Design optimization of steel portal frames

Santiago Hernandez*, Arturo N. Fontan, Juan C. Perezzan, Pablo Loscos

School of Civil Engineering, University of Coruna, Campus of Elvina, 15071 Coruna, Spain

Available online 12 May 2005

Abstract

A computer code written by the authors at the University of Coruna and aimed to produce optimum design of portal frames is

presented in this paper. The PADO software provides the least weight design for the most common types of planar frames. The code

incorporates the complete set of constraints considered in the Spanish code of practice for steel structures. The software has a graphical

interface to provide a user-friendly communication. Some application examples are presented in this paper to make clear the capabilities

of PADO code.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Design optimization; Steel structures; Computer codes; Portal frame design; Steel code of practice

1. Introduction

Optimum design of internal dimensions of

metal structures has been considered by several researchers

since many years ago. Sometimes this problem is one level

in a multilevel optimization of complex structural systems

[1,2]. Other contributions were pointed out in [3,4].

Lately, a lot of emphasis has been put into linking

computer graphics and numerical optimization techniques

as a way of overcoming the existing hurdles which prevent

practitioners from using optimization techniques. The

research conducted was in agreement with this approach.

2. Optimum design with wide flange steel bars

A general shape for wide flange steel bar is presented in

Fig. 1. The problem of obtaining optimum design

of structures with profile like this is equivalent to

identify values for thicknesses e, e1, width b and depth h

of each bar leading to the minimum amount of

material capable to overcome a set of combination of

internal forces.

0965-9978/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advengsoft.2005.03.006

* Corresponding author.

E-mail addresses: hernandez@udc.es (S. Hernandez), afontan@udc.es

(A.N. Fontan), jperezzan@udc.es (J.C. Perezzan), ploscos@ferrovial.es

(P. Loscos).

Steel code of practice in each country requires a set of

regulations to guarantee the safety of a steel element. In

summary the set of conditions is related to:

–

Von Mises stress.

–

Local instability of element parts.

–

Instability of element.

The set of conditions of the Spanish code of practice

[5] for a cross section having a combination of axial force

N*, bending moments M*x, M*

y and shear forces T�x , T�

y are:

–

Von Mises stress.

The value of Von Mises stress is evaluated at location 1

as:

s� ZjN�j

AC

jM�x j

Wx

CjM�

y j

Wy

%su

At location 2 Von Mises stress will be:

s� Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijN�j

AC

jM�x j

Ix

ðh K2e1Þ

2

� �2

C3T�

y

ðh K2e1Þe

� �2s

%su

and at location 3:

s� Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijN�j

AC

jM�x j

Ix

ðh Ke1Þ

2

� �2

C30:75T�

x

be1

� �2s

%su

Advances in Engineering Software 36 (2005) 626–633

www.elsevier.com/locate/advengsoft

Fig. 1. Wide flange steel profile.

S. Hernandez et al. / Advances in Engineering Software 36 (2005) 626–633 627

where A is the cross-section area, Ix, Iy the inertia moduli,

Wx, Wy the strength moduli and su the yield stress of the

material.

Values of Von Mises stress at these three locations need

to be evaluated at both ends of the bar and at medium span.

†

Local instabilities.

†

Local buckling of web.

This regulation requires two conditions:h Ke1

e%K

where

Yield stress su K

2600 kg/cm2 71.43

3600 kg/cm2 62.5

and he%0:6ly

ffiffiffiffiffiffiffiffi2400

su

qwith 0:6lyR45where ly is the mech-

anical slenderness on y axis.

†

Local buckling of flange.

The following constraint need to be accomplished:

b

2e1

%15

ffiffiffiffiffiffiffiffiffiffiffi2400

su

s

†

Global buckling of the element.

The condition related with global buckling of the element

is

s� ZjN�j

Au C

jM�x j

Wx

CjM�

y j

Wy

� �max; 004L central

%su

where

u Z2su

1:3su CsE Kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1:3su CsEÞ

2 K4susE

pwith sE Z p2E

l2

The mechanical slenderness l is: lZmax{lx,ly,lq}

being:

lx Zbxl

ix

ly Zbyl

iy

lq Z maxflx; lyg

ffiffiffiffiffiffiffiffiffiffiffiffiffii2x C i2

y

qiq

bx, by are slenderness coefficients along x and y axis.

ix, iy, iq are gyration radii and can be written as:

ix Z

ffiffiffiffiIx

A

r; iy Z

ffiffiffiffiIy

A

r; iq

Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIA

minfIx; Iyg

bi

0:8b0

�2

C0:039ðbilÞ2

Iq

minfIx; Iyg

s

bi is the slenderness coefficient for the axis having minimum

inertia modulus and bo is the warping slenderness

coefficient.

