CREATING MODELS OF TRUSS STRUCTURES WITH OPTIMIZATION

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CREATING MODELS OF TRUSS STRUCTURES WITH
OPTIMIZATION

• Authored by
• Jeffrey Smith, Jessica Hodgins,
• Irving Oppenheim, Andrew Witkin

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• Overview of the presentation
• Introduction
• Representation of Truss Structures
• Optimization of Truss Structures
• Results
• Conclusion

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Introduction

• An Overview on
• The Topic of the paper
• Trusses
• Optimization
• Techniques in Structural Engineering
• Authors Goal

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• A Recurrent Problem
• Creation of Realistic Models Of Complex
• Man-Made Structures
• Solutions
• Hand Models
• Or
• Computer Generated Models

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• Hand Models
• Difficult
• Time Consuming
• Tedious
• Difficult to Make them Virtually Realistic

• Computer Generated
• Models
• Easier Faster Generation
• Physically Realistic

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• The Problem of
• Automatic Generation of Man-Made Structures
• very Less work done so far
• very Few publications on this topic
• very Large scope for Research Study

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This Paper deals with a sub topic of this problem
Creating Computer Generated Models of Truss
Structures

• and presents,
• A Method based on,
• Practices in the field of Structural Engineering
• Simple Optimization Techniques
• Minimum User Effort,
• to quickly create novel and physically realistic
  truss structures
• such as bridges towers.

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• What Are Trusses?
• Triangular frameworks that hold up
• Roofs Floors Walls
• Structures which are composed of beams with axial
  forces
• connected concentrically with welded or bolted
  joints
• A broad category of man made structures
• Visually ComplexDifficult to Model
• Examples of Truss Structures
• Bridges Water Towers
• Cranes Roof Support Trusses
• Temporary Construction
• Framework

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Some Examples of Truss Structures
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Another Example of a Truss Structure

• More than 15000 Girders
• Connected at over 30000 points

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• Optimization techniques in Structural Engineering
• Cross Sectional Optimization
• Topology Optimization
• Geometry Optimization

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Cross Sectional / Size Optimization

• Fixed Topology Geometry
• (Number of beams joints,
• their connectivity locations)
• Design Variables
• Properties that affect the cross sectional area
  of a beam
• eg. The radius thickness of each tube
• in case of tubular elements.

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Topology Optimization
Design Variables The number of beams and joints
as well as their connectivity rather than their
individual shapes Members may be removed or
added to optimize a structure
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Geometry Optimization
Refines the position,
strength, topology of a truss structure
Optimization is done through an iterative process
wherein the continuous design variables are
optimized in one pass and the topology is changed
in the second pass Geometry of the Overall
Structure can be changed
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• The Ultimate Goal of Structural Optimization
• Highly Accurate Modeling of Reality
• Strict Physical Accuracy
• Concerns in the Computer Graphics field
• Speedy Solution
• Visual Impact
• The Authors Goal
• Optimization of the truss structures
• with the goals of
• Speed
• User Control
• Physical Realism

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Representing Truss Structures

• Constructing The Model

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• Authors Model of the Truss Structures
• The Truss
• A connected set of three dimensional particles,
• made up of rigid beams, pin connected at joints
  and
• exerting only axial forces
• Components
• Beams
• Every beam has exactly two end points
• Pin-Joints
• Accommodate any number of beams

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• Three Types of Pin Joints
• Anchors -
• Points where beams are joined to the earth.
• Loads -
• Points at which external loads are being
  applied.
• Free Joints -
• Pin Joints where beams connect, but which
• are not in contact with the earth and have
  
• no external loads.

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Purpose of a Structure A Bridge Support some
minimum weight along its span The Eiffel
Tower Support the Observation decks A Roof
Truss Support the Roofing material .
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The Authors Model Support requirements
External loads placed by the user, at user chosen
load points Anchors Specified by user Free
Joints Beams Automatically added connected
to all three sets of joints Whether the joints
are placed above or below the loads is
also specified by user
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An Initial Guess Used to Create the Model of a
Typical railroad Bridge
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A Typical Railroad Bridge
The Truss Bridge designed by the Authors
software
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Optimizing Truss Structures

• Forces Acting on Joints
• Objective Function
• Constraints
• Mass Functions
• Optimization

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Forces Acting on any joint, i
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Objective Function
Given an objective function G ( usually the
total mass ), Optimization of a truss structure,
subject to stability constraints is achieved by
,
Minimizing G(q) such that Fi (q) 0
i1Nj
where, Nj the number of joints q the vector of
design variables
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Constraints
Stability Constraints Force Balance
Constraints Inequality Constraints Symmetry
Constraints Obstacle Avoidance Constraints
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Mass Functions
Mass of a beam under tension, mT -kT ?j
lj Mass of a beam under compression, mC?
Ajlj kCsqrt( ?j) lj2 where, kT
scaling factor based on density tensile
strength of the material.5x10-6 kg/m3 for
steel kC scaling factor based on density
compressive strength of the
material.1.5x10-5 kg/m3 for steel
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Optimization Procedure
A multilevel design algorithm based on
sequential quadratic programming Step
1 Minimize the objective function using all the
constraints and equations described in the last
section. Derive a feasible structure. Step
2 Merge any pairs of joints that are connected
to one another by a beam less than minimum
allowable length Eliminate beams exerting less
than minimum workless force Step 3 Results
examined by user. If found unsatisfactory
for aesthetic or mass reasons, perform more
iterations, using this new structure as the
starting point.
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Results

• Bridges
• Eiffel Tower
• Roof Trusses
• Timing Information

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Bridges

• Initial Guess
• 22 free joints 163 beams
• Final Design
• 48 joints 144 beams
• Some of the joints members
• were merged/eliminated during
• the topology cleaning step.
• Constraints
• Volume above the deck to be kept clear
  
• Movement of free joints limited to
• vertical planes

A Warren Truss Railroad Bridge
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Bridges
A Bridge with all truss work underneath the deck

• Same initial structure
• Constraints
• No material to be placed
• above the deck
• Removal of constraint to
• keep joints in a vertical plane

This solution results in savings in cost of
materials
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Bridges
A cantilever bridge

• Constraints
• No Material above the deck
• Movement of joints only in
• vertical planes

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Bridges
A Through Deck Cantilever Bridge
Same starting point objective function as
previous but a different solution
Constraints Removal of both constraints
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Eiffel Tower

• Optimization of
• A tall tower similar to the upper two thirds
• of the Eiffel tower
• 4 user specified anchors
• 4 user specified external loads at the top
• Automatic generation of initial, rectilinear
• set of joints and beams
• Observation decks bottom thirds
• not synthesized since their design
• governed by aesthetic demands

A tower designed by using the authors software
The Eiffel tower
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Roof Trusses
Frameworks used to support the roofs of buildings
Cambered Fink Truss
Composite Warren Truss
Scissors Truss
Trusses generated for two different roof pitches
Same initial Geometry Different Objective
functions - Total Mass (for cambered Fink
Scissors) - Total Length of beams(for
composite Warren) - Different roof mass
(Scissors trusses have a roof that weighs
twice as much as that of thr other two types of
trusses)
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Timing Information

• Maximum total time for all the illustrated
  examples
• Less than 15 minutes
• on a 275 MHz R10000 SGI Octane
• Time taken to model these by hand
• At least more than an hour

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Conclusion
Speed in construction of truss structures
achieved Physical Realism achieved
Scope for Further Development
Consideration for More abstract aesthetic
criteria Load Envelope Stress Limits Actual Cost
of Construction
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Questions?