Numerical Approach for Torsion Properties of Built-Up Runway Girders

Transcription

1 Tamkang Journal of Science and Engineering, Vol. 12, No. 4, pp (2009) 381 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 (C w ) of the WC/SC girders are not provided in the AISC design manuals because the C w calculation for a WC/SC section is not a routine process but a tedious task. This study summarizes the theoretical C w 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 C w which in terms of numerical expressions is too complicated to be completed by hand-held calculators, the C w 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 C w will make engineers to better evaluate the elastic critical moment (M cr ) 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 (C w ). The C w 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 C w calculation of a WC/SC section is not a routine process but is a tedious and complicated task. *Corresponding author. dmlue@dragon.nchn.edu.tw 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 C w, which in terms of numerical expressions, is too complicated to be completed by hand-held calculators, the C w is set to be calculated 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 lipped section (refer to Figure 2) closely approximates the WC/SC section. The comparison results are quite compatible. The obtained C w values will make practicing engineers to calculate the elastic critical moment (M cr )of WC/SC girders much easier as compared with the ones provided by the current LRFD Specification [4].

2 382 Wei T. Hsu et al. Figure 1. Built-up WC and SC sections. 2. Integration Formulas for Warping Constant of WC/SC Section According to Galambos [5] and Heins [6], the integration formulas for warping constants of open thinwalled sections which include the WC/SC sections can be summarized (refer to Figures 3) and are given as follows. To calculate the value of C w, 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(X c, Y c ), refer to Figures 3, can be given as follows. The centroid of section: C(X c, Y c ) (1) 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 section S(X s, Y s ), refer to Figure 3, can be given as follows. The shear center of section: S(X s, Y s ) and y axes, respectively. The term w is the warping deformation of any point on the middle line a distance s from the edge o and the terms I wx and I wy are warping products of inertia about x and y axes, respectively. I wx and I wy can be obtained by performing the integration, some algebra and noting that cos ij = (x j x i )/L ij (refer to Figure 5). The warping constant of section: C w Figure 2. Section for Australian runway Beam. (3) (3.a) (3.b) (3.c) (2) (2.a) (2.b) The terms I x and I y are moments of inertia about the x Figure 3. Thin-walled open cross section.

3 Numerical Approach for Torsion Properties of Built-Up Runway Girders 383 W n the normalized unit warping, w s the unit warping 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 simplified by the fact that between points of intersection the unit warping properties w, w s, and W n vary linearly (Figure 5). The theoretical integration formulas based on Galambos [5] can be written in terms of numerical expressions and are given as follows. The centroid of section: C(X c, Y c ) (4) x i and y i are the x-coordinate and y-coordinate of plate element i, respectively, and A i is the area of plate element i. The shear center of section: S(X s, Y s ) have I xy = 0 and above equations of shear center can be rewritten as follows. (6) (6.a) (6.b) (6.c) The unit warping property w is the unit warping with respect to the centroid. x i and x j are the x coordinates of the ends of the element. y i and y j are the y coordinates of the ends of the element. w i and w j are the corresponding values of w at the ends of element. t the thickness of plate element, ij the distance between the tangent of element ij and the centroid and L ij the length of element ij. The warping constant of section: (7) (5) I xy is the product of inertia. Because the built-up sections of WC/SC are singly symmetrical sections, we (7.a) Figure 4. WC section. Figure 5. Distribution of warping deformation w on a plate element.

4 384 Wei T. Hsu et al. (7.b) (7.c) The unit warping properties w s and W n are the unit warping with respect to the shear center and the normalized unit warping, respectively. w s the unit warping with respect to shear center. w si and w sj are the corresponding values of w s at the ends of element, t ij is the thickness of plate element ij, ij the distance between the tangent of element ij and the centroid and L ij 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(X c, Y c ) (4) Section properties of C based on the AISC design manual. A =9.96in. 2, d =15.00in., t w =0.400in., b f =3.400in., t f = in. (1) The centroid of section : C(X c, Y c ) The coordinates and areas for each plate element (i) of the WC section are summarized as given in Table 1. Substitute these values (x i, y i, and A i included in Table 1) into the formula of centroid (Eq. 5) and the values of X c and Y c can be obtained. The coordinates for the centroid (X c, Y c )= (7.500 in., 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(X s, Y s ) The terms I x and I y are moments of inertia about the x and y axes, respectively. The I x and I y values of the builtup section can be calculated by using the tabulated values given in the AISC design manual and be calculated as below. I x = I x(w-shape) + I y(channel) + A w d A c d 2 2 = in. 4 I y = I y(w-shape) + I x(channel) = 583 in. 4 (2) The shear center of section: S(X s, Y s ) (3) The warping constant of section: C w (6) (7) A w theareaofw-shape,a c theareaofchannel,d 1 = the distance between the y coordinate of centroid of built-up section and the centroid of w-shape section, d 2 = 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 (I wx ), the parameters including x i, x j, w i, w j, t ij,andl ij are required and are listed in Tables 2, 3, and 4. Substitute these 5. Illustrated Example Proceed the steps specified in the previous section to evaluate the warping constant of a built-up section made of W and C The built-up section is divided 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 W based on the AISC design manual. A = 44.2 in. 2, d = in., t w = in., b f = in., t f =0.940in. Table 1. Coordinates and areas of plate elements element no. (i) x i (in.) y i (in.) A i (in. 2 )

