1. Lecture #7
Statically indeterminate structures.
Force method

2. CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Statically determinate
Statically indeterminate
Equilibrium equations could be directly solved, and thus forces could be calculated in an easy way
Equilibrium equations could be solved only when coupled with physical law and compatibility equations
Stress state depends only on geometry & loading
Stress state depends on rigidities
Not survivable, moderately used in modern aviation (due to damage tolerance requirement)
Survivable, widely used in modern aviation
(due to damage tolerance property)
Easy to manufacture
Hard to manufacture 2

3. WAYS TO SOLVE A SOLID MECHANICS PROBLEM3

4. (slope-deflection method)
METHODS TO SOLVE INDETERMINATE PROBLEM
Stiffness method
(slope-deflection method)
Flexibility method
(force method) 4

5. Slope-deflection method
METHODS TO SOLVE INDETERMINATE PROBLEM
Small degree
of statical indeterminacy
Force method
Slope-deflection method
Slope-deflection method
in matrix formulation
Numerical methods
Large degree
of statical indeterminacy 5

6. FLOWCHART OF SOLUTION USING FORCE METHOD6

7. 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

8. i, j – indexes varying from 1 to the degree of statical indeterminacy.
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:
i, j – indexes varying from 1 to the degree of statical indeterminacy. 8

9. SYSTEM OF CANONICAL EQUATIONS OF FORCE METHOD
Partial case for a singly statically indeterminate problem:
Partial case for a twice statically indeterminate problem: 9

10. SYSTEM OF CANONICAL EQUATIONS OF FORCE METHOD
Determination of coefficients for trusses:
In trusses, if the rod is removed in basic system, the term corresponding to this rod in unit system should not be omitted.
Coefficients for members subjected to bending: 10

11. i – index varying from 1 to the degree of statical indeterminacy;
TOTAL STRESS STATE
The total stress state is found as a sum of loaded state and unit states multiplied by corresponding constraint reactions:
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

12. Physical meaning of equations (revision)
THREE BASIC EQUATIONS
Physical meaning of equations (revision) 12

13. How are they implemented in force method
THREE BASIC EQUATIONS
How are they implemented in force method 13

14. EXAMPLE OF FORCE METHOD APPLICATION14

15. 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

16. SYSTEM OF CANONICAL EQUATIONS OF FORCE METHOD
Specific case for temperature actuated displacement:
Specific case when displacement at the constraint is given: 16

17. … Internet is boundless …
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

18. Statically indeterminate structures.
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