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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

9.1 Definition of trusses
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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 loads.

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) are :

  1. 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.

  2. 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 joint.


    Fig. 9.1a :  Example of 2-D Truss

  3. 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.
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.

Click here for a glimpse at some commonly employed trusses.

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9.2 Properties of 2-force members
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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 are acting.

As an example let's look at the member CE extracted from Fig. 9.1a


Figure 9.2a
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 three-dimensional trusses.


Figure 9.2b
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 zero-force members. Nominally these members do not carry any load but they prevent a member under compression from buckling by providing lateral support.

Figure 9.2c
Failure modes

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9.3 Method of Joints
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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 (three-dimensional) trusses this method results in a sequence of sets of two ( three) linear equations.

Prelims


Figure 9.3a
Method of joints
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.
Assume that the forces P1 and P2 are known as well.
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.

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

  1. 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.
  2. If the number of unknown forces is less than the number of available equations the truss will collapse.
  3. 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 :

A     B     C     D     E     F    

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) are unknown. 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 :

  1. Solve as many of the overall equilibrium equations as you can.

  2. Find zero-force members. Click here if you want to find out how to do that (might save you later ?!).
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).

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.

Problems

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

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9.4 Method of Sections
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Prelims

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 Joints.

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.


Figure 9.4a
Sample Truss
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.


Figure 9.4b
Sample Truss, 3 members removed
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.

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".

Problems

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

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9.5 Compound Trusses
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Compound Trusses are trusses which one can divide into two or more sub-trusses. This might help in the determination of internal forces.


Figure 9.5a
Compound Truss
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.

Figure 9.5b
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.

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9.6 Trusses in 3-D
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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 2-dimensional trusses.

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 per joint.

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.

Figure 9.6a
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 z-direction.

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 :

  1. Joint D : 3 equations for the three forces in the members AD, BD, and CD.

  2. Joint B : 3 equations for the single support force component and the forces in members AB and BC.

  3. Joint C : 3 equations for the two support force components and the force in member AC.

  4. 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.

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9.7 Summary
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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 :
  1. Consisting only of 2-force members.

  2. Loads and support forces act only on joints.

Two principle methods are available to obtain the desired forces :

  1. 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.

  2. 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.

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9.8 Self-Test and computer Program TRUSS
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Self-Test

The self-test is a multiple-choice test. It allows you to ascertain your knowledge of the definition of terms and your understanding of important results.

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 :

2-dimensional trusses                                     3-dimensional trusses

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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Zig Herzog, hgn@psu.edu
Last revised: 08/21/09