Chapter6

# Chapter6 - Chapter 6 Statically Indeterminate Frames 1...

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© K. H. Ha - ENGR 343 Chapter 6 - v1.2 6.1 Chapter 6 Statically Indeterminate Frames 1. General Procedure 2 2. Systems With One Redundant Force 4 Example 1 4 Example 2 9 Example 3 12 Exercise 1 16 3. Systems with Multiple Redundants 17 3.1 The Compatibility Equations 18 3.2 The Flexibility Coefficients 18 Example 4 19 4. Treatment of Supports’ Movements 22 4.1 Supports’ Movements at the Redundants 22 4.2 Supports’ Movements not at the Redundants 23

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© K. H. Ha - ENGR 343 Chapter 6 - v1.2 6.2 1. General Procedure The same procedure for analysis of trusses also works for frames as follows (see illustration in Example 1): 1. Identify and number the member forces Q M×1 and deformations q M×1 to take into account the action in individual members. The flexibility relation q M×1 = f M×M Q M×1 may include bending and/or axial deformations depending on the action of the individual members. 2. Choose redundant forces and the corresponding primary system. First, we identify the degree of statical indeterminacy ( I ) of the system. If ( I ) is greater than zero, select a primary system by suitable cuts, exposing ( I ) internal forces and/or reactions X I×1 = { X 1 , X 2 , . . X I } which are now considered as additional external forces acting on the primary system. Proper selection of the redundant forces will ensure that the primary system is both stable and statically determinate. 3. Analyse the primary system for ( I + 1) loading cases: (a) For each redundant in turn: Apply a load of one-unit at the redundant . With the unit load at the redundant X i , compute and save the member forces in the i th -column of matrix B X . The order of B X is M by I. Once matrix B X is complete, compute the flexibility matrix: F XX = B X T f B X ( 6 . 1 ) where F ij = relative displacement at X i caused by a pair of unit forces at X j . 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 F ij due to unit force at X j at X i Translation or rotation
© K. H. Ha - ENGR 343 Chapter 6 - v1.2 6.3 (b) The actual loading : Compute and save the member forces in vector Q L and compute the relative displacements (at the cut sections) r o , from Eq.2.18: r o = B X T ( f Q L + o q ) (6.2) Vector o q takes into account thermal expansion or member loading within individual members (Table 2.1). 4. Form ( I ) compatibility equations that restore the continuity at the redundants 1 : 0 or 0 0 0 2 2 1 1 2 2 22 1 21 2 2 1 2 12 1 11 1 1 = + = = + + + + = = + + + + = = + + + + = X F r r XX o X I II I I oI XI I I o X I I o X X F X F X F r r X F X F X F r r X F X F X F r r L L L L (6.3) r xi = total relative displacement at the redundant X i caused by environmental effects, actual loading as well as X 1 , X 2 , . . X I applied to the primary system. r o i = relative displacement at the redundant X i caused by environmental effects and actual loading only. The environmental effects, however, do not cause internal member forces in the primary system.

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Chapter6 - Chapter 6 Statically Indeterminate Frames 1...

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