Nonlinear Structures:

1
Nonlinear Structures

• Analyzing Stability

Colleen Duffy University of St. Thomas
2
Outline

• Background information
• Relevant info
• Irrelevant info
• Method 1 balancing forces
• Method 2 potential energy
• 2nd Derivatives Test
• Gradient Descent
• Conclusion
• Works Consulted

3
Abstract

• In any field of study idealizations make the
  world easier to deal with and easier to
  understand. It is common to use generalizations
  and simplifications to understand basic
  principles and get a fuzzy image of what is being
  dealt with. This is a necessary first step in
  attaining a sharper picture of the world, but
  there comes a point when we have to leave utopia
  and come back to the real world. This research
  project deals with determining the stability of
  structures, which are by nature nonlinear.
  Nonlinear systems, however, are messier and more
  difficult to work with. This presentation
  approaches the problem in a couple of different
  ways.

4
Glossary

• 1.      elongation amount of a change in length
  
• 2.      frame structure made up of rigid pieces
  each of which has 2 joints connecting it with
  other members (Lamb)
• 3.      Hookes Law - law of elasticity
  discovered by the English scientist Robert Hooke
  in 1660, which states that, for relatively small
  deformations of an object, the displacement or
  size of the deformation is directly proportional
  to the deforming force or load, F-kx
  (Britannica)
•  
• 4.      just rigid ceases to be rigid when any
  one bar is removed (Lamb)
•  
• 5.      node where two or more bars meet and
  are held together, a joint
•  

5
Glossary cont.

• 6.      potential energy the energy a body
  possesses by virtue of its position relative to
  some standard position and it is measured by the
  amount of work required to get it into its
  position or by the amount of work it can perform
  in returning to its original position (assume no
  dissipated work) (Scott)
•  
• 7.      stability when a structure maintains
  its basic location and shape, does not collapse,
  when subjected to moderate forcing
• 8.      stiff/rigid incapable of deformation
  without alteration of the length of at least one
  bar (Lamb)
•  
• 9. structure - made up of bars connected at
  joints. The bars can be slightly elongated, act
  like stiff springs, and can rotate to a degree
  at the nodes.

6
Linear vs. Nonlinear

• The main difference is in the equation we use for
  the elongation of a bar. In the linear case we
  use a linear approximation determined from the
  Taylor polynomial of the elongation. Now, we are
  using the actual equation.
• The ends of a bar are at ai(xi,yi,zi) and
  aj(xj,yj,zj). The length of this bar is
  ai-aj

• When a bar is stretched, it has a new length
  Le where u is the direction vector of the
  displacement and ? is the magnitude.
• The linear approximation of the elongation
  is
• The actual value is where

(ai ?ui ) - (aj?uj )
e
a ai - aj 2, b 2(ai - aj).( ui uj), and
c ui uj 2
7
Method 1 Balancing Act

• The bars obey Hookes Law the internal force is
  y-ce, c is the spring constant
• If the structure is stable, there will be no
  elongation of the bars (no external force). Or,
  the internal force must be equal to the external
  force at each node.
• Solutions give the displacements of the nodes.
  An infinite number signifies an unstable
  structure, a discrete number shows the possible
  stable configurations.

fijyij
8
Example 1 Compass
a3
f
y1
y2
a1
a2
We need to add up the forces at each node and set
each equal to the external force then solve these
equations simultaneously.
The shortcut is to take the elongation/internal
force for each bar and multiply by the unit
vectors of each node for that bar. This
eliminates the need for calculating the
elongation twice for each bar.
These are the solutions to the system of
equations. Notice that in the nonlinear analysis
there are 4 implicit solutions and in the linear
there is only one simplified solution. This
demonstrates the limitations of linear
approximation.
Nonlinear
Linear
9
This shows how the structure is moving in time.
As you can see two nodes are fixed while the top
one rotates in a circle. In this case the bars
are not being stretched.
Compass con.
We now fix nodes 1 and 2. The compass is still
unstable as can be seen below. Again we see a
pronounced difference between linear and
nonlinear results.
This is the linear picture. Instead of moving in
a circle, the top node moves along a straight
line. Also, the bars are not stretched when
considered as linear, but do stretch when
considered as nonlinear.
The nonlinear structure moves in a circle in the
x-z plane. This is a graph of the solutions
10
Example 2 arch
In our second example we see a structure that
gives no solution for an answer. This means that
no configuration leads to zero elongation. This
does not intuitively make sense because a zero
displacement should always be a solution. Using
other methods we will see that we are right later
on.
Unstable arch

When a bar is added to the arch, it becomes
stable. The solution then shows the four
possible configurations that the arch can have.
It will not, however, move between these
configurations as in its unstable form.
Stable arch
One of the solutions given above for zero
force/elongation
11
Method 2 Potential

• Potential energy for a spring is PE1/2 kx2. The
  spring constant, k, is still taken as one and x
  is e. Therefore, the potential energy for a
  frame is
• The frame is stable when potential energy is at a
  minimum. Just like in the last method this
  occurs when elongation is zero.
• To find the minimum we can take the partial
  derivatives of the potential energy function and
  then use the Second Derivatives Test or we can
  use the Gradient/Steepest Descent Method.

