Some of the most common constructions we see around us are edifices & A ; Bridgess. In add-on to these. one can besides sort a batch of other objects as “structures. ” For case: The infinite station Chassis of your auto Your chair. tabular array. bookshelf etc. etc. Almost everything has an internal construction and can be thought of as a “structure” . The aim of this chapter is to calculate out the forces being carried by these constructions so that as an applied scientist. you can make up one’s mind whether the construction can prolong these forces or non. Remember: External forces: “Loads” moving on your construction. Note: this includes “reaction” forces from the supports as good. Internal forces: Forces that develop within every construction that keep the different parts of the construction together.
In this chapter. we will happen the internal forces in the undermentioned types of constructions: Trusss Frames Machines
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6. 2-6. 3 Trusss
Trusss are used normally in Steel edifices and Bridgess. Definition: A truss is a construction that consists of All straight members connected together with pin articulations connected merely at the terminals of the members and all external forces ( tonss & A ; reactions ) must be applied merely at the articulations. Note: Every member of a truss is a 2 force member. Trusss are assumed to be of negligible weight ( compared to the tonss they carry )
Types of Trusss
Simple Trusss: constructed from a “base” trigon by adding two members at a clip.
Note: For Simple Trusses ( and in general statically determinate trusses ) m: members r: reactions n: articulations
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6. 4 Analysis of Trusss: Method of Joints
See the truss shown. Truss analysis involves: ( I ) Determining the EXTERNAL reactions. ( two ) Determining the INTERNAL forces in each of the members ( tenseness or compaction ) .
Read Example 6. 1
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Exercise 6. 13
Similarly. work out articulations C. F and B in that order and cipher the remainder of the terra incognitas.
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6. 5 Joints under particular loading conditions: Zero force members Many times. in trusses. there may be articulations that connect members that are “aligned” along the same line.
Similarly. from joint Tocopherol: DE=EF and AE=0
Exercise 6. 32 Identify the zero-force members.
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6. 6 Space Trusss
Generalizing the construction of planar trusses to 3D consequences in infinite trusses. The most simple 3D infinite truss construction is the tetrahedron. The members are connected with ball-and-socket articulations. Simple infinite trusses can be obtained by adding 3 elements at a clip to 3 bing articulations and fall ining all the new members at a point. Note: For a 3D determinate truss: N: articulations 3n = m+r m: members r: reactions
If the truss is “determinate” so this status is satisfied. However. even if this status is satisfied. the truss may non be determinate. Thus this is a Necessary status ( non sufficient ) for solubility 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 Trusss: Method of Sections
The method of articulations is good if we have to happen the internal forces in all the truss members. In state of affairss where we need to happen the internal forces merely in a few specific members of a truss. the method of subdivisions is more appropriate. Method of subdivisions: Imagine a cut through the members of involvement Try to cut the least figure of members ( sooner 3 ) . Draw FBD of the 2 different parts of the truss Enforce Equilibrium to happen the forces in the 3 members that are cut. For illustration. happen the force in member EF:
Read Examples 6. 2 and 6. 3 from the book. Exercise 6. 63 Find forces in the members EH and GI.
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6. 8 Compound Trusses ; Determinate vs. Indeterminate Trusses. Trusss made by fall ining two or more simple trusses stiffly are called Compound Trusses.
Excessively constrained. Indeterminate
Exercise 6. 69 Classify the trusses as: Externally: Wholly / Partially /Improperly constrained Internally: Determinate / Indeterminate. ( if wholly constrained )
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6. 9 – 6. 11 Frames
Frames are structures with at least one multi-force member. i. e. atleast one member that has 3 or more forces moving on it at different points.
Frame analysis involves finding: ( I ) External Reactions ( two ) Internal forces at the articulations
Note: Follow Newton’s 3rd Law
Frames that are non internally Rigid
When a frame is non internally stiff. it has to be provided with extra external supports to do it stiff. The support reactions for such frames can non be merely determined by external equilibrium. One has to pull the FBD of all the constituent parts to happen out whether the frame is determinate or indeterminate.
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Example 6. 4
Read examples 6. 5 and 6. 6 Exercise 6. 101
Exercise 6. 120
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6. 12 Machines
• Machines are constructions designed to convey and modify forces. Their chief intent is to transform input forces into end product forces. • Machines are normally non-rigid internally. So we use the constituents of the machine as a free-body. • Given the magnitude of P. determine the magnitude of Q.
Exercise 6. 143
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Determinate vs. Indeterminate Structures
Structures such as Trusss and Frames can be loosely classified as: Determinate: When all the terra incognitas ( external reactions and internal forces ) can be found utilizing “Statics” i. e. Drawing FBDs and composing equilibrium equations. Indeterminate: When. non all the terra incognitas can be found utilizing Statics. Note: Some/most terra incognitas can still be found. Structures can besides be classified as: Wholly restrained Partially restrained Improperly restrained For trusses. we have been utilizing “formulas” such as ( 2n = m+r ) for planar trusses. and ( 3n = m+r ) for infinite trusses to judge the type of construction. For frames. this can be much more complicated. We need to compose and work out the equilibrium equations and merely if a solution exists. we can reason that the construction is determinate. Otherwise the construction may be partly constrained or indeterminate or both. IMPORTANT: One of the best ways ( and mathematically right manner ) to reason determinacy of any construction is by utilizing Eigen-values. Eigen-values tell us how many independent equations we have and whether can or can’t work out a system of equations written in the signifier of Matrixs.
[ A ] ten = B
To make this. Pull the FBDs of all stiff constituents of the construction Write out the all the possible equilibrium equations. Case 1: Number of Equations ( E ) & lt ; Number of Unknowns ( U ) INDETERMINATE Case 2: Number of Equations ( E ) & gt ; Number of Unknowns ( U ) PARTIALLY RESTRAINED Case 3: Number of Equations ( E ) = Number of Unknowns ( U ) Find the figure of non-zero Eigen-values ( V1 ) of the square matrix [ A ] . Find the figure of non-zero Eigen-values ( V2 ) of the rectangular matrix [ A|b ] . Case 3 ( a ) : = & gt ; Unique Solution DETERMINATE Case 3 ( B ) : V1 & lt ; E = & gt ; Improperly constrained Number of INDEPENDENT equations = V1 & lt ; U Indeterminate & A ; Partially constrained ( I ) V1 = V2 Infinitely many solutions possible ( two ) V1 & lt ; V2 = & gt ; No solution exists Note: In this process. it is better non to cut down the figure of terra incognitas or figure of equations by utilizing belongingss of 2-force or 3-force members. V1 = E = U