Numerical Approach for Torsion Properties of

Built-Up Runway Girders

Wei T. Hsu, Dung M. Lue* and Bor T. Hsiao

Department of Civil Engineering, National Chung-Hsing University,

Taichung, Taiwan 402, R.O.C.

Abstract

It is a common practice in a crane runway girder to place a channel, open-side down, over the

top flange of a W-shape. The crane runway girder with section mentioned above called as WC

(W-shape with Channel Cap) or SC (S-shape with Channel Cap) girder by authors is an efficient and

economical one. The warping constants (Cw) of the WC/SC girders are not provided in the AISC

design manuals because the Cw calculation for a WC/SC section is not a routine process but a tedious

task.

This study summarizes the theoretical Cw formulas which are expressed in terms of

mathematical integration. The integration formulas can be written in terms of numerical expressions

by considering the fact that the section is made up of thin-walled plate elements. Since the Cw which in

terms of numerical expressions is too complicated to be completed by hand-held calculators, the Cw is

set to be calculated by computer. The accuracy of computer-assisted results is compared with the

Australian built-up sections of crane runway girder and the results are quite compatible. The calculated

values of Cw will make engineers to better evaluate the elastic critical moment (Mcr) of the girders with

WC/SC sections.

Key Words: Runway Girder, Warping Constants, WC/SC Girder

1. Introduction

The WC/SC section (refer to Figure1) resisting an

applied torsional moment is subjected to pure torsional

stresses, warping shear stresses, and warping normal

stresses. The evaluation of warping related stresses is not

an easy task for practicing engineers and the difficulty

comes from the calculation of warping constant (Cw).

The Cw values of open thin-walled section are provided

in the AISC design manuals (1989 [1], 1993 [2], 1999

[3], 2005 [4]) for some practical sections but not for

WC/SC sections. The Cw calculation of a WC/SC section

is not a routine process but is a tedious and complicated

task.

This study aims to manipulate theoretical formulas

which expressed in terms of mathematical integration.

Viewing the fact that the section is made up of open

thin-walled plate elements, the integration formulas can

be written in terms of numerical expressions.

Because the warping constant Cw, which in terms of

numerical expressions, is too complicated to be com-

pleted by hand-held calculators, the Cw is set to be calcu-

lated by computer. The accuracy of computer-assisted

results are checked and identified with the Australian

built-up section of crane runway girder, which is a lip-

ped section (refer to Figure 2) closely approximates the

WC/SC section. The comparison results are quite com-

patible. The obtained Cw values will make practicing en-

gineers to calculate the elastic critical moment (Mcr) of

WC/SC girders much easier as compared with the ones

provided by the current LRFD Specification [4].

Tamkang Journal of Science and Engineering, Vol. 12, No. 4, pp. 381389 (2009) 381

*Corresponding author. E-mail: dmlue@dragon.nchn.edu.tw

2. Integration Formulas for Warping

Constant of WC/SC Section

According to Galambos [5] and Heins [6], the inte-

gration formulas for warping constants of open thin-

walled sections which include the WC/SC sections can

be summarized (refer to Figures 3) and are given as fol-

lows. To calculate the value ofCw, the centroid of section

and the shear center of section had to be located first.

Based on the basic mechanics, the centroid of section

C(Xc, Yc), refer to Figures 3, can be given as follows.

The centroid of section: C(Xc, Yc)

(1)

where x and y are the x and y coordinates of dA along the

section and A is the area of open thin-walled section.

According to Galambos [5], the shear center of sec-

tion S(Xs, Ys), refer to Figure 3, can be given as follows.

The shear center of section: S(Xs, Ys)

(2)

(2.a)

where

(2.b)

The terms Ix and Iy are moments of inertia about the x

and y axes, respectively. The termw is the warping defor-

mation of any point on the middle line a distance s from

the edge o and the terms Iwx and Iwy are warping products

of inertia about x and y axes, respectively. Iwx and Iwy can

be obtained by performing the integration, some algebra

and noting that cosij = (xj xi) / Lij (refer to Figure 5).

The warping constant of section: Cw

(3)

where

(3.a)

(3.b)

(3.c)

382 Wei T. Hsu et al.

Figure 1. Built-up WC and SC sections.

Figure 2. Section for Australian runway Beam.

Figure 3. Thin-walled open cross section.

where Wn the normalized unit warping, ws the unit war-

ping with respect to shear center, t the thickness of plate

element, s the distance between the tangent and the

shear center, and A the area of section.

