Subject
Year
: S1014 / MECHANICS of MATERIALS
: 2008
Indeterminate Structure
Session 23-26
Indeterminate Structure
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What is Indeterminate ?
a structure is statically indeterminate
when the static equilibrium equations
are not sufficient for determining
the internal forces and reactions
on that structure …..
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What is Indeterminate ?
“Statically Indeterminate” means
the # of unknowns exceeds the
number of available equations of
equilibrium.
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What is Indeterminate ?
Statics (equilibrium analysis) alone
cannot solve the problem
nR = # of reactions (or unknowns)
nE = # of equilibrium equations
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What is Indeterminate ?
• If nR > nE: statically indeterminate
- too many unknowns, must invoke a
constraint such
as a deformation relation.
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What is Indeterminate ?
• If nR = nE: statically determinate
- forces
in each member only depend on
equilibrium.
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Statically Indeterminate Examples
Free body diagram
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Statically Indeterminate Examples
Free body diagram
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STATISTICALLY INDETERMINATE BEAMS AND SHAFTS (CONT.)
• Strategy:
The additional support reactions on the beam or shaft that are not needed to keep it
in stable equilibrium are called redundants. It is first necessary to specify those
redundant from conditions of geometry known as compatibility conditions.
Once determined, the redundants are then applied to the beam, and the remaining
reactions are determined from the equations of equilibrium.
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METHOD OF SUPERPOSITION
•
Necessary conditions to be satisfied:
1. The load w(x) is linearly related to the deflection v(x),
2. The load is assumed not to change significantly the original geometry of the
beam of shaft.
Then, it is possible to find the slope and displacement at a point on
a beam subjected to several different loadings by algebraically
adding the effects of its various component parts.
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STATISTICALLY INDETERMINATE BEAMS AND SHAFTS
• Definition: A member of any type is classified statically indeterminate if the number
of unknown reactions exceeds the available number of equilibrium equations.
e.g. a continuous beam having 4 supports
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USE OF THE METHOD OF SUPERPOSITION
Elastic Curve
• Specify the unknown redundant forces or
moments that must be removed from the beam in
order to make it statistically determinate and
stable.
• Using the principle of superposition, draw the
statistically indeterminate beam and show it equal
to a sequence of corresponding statistically
determinate beams.
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USE OF THE METHOD OF SUPERPOSITION
Elastic Curve
•The first of these beams, the primary beam,
supports the same external loads as the statistically
indeterminate beam, and each of the other beams
“added” to the primary beam shows the beam loaded
with a separate redundant force or moment.
• Sketch the deflection curve for each beam and
indicate the symbolically the displacement or slope
at the point of each redundant force or moment.
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USE OF THE METHOD OF SUPERPOSITION
Compatibility Equations
• Write a compatibility equation for the
displacement or slope at each point where
there is a redundant force or moment.
• Determine all the displacements or slopes
using an appropriate method
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USE OF THE METHOD OF SUPERPOSITION
Compatibility Equations
• Substitute the results into the compatibility
equations and solve for the unknown redundants.
• If the numerical value for a redundant is positive,
it has the same sense of direction as originally
assumed. Similarly, a negative numerical value
indicates the redundant acts opposite to its
assumed sense of direction.
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USE OF THE METHOD OF SUPERPOSITION
Equilibrium Equations
Once the redundant forces and/or
moments have been determined, the
remaining unknown reactions can be found
from the equations of equilibrium applied
to the loadings shown on the beam’s free
body diagram.
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Buckling
• Buckling is a mode of failure that does not depend on stress or strength, but
rather on structural stiffness
• Examples:
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More buckling examples…
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Buckling
• The most common problem involving buckling is the design of
columns
– Compression members
• The analysis of an element in buckling involves establishing a
differential equation(s) for beam deformation and finding the
solution to the ODE, then determining which solutions are stable
• Euler solved this problem for columns
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Euler Column Formula
•
c EI

2
L
2
Pcrit
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Pcrit 
 EI
2
2
e
L
Euler Column Formula
•
Where C is as follows:
C = ¼ ;Le=2L
Fixed-free
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Euler Column Formula
•
Where C is as follows:
C = 2; Le=0.7071L
Fixed-pinned
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Euler Column Formula
•
Where C is as follows:
C = 1: Le=L
Rounded-rounded
Pinned-pinned
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Euler Column Formula
•
Where C is as follows:
C = 4; Le=L/2
Fixed-fixed
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Buckling
• Geometry is crucial to correct analysis
– Euler – “long” columns
– Johnson – “intermediate” length columns
– Determine difference by slenderness ratio
• The point is that a designer must be alert to the possibility of
buckling
• A structure must not only be strong enough, but must also be
sufficiently rigid
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Buckling Stress vs. Slenderness Ratio
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Johnson Equation for Buckling
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Solving buckling problems
Find Euler-Johnson tangent point with
2 E


Sy
Le
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2
Solving buckling problems
For Le/ < tangent point (“intermediate”), use Johnson’s Equation
Scr 
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 E
2
 Le 


  
2
Solving buckling problems
For Le/ > tangent point (“long”), use Euler’s equation:
2
 Le 
Scr  S y  2  
4 E   
Sy
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2
Solving buckling problems
For Le/ < 10 (“short”)
Scr = Sy
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Solving buckling problems
If length is unknown, predict whether it
is “long” or “intermediate”, use the
appropriate equation, then check using
the Euler-Johnson tangent point once
you have a numerical solution for the
critical strength
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Special Buckling Cases
• Buckling in very long Pipe
Pcrit
c 2 EI

L2
Note Pcrit is inversely related to length squared
A tiny load will cause buckling
L = 10 feet vs. L = 1000 feet:
Pcrit1000/Pcrit10 = 0.0001
•Buckling under hydrostatic Pressure
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Pipe in Horizontal Pipe Buckling Diagram
•
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Far End vs. Input Load with Buckling
•
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Buckling Length: Fiberglass vs. Steel
•
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