This article is an author-created prior version for The Fourth China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems, Kunming, pp.89-94, Nov. 6-9, 2006, China

A SIMPLE ESTIMATION OF FABRICATION COST AND MINIMUM COST DESIGN FOR STEEL FRAMES

Kiichiro Sawada1* , Hitoshi Shimizu2 , Akira Matsuo1, Takaichi Sasaki1, Takashi Yasui3 and Atsushi Namba3 1Social and Environmental Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-hiroshima, Japan

* Corresponding author:kich@hiroshima-u.ac.jp 2Takenaka Corporation Hiroshima office, 10-10, Hashimoto-cho, Hiroshima,Japan

3 Minami Kogyo,10883-40 ,Hara, Hachihonmatsu ,Higashi-hiroshima , Japan

Abstract In this study, the steel fabrication cost functions are first shown. Next, the minimum cost frames are designed by the

optimization method and the presented cost functions and are compared with the minimum weight frames. The problem solved here is to determine the cross section of each member, which minimizes the sum of the fabrication cost and material cost under the constraints of the aseismic design in Japan. It includes the constraints on the member stress, story drift and collapse load for the building subjected to the vertical load and the horizontal load. This problem is solved by Genetic Algorithm based on the ranking selection. Keywords: Building Structures, Fabrication Cost, Minimum Cost Design

1 Introduction There are a lot of studies on structural optimization of buildings. Many of these studies solve the minimum weight

design problems [1,2]. At the initial stage of the structural steel design in Japan, the fabrication cost of the building is often predicted based on the structural weight. However, structural weight cant necessarily predict the fabrication cost exactly. One of the reasons is because the fabrication cost for steel members depends upon the complexity of the connections rather than the structural weight.

We have already shown that the fabrication cost of steel structures depends upon the number of parts and the jointed sectional area from the questionnaires on fabrication time to fabricating workers [3]. In addition, we have derived the fabrication cost functions from the estimated cost data of the fabricating company and the questionnaires [3].

In this study, the fabrication cost functions are first shown. Next, the minimum cost frames are designed by Genetic Algorithm based on the ranking selection [4,5], and are compared with the minimum weight frames.

2 The Typical Beam-to-Column Connection in Japan Figure1 shows the typical H-beam-to-RHS-column connections in Japan. In the fabricating company, the beam to

column connection is first welded to two through diaphragms by full-penetration welds as shown in Fig.2. In this paper, the beam to column connection consisting of these three parts as shown in Fig.2 is named the connection box. The connection box is welded to the flanges of the bracket by full-penetration welds and to the web of the bracket by fillet welds as shown in Fig.3. Finally, the columns are welded to the connection box by full-penetration welds as shown in Fig.4. The bracket is connected to the beam by high strength bolts in the field. This study deals with the buildings having the beam-to-column connection shown in Fig.1.

3 Fabrication Cost Functions In this study, the following function is presented to predict the steel fabrication cost.

CF=CP+CB+CW+CI (1) where CF represents the steel fabrication cost, CP represents the preparatory process cost, CB represents the assembly cost, CW represents the welding cost, and CI represents the information cost such as making shop drawings.

Figure 1. Typical Beam to column connection

Figure 2. Figure 3.

Figure 4.

The preparatory process is comprised of marking, drilling, blasting and flange bevels of diaphragms, beams and brackets. It has been observed from the questionnaires [3] that the preparatory process cost depends on the number of parts such as diaphragms, girders and brackets rather than structural weight. The following function is proposed to estimate the preparatory process cost, CP.

)(1i

BBP

nj

iD NPNPKPCP

(2) where NPDi represents the number of diaphragms for beam to column connection i, nj represents the number of beam to column connections, NPB represents the number of beams and brackets, PB and KP represent the coefficients to evaluate the preparatory process cost.