3. Numerical Formulas for Warping Constant

of WC/SC Section

Because the WC/SC sections are made up of open

thin-walled plate elements (Figure 4), the computation

of the torsional section properties can be greatly simpli-

fied by the fact that between points of intersection the

unit warping properties w, ws, and Wn vary linearly (Fig-

ure 5). The theoretical integration formulas based on

Galambos [5] can be written in terms of numerical ex-

pressions and are given as follows.

The centroid of section: C(Xc, Yc)

(4)

where xi and yi are the x-coordinate and y-coordinate of

plate element i, respectively, and Ai is the area of plate

element i.

The shear center of section: S(Xs, Ys)

(5)

where Ixy is the product of inertia. Because the built-up

sections of WC/SC are singly symmetrical sections, we

have Ixy = 0 and above equations of shear center can be

rewritten as follows.

(6)

where

(6.a)

(6.b)

(6.c)

The unit warping property w is the unit warping with

respect to the centroid. xi and xj are the x coordinates of

the ends of the element. yi and yj are the y coordinates of

the ends of the element. wi and wj are the corresponding

values ofw at the ends of element. t the thickness of plate

element, ij the distance between the tangent of element

ij and the centroid and Lij the length of element ij.

The warping constant of section:

(7)

where

(7.a)

Numerical Approach for Torsion Properties of Built-Up Runway Girders 383

Figure 4. WC section.Figure 5. Distribution of warping deformation w on a plate

element.

(7.b)

(7.c)

The unit warping properties ws and Wn are the unit

warping with respect to the shear center and the normal-

ized unit warping, respectively. ws the unit warping with

respect to shear center. wsi and wsj are the corresponding

values of ws at the ends of element, tij is the thickness of

plate element ij, ij the distance between the tangent of el-

ement ij and the centroid and Lij the length of element ij.

4. Numerical Steps for Calculation of

Warping Constant

The numerical steps for the calculation of WC/SC

warping constant is summarized as follows.

(1) The centroid of section: C(Xc, Yc)

(4)

(2) The shear center of section: S(Xs, Ys)

(6)

(3) The warping constant of section: Cw

(7)

5. Illustrated Example

Proceed the steps specified in the previous section to

evaluate the warping constant of a built-up section made

of W36 150 and C15 33.9. The built-up section is di-

vided into eleven plate elements and their related joint

numbers are given as shown in Figure 4.

Solution: unit conversion 1 in. = 2.54 cm

Section properties of W36 150 based on the AISC de-

sign manual.

A = 44.2 in.2, d = 35.85 in., tw = 0.625 in., bf = 11.975 in.,

tf = 0.940 in.

Section properties of C15 33.9 based on the AISC de-

sign manual.

A = 9.96 in.2, d = 15.00 in., tw = 0.400 in., bf = 3.400 in.,

tf = 0.650 in.

(1) The centroid of section : C(Xc, Yc)

The coordinates and areas for each plate element (i)

of the WC section are summarized as given in Table 1.

Substitute these values (xi, yi, and Ai included in Table 1)

into the formula of centroid (Eq. 5) and the values of Xc

and Yc can be obtained. The coordinates for the centroid

(Xc, Yc)= (7.500 in., 21.146 in.) with respect to the lower

left corner, the origin of the coordinates as shown in

Figure 4.

(2) The shear center of section: S(Xs, Ys)

The terms Ix and Iy are moments of inertia about the x

and y axes, respectively. The Ix and Iy values of the built-

up section can be calculated by using the tabulated va-

lues given in the AISC design manual and be calculated

as below.

Ix = Ix(w-shape) + Iy(channel) + Aw d12 + Ac d2

2 = 11372 in.4

Iy = Iy(w-shape) + Ix(channel) = 583 in.4

where Aw the area of w-shape, Ac the area of channel, d1

= the distance between the y coordinate of centroid of

built-up section and the centroid of w-shape section, d2

= the distance between the y coordinate of centroid of

built-up section and the y coordinate of centroid of

channel.

