Lecture #8 Statically indeterminate structures. Force method CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Statically determinate Statically indeterminate Equilibrium equations could Equilibrium equations could be directly solved, and thus be solved only when forces could be calculated coupled with physical law in an easy way and compatibility equations Stress state depends only Stress state depends on on geometry & loading rigidities Not survivable, moderately Survivable, widely used in modern aviation used in modern aviation (due to damage tolerance (due to damage tolerance requirement) property) Easy to manufacture Hard to manufacture 2 WAYS TO SOLVE A SOLID MECHANICS PROBLEM Displacements are set as unknowns Strains are derived Stresses are derived Equilibrium equations are solved Compatibility equations Constitutive equations Stresses are set as unknowns Constitutive equations Strains are derived Equilibrium equations Compatibility equations Equilibrium equations Compatibility equations are solved 3 METHODS TO SOLVE INDETERMINATE PROBLEM Stiffness method (slope-deflection method) Flexibility method (force method) Displacements are set as unknowns Stresses are set as unknowns Equilibrium equations are solved Compatibility equations are solved 4 METHODS TO SOLVE INDETERMINATE PROBLEM Small degree of statical indeterminacy Force method Slope-deflection method Slope-deflection method in matrix formulation Large degree of statical indeterminacy Numerical methods 5 FLOWCHART OF SOLUTION USING FORCE METHOD Classification of the problem Basic system Loaded and unity states Canonical equations Redundant constraints are removed In loaded state, external load is applied. In unity states, unit force is applied instead of constraint. Displacements corresponding to removed constraints are determined for each state Forces in removed constraints are determined Total stress state 6 BASIC (PRIMARY) SYSTEM OF FORCE METHOD Two major requirements exists: - basic system should be stable; - basic system should be statically determinate. Finally, basic system should be chosen in such a way to simplify calculations as much as possible. For example, for symmetrical problem it is essential to choose a symmetrical basic system. 7 SYSTEM OF CANONICAL EQUATIONS OF FORCE METHOD The number of equations in the system is equal to the degree of statical indeterminacy The canonical equation states that the displacement corresponding to removed constrain is zero. General view of the system is: iF ij Xj 0 j i, j – indexes varying from 1 to the degree of statical indeterminacy. 8 SYSTEM OF CANONICAL EQUATIONS OF FORCE METHOD Partial case for a singly statically indeterminate problem: 1 F 11 X 1 0 Partial case for a twice statically indeterminate problem: 1 F 11 X 1 1 2 X 2 0 2 F 21 X 1 22 X 2 0 9 SYSTEM OF CANONICAL EQUATIONS OF FORCE METHOD Determination of coefficients for trusses: 1F N F N 1 ; 11 N 1 N 1 EA EA k In trusses, if the rod is removed in basic system, the term corresponding to this rod in unit system should not be omitted. k Coefficients for members subjected to bending: F 1F Mz 1 Mz EIz 1 dx ; 11 Mz 1 Mz dx EIz 10 TOTAL STRESS STATE The total stress state is found as a sum of loaded state and unit states multiplied by corresponding constraint reactions: Fk F (F ) k F (1 ) k Xi i i – index varying from 1 to the degree of statical indeterminacy; k – index of the force factor F; F could represent any force factor (normal force, bending moment, constraint reaction etc.). 11 THREE BASIC EQUATIONS Physical sense of equations (revision) Equilibrium equations This is not only the sum of forces or moments, but applies for elementary volume as well Constitutive equations Physical law, expresses the relation between stress and strain Compatibility equations Solid body should remain continuous while being deformed 12 THREE BASIC EQUATIONS How are they implemented in force method Equilibrium equations Equilibrium equations for basic system in loaded and unit states Constitutive equations Through Young’s moduli in Mohr’s integrals Compatibility equations Canonical equations of force method 13 EXAMPLE OF FORCE METHOD APPLICATION 14 EXAMPLE OF FORCE METHOD APPLICATION 0.280 Analytically derived values are shown near numerical results. 0.604 -0.427 -0.396 0.280 0.280 15 SYSTEM OF CANONICAL EQUATIONS OF FORCE METHOD Specific case for temperature actuated displacement: 1F T N 1 k Specific case when displacement at the constraint is given: 1 F 11 X 1 constraint 16 WHERE TO FIND MORE INFORMATION? For force method, I recommend Kassimali. Structural Analysis. 3rd ed. 2005 Chapter 13 For Mohr’s integral, refer to your Mechanics of Materials course. … Internet is boundless … 17 TOPIC OF THE NEXT LECTURE Statically indeterminate structures. Fuselage frames All materials of our course are available at department website k102.khai.edu 1. Go to the page “Библиотека” 2. Press “Structural Mechanics (lecturer Vakulenko S.V.)” 18

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