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Chapter 9 : Analysis of Trusses
- 9.1 Definition of trusses
- 9.2 Properties of 2-force members
- 9.3 Method of Joints
- 9.4 Method of Sections
- 9.5 Compound Trusses
- 9.6 Trusses in 3-D
- 9.7 Summary
- 9.8 Self-Test and computer Program TRUSS
Trusses are structures consisting of two or more straight, slender members
connected to each other at their endpoints. Trusses are often
used to support roofs, bridges,
power-line towers, and appear in many other applications.
Here is a collection of various
structures involving trusses I have come across.
Object of our calculations is to determine the external support forces
as well as the forces acting on each of the members for given external
In order to make calculations possible a few assumptions are made which
in most cases reflect reality sufficiently close so that our
theoretical results match experimentally determined ones sufficiently
accurate. These assumptions pertain to two- as well as three-dimensional
trusses. The three assumptions (or maybe better called idealizations)
If doubt arises that for a given design any of the three
assumptions may not reflect reality accurately enough
a more advanced analysis should be conducted.
- Each joint consists of a single pin to which the respective
members are connected individually.
In reality we of course find that members are connected
by a variety of means : bolted, welded, glued, rivited
or they are joined by gusset plates.
Here are some photos
of real-life joints.
- No member extends beyond a joint.
In Fig. 9.1a the schematic of a 2-dimensional truss
is shown. That truss consists of 9 members and 6 joints.
There is a member from joint A to B, another from joint B
to C, and a third from joint C to D.
In reality we may have a single beam extended all the way from
joint A to D, but if this beam is slender (long in comparison
to a lenght representing the size of its cross section)
it is permissible to think of this long beam being
represented by individual members going just from joint to
Fig. 9.1a :  Example of 2-D Truss
- Support forces (R1 and R2) and external loads
(P1 and P2) are only applied at joints.
In reality this may not quite be the case. But if for example
the weight of a member has to be taken into account we could
represent that by two forces each equal to half the weight
acting at either end point. In similar fashion one can assign
snow loads on roofs to single forces acting at the joints.
Click here for a glimpse
at some commonly employed trusses.
The three assumptions brought in the previous chapter render each individual
member of a truss to be what is called a "2-force member", that is a member
with only two points (usually the end-points) at which forces
As an example let's look at the member CE extracted from Fig. 9.1a
2-force member plus forces
as shown in Fig. 9.2a. I also show the joints C and E with the arrows
representing the forces exerted by the connected members onto each joint.
In red are entered the forces exerted by the member CE onto the two joints.
Acting on member CE we have the two
forces FCE and FEC, respectively, which
by the principle of action=reaction, are exactly equal but oppositely directed
to the (red) forces the member exerts on the two joints it is connected to.
Member CE has to be in equilibrium and therefore :
- In order for the sum of the moments about point C to be zero, the
line of action of force FEC has to go through point C.
- In order for the sum of the moments about point E to be zero, the
line of action of force FCE has to go through point E.
- In order for the sum of the forces in the direction of line CE to
be zero the two forces, FCE and FEC,
have to be equal but oppositely directed.
Note that the three points mentioned above pertain equally to two- and
2-force member with forces
Fig. 9.2b shows this in graphical form. The two forces acting on member CE
either pull or push at either endpoint in opposite direction with equal
strength. If they pull we say that the member is under tension, if they
push, it is said to be under compression. For the case inbetween, when
the forces at either endpoint are zero, we speak of a zero-force member.
This distinction is of great importance and you never should forget
to indicate tension, compression, and zero-force clearly for each member of
a truss when asked to determine the forces.
The reason for this distinction is a consequence of the different ways
a particular member of a truss can fail.
If a member is under tension the only failure mode occurs when the
forces trying to pull so hard that somewhere along the beam adjacent molecules/atoms cannot hold onto each other any longer and separate.
If a member is under compression two different types of failures can
occur : if the member is somewhat short and stubby molecules/atoms will
not be able to resists the external forces and the member will start
to crumble or deform to a shorter piece of material.
If on the other side the member is long and slender a
phenomenon called buckling may set in way before "crumbling" occurs. The
member simply does not want to stay straight anymore.
