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Lecture #6 Classification of structural analysis problems. Statical determinacy. CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS. 2. CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS. 3. CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS. - PowerPoint PPT Presentation
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Lecture #6Classification of structural analysis problems. Statical determinacy
Structures
Constrained (fixed)
External loads are balanced by constraint
forces
Unconstrained (free)
External loads are balanced by inertia forces
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
2
Structures
Constrained (fixed) a) stable (immovable)
b) unstable (movable)
Unconstrained (free)a) stable (invariable)
b) unstable (variable)
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CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
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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
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
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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.
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
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CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Equilibriumequations
Constitutiveequations
Compatibilityequations
This is not only the sum offorces or moments, but appliesfor elementary volume as well
Physical law, expressesthe relation between
stress and strain
Solid body shouldremain continuous
while being deformed
Three basic equations
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Geometrical nonlinearity
Physical nonlinearity
---
Equilibrium conditions depend on displacement values
Plastic effects are taken into account (nonlinear Physical law)
Usually, only linear case of Compatibility conditions is studied
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMSTwo basic nonlinearities
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Question #1: is problem stable or not?We must determine which science to use for analysis, and should we consider the geometrical nonlinearity.
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Question #2: is structure statically determinate or not?The answer is required to choose the proper method of structural mechanics.
… and if structural analysis could be applied for a given problem, we get …
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The structure is statically determinate if internal forces in all members and all constraint forces could be determined using equations of equilibrium only.
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
statically determinate
statically indeterminate
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EXAMPLES OF TRUSSES USED IN BRIDGES
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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CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Is the structure simple?
Use kinematic analysis, to check for stability and suppose the determinacy. This
is necessary but not sufficient method
Is the structure stable?
Use structural analysis, to find both stability and determinacy
Use statical analysis, which is sufficient to confirm the results of
kinematic analysis
yes
yes
no
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To analyze the structure for kinematic stability and static determinacy, three methods are used:
METHODS TO CLASSIFY THE PROBLEM
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• 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.
BASIC DEFINITIONS
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Degrees of freedom (DOF) – independent parameters which determine the position of the member.
BASIC DEFINITIONS
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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Two approaches are used:composition and decomposition.
STRUCTURAL ANALYSIS
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.
Number of DOFs in system is calculated.
KINEMATICAL ANALYSIS
Formulas for trusses:
a) for 2d:
b) 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.
2i r c n 3i r c n
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KINEMATICAL ANALYSIS
Formulas for general structures:
a) for 2d:
b) 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.
2 2 3i r c h n d 3 3 6i r c h n d
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Results of kinematical analysis:
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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• 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.
STATICAL ANALYSIS
Kinematical analysis supposes that structure is once statically indeterminate:
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STATICAL ANALYSIS - EXAMPLE
2 2 3
3 4 2 0 2 0 3 2 1
i r c h n d
Statical analysis claim that structure is not stable!21
STATICAL ANALYSIS - EXAMPLE
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STATICAL ANALYSIS - EXAMPLE
Is the structure simple?
Use kinematic analysis, to check for stability and suppose the determinacy. This
is necessary but not sufficient method
Is the structure stable?
Use structural analysis, to find both stability and determinacy
Use statical analysis, which is sufficient to confirm the results of
kinematic analysis
yes
yes
no
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To analyze the structure for kinematic stability and static determinacy, three methods are used:
METHODS TO CLASSIFY THE PROBLEM
TOPIC OF THE NEXT LECTURE
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Statically indeterminate structures. Method of forces
All materials of our course are availableat department website k102.khai.edu
1. Go to the page “Библиотека”2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”