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Chapter 10 : Frames, tools, and other devices

10.1 Introduction
10.2 Principle of Analysis
10.3 Example with simple joints only
10.4 Complex-joint, method of primary member
10.5 Complex-joint, method of joint
10.6 Method of sections
10.7 Numerical example using a linear equation solver
10.8 Summary
10.9 Self-Test, computer programs and problems

10.1 Introduction
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In this chapter we are dealing again with multi-member structures. The difference in comparison to the trusses in chapter 09 is that at least one of its members is NOT a 2-force member ! Or in other words, at least one member of such a structure is exposed to forces at more than 2 points. We still keep the assumptions that members are connected by pin/hole type like connection and that all structures are 2-dimensional. Examples : (Click on your choice)

A pair of pliers

An oil rig

A backhoe

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10.2 Principle of Analysis
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Here are the basic rules which will help you determine the forces acting on each member :

For each individual member draw the Free Body Diagram showing all the joints at which forces are acting.

Keep in mind that in whatever way 2 members are connected they always exert onto each other forces which are equal in magnitude and exactly opposite in direction.

If gravity has to be taken into account it can be represented in terms of single force(s) acting at the respective center(s) of mass.

It can be shown that writing out the equilibrium equation for each member will result in enough algebraic equations to solve for all unknown forces. The equilibrium equation(s) of the entire structure must of course always be satisfied as well, a fact which can be used to check on the forces obtained.

The collection of the individual Free Body Diagrams is often referred to as the disassembled or exploded Free Body Diagram.

1. Hint :
The identification of 2-force members with equal but oppositely directed forces on either end can be very useful (although is not necessary) because it reduces the number of unknowns.

2. Hint : Forces of which magnitude and direction are unknown are best represented by their x- and y-components.

4. Hint : Sometimes it is of advantage to use the equations for the equilibirium of the entire structure to determine unknown forces. In such a case there will be left equilibrium equations on individual members/pins to serve as checkup.

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10.3 Example with simple joints only
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Figure 10.3a
Simple-joints example
The example shown in Fig. 10.3.a is a structure in which all joints are simple , that is, at each joint only two items come together and exert forces onto each other.
  • At joint A the member AC and the support
  • At joint B the member BDC and the support
  • At joint C the members AC and BDC
  • At joint D the member BDC and the known external force P
Another observation is that the member AC is a 2-force and the member BDC is a 3-force member.

Assume that the force P and the lengths a, b, c, and h are given. The task is to determine the support forces at the joints A and B and the force the two members exert onto each other at joint C.

Analysis

Of course we start with drawing the exploded Free Body Diagram.

As mentioned earlier the identification of 2-force members inside a structure will reduce the number of unknown forces. To demonstrate that, you might want to look at my two approaches, one playing dumb and one making use of the fact that member AC is a 2-force member.

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10.4 Complex-joint, method of primary member
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Figure 10.4a
Example with complex joint
The example 2 as shown in Fig. 10.4.a has some similarity with example 1. The major difference is not so much that is has one more member but that we now have a joint D at which two members are connected to each other and an external force is applied. Assume that the force P (magnitude and direction) and the lengths a, b, c, and h are given. The task is to determine the support forces at the joint A and B and the forces each of the three members is experiencing.

Analysis

Again, drawing the Free Body Diagrams is our primary goal.
The new problem we'll have to deal with occurs at joint D where two members are connected and the external force P is acting.

A joint at which either more than 2 members are connected or two members plus external force(s) is called a complex joint.

One method to deal with such a situation is the method of primary member another the method of joints, see section 10.5.

At a complex joint choose one of the members to be the primary member assuming that the external force(s) and ALL OTHER member(s) are connected to it individually.

After finding all forces acting at the complex joint onto the primary member you must find the resultant of thes forces.

As far as the example is concerned, if your primary member is the member BDC in Figure 10.4a you can compare your Free Body Diagrams and solution to mine by clicking here. Choosing the member DE as the primary member is equally legal.

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10.5 Complex-joint, method of joint
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The method of joint takes an alternative (to the method of primary member) point of view of a complex joint.

At a complex joint imagine that all members and external force(s) are connected individually to a single pin.

You must include the equilibrium equations (sum of forces in x- and y-direction) for this pin into your analysis.

The advantage of the method of joint is that it yields directly the forces acting on each of the involved members at the location of the complex joint. Choosing a primary member on the other hand results in a multitude of forces - one from each connected member and each external force - acting at the location of the complex joint on the primary member. The resultant of these forces has to be determined subsequently.
The disadvantage of the method of joint is that you get 2 more unknowns for each such pin.
When 3 or more 2-force member form a complex joint the method of joint is a must, joint 9 of the backhoe is a good example for this.


To contrast the method of primary member against the method of joint I have chosen to analysis the same example as brought in section 4 of this chapter, see Figure 10.4a. Here is my exploded FBD and solution.