To prevent buckling we often employ
Nominally these members do not carry
any load but they prevent a member under compression from buckling by
providing lateral support.
The Method of Joints makes use of the properties of 2-force members as derived
in section 9.2 in an interesting way which I demonstrate
using the sample truss from section 9.1. For two-dimensional
this method results in a sequence of sets of two ( three) linear equations.
Fig. 9.3a shows this truss again with its geometry given in terms of
the angles alpha, beta, and gamma as well as the length
a,b, and c. The members AB, BC, CD, and EF are parallel to each other.
Method of joints
Assume that the forces P1 and P2 are known
I also entered the as of yet unknown support forces. Because at point A we
have a roller-type connection the support force R1 has
only a vertical component. At point D we have a pin/hole type connection
which gives rise to a vertical as well as horizontal component for
the support force R2.
Furthermore, I entered all forces ( in purple ) the 2-force members
exert on their respective joints. Remember that each member pulls/pushes
with equal force on its two joints. In the figure I labelled these forces
according to the labels of the joints involved and assumed that each member
pulls on each joint. I have done this just for the purpose of easy
book-keeping. For those members actually under compression the value
for the respective force will then come out to be negative.
(no need to go back into the drawing and change the direction of the arrow,
everybody in the business will see the negative sign of the answer and look
at your drawing and knows what's going on.)
Principle of Method
In the Method of Joints we consider now the equilibrium of each joint.
- For a 2-dimensional truss as shown here that gives us two equations for
each joint :
sum of the forces in horizontal and sum of the forces in the vertical
direction for example.
In the above example we have 6 joints and therefore get a total of
For a 3-D truss we have to satify 3 equilibrium equations for each joint.
- As far as unknowns is concerned we have one unknown force for each
of the 9 members and 3 unknown support forces for a total of 12
unknowns for our example.
A truss (2-D or 3-D) is statically determined only if the number of unknown
forces (one per member plus unkonowns stemming from the support forces) is
equal to the number of available equations ( 2 (3) times
the number of joints).
3 foot notes
- If the number of unknown forces exceeds the number of available equations
the truss is said to be statically undetermined, one needs more information
(usually about the way individual members deform under influence of forces)
to determine the forces.
- If the number of unknown forces is less than the number of available
equations the truss will collapse.
- On first sight one is tempted to think that by considering
the equilibrium of the entire truss more equations
can be derived and hence the number
of unknowns can be increased correspondingly. Unfortunately, as it
turns out, these new equations
are linearly dependent on the equilibrium equations on all
joints and therefore are automatically satified once the equilibrium
equations on all joints are satisfied.
On the good side, this redundancy can be used to tests your calculations
and/or to solve the system of equations faster.
Feel free to test your abilities to write out such equilibrium equations
and check against mine. For the truss shown in Fig. 9.3a I looked at the
equilibrium of each joint individually, just click on the latter in the
following list and compare my sketches and equations with yours :
Solving the Equation System
As an example I have summarized all 12 equations representing the equilibrium
conditions on the joints of the truss shown in Fig. 9.1a.
Click here for a closer look.
Mathematicians would classify this system as a system of linear
equations with constant coefficients ( the values of cos , sin of the various
angles) in which the forces are the unknowns. To solve such equation systems
various methods are available, many of them based on the Gauss-elimination
method or various matrix methods. I have written such a program in a web-based
format. ( Program Truss , 2-D version ,
3-D version ).
For many trusses, the example in Fig. 9.1a being no exception, it is possible
to solve for the unknowns forces "manually" by considering the joints in
a particular order which can be detected by inspection. Often it is necessary
to involve also the equilibrium equations for the entire truss
as shown here.
The principle of this method is to find by inspection ( of
Fig. 9.3a if you like
to work along ) a joint
which is acted upon by forces of which at most 2 forces (3 forces in 3-D)
Solve the equlibrium equations for this joint and repeat.
If you are lucky you can solve for all the unknown forces and then use
the equlibrium equations of the entire truss to check up on your results.