It has also been observed from the questionnaires [3] that the assembly cost depends on the number of parts rather than structural weight. The following fucntion to estimate the assembly cost, CB is proposed.

)(1i

0 CBC

nj

i NBNBKBCB

(3) where NB0i represents the number of parts in connection boxes and brackets for beam to column connection i, NBC represents the number of columns, BC and KB represent the coefficients to evaluate the assembly cost.

It has been observed from the questionnaires [3] that the welding cost depends on sum of jointed sectional areas. The following function is proposed to estimate the welding cost, CW.

nbb

iiBB

nj

iiD AAKWCW

11 (4)

,where ADi represents the jointed sectional area between the column and the diaphragm for beam to column connection i, nbb represents the number of brackets, KW represent the coefficient to evaluate the welding cost. The information cost represents the cost for shop drawings and full scaling. Since the information cost depends on the number of sheets of shop drawings, the following function based on the number of columns and beams is proposed. WKIgNIbKIbNIcKIcCI (5) ,where NIc represents the number of shop fabricated column trees, NIb represents the number of beams having different cross sectional size, W represents the total structural weight of the frame, KIc, KIb, KIg represent the coefficient to evaluate the information cost. Computational examples of NPDi in Eq.(2), NB0i in Eq.(3) and ADi in Eq.(4) for beam to column connection i

NPDi in Eq.(2), NB0i in Eq.(3) and ADi in Eq.(4) for beam to column connection i are computed as follows. For a standard beam to column connection shown in Fig.5, NPDi=2, NB0i=7, ADi=4Aci Aci: the cross sectional area of the column adjoining to connection i

Bracket

Connection box

For a beam to column connection including an internal diaphragm shown in Fig.6, NPDi=3, NB0i=8, ADi=5Aci For a beam to column connection having different upper and lower column depths shown in Fig.7, NPDi=2, NB0i=10, ADi=4Aci+4HbiTci Hbi: the cross sectional depth of the beam adjoining to connection i Tci: the thickness of the lower column adjoining to connection i

4 Cost Coefficients The values of PB and BC in Eqs.(2) and (3) were computed from questionnaires on fabrication time as follows [3]. PB=15, BC=3 The values of KP, KB, KW, KIb, KIc, KIg in Eqs.(2),(3),(4) and (5) were computed from the least square approximation based on estimated cost data of the fabricating company as follows [3]. KP=196 yen, KB=1150yen, KW=16.4yen/cm2, KIb=3550yen, KIc=12200yen, KIg= 999yen/t

6 The Minimum Weight Design Problem and The Minimum Cost Design Problem The minimum weight design problem of steel structural frames as shown in Fig.8 can be formulated as follows.

)1(

),...,1(,,),,...,1(),,...,1(),,...,1(

1

M

iii

ibibibidb

icidc

LAWminimizewhich

NBibTfTwBNDBidbHNCicTNDCidcDFind

tosubjected

1

)....,2,1(1200/1/

)...,2,1(1

PP

kkkD

jMj

j

jNj

jjS

g

NFkHg

NMjfZ

MfA

Ng

(2a-c)

where W,,Ai, and Li denote respectively the structural weight, weight per unit volume of steel, the cross-sectional area, and member length; NM, NF denote respectively the number of members and the number of stories. Nj, Mj, fNj, fMj,k, and Hk denote respectively the axial force, bending moment, allowable stress for the axial force and bending moment, interstory drift of story k, and height of story k. p is the collapse load factor. The design variables, Didc, Tic, Hidb, Bib,

beam

column

through diaphragm

internaldiaphragm

Fig.5 Fig.6 Fig.7

Twib, Tfib are chosen from the list of standard section sizes [1]. The abovementioned constraints are based on the Japanese building standard law [4] and the Japanese Design Standard for Steel Structures [5]. In this study, the stiffness matrix method is used for elastic constraints, and compact procedure [6], one of limit analysis methods, is used for plastic constraints. The minimum cost design problem of steel frames can be formulated as follows.

