1. Lecture #6
Classification of structural analysis problems. Statical determinacy

2. CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS2

3. CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS3

4. internally deficient externally deficient
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Kinematically unstable structures could not be analyzed by methods of structural mechanics.
They represent mechanisms and are studied by engineering mechanics.
Before starting the force analysis, one should check if the structure kinematically stable or not.
The reason of instability could be internal or external.
internally deficient
externally deficient 4

5. CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Instability could be instantaneous and permanent.
Usually, structures which are unstable instantaneously, could be analyzed as geometrically nonlinear problems, but this is a special part of structural mechanics science. 5

6. CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Three basic equations6

7. Two basic nonlinearities
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Two basic nonlinearities 7

8. CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Question #1: is problem stable or not?
We must determine which science to use for analysis, and should we consider the geometrical nonlinearity.
… and if structural analysis could be applied for a given problem, we get …
Question #2: is structure statically determinate or not?
The answer is required to choose the proper method of structural mechanics. 8

9. statically determinate
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
The structure is statically determinate if internal forces in all members and all constraint forces could be determined using equations of equilibrium only.
statically determinate
statically indeterminate 9

10. EXAMPLES OF TRUSSES USED IN BRIDGES10

11. 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 11

12. METHODS TO CLASSIFY THE PROBLEM
To analyze the structure for kinematic stability and static determinacy, three methods are used: 12

13. Disk (ABD) – any general bar, excluding rods.
BASIC DEFINITIONS
Rod (AC, CB, CD) – bar which works only in tesion/compression. Wires and columns are partial cases.
Disk (ABD) – any general bar, excluding rods.
Node (A, C, D) – joint of rods, including nodes at supports.
Hinge (none at this figure) – hinge between disks. 13

14. Disk has 3 DOFs in plane and 6 DOFs in space.
BASIC DEFINITIONS
Degrees of freedom (DOF) – independent parameters which determine the position of the member.
Disk has 3 DOFs in plane and 6 DOFs in space.
Node has 2 DOFs in plane and 3 DOFs in space.
Each type of support constrains certain number of DOFs. 14

15. Two approaches are used: composition and decomposition.
STRUCTURAL ANALYSIS
Two approaches are used:
composition and decomposition.
Members satisfying structural rules for planar systems:
node of two not collinear rods;
disk connected by three rods, not parrallel and not crossing in one point;
disk connected by a hinge and a rod which do not pass through the hinge.
Members satisfying structural rules for spatial systems:
node of three rods not liying in one plane;
disk connected by six rods, neither two of them are collinear. 15

16. i – degree of indeterminacy; r – number of rods;
KINEMATICAL ANALYSIS
Number of DOFs in system is calculated.
Formulas for trusses:
for 2d:
for 3d:
i – degree of indeterminacy;
r – number of rods;
c – number of constrained DOFs (or number of DOFs for free body if structure is free);
n – number of nodes. 16

17. i – degree of indeterminacy; r – number of rods;
KINEMATICAL ANALYSIS
Formulas for general structures:
for 2d:
for 3d:
i – degree of indeterminacy;
r – number of rods;
c – number of constrained DOFs (or number of DOFs for free body if structure is free);
h – number of hinges which are not nodes;
n – number of nodes;
d – number of disks. 17

18. i < 0 – unstable problem; i = 0 – statically determinate problem;
KINEMATICAL ANALYSIS
Results of kinematical analysis:
i < 0 – unstable problem;
i = 0 – statically determinate problem;
i > 0 – statically indeterminate problem.
If kinematical analysis shows that problem is stable, the result should be checked by statical analysis. 18

19. rang(A)=min(m,n) STATICAL ANALYSIS
Matrix of coefficients A(m,n) of static equilibrium equations is calculated.
The single condition is that
rang(A)=min(m,n)
Despite the simplicity of formulation, statical analysis is most complex and comprehensive.
Statical analysis is sufficient by itself, but is usually used as a last step for complex problems. 19

20. STATICAL ANALYSIS - EXAMPLE
Kinematical analysis supposes that structure is once statically indeterminate: 20

21. Statical analysis claim that structure is not stable!
STATICAL ANALYSIS - EXAMPLE
Statical analysis claim that structure is not stable! 21

22. STATICAL ANALYSIS - EXAMPLE22

23. METHODS TO CLASSIFY THE PROBLEM
To analyze the structure for kinematic stability and static determinacy, three methods are used: 23

24. Statically indeterminate structures. Method of forces
TOPIC OF THE NEXT LECTURE
Statically indeterminate structures. Method of forces
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.)” 24