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

Determinate or Indeterminate ? Yes Structure Equilibrium No

Why Indetermiante ? Advantages Smaller stresses and deflections

Why Indetermiante ? Advantages Fail safe 1995 Oklahoma City bombing

Prices Disadvantages Stresses due to support settlement

Prices Disadvantages Stresses due to temperature changes

Indeterminate Structures

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

Symmetrical Structures

Symmetrical Structures

Symmetrical Loadings

Symmetrical Loadings

Anti-Symmetrical Loadings

Anti-Symmetrical Loadings

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

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

Decomposition of Loadings

Decomposition of Loadings

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

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

Symmetrical Structures under Symmetrical Loads

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

Symmetrical Structures under Anti-symmetrical Loads

Analysis

Analysis 6 degrees of indeterminacy 4 degrees of indeterminacy

Analysis

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

Compatibility DB=0 By

Compatibility

Compatibility

Compatibility

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

Compatibility

Compatibility Settlement

Compatibility Settlement

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

Least Work Method DF F P 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

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

Least Work Method Strain energy F1 F2 Fn P1 P2 Pm Forces that do work Forces that do not do work