MDME: MANUFACTURING, DESIGN, MECHANICAL ENGINEERING

TRUSSES

Trusses are very efficient way to make a structure. We will study planar trusses where every joint is a pin joint. Such a truss is built completely of 2-force members (struts), so they can only ever be in pure tension or compression.

Lecture Notes 1 Method-of-Joints.pdf    Method-of-Joints.one

Lecture Notes 2 Method-of-Sections.pdf    Method-of-Sections.one

There are many ways to study trusses, but they mostly fall into 2 methods: The Method of Joints and the Method of Sections.

Method of Joints

This can be a slow method for a large truss, but it is very simple to understand.

We simply pick a Joint that has no more than 2 unknowns, then solve it using the rules of equilibrium:

(Although, since each joint is a CONCURRENT force problem, we do not need to do moments)

Now that we know everything about this joint we can move on to the next joint, and the next etc.

This is what happens if you don't have equilibrium at every joint...

How to do Method of Joints

In this method you need a Free Body Diagram of each joint and solve for the unknown member forces at that joint. Once this joint is completed, move joint to joint through the truss to solve for the force in all members.

Procedure
1. Draw a Free Body Diagram of the whole truss and find the external reactions if required (using Moments).
2. Choose a joint with only 2 unknowns (probably starting at the roller joint).
3. Draw a Free Body Diagram of that joint and find internal forces (by a force polygon).
4. Transfer these forces to adjacent joints (compression = pushing both ends, tension = pulling both ends).

Notes
• If you can't guess, assume tension (pulling on joint). If you get a negative answer it must be in compression.
• Give your answer for the force in each member as a positive number (with a T for tension) OR a negative number (with a C for compression).
• Once a force is known on one end of the member, the same force is then OPPOSITE on the other end. (rotated by 180 degs).

The Method of Joints will solve any truss, but sometimes is might be doing it the long way - especially if you want to know what is happening in the middle of a complex truss. However, if you are designing the thing, you probably want to know the forces in every member anyway, so this method is usually suitable. Also, this method is self-checking. By the time you work all the way through the truss you should have forces that match the reactions at the other end. So you know if you did it right.

Example. (See example 9.1, page 129 of text; Ivanoff Engineering Mechanics). Note that Ivanoff uses Bow's notation which can be a little awkward at first. (See Labelling of Trusses - below).

From L.J.Miriam; Statics SI Version Vol 1; John Wiley & Sons, 1980.
Notes on the method of joints;
A worked example:

Worked Example (Method of Joints)

Worked Example:  Truss-MoJ.pdf    Truss-MoJ.one    Truss-MoJ.dwg

Animation: Truss by Method of Joints (Tim Lovett 2013)

Worked Example with audio: Trusses: Method of Joints (Tim Lovett 12-May-2014)

Method of Sections

This method is a quick way to find the stresses somewhere in the middle of a complex truss, without needing to solve every joint. It relies on the fact that if the truss is in equilibrium, then ANY section of the truss must be in equilibrium - including half the truss if you want!

So we can just cut the truss in half and make a FBD of one of the halves (making sure the other half we threw away has been replaced by the forces it applied TO THE BODY).

Then solve these forces using the equilibrium equations (as usual).

Sounds easy enough, but in practice we have to be a little clever to make sure we can solve the equilibrium equations. We do this by careful choice of where to take moments.

How to do Method of Sections

Chop the truss in half and solve for the severed members.

Procedure

1. Solve the support reactions of the entire truss (if necessary).
2. Cut the truss in half right through a member that you want to know. Draw a FBD for a sectioned half, replacing the severed members with forces. (You usually cut thru 3 members. Pick the easy side too)
3. Write moment equation by taking moments about a point of intersection of 2 of the three unknown forces in order to find the third one. (Sometimes these pivot points are not even on the body! Usually in a truss you will cut 3 members - it there were 4 you might be in trouble unless you can find an intersection of 3, to leave 1 for the moment equation).
4. Solve equilibrium equations to find the rest of the members through the section (if necessary).

Note

Like any non-concurrent force problem, you only have to do the moment equation once. When we are left with two unknown forces (of known direction), we can solve by equilibrium of forces (force polygon - by X,Y components, trigonometry or CAD).

Note 1: The Method of Sections is typically used when you want to analyse members that are in the middle of a complex truss. The principle is very important though, because it demonstrates how a FBD can be defined any way you want. By chopping the truss in half (i.e. making a section through the truss) you are actually splitting the original body (the whole truss) into 2 separate bodies (Left and right halves of the truss) - and then solving for equilibrium of non-concurrent forces. This requires taking moments about different points until you have enough equations to solve all the unknowns (which are the chopped members).

