Chapter 6: Analysis of Structures
Some of the most common structures we see around us are buildings & bridges. In addition to these, one can also classify a lot of other objects as "structures." For instance: The space station Chassis of your car Your chair, table, bookshelf etc. etc. Almost everything has an internal structure and can be thought of as a "structure". The objective of this chapter is to figure out the forces being carried by these structures so that as an engineer, you can decide whether the structure can sustain these forces or not. Recall: External forces: "Loads" acting on your structure. Note: this includes "reaction" forces from the supports as well. Internal forces: Forces that develop within every structure that keep the different parts of the structure together.
In this chapter, we will find the internal forces in the following types of structures : Trusses Frames Machines
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Trusses are used commonly in Steel buildings and bridges. Definition: A truss is a structure that consists of All straight members connected together with pin joints connected only at the ends of the members and all external forces (loads & reactions) must be applied only at the joints. Note: Every member of a truss is a 2 force member. Trusses are assumed to be of negligible weight (compared to the loads they carry)
Types of Trusses
Simple Trusses: constructed from a "base" triangle by adding two members at a time.
Note: For Simple Trusses (and in general statically determinate trusses) m: members r: reactions n: joints
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6.4 Analysis of Trusses: Method of Joints
Consider the truss shown. Truss analysis involves: (i) Determining the EXTERNAL reactions. (ii) Determining the INTERNAL forces in each of the members (tension or compression).
Read Example 6.1
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Similarly, solve joints C, F and B in that order and calculate the rest of the unknowns.
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6.5 Joints under special loading conditions: Zero force members Many times, in trusses, there may be joints that connect members that are "aligned" along the same line.
Similarly, from joint E: DE=EF and AE=0
Exercise 6.32 Identify the zero-force members.
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6.6 Space Trusses
Generalizing the structure of planar trusses to 3D results in space trusses. The most elementary 3D space truss structure is the tetrahedron. The members are connected with ball-and-socket joints. Simple space trusses can be obtained by adding 3 elements at a time to 3 existing joints and joining all the new members at a point. Note: For a 3D determinate truss: n: joints 3n = m+r m: members r : reactions
If the truss is "determinate" then this condition is satisfied. However, even if this condition is satisfied, the truss may not be determinate. Thus this is a Necessary condition (not sufficient) for solvability of a truss. Exercise 6.36 Determine the forces in each member.
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Similarly find the 3 unknowns FBD, FBC and BY at joint B.
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6.7 Analysis of Trusses: Method of Sections
The method of joints is good if we have to find the internal forces in all the truss members. In situations where we need to find the internal forces only in a few specific members of a truss, the method of sections is more appropriate. Method of sections: Imagine a cut through the members of interest Try to cut the least number of members (preferably 3). Draw FBD of the 2 different parts of the truss Enforce Equilibrium to find the forces in the 3 members that are cut. For example, find the force in member...