Quite often you will get "stuck" though (or even don't get started in the
first place). Don't dispair, here are two tricks which might help
you out and "deliver" a joint with only two unknown forces :
If you employ one or both of the above tricks and then solve subsequently
for the remaining unknown
forces you will be left with at least one joint the equilibrium of which
you do not need to consider. My recommendation : check the equilibrium
of this final joint anyway with your previously obtained values of the forces.
(Hey, that little bit of checking is better
than a bridge collapsing).
- Solve as many of the overall equilibrium equations as you can.
- Find zero-force members. Click here
if you want to find out how to do that (might save you later ?!).
If you like, click here to see the order
in which I would solve for the forces of the truss in Fig. 9.1a and read
some more useful info.
Well, does the manual version of the Method of Joints, including the
two tricks, always work ?
The answer is unfortunately NO, and
here is an example.
Problem 9.3a : 2-member truss
Problem 9.3b : 4-member truss
Problem 9.3c : 7-member truss
Problem 9.3d : Roof-truss, Fink, snow load
Problem 9.3e : Roof-truss, Howe, snow load
One disadvantage of the Method of Joints when
employed without the help of computer programs like
Program TRUSS is its sequential nature. That is,
in order to calculate forces based on the equilibrium equations on a particular
joint we have to use results of preceeding calculations. Hence errors
propagate and way too often get magnified in the process. In contrast to that,
the Method of Sections aims at calculating
the force of selected members directly and can therefore be used to
check results obtained by the Method of Joints (my favorite usage).
Additionally, in the absence of computer programs you find yourself
sometimes in the position that you have to jump-start the Method of
Principle of Method
The Method of Joints was used to analyze the forces in a truss by looking
at the equilibrium of its individual members ( discovering the properties
of two-force members ) and individual joints (to find equations to be solved
for the values of the forces individual members exert and the forces
supporting the truss).
In the Method of Sections we consider the equilibrium of a selected
part of a truss consisting of any number of members and joints. Often
this is done after the overall equilibrium equations have been solved.
Here I describe the method as it applies to two-dimensional trusses which
usually means that you will have to solve three equilibrium equations
which still can be done "manually". For three-dimensional
trusses this would result in six such equations.
As example, assume that our task is to find the force in the member CE
of the sample truss shown in Fig. 9.4a.
Also, assume that the geometry of the truss, the external loads and support
forces are known.
Our strategy is now to "mentally" remove three members
according to the following two rules :
- One of the members is the one the force of which you
wish to calculate.
- The removal of the three members has to divide the truss
into two separate sections.
Often you will have several equivalent choices. For the truss in Fig. 9.4a
there is only one, namely removal of the three members BC, CE, and EF.
You also might think of these three members as pieces which hold the
two sections together and exert onto them just enough forces ( again only
in the direction of these members) to hold each section in equilibrium.
In Fig. 9.b we see the two resulting parts in terms of their respective
Free-Body-Diagrams. Each part is exposed to the external loads/support forces
as well as the forces the three members exert onto it.
Sample Truss, 3 members removed
Solving now the equilibrium equations of either part ( the choice is yours )
you obtain the forces in the three removed members.
In the above example we could look at the sum of the force in vertical
direction on the left section of the truss :
R1 - P1 - FCE cos( β ) = 0
If you happen to be interested in the force FBC the sum of the
moments about point E (of either the left or the right part) would be
just fine because it contains only FBC as unknown.
And for FEF ?
FOOTNOTE : In many text books you find instead of "removing three members"
the phrases "cut three members" or "section the truss". The latter is
probably the origin of the title "Method of Section".
Problem 9.4a : Roof-truss, Fink, snow load
Problem 9.4b : Roof-truss, Howe, snow load
Problem 9.4c : Escalator Support
Problem 9.4d : Stadium Roof, I
Problem 9.4e : Stadium Roof, II
Compound Trusses are trusses which one can
divide into two or more sub-trusses. This might help in
the determination of internal forces.
Whether a truss is a compound truss depends
very much on who is looking.
Fig. 9.5a is an example of a compound truss. The members 12, 13, 23,
24, and 34 could be viewed as comprising one sub-truss, let's call
this the sub-truss 1234. The other
members making up a second sub-truss, called 4567.
This division can help us in this case because each of the two sub-trusses
is actually a 2-force member, that is each sub-truss has only two points
at which forces are acting (joint 1 and 4 for the left sub-truss and
joint 7 and 4 for the right sub-truss.