)3(

),...,1(,,,),,...,1(,

11

nj

iDi

nd

iii

ibibibibicic

VLAKSCFC

inimizemwhichnbib

TfTwHBncicTDFind

)2()2.( caEqtosubjected where CF represents the fabrication cost shown in Eq.(1), KS represents steel material cost per unit weight, VDi represents the total volume of diaphragm in beam to column connection i, nj represents the number of beam to column connections.

7 Design Examples The minimum weight design and the minimum cost design are executed for the center plane frame of five-story building shown in Fig.9. Youngs modulus E, the yield stress F, and the steel weight per unit volume are specified as follows. E= 2.06105(N/mm2), F=235(N/mm2), =76.93(N/cm3) The design load for the elastic constraints, Eq.(2a) and (2b) is shown in Fig.10. The design horizontal load for the plastic constraints, Eq.(2c), is twice as much as the one shown in Fig.10. The design variables of the members are also shown in Fig.10. Genetic Algorithm (GA) is applied for both the minimum weight design and the minimum cost design. GA control parameters are population of 100, crossover probability of 1.0 and mutation probability of 0.01. The following computational results are the solutions having the minimum objective function for 10 solutions obtained by using a different initial random number each time. Table 1(A),(B) show the minimum cost solutions for KS=40000yen/t and KS=80000yen/t. Table 1(C) shows the minimum weight solution. The total cost C of the minimum cost solution for KS=40000yen/t and KS=80000yen/t is less than that of the minimum weight solution, respectively. Figure 11 shows the fabrication cost and material cost for the minimum weight frame and the minimum cost frame. It is observed from this figure that the minimum cost frame has lower fabrication cost than the minimum weight frame, while both frames have almost same material cost. Figure12(A),(B) show the cross sectional area diagram of the minimum cost solution for KS=40000yen/t and the minimum weight solution. The numerical values in these figures represent cross sectional depth. It is observed from these figures that the minimum cost frames consist of several unified member depths, while most of the member depths of the minimum weight frames are different.

Column

Beam

Column section Beam section

f

w

Fig.8(A) Steel structural frame Fig.8(B) Column section and beam section

8m 6m 8m

8m6m

8m

4m3.5m3.5m3.5m3.5m

Fig.9 Five-story building

-D1T1

-D2T2

H-H1B1Tw1Tf1

-D1T1

-D2T2

-D2T2

-D1T1

-D1T1H-H3B3Tw3Tf3

H-H5B5Tw5Tf5

H-H7B7Tw7Tf7

H-H9B9Tw9Tf9

-D2T2

-D2T2

-D2T2

H-H2B2Tw2Tf2

H-H4B4Tw4Tf4

H-H6B6Tw6Tf6

H-H8B8Tw8Tf8

H-H10B10Tw10Tf10 -D2T2

-D2T2

-D2T2

-D1T1

-D1T1H-H1B1Tw1Tf1

H-H3B3Tw3Tf3

H-H5B5Tw5Tf5

H-H7B7Tw7Tf7

H-H9B9Tw9Tf9

94.1kN 94.1kN188kN 165kN 141kN 165kN 188kN

343kN

211kN

164kN

125kN

88.6kN

Fig.10 Five-story plane frame

Sectional size (mm) A (cm2)Column

5F -40016 234.84F -40016 234.83F -40016 234.82F -50016 298.81F -50016 298.8

Outer girderRF H-500250916 123.65F H-500200922 130.54F H-500250916 123.63F H-7002501225 205.82F H-5002001219 132.9

Inner girderRF H-500200912 92.295F H-500200912 92.294F H-5002001222 144.23F H-7002501222 191.52F H-5002001222 144.2

