Date post: | 14-Apr-2018 |
Category: |
Documents |
Upload: | nitin-chandra |
View: | 221 times |
Download: | 0 times |
of 20
7/27/2019 Bolt Pattern Optimization
1/20
Nitin Chandra 2010127
Kunal Ji Baranwal 2010098
Bolt Pattern Design Optimization
Project
7/27/2019 Bolt Pattern Optimization
2/20
Optimization Problem
The goal of this problem is to find the boltlocations that maximize the force P that can be
carried by the bolted joint before it fails.
Failure occurs when the shear stresses in any
one of the bolts exceeds the yield stress of the
bolt.
In order simplify the calculations, we will
reformulate this problem as an equivalent
optimization problem, which is to find the bolt
locations that minimize the fraction of the force P
that is felt by any one of the bolts.
7/27/2019 Bolt Pattern Optimization
3/20
7/27/2019 Bolt Pattern Optimization
4/20
Formulation of Objective
Function
The shear force, from the applied load, on eachof the bolts:
The applied force vector for the force pulling the
plates apart is:
= 1/ 2 1/ 2
The direct shear load is distributed evenly
between the three bolts in the direction of theapplied force. This force on each bolt is given by
the equation:
=/3 1/ 2 1/ 2
7/27/2019 Bolt Pattern Optimization
5/20
Forces induced by the effective moment between
and the centroid of the bolt pattern also exist
(). In order to find these forces, the moment vector,
=
Where is moment arm, or position vector fromthe centroid of the bolt pattern to the applied load:
= +
=
++
and =
++
Finally, substituting and we get:
=
[ + ]
7/27/2019 Bolt Pattern Optimization
6/20
we can now calculate how it is distributed among
the three bolts. This can be done by first finding the
moment arm between each and the centroid ofthe bolt pattern:
= +
These moment arms are then crossed with the
vector to find the direction of the moment forcefor each bolt:
=
( )
=
+
[ ]
Additionally, we must know the magnitude of each
of these forces, which can be found by using the
following equation: .
7/27/2019 Bolt Pattern Optimization
7/20
Now that we have both the magnitude and
direction, we can use their product to find
:
=
With both and known, we can now find themagnitude of the resultant force on each of the
bolts:
= +
= ||
7/27/2019 Bolt Pattern Optimization
8/20
Constraints
To ensure that the bolt pattern does not fail bytear out (the bolts being too close to the edge of
the plate), a safety factor of 1.25 diameters
should be used.
1.25d < , < W1.25d
Additionally, to ensure that the bolt heads do not
interfere with each other, we will add a constraint
that the distance between any two bolts must beat least as great as the diameter of the bolt
heads:
> for i , j =
1 2 3
7/27/2019 Bolt Pattern Optimization
9/20
Data
= 1.0 in
= 4.75 in
P =1 (assume unit force for optimization problem)
Length of steel plates, L = 5:5 in Width of steel plates, W = 2 in
Bolt diameter, d = 0:25 in
Bolt head diameter,
= 0:5 in
Minor bolt diameter area, A = 0:0269
Minimum bolt proof strength, Sy = 85kpsi
7/27/2019 Bolt Pattern Optimization
10/20
Solution Methods
Gradient Based Methods Generalized Reduced Gradient Method ( GRG )
Evolutionary Methods
Particle Swarm Optimization ( PSO )
Genetic Algorithms
7/27/2019 Bolt Pattern Optimization
11/20
GRG
7/27/2019 Bolt Pattern Optimization
12/20
PSO
Our Code
MATLABs PSO toolbox
7/27/2019 Bolt Pattern Optimization
13/20
PSO Algorithm ( Pseudo-Code )
7/27/2019 Bolt Pattern Optimization
14/20
Genetic Algorithms
Our Code
MatlabsGA toolbox
Fitness: To evaluate the fitness, each design must be analyzed to
evaluate the objective f (minimized) and constraints gi
7/27/2019 Bolt Pattern Optimization
15/20
GA: New Generations The genetic algorithm goes through a four-step
process to create a new generation from the currentgeneration:
1) Selection1) Tournament Selection
2) Roulette Wheel Selection
2) Crossover1) Blend crossover
3) Mutation1) To introduce diversity into the population of designs.
4) Elitism1) Children must compete with their parents to survive to
the next generation.
7/27/2019 Bolt Pattern Optimization
16/20
GA: New Generations
The genetic algorithm goes through a four-step process to createa new generation from the current generation:
1) Selection1) Suppose our tournament size is three. We randomly select three designs from the
current generation, and the most fit of the three becomes the mother design. Then werandomly select three more designs from the current generation, and the most fit of thethree becomes the father design. One may vary the fitness pressure by changing thetournament size.
2) Crossover
1) A crossover probability is specified by the user. A random number between zero andone is generated, and if it is less than the crossover probability, crossover isperformed. Otherwise, the mother and father designs, without modification, become
the two children designs.
2) 3) Mutation
1)
4) Elitism1) The new generation is combined with the previous generation to produce a combined generation of
7/27/2019 Bolt Pattern Optimization
17/20
Results
MATLAB Tool Box Results
Genetic Algorithm (0.860372 seconds) PSO (1.042512
seconds)
[1.6875 0.3125 1.6061 0.3125 1.6875[ 1.6875 0.3126 1.5898 0.3132 1.6875
1.5867]
7/27/2019 Bolt Pattern Optimization
18/20
Results
Our Code
[ 1.55216 0.57395 1.47057 0.40754 1.53735 1.68749]
PSO
7/27/2019 Bolt Pattern Optimization
19/20
References
An improved particle swarm optimizer formechanical design optimization problems
(2007)
S. He , E. Prempain & Q. H. Wua
Department of Electrical Engineering andElectronics,
The University of Liverpool, Liverpool
7/27/2019 Bolt Pattern Optimization
20/20
Thank You