I tried to convey this in Fig. 9.5b. For known load P and geometry
we now can determine the forces FL and FR
from the equilibrium equation on joint 4.
Forces in Compound Truss
After determining FL and FR by analyzing the
equilibrium of joint 4 all external forces on the two sub-trusses
are known and each sub-truss can be analyzed separately.
The analysis of 3-dimensional trusses (extremely wide-spread in practice)
is usually not content of an introductory course into statics although the
underlying principles for their analysis are identical to that of
We have the same restrictions on the location of the loads, joints are now
of ball/socket type and support forces may have now 3,2, or only 1 unknown
component depending on the type of support employed.
All members are still 2-force members with the forces they exert on the
joints at their two endpoints stil equal but oppositely directly and in line
with the line connecting the two endpoints. Hence, these forces have now
in general three components and we have three equlibrium equations
A 3-dimensional truss is statically determined only if the number of unknown
forces (one per member plus support forces) is equal to the number
number of available equations ( 3 times
the number of joints).
Setting up the equilibrium equations and solving them is though an order of
magnitude (at least) more tedious than for 2-dimensional trusses.
Fig. 9.6a is a simple example where the truss consists of a single
tetrahedron with vertices A, B, C, and D. A single load P
(having x-, y-, and z-components) is applied at joint D.
3-D Truss, example
The support forces are chosen such that the tetrahedron (think of it
as a solid body) cannot move away nor rotate in any which way.
In 3 dimensions this necessitates 6 components of support forces.
At joint A I have specified a ball/socket connection ( 3
unknown components), at point C we have a roller-type connection
(2 unknown components) and at point B single component in the
We can solve for the unknown forces in the 6 members and the 6 components
of support forces by applying the method of joints in the following order :
- Joint D : 3 equations for the three forces in the members
AD, BD, and CD.
- Joint B : 3 equations for the single support force
component and the forces in members AB and BC.
- Joint C : 3 equations for the two support force components
and the force in member AC.
- Joint A : 3 equations for the three support force components.
You then can use the overall equilibirum equations for a check-up.
It is nearly impossible to do these calculations without vector notation.
Some more examples of 3-dimensional trusses can be found as sample
cases for a
3-D truss program.
In this chapter we were concerned with the determination of support forces
and forces internal members are exposed to.
The structures we could investigate were called trusses which have the
properties of :
- Consisting only of 2-force members.
- Loads and support forces act only on joints.
Two principle methods are available to obtain the desired forces :
- The Method of Joints which provides us with two ( three in 3-D cases) equations per joint leading to a system of linear equations
for the unknown forces. If the truss is statically determined this
system can always be solved by a computer program (like
Program TRUSS) or in many cases by
inspecting the truss as to the order in which these equations must
be solved. Depending on the truss geometry this approach is not
always possible but solving the overall equilibrium equations and/or
looking at the truss as a compound truss might help.
When solving for the forces without a computer the sequential nature
of the Method of Joints is a disadvantage because errors made
initially affect subsequent calculations.
- The Method of Sections can also be used to "jump-start" the method
of joints. It is very useful when the force of only a few internal
members are to be determined. The principle is here to remove 3 members
(in 2-dimensional cases) with one member being the one of which we wish
to determine the forces. The removal of the 3 members has to divide
the truss into two separate parts.
The study of the equilibrium of either
part yields the forces of the 3 removed members.
The self-test is a multiple-choice test. It allows you to ascertain your
knowledge of the definition of terms and your understanding of
Click here to do the test.
Computer Program TRUSS
This program is based on the Method of Joints. The user specifies the geometry
of the truss in terms of the location of all joints and how these are connected
by members and then specifies given external forces and finally
provides information concerning the support forces acting on the truss.
For more information follow the links below :
A warning in particular to my students. Usage of a computer program
(except for special parameter studies of which we will do one or the other)
does not teach you anything
more than just how to use that particular program. The real juice lies
in the understanding of the different methods employed and evaluating
whether the obtained results make sense.
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Send a Note to Zig
Zig Herzog; email@example.com
Last revised: 08/21/09