†

Maximum value of mechanical slenderness.

With N* of compression l%200.

With N* of traction l%300

†

Local buckling.

Bending moment M* at each cross-section need to

accomplish:

M�%Mcr

being

Mcr Zk

l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiEGIyIq

pwhere E, G are the longitudinal and transversal elastic moduli

and Iq the warping modulus. Parameter k takes value kZ3.14

if the bar is built-in at both ends. In general k depends on the

boundary conditions and loading type and can be approxi-

mated by a fifth order expression according a:

kðaÞ Z a5a5 Ca4a4 Ca3a3 Ca2a2 Ca1a Ca0

where a is a non-dimensional parameter defined by:

a ZEIyh2

4GIql2

Table 1

Coefficients ai values

Loading Location a5 a4 a3 a2 a1 a0

M at free end – K1!10K7 66.667 K20 K10.167 7.55 1.53

Isolated load at free end Upper flange 26,400 K19267 5103.3 K567.83 19.277 4.01

Central K8266.7 6200 K1490 32.5 43.177 4.01

Lower flange K5600 K5800 4210 K877.5 111.91 4.01

Distributed load Upper flange K67.467 37933 K6803.3 389.17 K1.37 6.43

Central K70,133 36600 K46,36.7 K387.5 138.23 6.43

S. Hernandez et al. / Advances in Engineering Software 36 (2005) 626–633628

where h is the distance between flanges in the cross

sections and the coefficients ai (iZ1,5) can be obtained

from Table 1.

3. PADO software

PADO is a computed code aimed to produce the

least weight design of a portal frame designed with wide

flange steel bars. Three types of bars can be considered by

PADO:

Fig. 2. Blocks of PA

Fig. 3. Windows related to input data module. (a) Frame typologies. (b) Frame supp

loading data. (f) Constraint considerations.

(a)

DO

ort c

Standard Spanish profiles.

(b)

Optimized standard profiles as provided by the code

SAFO [6].

(c)

Tailor made profiles.

The design variables consists only on the area of the

profiles in cases (a) and (b) because the set of mechanical

parameters are related to them using correlation functions.

For tailor made bars the internal dimensions e, e1, h, b are

chosen as design variables.

software.

onditions. (c) Frame geometry. (d) Cross section of elements. (e) Set of

Fig. 3 (continued)

S. Hernandez et al. / Advances in Engineering Software 36 (2005) 626–633 629

The set of constraints are the whole list of conditions

related to linear and non-linear behaviour contained in the

Spanish set of practice. Stress constraints are defined at both

end of the bar and also in the cross-section at medium length.

The set of loading considered in the structural analysis is

composed by: self weight, maintenance load at roof, thermal

gradient, snow loads, transversal and longitudinal moving

crane and transversal and longitudinal wind pressure.

PADO is composed of the modules shown in Fig. 2.

(a)

Module of data input. Requires information on the

following topics:

– Frame typology.

– Frame geometry and support conditions.

– Cross-section of elements.

– Set of loading data.

– Limit value of the set of constraints.

PADO has a graphical interface that allows an easy

interaction with the user. The figures (Fig. 3) show some of

PADO windows.

(b)

Module of structural analysis. This block carries out the

structural analysis of the portal frame with constant or

Fig. 3 (continued)

S. Hernandez et al. / Advances in Engineering Software 36 (2005) 626–633630

tapered cross-section bars using the finite element method.

PADO uses an in-home code which has been tested versus

other structural analysis codes like SAP90 [7].

(c)

Module of structural optimization. This part of the code

arranges the set of constraints to be included in the

problem. The design optimization problem is prepared

as a minimization problem.

min Fð�xÞ

subject to:

gjð �xÞ%0 j Z 1;.; n

where x is the set of design variables as indicated in previous

paragraphs, gj(x) is the set of constraints and F(x) is the

objective function which represents the amount of material

of the frame.

(d)

Module of optimization. This optimization problem

posed is solved by using ADS software [8].

The optimization algorithm selected is the modified

method of feasible directions. PADO allows several

values to define the convergence parameter in the

interval 0.01–0.05.

(e)

Module of results display. PADO has graphical

information of the solution of the design optimization

problem, including differencies between initial and

optimal solutions and information on the evolution of

the objective function values. The Fig. 4 shows some

windows with this information.

4. Application example 1

The first example is the frame presented in Fig. 5, having

built-in supports. The external columns are tapered elements

and the internal columns change their cross-section at the

location of the crane. The set of data required to identify

loads is the following:

–

Length between frames: 6 m

–

Length between cover beams: 2 m

–

Cover service load: 200 N/m2

–

Cover beams weight: 50 N/m2

–

Topographical altitude: 600 m

–

Wind classification area: Z

–

Topographical conditions: adverse

–

Longitudinal walls: without opening

–

Transversal walls: more than 33% openings

–

Crane loads:

Maximal vertical reaction: 50 kN

Minimal vertical reaction: 20 kN

Friction coefficient: 0.15

Fig. 4. Windows of graphical results. (a) Evolution of standard cross sections.

