Lecture #6 Classification of structural analysis problems. Statical determinacy

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

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Structures

Constrained (fixed) a) stable (immovable)

b) unstable (movable)

Unconstrained (free)a) stable (invariable)

b) unstable (variable)

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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


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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.


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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.


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.


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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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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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:


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


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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.)”

http://k102.khai.edu/

http://k102.khai.edu/ru/site/library.html

http://k102.khai.edu/ru/site/structural-mechanics.html



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Lecture #6 Classification of structural analysis problems. Statical determinacy