Chapter4

# Chapter4 - Chapter 4 Statically Indeterminate Beams 1...

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© K. H. Ha - ENGR 343 Chapter 4 - v2.2 4.1 Chapter 4 Statically Indeterminate Beams 1. Consistent Deformation Method. ............................................................... 2 1.1 One Redundant Systems . ................................................................ 2 1.2 Alternative Primary Systems. ........................................................... 3 1.3 Standard Form of the Compatibility Equations . ................................... 5 2. Fixed-End Moments . ............................................................................... 6 Example 1 . .......................................................................................... 6 Example 2 . .......................................................................................... 8 Example 3 . .......................................................................................... 9 Exercise 1 . ........................................................................................... 11 3. Fixed-End Moments by Member’s Flexibility Relation. ................................. 12 4. Three-Moment Equation for Continuous Beam Analysis . ............................. 15 4.1 The Basis for Three-moment Equation. ............................................ 15 Example 4 . ........................................................................................ 16 Exercise 2 . ........................................................................................... 19 Exercise 3 . ........................................................................................... 20 4.2 The Classical Three-moment Equation . ............................................ 21 Example 5 . ........................................................................................ 22 4.3 Effects of Support Movements. ....................................................... 24 Example 6 . ........................................................................................ 25 Exercise 4 . ........................................................................................... 28 Exercise 5 . ........................................................................................... 28

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© K. H. Ha - ENGR 343 Chapter 4 - v2.2 4.2 1. Consistent Deformation Method 1.1 One Redundant Systems Objective For the 2-span continuous beam in Fig.4.1(a), we want to select a suitable redundant force and set up the corresponding compatibility equation for its solution. w a b c L /2 L w a b c a b c X 5 wL 4 384 EI L 3 48 X (c) Actual loading on primary system (d) Redundant force on primary system w a b c L L X (a) Actual system and loading (b) Primary system and loading r x = 0 ? Figure 4.1 Degree of statical indeterminacy The beam is indeterminate to the first degree 1 since it is a stiff body supported by 4 reaction components. The number of required redundant forces is I = 1. Primary system and redundant forces Let us choose the vertical reaction at the support b as the redundant force X . Next, expose X as an external force on the primary system (b). The primary system is subject to the actual loading as well as to the redundant force X which is now considered as an external force. Note that the primary system must be statically determinate and stable . 1 Considering bending action only: M = 2, N = 1, I = 1.
© K. H. Ha - ENGR 343 Chapter 4 - v2.2 4.3 The criterion for solution: Compatibility equations The two systems (a) and (b) are identical when the value of X is such that the vertical displacement at X , say r x , of point b is zero. The equation r x = 0 is the compatibility condition that makes the two systems the same. Setting up the compatibility equations It is necessary that the primary system (b) be statically determinate and stable 2 so that the displacement 3 r x can be computed in terms of the actual loading w and X. Using virtual work method, or from beam tables, we get 0 48 384 5 3 4 = + = X EI L EI wL r x ( 4 . 1 ) The first term in r x is due to the actual loading w acting alone on the primary system as shown in Fig.(c). The second terms is due to X acting alone as shown in Fig.(d). They can be found from beam tables, or computed separately and combined. Solving for X from the compatibility equation gives: 8 5 wL X = Complete solution With X known, the complete solution for the original indeterminate System (a) may be easily found by analysing the determinate System (b).

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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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Chapter4 - Chapter 4 Statically Indeterminate Beams 1...

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