To obtain the warping product of inertia (Iwx), the

parameters including xi, xj, wi, wj, tij, and Lij are required

and are listed in Tables 2, 3, and 4. Substitute these

384 Wei T. Hsu et al.

Table 1. Coordinates and areas of plate elements

element no. (i) xi (in.) yi (in.) Ai (in.2)

01 04.506 00.470 5.628

02 07.500 17.925 21.2310

03 04.506 35.580 8.023

04 01.512 35.580 0.000

05 00.756 36.050 0.605

06 00.325 34.350 1.950

07 10.494 00.470 5.628

08 10.494 35.580 8.023

09 13.488 35.580 0.000

10 14.244 36.050 0.605

11 14.675 34.350 1.950

Numerical Approach for Torsion Properties of Built-Up Runway Girders 385

Table 2. Joint coordinates, lengths, and thicknesses of plate element

element no. joint no. xi / xj (in.) yi / yj (in.) Lij (in.) tij (in.)

1(i) -5.988- -20.676-01

2(j) 0.000 -20.676-

05.988 0.940

2(i) 0.000 -20.676-02

3(j) 0.000 14.434

35.110 0.625

3(i) 0.000 14.43403

4(j) -5.988- 14.434

05.988 1.340

4(i) -5.988- -14.434-04

5(j) -5.988- 14.904

00.470 0.000

5(i) -5.988- 14.90405

6(j) -7.175- 14.904

01.188 0.400

6(i) -7.175- 14.90406

7(j) -7.175- 11.704

03.200 0.650

8(i) 5.988 -20.676-07

2(j) 0.000 -20.676-

05.988 0.940

9(i) 5.988 14.43408

3(j) 0.000 14.434

05.988 1.340

10(i) 5.988 14.90409

9(j) 5.988 14.434

00.470 0.000

11(i) 7.175 14.90410

10(j) 5.988 14.904

01.188 0.400

12(i) 7.175 11.70411

11(j) 7.175 14.904

03.200 0.650

Table 3. Unit wrappings wi and wj

element no. ij (in.) Lij (in.) wij (in.2) joint no. wi (in.

2) wj (in.

2)

1(i) 0.00001 20.676 5.988 123.800

2(j) 123.800

2(i) 123.80002 0.000 35.1100 0.000

3(j) 123.800

3(i) 123.80003 14.434 5.988 86.421

4(j) 210.221

4(i) 210.22104 -5.988 0.470 -2.814

5(j) 207.407

5(i) 207.40705 14.907 1.188 17.698

6(j) 225.105

6(i) 225.10506 7.175 3.200 22.960

7(j) 248.065

8(i) 247.60007 -20.676 5.988 -123.80

2(j) 123.800

9(i) 37.37808 14.434 5.988 86.421

3(j) 123.800

10(i) 40.19209 -5.988 0.470 -2.814

9(j) 037.378

11(i) 22.49410 14.904 1.188 17.698

10(j) 040.192

12(i) -0.46511 7.175 3.200 22.960

11(j) 022.494

parameters (xi, xj, wi, wj, tij, and Lij) into the given for-

mulae (Equations 6, 6.a, 6.b, and 6.c) and the value of Iwx

can be obtained. The shear center of section is then deter-

mined by using Eq. (6) and

Ys = Iwx / Iy = 3932.55 / 583.13 = 6.744 in.

Therefore, the shear center is S (Xs, Ys) = (0 in., 6.74 in.)

or S (Xs, Ys) = (7.50 in., 27.89 in.) with respect to left

lower corner as shown in Figure 4.

(3) The warping constant of section: (Cw)

The required parameters including wsi, wsj, Wni, Wnj,

tij, and Lij are listed in Tables 5, 6, 7, and 8. Substitute

these values (wsi,wsj,Wni,Wnj, tij, Lij and refer to Figure 4)

386 Wei T. Hsu et al.

Table 4. Calculation for warping product of inertia (Iwx)

element no. wi xi wj xj tij Lij1

3( + ) wi xj wj xi

1

6( + )

