Classification of structural analysis problems. Statical determinacy

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Lecture 4 Classification of structural analysis
problems. Statical determinacy
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CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
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CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
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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 (revision)
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CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Two basic nonlinearities
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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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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
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METHODS TO CLASSIFY THE PROBLEM
To analyze the structure for kinematic stability
and static determinacy, three methods are used
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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 (-) 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
• for 2d
• for 3d
• i degree of indeterminacy
• r number of rods
• c number of constrained DOFs
• n number of nodes.

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KINEMATICAL ANALYSIS

• Formulas for general stuctures
• for 2d
• for 3d
• i degree of indeterminacy
• r number of rods
• c number of constrained DOFs
• 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 lt 0 unstable problem i 0 statically
determinate problem i gt 0 statically
indeterminate problem.
If kinematical analysis shows than 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
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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
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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 ?????????? 2. Press Structural Mechanics
(lecturer Vakulenko S.V.)
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