12
Compass checking answers

• We will now analyze the compass using its
  potential energy. Hopefully we will get the same
  results.
• Here is its potential energy function
• By taking the partials with respect to x, y, and
  z, setting them all equal to zero, and solving
  simultaneously, we get the critical points.
• To test them we have to make sure each of the
  determinants of the partials is greater than zero.

0, 0,
0, , and 0
13
Compass con.

• The test tells us that 3 of the equilibrium
  points are maximums and the other ones are
  indeterminate using the test. They do, however,
  satisfy the first two implicit answers.
• The first two are the same answers we got from
  the first method, so this method checks.
• We can also graph the potential energy function
  to see if we can tell what its minimums are
  graphically.
• The minimums are a circle. This verifies that
  the critical points we got are indeed minimums
  and correspond to the solution from method 1.

14
Gradient Descent

• Finding the minimums of the potential energy is a
  long process done the way just explained. An
  easier way (with a computer) is to use the
  gradient descent method.
• The gradient of a function is
• - points in the direction of steepest
  descent, so small steps can be taken along the
  gradient until a local minimum is found. This is
  similar to the idea of Newtons Method.
• Disadvantages 1. You have to have an idea of
  where the local minimums lie. 2. You cant get
  complete answers i.e. in an unstable structure
  when there are an infinite number of minimums you
  cannot find them all or a function that describes
  all of them.
• Advantages 1. You can find an answer where
  solving a system of equations led to no solution
  before. 2. It is easier and you do not have to
  worry about indeterminate critical points.

15
Arch solution exists

• Recall that by using method 1 no solution was
  found for the unstable arch. (You can use
  Newtons Method to find the roots of the system
  of equations but the results are not very
  accurate.)
• We can make a table of values of the minimums
  using the gradient descent method. We can only
  get a sampling this way, but at least we get some
  idea.
• We can graph each of the solutions and then
  animate them to see how the arch moves.

16
Conclusions

• Why have two methods?
• One may work when the other does not
• Check answers
• One may be easier to conceptualize or to do
• Both methods tell us when a structure is stable
• Nonlinear analysis is more complex and gives
  messier answers, but they are a lot more accurate
  than using linear approaches. The linear
  solutions often gave one possibility where many
  exist.
• Where to go from here
• Try to make the nonlinear analysis doable for
  more complex structures
• See how various external forces affect the frames.

17
Questions?
18
Works Consulted

• Lamb, Horace. Statics, 3rd Ed. Cambridge
  University Press, 1946.
• Mathematica.
• Olver, Peter J. and Chehrzad Shakiban. 6.3
  Structures in Equilibrium. 2000.
• Saaty, Thomas L. and Joseph Bram. Nonlinear
  Mathematics. New York
• McGraw-Hill Book Company, Inc., 1964.
• Scott, Merit. Mechanics Statics and Dynamics.
  New York McGraw-Hill Book
• Company, Inc., 1949.

19
Math Jokes

• What is "pi"?
• Mathematician
• Pi is the number expressing the relationship
• between the circumference of a circle and its
  diameter.
• Physicist
• Pi is 3.1415927plus or minus 0.000000005
• Engineer
• Pi is about 3.

A Mathematician (M) and an Engineer (E) attend a
lecture by a Physicist. The topic concerns
Kulza-Klein theories involving physical processes
that occur in spaces with dimensions of 9, 12 and
even higher. The M is sitting, clearly enjoying
the lecture, while the E is frowning and looking
generally confused and puzzled. By the end the E
has a terrible headache. At the end, the M
comments about the wonderful lecture. E "How do
you understand this stuff?" M "I just visualize
the process" E "How can you POSSIBLY visualize
something that occurs in 9-dimensional space?"
M "Easy, first visualize it in N-dimensional
space, then let N go to 9" 
"Algebraic symbols are used when you do not know
what you are talking about."
Clearly I don't want to write down all the
"in-between" steps. Trivial If I have to show
you how to do this, you're in the wrong class.
It can easily be shown No more than four hours
are needed to prove it. Check for yourself This
is the boring part of the proof, so you can do it
on your own time. Hint The hardest of several
possible ways to do a proof.Brute force Four
special cases, three counting arguments and two
long inductions.Elegant proof Requires no
previous knowledge of the subject matter and is
less than ten lines long.Similarly At least one
line of the proof of this case is the same as
before.Two line proof I'll leave out everything
but the conclusion, you can't question 'em if you
can't see 'em. Briefly I'm running out of time,
so I'll just write and talk faster. Proceed
formally Manipulate symbols by the rules without
any hint of their true meaning. Proof omitted
Trust me, It's true.
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