Structural Weight 26.88 (t)Total Cost 2,288,100yen

Sectional size (mm) A (cm2)Column

5F -40016 234.84F -40016 234.83F -40016 234.82F -45016 266.81F -45016 266.8

Outer girderRF H-400200916 98.575F H-500200916 107.64F H-500250922 152.63F H-6002501222 178.22F H-7002001225 180.8

Inner girderRF H-400200912 83.295F H-500250919 138.04F H-500250916 123.63F H-6002501219 163.92F H-7002001222 169.5

Structural Weight 26.26 (t)Total Cost 3,347,200yen

Sectional size (mm) A (cm2)Column

5F -40016 234.84F -40016 234.83F -40016 234.82F -45016 266.81F -45016 266.8

Outer girderRF H-400200916 98.575F H-400200919 110.04F H-500200919 119.03F H-7002001225 180.82F H-7003001222 213.5

Inner girderRF H-400200919 110.05F H-500200922 130.54F H-7003001219 196.23F H-400200912 83.292F H-400200912 83.29

Structural Weight 25.94 (t)Total Cost 2,340,000yen (KS=40,000yen/t)

3,450,000yen (KS=80,000yen/t)

Table 1(A) The minimum cost design (KS=40000yen/t)

Table 1(B) The minimum cost design (KS=80000yen/t)

Table 1(C) The minimum weight design

0.00E+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06 2.50E+06

minimum costdesign

minimum weightdesign

fabrication costmaterial cost

0.00E+00 1.00E+06 2.00E+06 3.00E+06 4.00E+06

minimum costdesign

minimum weightdesign

fabrication costmaterial cost

Fig.11(A) Costs for the minimum weight frame and the minimum cost frame (KS=40000yen/t)

Fig.11(B) Costs for the minimum weight frame and the minimum cost frame (KS=80000yen/t)

Cost (yen)

Cost (yen)

8 Conclusions In this study, the steel fabrication cost functions have been first shown. Next, the minimum cost frames have been

designed by Genetic Algorithm based on the ranking selection, and compared with the minimum weight frames. The following remarks have been obtained in computational results. (1) The total cost C of the minimum cost solution is less than that of the minimum weight solution under the same condition. The minimum cost frame has lower fabrication cost than the minimum weight frame, while both frames have almost same material cost. (2) The minimum cost frames consist of several unified member depths, while most of member depths of the minimum weight frames are different.

References 1. Sawada K., Matsuo A. : An Exact Algorithm and Approximate Algorithms for Discrete Optimization of Steel

Building Frames, CJK-OSM3, 663-668, 2004.10 2. Uetani, K., Tsuji, M. and Takewaki, I., Application of an optimum design method to practical building frames with

viscous dampers and hysteretic dampers, Engineering Structures 25, 579-592, 2003 3. Shimizu H., Sawada K., Matsuo A., Sasaki T. and Namba T. : A Study on Fabrication Cost Estimation for Steel

Frames, Journal of constructional steel, Vol.14, 2006.11(In Japanese) 4. Jenkins W.M. Plane frame optimum design environment based on genetic algorithm, ASCE,

Vol.118(11),pp3103-3112,1992 5. Ohsaki, M., Genetic Algorithm for Topology Optimization of Trusses, Computers & Structures, Vol. 57. No. 2. pp.

219-225. 1995.1 6. The Ministry of Construction of Japan, The Building Standard Law of Japan, 1994(In Japanese) 7. Architectural Institute of Japan, Design Standard for Steel Structures, 2002(In Japanese) 8. Livsley, R.K., A Compact FORTRAN Sequence for Limit Analysis, Int. J. of Num. Meth. Eng. Vol.5, No.3,

pp.446-449, 1973

500 500

500 500

500

500

700 700 700

500 500 500

500 500 500

400

500

Fig.12(A) Cross sectional area diagram for the minimum cost frame (KS=40000yen/t)

500400

500 700

400

500

700 400 700

700 400 700

400 400 400

400

450

Fig.12(B) Cross sectional area diagram for the minimum weight frame