Comparison of robustness of metaheuristic algorithms for

steel frame optimization

RYAN ALBERDI

KAPIL KHANDELWAL

Dipak Prasad

16CE65R10

Algorithmic robustness: -

It is defined as the ability of algorithms to consistently

converge to low cost design irrespective of variable space

and initial starting point.

Diversification: -

It is the ability of algorithms to search new regions of

variable space by incorporating randomness in variable

space.

This helps to escape local optima by thoroughly

searching variable space.

It may cause convergence issue as it fails to converge at

one solution.

Intensification: -

Intensification is the use of memory to seek out regions of the

variable space that have been shown to produce good

solutions.

Intensification encourages convergence by continually

drawing solutions from favorable regions of the variable

space, but will increase the likelihood of converging to local

optima.

The performance of metaheuristic algorithms depends on

a balance between the conflicting characteristics of

diversification and intensification.

These algorithms provide solutions by searching through

discrete variable space of steel sections while incorporating design

codes and other practical constraints of serviceability and

aesthetic aspects.

Problems of steel frame design optimization

1. Large no. of variable space makes search of global optimum

difficult.

2. Different properties inherent to steel sections as well as lack of

knowledge of governing serviceability.

These challenges defined earlier may cause hindrance to

robustness of algorithms.

Robustness of an algorithm may vary with the size of variable

space i.e. an algorithm may loss its robustness with increasing no.

of design variables.

Due to large no. of variable spaces and different governing

metric mean may cause an algorithm to converge to two different

solutions.

Problem formulation

Here we assume design vector x of size ng where ng is total no. of

group in problem.

These groups are formed in such a manner that each member in

group will be assigned same cross-section.

If nk are the possible choices of cross-section for group no. k

than from combinatorics total no. of solution set is given by

N= n1 x n2 x…x nng

Problem formulation

strength constraints from AISC-LRFD

Where

r=1,2,…n, n is total no. of elements in frame

Pur=Required tensile or compressive strength

Pnr=Nominal tensile or compressive strength

φ = 0.9 for tension and 0.85 for compression

Muxr/Muyr= Required flexure strength about major/minor axis

Mnxr/Mnyr=Nominal flexure strength about major/minor axis

Φb= Resistance factor for flexure = 0.9

Inter story drift constraint

Top story drift constraint

Where

r= 1,2,…n , no. of stories

δr= maximum drift at story r

δa= allowable maximum displacement

δT= maximum displacement at top floor

δaT= allowable maximum displacement

δa or δaT may vary with problem depending on materials used in

construction.

Constructability constraints

Where

r= 1,2 ,…n,

n=total no. of constructability constraints

bbf= flange width of beam

bcf= flange width of column

Objective function

minimize fp(x)

These auxiliary functions are as follows

Where

fp(x) is penalized objective function

f(x) is weight of frame

ρ= density of steel

Aj= cross sectional area of member assigned to jth group

Li= length of ith member of group

αs , αd , αt , αc are penalty factor corresponding to strength,

drift , top displacement, constructability violations respectively.

Drift dominated member

These are the members which violates or comes near to violating

drift constraints but does not violate strength constraints.

Strength dominated member

These are the members which violates or comes closer to violating

strength constraints but does not violate drift constraints.

Table shows neighborhood matrix of a

particular steel section based on plastic

section modulus, moment of inertia and

radius of gyration.

Depending on the behavior of member

algorithms can use any of column for

searching optimum cross-section.

Algorithms involved in study of Robustness

1. Ant colony optimization (ACO)

2. Genetic algorithms (GA)

3. Harmony search (HS)

4. Design driven harmony search (DDHS)

5. Adaptive harmony search (AHS)

6. Particle swarm optimization (PSO)

7. Simulated annealing (SA)

8. Improved simulated annealing (ISA)

9. Tabu search (TS)

Benchmark design problems

Five steel frames are created in order to test the optimization algorithms.

Linear static analysis is performed and all frames use ASTM992.

ASTM992 is a structural steel alloy used for wide flanges and I- sections

in USA.

It has following properties: -

Density = 7850 kg/m3

Tensile yield strength = 345MPa (50ksi)

Ultimate tensile strength = 450Mpa (65ksi)

Strain to rupture = 18% for 200 mm long test specimen;

21% for 50 mm long test specimen.

Generally represented as ‘ASTM992(Fy=50ksi,Fu=65ksi)’

Planar moment frame – 3 bay X 3 story

Here 3 design variables had been

used for optimization.

First group consists of all inside

columns.

Second group consists of all outside

columns.

Third group consists of all beams.

The variable space used for column

consists of all 18 W10 and 29 W12

sections.

The variable space for beam

consists of all 267 wide-flange

sections.

To investigate robustness each algorithm is run 500 times for

planar frame and 200 times for space frame.

Result table indicates the following facts

1. Avg. weight is mean of total wt. of structure from different run.

2. Std. weight is standard deviation of total wt. of structure from

different run.

3. Percentage feasible indicates the percentage of run generating

feasible result.

4. Percentage optimal indicates percentage of run generating

exact optimal solution.

The optimal solution for 3bay-3story planar frame consists of

W21X44 section for beam group,W12X120 section for inside columns

and W12X53 section for outside columns. The weight of this globally

optimal design is 107.32 KN.

Essential facts from result: -

DDHS finds the optimal solution in every run.

Only ISA generates some non feasible results.

ISA have highest standard deviation for its solutions.

After DDHS, ISA generates the global optimum solution maximum times.

PSO could not produce actual result for a single run. Also its results’ avg.

weight is much far from actual one.

Planar moment frame- 5 bay X 14 story

Here 20 design variables are

involved in analysis 12 column

group and 8 beam group.

The variable space used for

column group consists of all

W12,W14,W18,W21,W24,W27

sections while all 267 sections

have been used for beam group.

The total variable space consists of

1.34e45 permutations.

Planar braced frame 5 bay X 14 story

Here 27 design variables are

involved. These are 12 column, 8

beam and 7 bracing groups.

The variable group for column

consists of W12, W14, W18, W21,

W24,W27 sections in AISC manual.

The variable group for beam consists

of all 267 sections and for brace

groups only W8,W10,W12,W14

sections are considered.

Total no. of variable space consists of

1.09e59 permutations.

Space moment frame 5 bay X 5 bay X 20 story

16 column and 10 beam group

W12,W14,W18,W21,W24,W27 for column

All 267 sections for beam group

3.58e58 permutations in variable space

Space braced frame 5 bay X 3 bay X 25 story

20 column group, 10 beam

group and 10 brace group

All 267 sections for all groups

1.15e97 permutations for

variable space.

Algorithms comparison for each cases are as follows

Conclusion

Design driven harmony search is the most robust overall obtaining the lowest

average weight and finding 100% feasible design.

Tabu search, harmony search, adaptive harmony search perform well as

variable space is increased.

Genetic algorithm and ant colony optimization consistently find feasible

solution but unable to converge to optimal design.

Particle swarm optimization is not appropriate in its current form for steel

frame optimization but its performance can be improved by using metric

information by appropriately organizing variable space.

Simulated annealing and improved simulated annealing can achieve results

comparable to DDHS but only for small variable space.

In this case only gravity and wind loads are taken into account so further study

with seismic load is necessary.