bolt pattern optimization

7/27/2019 Bolt Pattern Optimization

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Nitin Chandra 2010127

Kunal Ji Baranwal 2010098

Bolt Pattern Design Optimization

Project


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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.


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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


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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:

=

[ + ]


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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: .


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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:

= +

= ||


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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


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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


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Solution Methods

Gradient Based Methods Generalized Reduced Gradient Method ( GRG )

Evolutionary Methods

Particle Swarm Optimization ( PSO )

Genetic Algorithms


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GRG


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PSO

Our Code

MATLABs PSO toolbox


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PSO Algorithm ( Pseudo-Code )


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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


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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.


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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


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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]


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Results

Our Code

[ 1.55216 0.57395 1.47057 0.40754 1.53735 1.68749]

PSO


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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


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Thank You

bolt pattern optimization

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