Saturday, December 8, 2012

Structural mechanics - Trusses -Perfect and Imperfect Frames

Trusses are the structural elements which are formed by joining different members or bars with the help of joints at their ends. Trusses are used to transfer the loads safely along  the girders/horizontal section on the portals to increase the stiffness.

Trussed bridge
 They can also be seen on the bridge portals or roofs etc. In actual practice members are generally welded/riveted to make the rigid joints and sometimes they may be bolted too.


To analyse a pin jointed frame we make some assumptions:

1) The joints are hinged(Pinned joints) so the net moment at any joint is zero.
2) The loads are applied only at the joints.
3) Only axial stresses are induced in the members.
4) Moments are the joints are zero.
5) Only axial forces(tensile and compressive) are induced in the member.

 In structural Engineering the frames are classified in two broad categories:
1) Perfect frames and    2)Imperfect frames.

Perfect frames are the frames which can be analysed to get the internal member forces and external support reactions by using the three conditions of static equilibrium. A simple example of a perfect frame is a triangular frame which is formed by joining the three member with the help of three joints.

Similarly Imperfect frames are the frames which can not be analysed to get the internal member forces and external support reactions by using the three conditions of static equilibrium.

Types of frames- Perfect and Imperfect(Deficient and redundant)

We know that in a triangular frames there are only three joints and there are three members.
For any structure to be in equilibrium the net force on the structure in the in either direction should be zero and also the net moment of all the forces at any joint should be equal to zero.

We can check it by taking the algebraic sum of all the forces in the two different directions for a plane frame(2-D structure) and it should be equal to zero.

Also take the moment of all the forces about a single joint and check if it is also equal to zero.So there are three conditions of equilibrium.
An indeterminate frame is a frame for which the three equations of the static equilibrium are not sufficient to analyze for its member forces and the external support reactions.

To check if the frame is perfect/determinate or imperfect remember the following:
For perfect frame:
m  = 2j -3
m=nos. of members in a frame
j = nos. of joints.
For a triangular frame m =3 and j = 3 so when we put these values in the equation above we get both the sides equal so, a triangular frame is a simple example of a perfect frame.

Now if m<2j-3 then the frame is called a deficient frame and the frame is unstable, i.e. the frame will deform if the external load is applied.

And if m>2j-3 then the frame is redundant. It is stable but we can not analyse the frame with the help of the conditions of the static equilibrium. and the difference between the m and 2j-3 is know as the redundancy of the frames.
For determinate frames/perfect frames, we can easily find out the external support reaction using the three equations of static equilibrium. To determine the forces in the members of the trusses,

There are two analytic methods to analyse the forces in the members of the trusses:

  1.  Method of Joints.
  2. Method of sections.

A) Method of Joints:

As per the assumptions made above the joints are hinged/pinned, and they loads are applied at the joints only. For the joints to be in static equilibrium, it must follow that the net force on a hinge must be zero.

In two dimensional frames/plane frames, we can further say,
1) Sum of all the horizontal forces is zero.
2) Sum of all the vertical forces is zero.

So if we follow the above two equations, we can find any two unknown forces in the members meeting at a joint. So to apply the method of joint one must remember the following statement:
Method of joint should be applied at the joints where the numbers of unknown member forces is two or less than two.

In a perfect frame to apply method of joints, you must start with a joint having two or less than two unknown member forces. Afterwards you can further move to the joints which earlier had more than two unknown member forces and now they will have two or less than two.


B) Method of Sections:

Method of section is based on the third equation of the static equilibrium discussed above, Sum of all the moments about a hinge is zero. We can analyze a section containing three or less than three unknown member forces, using the method of section.

When we cut a section(either to left or right) along three members, in perfect frames, two of them will meet at same joint.
Take the moment of all the forces about that joint and the moment of the two unknown forces will become zero(no perpendicular distance), so there will be the third unknown force in the equation of equilibrium.

Similarly, take the moment of all the forces at some other joint where other two forces(one previous one, and one just determined) meet. Automatically again there will be one unknown in the equation, so find that, and similarly you can go for third one.
Remember, for applying method of section, the section must contain three or less than three unknown member forces.

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