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CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

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Determinate or Indeterminate ? Structure Equilibrium Determinate Indeterminate Yes No

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Why Indetermiante ? Advantages Smaller stresses and deflections

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Why Indetermiante ? Advantages Fail safe 1995 Oklahoma City bombing

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Prices Disadvantages Stresses due to support settlement

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Prices Disadvantages Stresses due to temperature changes

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Indeterminate Structures

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Symmetric Structures StructureReflection Axis of symmetry Identical in geometry, supports and material properties

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Symmetrical Structures

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Symmetrical Loadings

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Anti-Symmetrical Loadings Anti-Symmetrical

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Anti-Symmetrical Loadings

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Decomposition of Loadings (A) (B) = (A)/2 (C) = Reflection of (B) (B)+(C) (B)-(C) Symmetrical Anti-symmetrical sum

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Decomposition of Loadings Loadings = Symmetrical + Anti-symmetric Loads + =

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Decomposition of Loadings

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Analysis of Symmetrical Structures Loading Anti-symmetrical Loading Symmetrical Loading Symmetrical Structure Response 1 Response 2 + Response

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Symmetrical Structures under Symmetrical Loads L aa PP Moment & vertical displacement ≠ 0 Slope & axial displacement = 0 P V ≠ 0 slope = 0 M ≠ 0

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Symmetrical Structures under Symmetrical Loads

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Symmetrical Structures under Anti-symmetrical Loads L/2 a a P P Slope ≠ 0 Moment & vertical displacement = 0 P L/2 Slope ≠ 0 M = 0 V = 0

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Symmetrical Structures under Anti-symmetrical Loads

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Analysis

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6 degrees of indeterminacy 4 degrees of indeterminacy

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Analysis

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Analysis of Statically Indeterminate Structures Force methods This chapter Displacement methods Next two chapters

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Compatibility ByBy B =0

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Compatibility

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A B P C D C D P 1 1 = +

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Settlement

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Compatibility Settlement

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Least Work Method Castigliano’s theory P P F U(P,F) M(P,F)

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Least Work Method P FF F M(P,F) U(P,F) Obtain F The magnitude of redundant force must be such that the strain energy stored in the structure is a minimum

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Least Work Method Virtual work principle P F PP P F M(P,F) Castigliano’s theorem Least work principle Note: F does not do any work !

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Least Work Method P1P1 FnFn F2F2 F1F1 PmPm P2P2 Strain energy Forces that do not do work Forces that do work

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