Analysis of Structures - Trusses, Method of Joints and Sections - A Tutorial with Solved Problems


                                     ANALYSIS OF STRUCTURES                  

Introduction

Unlike the previous chapter in this unit we will be dealing with equilibrium of supporting structure. The structures may consist of several sections. They form the supporting structures of bridges, pillars, roofs etc. It is important to have a basic knowledge of this topic as it concerns with the safety and stability of a several important structures. We will be studying about the various internal forces responsible for keeping the structures together.


Following figure gives a basic idea of what we are going to study. The given figure is a normal diagram of a book shelf. The second figure shows the role of internal forces in maintaining the system equilibrium. Free body diagram of various components are shown. It is clear from the diagram that the forces of action and reaction between various parts are equal in magnitude and opposite in direction.

             


Definition of a Truss

Am truss is a network of straight slender members connected at the joints. Members are essentially connected at joints. The every member has force only at extremities. Further for equilibrium the forces in a member reduce to two force member. Thus no moments only two force member. In general trusses are designed to support. Trusses are designed to support weight only in its plane. Therefore trusses in general can be assumed to be 2-dimensional structures. Further in case weight of individual member is to be taken into consideration, half of them are to be distributed at each of the pinned ends. Figure below shows a sample truss. There are nine individual members namely DE, DF, DC, BC, BF, BA, CF, EF, FA. Structure is 2-dimensional structure, supported by pin joints at A and E.

Sometime a member may be given of a shape like: In such a case take the line the line joining their ends as the line of action of force.


Analysis of a Truss

A truss needs to be stable in all ways for security reasons. Simplest stable truss ABC is shown in figure.

The second diagram depicts that how instable the truss structure is. The truss ABCD can easily be deformed by application of the force F. Trusses constructed by adding triangles such as arms AC and CD to the above stable truss ABC are called simple trusses. No doubt simple trusses are rigid(stable). Further it is not always necessary that rigid trusses will necessary be simple. Let m be no. of members and n be number of joints. For a truss with m<2n-3 deficiency of members, unstable ,fewer unknowns than equation.
 m= 2n-3 uses all the members, statically determinate 
 m>2n-3 excess member , statically indeterminate, more unknown than equation

Method of Joints

In the following section we will consider about the various aspects of trusses. Distribution of forces, reactions forces at pins, tension and compression etc.
Step 1. Find the reaction at supporting pins using the force and the moment equations.
Step 2. Start with a pin, most preferably roller pin,wher there are 2 or less than two unknowns.
Step 3. Proceed in a similar way and try to find out force in different members one by one.
Step 4. Take care of while labelling forces on the members. Indicate compression and tension clearly.
Step 5. Finally produce a completely labelled diagram.
Step 6.Try to identify the zero force members. It makes the problem simple.

Above shown are the conditions of compression or tension, deceided as per the direction of force applied by the pin joints to the members.


Example: Find the forces in the members AF, AB , CD, DE, EC and the reactin forces at A and D. CD = 3m.



Sol:




As per the type of joint the reaction forcces are shown below.Clearly Ax = O (balancing forces horizontally) Ay + Dy = 10 KN(balancing forces vertically) taking moment about D.


MD = 10*3 – Ay *9 = 0 (zero for equlibrium) , therefore Ay = 10/3 KN , and Dy = 20/3 KN(10-10/3)

Now we have drawn the free body diagram of the pin A. We have assumed force at pin A due to the members in some direction. From the given data we can conclude tanɵ= 4/3 , sinɵ= 4/5 , cosɵ = 3/5. Balancing forces vertically. FAF sin = 10/3 . FAF = 25/6 kN .

Balancig forces horizontally,FAB = FAF cos = 2.5 kN .

Note the direction of the indicated forces are those applied be members to the pin. Force applied by pin onto the members will have the same magnitude but in opposite direction. Therefore we can easily state that member AF is in compression and member AB is in tension. Further each member is a two force member implying that it will exert the same amount of force to the pin on the other end but will be opposite in direction.

Now considering the joint D.

Balancing forces vertically. FDE sin = 20/3 . FAF = 25/3 kN .

Balancig forces horizontally,FCD = FDE cos = 5 kN .

Therefore we can easily state that member DE is in compression and member CD is in tension.


Now considering the joint C.



Balancing forces vertically. FEC = 10kN .

Balancing forces horizontally. FBC = 5kN .

Therefore we can easily state both the members EC and BC are in tension.

In case in the above given problem 10kN was placed some where else, then ae per the FBD at joint C there would be no vertical force to balance FEC. Hence force in EC would be zero. It is good to analyse the problem before hand and eliminate the zero force members, as they contribute nothing to the system.


Method of Sections

As the name suggests we need to consider an entire section instead of joints. When we need to find the force in all the members, method of joint is preferrable. For finding forces in few of the specific members method of joints is preferrable. Let us consider the same diagram as before.




We had been provided with the given system. We draw a axis aa’.

The axis should at max intersect three members. Then we separate the two sections apart. We can select any one of the part. We have just assumed he member to be in tension. We can find the reaction at supports. Now what we have done is divided the whole structure into two parts and taking into consideration various external reactions and member forces acting of one part.Suppose we have to find FEF. It is sufficient to write the equation MB =0(for equlibrium). To find FBC it is sufficient to write the equation of ME =0. Similarly we can also use the equatons Fx =0 ,Fy =0 , for the equlibrium of the section under consideration.




Here's a quick look at a few of the problems solved in this tutorial :

Q:
Following is a simple truss. Find the forces in the all the members by method of joints.




Q:
Find the reaction components at A and B. Also find the forces in each individual member, specify compression or tension.
     
Q:
Find the reaction components at A and C. Also find the forces in each individual member, specify compression or tension. Given AD=10in ,DC=7in , BD=8in.


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