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10.6 Method of Sections
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The method of sections can be used to directly find forces on selected members inside a structure without determining all forces on all members -- the situation being similar to the one we encountered when using this method to analyze trusses.

Try to break the structure into two section ( or sub-structures if you like) such that :

  • the forces these sections exert onto each other do NOT give rise to more than 3 unknowns.

  • among these forces is the one of interest to you.

Solve the equilibrium equations of one of the sections.

Because of being able to tolerate at most three unknowns you cannot expect to be able to use the method of sections to determine whatever force you wish to. Here is an example of a successful application of the method of section.

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10.7 Numerical example using a linear equation solver
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Figure 10.6a
Arial lift
The example shown in Fig. 10.6.a does not offer much new in terms of statics, although it is a nice training example. The figure exhibits a simplified version of an arial platform. The member CD is a hydraulic cylinder which when activated will make the platform GHJ rise or move downwards. Our task is again to determine all forces on all members. Main purpose of this example is to use a more advanced method to solve the resulting linear equation system utilizing a program available on the web and learn a little bit about the mathematics behind it (unless of course you know that already otherwise).

Free Body Diagram

You are more than welcome to try to sketch the exploded Free Body Diagram for the arial platform yourself. Here is mine together with some comments you might find useful. I also entered all relevant lengths needed to obtain all necessary equilibrium equations.

Equilibrium Equations

Depending on how many and which moment equations you utilize your set of equilibrium equations might look different although the number of equations should be the same.

+GX                                                       = 0
           +GY    +JY                                     = 2500
               2.6*JY                                     = 2500*1.2

           -GY                    -EY +BY                 = 0
+3.5*GX+2.6*GY        +1.75*EX+1.3*EY     +0.8*CD         = 0
    -GX                    -EX                -CD         = 0

                           +EX                +CD +AX     = 0
                  -JY             +EY                 +AY = 0
              -2.6*JY -1.75*EX+1.3*EY     -0.8*CD         = 0

These are nine equations for the nine unknowns :

GX,GY,JY,EX,EY,CD,BY,AX, and AY.

In preparation for things to come the equations are displayed so that in all equations the unknown forces appear in the same order and that corresponding terms are aligned in the vertical direction. There is no secret why the GX terms appear in the first column, GY terms in the second etc. You can choose any order you like. But, all terms containing only known values are written on the right-hand side of each equation.
Although I have written the equations in one particular order ( first 3 equations reflect the equilibrium of the platform GHJ, the next 3 equations the equilibrium of the member BDEG ) you can list them in any other order as well.

Equilibrium Equations in Matrix Form

In matrix form the above equation system is written in the following form:



The unknown forces GX,GY, etc. have here become the elements of an unknown vector (a column vector to be precise, because its elements are written one beneath the other). The numerical values on the right hand side constitute a known vector (here with nine elements) often referred to as the free vector, while the coefficients on the far left-hand side are called the elements of a square matrix (square, because it contains as many columns as rows).

A feature typical for the analysis of structures is that most of the elements of the matrix are equal to zero (sparse matrix).

You can recover the original way of writing this equation system in the following way :
To obtain the n'th (n = any number from 1 to 9 ) equation evaluate the dot-product of the vector of unknown forces with the vector containing all elements of the n'th row of the coefficient-matrix and set the result equal to the n'th element (from the top) of the free vector.

There are many numerical schemes known to solve above equation system classified as substitution schemes (Gauss-Seidel for example), matrix inversion, iterative methods, and relaxation methods. At the moment the details don't matter. The program available here uses the Gauss-Jordan elimination method.

Please click here as to how to present the above equation system to the available program and how to interpret its output.

If you are familar with the program and wish to use it for other problems of your own choosing you may click here.

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10.8 Summary
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Whenever you want to analyze the forces inside a structure which contains at least one multi-force member ( a member with forces acting at more than two points ) the methods introduced in this chapter must be employed.

The basic strategy on which all these methods rely starts with the development of the Free Body Diagrams of all individual members and the structure as a whole. If the problem is statically determinate the equilibrium equations on the individual members will yield enough equations to solve for all unknown forces and the equilibirium equations of the entire structure can serve as a check-up. Often it is of advantage though to start with the equilibrium equations of the entire structure and have some of the "internal" equations available for check-up.

Of advantage (reducing the number of unknown forces) is the discovery of 2-force members of the structure to be investigated by making use of the special properties of such a member.

Important is also the identification of complex joints ( joints at which more than 2 forces are acting). A decision has to be made to employ here either the method of the dominant member or the method of joints.

For even mildly complicated structures the number of unknown forces increases and often the resulting linear equation system should be tackled using corresponding computer programs.

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10.9 Self-Test and problems
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Self-Test

Sorry, still to come.

Computer Programs

None available for this chapter. Sorry.

Problems

Please, try them all.

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Zig Herzog; hgnherzog@yahoo.com

Last revised: 08/21/06