01 0 0 5.629 0 0 0-741.31 -695.47

02 0 0 21.9440 0 0 0 0

03 0 -1258.8 8.024 -3366.87 0-741.31 0 -991.38

04 -1258.8 -1241.9 0 0 -1258.80 -1241.95 0

05 -1241.9 -1615.1 0.475 0-452.36 -1488.15 -1347.93 -224.52

06 -1615.1 -1779.9 2.080 -2353.87 -1615.13 -1779.87 -1176.93-

07 -1482.6 0 5.629 -2781.85 0 -0741.31 -695.47

08 -0223.8 0 8.024 -0598.59 0 -0741.31 -991.38

09 -0240.6 -0223.8 0 0 -0240.67 -0233.82 0

10 -0161.4 -0240.6 0.475 -0063.65 -0134.70 -0288.38 -033.49

11 000-3.3 -0161.4 2.080 -0109.62 000-3.33 -0161.40 -054.80

-2619.39 -1313.16-

Iwx = + = -3932.55 in.5

Table 5. Unit wrappings wsi and wsj

element no. sij (in) Lij (in) sij Lij (in2) joint i, j wsi wsj

1(i) 000.00001 27.4200 5.988 164.179

2(j) 164.179

2(i) 164.17902 0.000 35.1100 0.000

3(j) 164.179

3(i) 164.17903 7.690 5.988 46.042

4(j) 210.221

4(i) 210.22104 -5.988- 0.470 -2.814

5(j) 207.407

5(i) 207.40705 8.160 1.188 9.690

6(j) 217.097

6(i) 217.09706 7.175 3.200 22.960

7(j) 240.057

8(i) 328.35707 -27.420-0 5.988 -164.179

2(j) 164.179

9(i) 118.13608 7.690 5.988 46.042

3(j) 164.179

10(i) 120.95009 -5.988- 0.470 -2.814

9(j) 118.136

11(i) 111.26110 8.160 1.188 9.690

10(j) 120.950

12(i) 088.30111 7.175 3.200 22.960

11(j) 111.261

Numerical Approach for Torsion Properties of Built-Up Runway Girders 387

Table 6. Values of Wni + wsi or Wnj + wsj

element no. tij (in.) Lij (in.) joint i, j wsi wsj tij Lij A (in.2) (CV)i

*(in.

2)

1(i) 000.00001 0.940 5.988

2(j) 164.1795.629 54.357 0924.164

2(i) 164.17902 0.625 35.1100

3(j) 164.17921.9440 54.357 7205.488

3(i) 164.17903 1.340 5.988

4(j) 210.2218.024 54.357 3004.185

4(i) 210.22104 0.000 0.470

5(j) 207.4070 54.357 0

5(i) 207.40705 0.400 1.188

6(j) 217.0970.475 54.357 0201.639

6(i) 217.09706 0.650 3.200

7(j) 240.0572.080 54.357 950.88

8(i) 328.35707 0.940 5.988

2(j) 164.1795.629 54.357 2772.485

9(i) 118.13608 1.340 5.988

3(j) 164.1798.024 54.357 2265.295

10(i) 120.95009 0.000 0.470

9(j) 118.1360 54.357 0

11(i) 111.26110 0.400 1.188

10(j) 120.9500.475 54.357 0110.300

12(i) 088.30111 0.650 3.200

11(j) 111.2612.080 54.357 0415.087

(CV)i 17849.52 in2

( ) (refer to Eqs. 7.a and 7.b)i ni si nj sjCV W w W w ;

2

1 1

1 1( ) ( ) 164.179 .

2 2

n n

si sj ij ij iCV w w t L CV inA A

Table 7. Values of Wni and Wnj

element no. joint i, j wsi wsj CV (in.2) Wni Wnj

1(i) 000.000 -164.17901

2(j) 164.179164.179

0

2(i) 164.179 002

3(j) 164.179164.179

0

3(i) 164.179 003

4(j) 210.221164.179

-46.042

4(i) 210.221 0-46.04204

5(j) 207.407164.179

-43.228

5(i) 207.407 0-43.22805

6(j) 217.097164.179

-52.918

6(i) 217.097 0-52.91806

7(j) 240.057164.179

-75.878

8(i) 328.357 -164.17907

2(j) 164.179164.179

0

9(i) 118.136 -046.04208

3(j) 164.179164.179

0

10(i) 120.950 -043.22809

9(j) 118.136164.179

-46.045

11(i) 111.261 -052.91810

10(j) 120.950164.179

-43.228

12(i) 088.301 -075.87811

11(j) 111.261164.179

-52.918

into the formula of warping constant (Eqs. 7, 7.a, 7.b,

and 7.c) and

Therefore, the warping constant of the built-up sec-

tion equal to 132128 in.6.

6. Warping Constant of Australian Crane

Runway Girder

The following equation for warping constant is pro-

vided by the Australian Institute of Steel Construction

[7] for the calculation of rolled crane runway beam. (re-

fer to Figure 2)

(8)

where

h = D (Te Tb) / 2 (8.a)

Te = Ac / dc (8.b)

where Iyt the moment of inertia of top flange about

y-axis, Iyb the moment of inertia of bottom flange about

y-axis, D the depth of the beam, Te the effective thick-

ness of top flange, Tb the thickness of bottom flange, Ac

the area of channel, and dc the depth of channel.

