Chapter5

# Chapter5 - Chapter 5 Statically Indeterminate Trusses...

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© K. H. Ha - ENGR 343 Chapter 5 - v2.0 5.1 Chapter 5 Statically Indeterminate Trusses Chapter 5 1 Statically Indeterminate Trusses 1 1. Systems with One Redundant Force 2 1.1 The Matrix of Members’ Flexibilities 2 1.2 Primary System and Redundant Force 2 1.3 Static Analysis of the Primary System 3 1.4 Compatibility Equations 4 1.5 Solution for X 5 1.6 Member Forces 5 Example 1 5 2. Systems with Multiple Redundant Forces 8 Example 2 10 3. Movements of Supports 14 Example 3 15 Example 4 16 Exercise 1 19

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© K. H. Ha - ENGR 343 Chapter 5 - v2.0 5.2 1. Systems with One Redundant Force 1.1 The Matrix of Members’ Flexibilities The matrix of members’ flexibilities f converts member forces Q into member deformations q : q = f Q + o q For truss members, matrix f is a diagonal matrix (Eq. 2.4): = M M M E A L E A L E A L 0 0 0 0 0 0 0 2 2 2 1 1 1 M O M M L L f ( 5 . 1 ) and o q contains member deformations due to temperature and member loading (see Table 2.1). 1.2 Primary System and Redundant Force Consider the truss system in Fig.5.1. It is statically indeterminate to the first degree (M = 6, N = 5, I = M-N = 1). This indeterminacy is due to the internal arrangement of members since the reactions are determinate 1 . An externally determinate system would become unstable if any one of its support restraint is removed, and thus, no reaction can be used as a redundant force. We must then choose a member force to be the redundant force. Let us choose, say, the axial force in member bc as the redundant X (i.e. X = F bc ). This requires that a cut be made to member bc in order to expose its force as a pair of opposing forces X external to the primary system. The cut may be made anywhere in member bc , but preferably at the end of the member as shown in Fig.5.1(b) in order to avoid cutting member bc into two parts. 1 Its three reactions can be found by equilibrium: = = = 0 , 0 , 0 y a x F M F .
© K. H. Ha - ENGR 343 Chapter 5 - v2.0 5.3 a d b c b c X X r x Displacement of node b Displacement of end b (a) Internally indeterminate system (b) Relative displacement r x P 1 P 2 Overlap of node and end b Figure 5.1 The force X itself will stretch member bc by the amount XL bc / EA bc . This stretching contributes to the relative displacement r x between the two faces of the cut section. 1.3 Static Analysis of the Primary System The primary system is subject to the actual external effects as well as the redundant force X . We need to analyse it for two separate loading cases (see Example 1): (a) Primary system subject to a pair of unit forces at the redundant X : Compute the member forces (by statics as shown in Chapter 1) and save them in vector B X . Next, compute F XX = B X T f B X ( 5 . 2 ) where F xx is the relative displacement at X (caused by the pair of unit redundant forces 2 ) at the cut section (see Eq.2.29). A typical flexibility coefficient such as F ij is the relative displacement at the redundant X i due to a unit force at the redundant X j . That is 2 The pair of unit forces cause the member forces B X , which cause deformations f B X , which cause displacement F XX = B X T f B X

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© K. H. Ha - ENGR 343
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## This note was uploaded on 10/21/2008 for the course BCEE 343 taught by Professor Dr.ha during the Winter '08 term at Concordia Canada.

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Chapter5 - Chapter 5 Statically Indeterminate Trusses...

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