5 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. x i/ x j (in.) y i/ y j (in.) L ij (in.) t ij (in.) 01 1(i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) Table 3. Unit wrappings w i and w j element no. ij (in.) L ij (in.) w ij (in. 2 ) joint no. w i (in. 2 ) w j (in. 2 ) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j)

6 386 Wei T. Hsu et al. Table 4. Calculation for warping product of inertia (I wx ) 1 element no. w i x i w j x j t ij L ij 3 ( + ) w 1 i x j w j x i ( + ) I wx = + = in. 5 parameters (x i, x j, w i, w j, t ij,andl ij ) into the given formulae (Equations 6, 6.a, 6.b, and 6.c) and the value of I wx can be obtained. The shear center of section is then determined by using Eq. (6) and Y s = I wx / I y = / = in. Therefore, the shear center is S (X s, Y s ) = (0 in., 6.74 in.) or S (X s, Y s ) = (7.50 in., in.) with respect to left lower corner as shown in Figure 4. (3) The warping constant of section: (C w ) The required parameters including w si, w sj, W ni, W nj, t ij, and L ij are listed in Tables 5, 6, 7, and 8. Substitute these values (w si, w sj, W ni, W nj, t ij, L ij and refer to Figure 4) Table 5. Unit wrappings w si and w sj element no. sij (in) L ij (in) sij L ij (in 2 ) joint i, j w si w sj (i) (j) (i) 3(j) (i) 4(j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j)

7 Numerical Approach for Torsion Properties of Built-Up Runway Girders 387 Table 6. Values of W ni +w si or W nj +w sj element no. t ij (in.) L ij (in.) joint i, j w si w sj t ij L ij A (in. 2 ) (CV) * i (in. 2 ) (i) (j) (i) 3(j) (i) 4(j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (CV) i in 2 ( CV ) W w W w (refer to Eqs. 7.a and 7.b) ; i ni si nj sj n n 1 1 CV ( w w ) t L ( CV ) in. 2A 2 si sj ij ij i 1 A 1 2 Table 7. Values of W ni and W nj element no. joint i, j w si w sj CV (in. 2 ) W ni W nj 01 1(i) (j) (i) 0 3(j) (i) 0 4(j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j) (i) (j)

8 388 Wei T. Hsu et al. Table 8. Calculation for warping constant of built-up WC section 2 2 element no. t ij L ij joint i, j W ni W nj Wni W ni W nj Wnj (i) - 2(j) (i) 0 3(j) (i) 0 4(j) (i) (j) (i) (j) (i) (j) (i) - 2(j) (i) (j) (i) (j) (i) (j) (i) (j) Warping Constant (C w ) = in. 6 C (w)i into the formula of warping constant (Eqs. 7, 7.a, 7.b, and 7.c) and Therefore, the warping constant of the built-up section equal to in Warping Constant of Australian Crane Runway Girder The following equation for warping constant is provided by the Australian Institute of Steel Construction [7] for the calculation of rolled crane runway beam. (refer to Figure 2) (8) T e = A c / d c (8.b) I yt the moment of inertia of top flange about y-axis, I yb the moment of inertia of bottom flange about y-axis, D the depth of the beam, T e the effective thickness of top flange, T b the thickness of bottom flange, A c theareaofchannel,andd c the depth of channel. The C w value for the Australian Built-up Section of Crane Runway beam is calculated by using the above formula (Equation 8) is given as follows. I yt = A i x i 2 + I yy = 2(15 1.6)(20.8) 2 +2 (1/12) (1/12) = cm 4 I yb = 1.6 (30) 3 / 12 = cm 4 A c = = 128 cm 2 h = D (T e T b ) / 2 (8.a) T e =A c /d c = 128 / 43.2 = 2.96 cm h = D (T e T b ) / 2 = 120 ( ) / 2 = cm

9 Numerical Approach for Torsion Properties of Built-Up Runway Girders 389 C w =(h 2 I yt I yb )/(I yt + I yb ) = ( ) / ( ) = cm 6 The C w value evaluated using the proposed steps specified in this study is equal to in. 6 = cm 6. The difference is 3.45% and its calculation is given by ( ) / = -3.45%. It can be concluded that the accuracy of the evaluated C w 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 proposed steps in this study. 2. The accuracy of computer-assisted results is compared 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 Construction, 9 th Edition, American Institute of Steel Construction, Chicago, Illinois (1989). [2] AISC, Load and Resistance Factor Design Manual of Steel Construction, 2 nd Edition, American Institute Steel Construction, Chicago, Illinois (1993). [3] AISC, Load and Resistance Factor Design Specification for Structural Steel Buildings, 3 rd Edition, American Institute of Steel Construction, Inc., Chicago, Illinois (1999). [4] AISC, Design Specification for Structural Steel Buildings, 13 th edition, American Institute of Steel Construction, Inc., Chicago, Illinois (2005). [5] Galambos, T. V., Structural Members and Frames, Prentice-Hall Inc., Englewood Cliffs, NJ, pp (1968). [6] Heins, C. P., Bending and Torsional Design in Structural Members, D. C. Heath Co., pp (1969). [7] AISC, Crane Runway Girders, Australian Institute of Steel Construction, Milsons Point, NSW, Australia, pp (1983). Manuscript Received: Jul. 3, 2008 Accepted: Mar. 5, 2009