Note 2: The Method of Sections is a great way to double-check your calculations. At any time during the Method of Joints you can cut the truss and see if you get the same answers using the Method of Sections.

Note 3: We have assumed trusses are pin jointed, which is usually an underestimation.
Welded or tightly bolted joints would usually be stronger.

Example. (See example 9.4, page 137 of text; Ivanoff Engineering Mechanics)

From L.J.Miriam; Statics SI Version Vol 1; John Wiley & Sons, 1980.
Worked examples:

Summary

Both the Methods of Joint and Methods of Sections are really nothing more than equilibrium. In fact, selecting a free body and doing equilibrium is all we ever do in this unit!

And when it comes to equilibrium we have two tools to use;

* The moment equation (carefully placed to illiminate all forces except one) This used for Non-Concurrent force body - such as in the Method of Sections.

* The force polygon (able to handle up to 2 unkown forces), which we can use any time we have 1 or 2 unkown forces.

.

Worked Example (Method of Sections)

Worked Example (Method of Joints):  Truss-MoS.pdf    Truss-MoS.one    Truss-MoS.dwg

Animation: Truss by Method of Sections (Tim Lovett 2013)

Worked Example with audio: Trusses: Method of Sections (Tim Lovett 9-May-2014)

Labeling of Trusses

There are three main ways trusses are labeled - by joints, by members and by spaces (Bow's Notation)

Labeling by JOINTS. Members at Joint A are called AB, BC, AC.. etc. We will just to this method.

Labeling by MEMBERS. The left support would be called Joint AB

Labeling by SPACES (Bow's Notation). This time the spaces are labeled (using letters, and usually in a clockwise direction). The members are BF, AF, FG etc.  The Joints (nodes) are abfa, fgeaf, etc. This method looks cumbersome but it is an essential step in the Maxwell Diagram - a method of solving trusses with one graphical construction. (For this chapter, Bow's Notation is OPTIONAL, and you will not be tested on Bow's Notation. We will use the more basic method of labeling the joints or members).

Indeterminate Trusses

Some trusses cannot be solved using the above method. A typical example is when members are criss-crossed. This means there are excess members, so the loads are shared between several members (such as a pair of diagonals). A determinate truss has just enough members - take one out and it will become a mechanism (it will move), and add one in and it will become indeterminate.

We can check whether a structure overconstrained or under-constrained;

m < 2j - 3 The truss will move (mechanism or under-constrained)
m = 2j - 3 The truss uses every member (determinate)
m > 2j - 3 The truss has excess members (indeterminate or over-constrained)

Where: m = no of members and j = number of joints.
Note: This equation only works for a 2 dimensional structure.

An indeterminate truss using pre-tensioned cables for diagonals. (Wright Brothers patent 1911)

Calculating determinacy; s = 26, k = 12
For 12 joints, a determinate truss would need this many members: s =  2*12 - 3  =  21 members.
So there are 5 extra members (which are the criss-crossed diagonals), therefore the structure is indeterminate.
(Can't work it out - simply)

In some cases the above rules do not apply. An apparently indeterminate truss can sometimes be determinate (i.e. every member is needed). The most common case is when criss-crossed diagonals are used, but they are not tensioned (as cables usually are). The best example of a determinate criss-crossed truss is where the diagonals are made from flat bar.

Zero Force Members

In certain trusses it is possible to have a member that carries no force. This only happens at certain loading conditions, and when the weight of the members is ignored. One classic example is the unloaded "T" joint.

Join G is an unloaded "T" joint connecting members EG, FG and IG. Since the horizontal members EG and IG cannot take any vertical forces, then FG cannot have a vertical force component. Hence FG is a zero-force member and does nothing in this loading arrangement. However, it we hung a load from point G, FG would now be taking that load.

Are there any other zero force members in this truss?

Questions:

Notes & Questions (From L J Miriam - Engineering Mechanics)
Homework Assignment:
• Do all questions 9.1 to 9.5 (page 133-134: Method of Joints). Note that the author uses Bow's Notation here, which is a special way of labeling the forces, members and joints of a truss. Bow's notation is essential for the Maxwell Diagram (which we are not using). So Bow's Notation is OPTIONAL, you will not be tested on Bow's Notation. We will use the more basic method of labeling the joints.
• Do questions 9.11, 9.12  (page 138-139: Method of Sections).

Exam Rules
:
Permitted: Open Book, Internet, Calculator, CAD
Not Permitted: Excel, any dedicated truss analysis software, pre-programmed solutions - including VisualBasic etc.