ∑ Fx = 0 and ∑ Fy = 0,
3.
Proceed to the rest of the joints and again concentrating on
joints that have very minimal of unknowns,
4.
Check member forces at unused joints with ∑ Fx = 0 and ∑ Fy
= 0,
5.
Tabulate the member forces whether it is in tension (+ve) or
compression (-ve) reaction.
6.
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4. The figure showing 3 selected joints, at B, C, and E. The
forces in each member can be determined from any joint or
point. The best way to start is by selecting the easiest joint
like joint C where the reaction Rc is already obtained and
with only 2 unknown, forces of FCB and FCD. Both can be
evaluated with ∑ Fx = 0 and ∑ Fy = 0 rules. At joint E, there
are 3 unknown, forces of FEA, FEB and FED, which may lead
to more complex solution compared to 2 unknown values.
For checking purposes, joint B is selected to show that the
equation of ∑ Fx is equal to ∑ Fy which leads to zero value, ∑
Fx = ∑ Fy = 0. Each member’s condition should be indicated
clearly as whether it is in tension (+ve) or in compression
(-ve) state.
Trigonometric Functions:
Taking an angle between member x and z…
Cos θ = x / z1.
Sin θ = y / z2.
Tan θ = y / x3.
2. Method of Sections for Truss Analysis
The section method is an effective method when the forces in all
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members of a truss are to be determined. If only a few member
forces of a truss are needed, the quickest way to find these forces
is by the method of sections. In this method, an imaginary cutting
line called a section is drawn through a stable and determinate
truss. Thus, a section subdivides the truss into two separate parts.
Since the entire truss is in equilibrium, any part of it must also be in
equilibrium. Either of the two parts of the truss can be considered
and the three equations of equilibrium ∑ Fx = 0, ∑ Fy = 0, and ∑ M =
0 can be applied to solve for member forces.
Using the same model of simple truss, the details would be
the same as previous figure with 2 different supports profile.
Unlike the joint method, here we only interested in finding
the value of forces for member BC, EC, and ED.
Few simple guidelines:
Pass a section through a maximum of 3 members of the
truss, 1 of which is the desired member where it is dividing the
truss into 2 completely separate parts,
1.
At 1 part of the truss, take moments about the point (at a joint)
where the 2 members intersect and solve for the member
force, using ∑ M = 0,
2.
Solve the other 2 unknowns by using the equilibrium equation
for forces, using ∑ Fx = 0 and ∑ Fy = 0.
3.
Note: The 3 forces cannot be concurrent, or else it cannot be
solved.
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6. A
virtual cut is introduced through the only required members which is
along member BC, EC, and ED. Firstly, the support reactions of Ra
and Rd should be determined. Again a good judgment is required to
solve this problem where the easiest part would be to consider
either the left hand side or the right hand side. Taking moment at
joint E (virtual point) clockwise for the whole RHS part would be
much easier compared to joint C (the LHS part). Then, either joint D
or C can be considered as the point of moment, or else using the
joint method to find the member forces for FCB, FCE, and FDE.
Note: Each value of the member’s condition should be indicate
clearly as whether it is in tension (+ve) or in compression (-ve)
state.
3. Graphical Method of Truss Analysis
(Maxwell’s Diagram)
The method of joints could be used as the basis for a graphical
analysis of trusses. The graphical analysis was developed by force
polygons drawn to scale for each joint, and then the forces in each
member were measured from one of these force polygons.The
number of lines which have to be drawn can be greatly reduced,
however, if the various force polygons are superimposed. The
resulting diagram of truss is known as the Maxwell’s Diagram.
In order to draw the Maxwell diagram directly, here are the simple
guidelines:
Solve the reactions at the supports by solving the equations of
equilibrium for the entire truss,
1.
Move clockwise around the outside of the truss; draw the
force polygon to scale for the entire truss,
2.
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7. Take each joint
in turn
(one-by-one),
then draw a
force polygon
by treating
successive
joints acted
upon by only
two unknown
forces,
3.
Measure the
magnitude of
the force in
each member
from the diagram,
4.
Lastly, note that work proceeded from one end of the truss to
another, as this is use for checking of balance and connection
to the other end.
5.
A simple triangle truss with degree of angle, θ is 60°on every angle
(a equilateral) and same member’s length, L on 2 types of support.
Yet again, evaluating the support reaction plays an important role in
solving any structural problems. For this case, the value of Hb is
zero as it is not influenced by any horizontal forces.The procedure
for solving this problem could be quite tricky and requires
imagination. It starts by labeling the spaces between the forces and
members with an example shown above; reaction Ra and applied
force, P labeled as space 1 and continue moving clockwise around
the truss. For each member, take example between space 1 and 5
would be the member AC and so forth. Note: Choose a suitable
scale for drawing the Maxwell diagram.
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8. In conclusion, the truss internal reactions as well as its member
forces could be determined by either of these 3 methods.
Nonetheless, the methods of joints becomes the most preferred
method when it comes to more complex structures.
For more information on the topic:
☞Analysis of 2D Truss Structure in SAP 2000
☞Statically Determinate Trusses
☞Solved Examples for Method of Joints for Truss Analysis
☞Truss Analysis
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