The Cw value for the Australian Built-up Section of

Crane Runway beam is calculated by using the above

formula (Equation 8) is given as follows.

Iyt = Ai xi2 + Iyy

= 2(15 1.6)(20.8)2 + 2 (1/12) 15 1.63

+ (1/12) 2 403 = 31.443 103 cm4

Iyb = 1.6 (30)3 / 12 = 3.6 103 cm4

Ac = 40 2 + 2 15 1.6 = 128 cm2

Te = Ac / dc = 128 / 43.2 = 2.96 cm

h = D (Te Tb) / 2 = 120 (2.96 + 1.6) / 2

= 117.72 cm

388 Wei T. Hsu et al.

Table 8. Calculation for warping constant of built-up WC section

element no. tij Lij joint i, j Wni Wnj2

niW Wni Wnj2

njW C(w)i

1(i) -164.17901 5.629

2(j) 026954.70 0 0 50576

2(i) 002 21.9440

3(j) 00 0 0 0

3(i) 003 8.024

4(j) -46.0420 0 2119.9 5670

4(i) 0-46.04204 0

5(j) -43.2282119.9 1990.3 1868.7 0

5(i) 0-43.22805 0.475

6(j) -52.9181868.7 2287.5 2800.3 1101

6(i) 0-52.91806 2.080

7(j) -75.8782800.3 4015.3 5757.5 8717

8(i) -164.17907 5.629

2(j) 026954.70 0 0 505760

9(i) 0-46.04208 8.024

3(j) 02119.9 0 0 5670

10(i) 0-43.22809 0

9(j) -46.0451868.7 1990.4 2120.1 0

11(i) 0-52.91810 0.475

10(j) -43.2282800.3 2287.5 1868.7 1101

12(i) 0-75.87811 2.080

11(j) -52.9185757.5 4015.3 2800.3 8717

Warping Constant (Cw) = 132128 in.6

Cw = (h2

Iyt Iyb) / (Iyt + Iyb) = (117.722

31.443

103 36.0 102) / (31.443 103 + 3.6 103)

= 44763672 cm6

The Cw value evaluated using the proposed steps speci-

fied in this study is equal to 132128 in.6 = 46366900

cm6. The difference is 3.45% and its calculation is gi-

ven by (44763672 46366900) / 46366900 = -3.45%.

It can be concluded that the accuracy of the evalu-

ated Cw is quite compatible. with the Australian rolled

section of crane runway girder.

7. Conclusion

This research summarizes the integration formulas

and the formulas in terms of numerical expressions for

crane runway girders made of WC or SC sections.

1. This research gives all warping constant values of

listed sections in the AISC design manuals (ASD and

LRFD). The warping constants evaluated using the pro-

posed steps in this study.

2. The accuracy of computer-assisted results is com-

pared with the Australian built-up section of crane

runway girder and the result is quite compatible.

3. This research provides a better evaluation for the WC

or SC sections when involve the warping constants of

section.

References

[1] AISC, Allowable Stress Design Manual of Steel Con-

struction, 9th Edition, American Institute of Steel Con-

struction, Chicago, Illinois (1989).

[2] AISC, Load and Resistance Factor Design Manual of

Steel Construction, 2nd Edition, American Institute

Steel Construction, Chicago, Illinois (1993).

[3] AISC, Load and Resistance Factor Design Specifica-

tion for Structural Steel Buildings, 3rd Edition, Ameri-

can Institute of Steel Construction, Inc., Chicago, Illi-

nois (1999).

[4] AISC, Design Specification for Structural Steel Build-

ings, 13th edition, American Institute of Steel Const-

ruction, Inc., Chicago, Illinois (2005).

[5] Galambos, T. V., Structural Members and Frames,

Prentice-Hall Inc., Englewood Cliffs, NJ, pp. 2753

(1968).

[6] Heins, C. P., Bending and Torsional Design in Struc-

tural Members, D. C. Heath Co., pp. 580 (1969).

[7] AISC, Crane Runway Girders, Australian Institute of

Steel Construction, Milsons Point, NSW, Australia,

pp. 4547 (1983).

Manuscript Received: Jul. 3, 2008

Accepted: Mar. 5, 2009

Numerical Approach for Torsion Properties of Built-Up Runway Girders 389