```Lecture #6
Classification of structural analysis problems.
Statical determinacy
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Structures
Constrained (fixed)
Unconstrained (free)
balanced by constraint
forces
balanced by inertia forces
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CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Structures
Constrained (fixed)
a) stable (immovable)
Unconstrained (free)
a) stable (invariable)
b) unstable (movable)
b) unstable (variable)
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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
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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.
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CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Three basic equations
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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CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Two basic nonlinearities
Geometrical
nonlinearity
Equilibrium conditions depend on
displacement values
Physical
nonlinearity
Plastic effects are taken into
account (nonlinear Physical law)
---
Usually, only linear case of
Compatibility conditions is studied
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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.
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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
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EXAMPLES OF TRUSSES USED IN BRIDGES
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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
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
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METHODS TO CLASSIFY THE PROBLEM
To analyze the structure for kinematic stability and
static determinacy, three methods are used:
Is the
structure
simple?
no
Is the
structure
stable?
yes
Use structural analysis, to find
both stability and determinacy
Use kinematic analysis, to check for
stability and suppose the determinacy. This
is necessary but not sufficient method
yes
Use statical analysis, which is
sufficient to confirm the results of
kinematic analysis
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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.
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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.
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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.
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KINEMATICAL ANALYSIS
Number of DOFs in system is calculated.
Formulas for trusses:
a) for 2d:
i  r  c  2n
b) for 3d:
i  r  c  3n
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.
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KINEMATICAL ANALYSIS
Formulas for general structures:
a) for 2d:
i  r  c  2h 2n  3d
b) for 3d:
i  r  c  3h 3n  6d
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.
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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.
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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.
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STATICAL ANALYSIS - EXAMPLE
Kinematical analysis supposes that structure is once
statically indeterminate:
i  r  c  2h 2n  3d 
 3 4 20 20 32  1
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STATICAL ANALYSIS - EXAMPLE
Statical analysis claim that structure is not stable!
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STATICAL ANALYSIS - EXAMPLE
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METHODS TO CLASSIFY THE PROBLEM
To analyze the structure for kinematic stability and
static determinacy, three methods are used:
Is the
structure
simple?
no
Is the
structure
stable?
yes
Use structural analysis, to find
both stability and determinacy
Use kinematic analysis, to check for
stability and suppose the determinacy. This
is necessary but not sufficient method
yes
Use statical analysis, which is
sufficient to confirm the results of
kinematic analysis
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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.)”
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