(b) Evolution of tailor made cross sections. (c) Evolution of objective function.

Fig. 5. Geometry of frame of example 1.

S. Hernandez et al. / Advances in Engineering Software 36 (2005) 626–633 631

The initial design chosen was:

External support

Lower cross section Upper cross section

h b e e1 h b e e1

350 150 10 10 400 200 10 15

Internal support Beam

Lower cross

section

Upper cross

section

h b e e1

HEB 300 HEB 200 350 200 10 10

The results provided by PADO corresponding to the

optimum solution were:

External support

Lower cross section Upper cross section

h b e e1 h b e e1

352 121 9 5 436 222 10 9

Internal support Beam

Lower cross

section

Upper cross

section

h b e e1

HEB 240 HEB 120 319 172 8 6

The graphical information windows are presented in

Fig. 6.

5. Application example 2

This example deals with a two bay frame having built-in

supports (Fig. 7). A movable crane is considered on a bay and

the columns change their cross-section at the crane height.

The set of data required to identify loads is the following.

–

Length between frames: 6 m

–

Length between cover beams: 2 m

–

Cover service load: 200 N/m2

–

Cover beams weight: 50 N/m2

–

Topographical altitude: 200 m

–

Wind classification area: W

–

Topographical conditions: adverse

–

Longitudinal walls: without opening

–

Transversal walls: without openings

–

Crane loads:

Maximal vertical reaction: 50 kN

Minimal vertical reaction: 20 kN

Friction coefficient: 0.15

Fig. 6. Windows related to optimization results. (a) Evolution of beam

shape. (b) Evolution of objective function.

Fig. 7. Geometry of frame of example 2.

S. Hernandez et al. / Advances in Engineering Software 36 (2005) 626–633632

The initial design chosen was:

External column

Lower cross-section Upper cross-section

h b e e1 h b e e1

300 300 10 10 400 300 10 10

Fig. 8. Windows related to optimiza

cross-section of external column. (b)

tion resul

Evolution

ts. (a) Ev

of object

olution o

ive functi

Inner column

Lower cross-sections Upper cross-section

h b e e1 h b e e1

300 300 10 10 400 300 10 10

External beam Central beam

h b e e1 h b e e1

500 350 15 15 500 350 15 15

f lower

on.

S. Hernandez et al. / Advances in Engineering Software 36 (2005) 626–633 633

The results provided by PADO corresponding to the

optimum design were:

External column

Lower cross-section Upper cross-section

h b e e1 h b e e1

223 190 6 7 492 281 12 11

Inner column

Lower cross-sections Upper cross-section

h b e e1 h b e e1

300 300 10 10 400 300 10 10

346 300 8 12 309 119 8 5

External beam Central beam

h b e e1 h b e e1

500 350 15 15 500 350 15 15

298 148 7 6 311 197 8 7

The graphical information windows are presented in

Fig. 8.

6. Conclusions

Some conclusions can be extracted from this research:

–

The complete set of regulations included in any code

of practice can be incorporated in optimization

software.

–

Graphical interfaces can help in attracting users to feel

comfortable with computer codes.

–

The PADO software is a reliable code for

structural optimization of portal frames and includes

the whole set of considerations required by real life

problems.

–

Special purpose optimization software can be a efficient

way to encourage engineers to include optimization

techniques in their design methodologies.

References

[1] Kim D. Multilevel multiobjective optimization for engineering

synthesis. PhD Thesis. University of California in Santa Barbara;

1989.

[2] Sobiesczanski-Sobieski J. Multilevel structural optimization.

Computer aided optimal design, NATO/ASI seminar. vol. 3 1986

p. 7–28.

[3] Hernandez S. Applications of design optimization in steel and concrete

industry. In: Belegundu AD, Mistree F, editors. Optimization in

industry. New York: ASME; 1998. p. 81–8.

[4] Hernandez S, Fontan AN. Practical applications of design optimization.

Southampton Boston: WIT Press; 2002.

[5] EA-95. Norma basica de la edificacion. Estructuras de acero en

edificacion. Ministerio de fomento 1995 [In Spanish].

[6] Fontan AN, Hernandez S. Shape optimization of steel standard profiles.

In: Hernandez S, Brebbia CA, editors. Computer aided optimum design

of structures VII, 2001. p. 221–34.

[7] SAP90 User’s Manual. Computers and Structures Inc., 1990.

[8] ADS. Fortran program for automated design synthesis. Colorado

Springs: VMA Engineering; 1998.