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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

EA
EA
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
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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
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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
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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 …
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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.)”
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