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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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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.
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. |
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.
| 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. |
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.
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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.
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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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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.
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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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Try to break the structure into two section ( or sub-structures if you
like) such that :
Solve the equilibrium equations of one of the sections.
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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.
| 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). |
+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.
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.
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.