SECONDARY BENDING MOMENT OF TRAPEZOID WEB BEAM UNDER SHEAR LOADING FONG SHIAU WEEN

ii SECONDARY BENDING MOMENT OF TRAPEZOID WEB BEAM UNDER SHEAR LOADING FONG SHIAU WEEN A project report submitted in partial fulfillment of the requirements for the award of the degree of Master of Engineering (Civil - Structure) Faculty of Civil Engineering Universiti Teknologi Malaysia NOVEMBER 2006

UNIVERSITI TEKNOLOGI MALAYSIA PSZ 19:16 (Pind. 1/97) BORANG PENGESAHAN STATUS TESIS υ JUDUL: SECONDARY BENDING MOMENT OF TRAPEZOID WEB BEAM UNDER SHEAR LOADING SESI PENGAJIAN : 2006/2007 Saya FONG SHIAU WEEN (HURUF BESAR) mengaku membenarkan tesis (PSM/Sarjana/Doktor Falsafah)* ini disimpan di perpustakaan Universiti Teknologi Malaysia dengan syarat-syarat kegunaan seperti berikut : 1. Tesis adalah hakmilik Universiti Teknologi Malaysia. 2. Perpustakaan Universiti Teknologi Malaysia dibenarkan membuat salinan untuk tujuan pengajian sahaja. 3. Perpustakaan dibenarkan membuat salinan tesis ini sebagai bahan pertukaran antara institusi pengajian tinggi. 4. ** Sila tandakan ( ) SULIT TERHAD (Mengandungi maklumat yang berdarjah keselamatan atau kepentingan Malaysia seperti yang termaktub di dalam AKTA RAHSIA RASMI 1972) (Mengandungi maklumat TERHAD yang telah ditentukan oleh organisasi/badan di mana penyelidikan dijalankan) TIDAK TERHAD Disahkan oleh (TANDATANGAN PENULIS) (TANDATANGAN PENYELIA) Alamat Tetap: 58, TAMAN BUKIT SIPUT, 85020 SEGAMAT, JOHOR. Tarikh : 15 NOVEMBER 2006 Tarikh: ASSOCIATE PROF. IR. DR. MOHD HANIM OSMAN Nama Penyelia 15 NOVEMBER 2006 CATATAN: * Potong yang tidak berkenaan * * Jika tesis ini SULIT atau TERHAD, sila lampirkan surat daripada pihak berkuasa/organisasi berkenaan dengan menyatakan sekali sebab dan tempoh tesis perlu dikelaskan sebagai SULIT atau TERHAD. Tesis dimaksudkan sebagai tesis bagi Ijazah Doktor Falsafah dan Sarjana secara penyelidikan, atau disertasi bagi pengajian secara kerja kursus dan penyelidikan, atau Laporan Projek Sarjana Muda (PSM)

I hereby declare that I have read this project report and in my opinion this report is sufficient in terms of scope and quality for the award of the degree of Master of Engineering (Civil- Structure) Signature :. Name of Supervisor : Associate Professor Ir. Dr. Mohd. Hanim Osman Date : 15 November 2006

ii I declare that this project report entitled Secondary Bending Moment of Trapezoid Web Beam Under Shear Loading is the result of my own research except as cited in the references. The report has not been accepted for any degree and is not concurrently submitted in candidature of any other degree. Signature :. Name : FONG SHIAU WEEN Date : 15 November 2006

To my beloved family iii

iv ACKNOWLEDGEMENTS I wish to take this opportunity to extend my sincere gratitude and heartfelt appreciation to my supervisor, Associate Professor Ir. Dr. Mohd. Hanim Osman, for his invaluable guidance, encouragement, support, concern and advice throughout the preparation of this study. I am indeed deeply indebted to Dr. Robiah Aznan, Faculty of Mathematical Sciences, University of Technology Malaysia, Skudai for her help in clearing my confusions and doubts in the process of doing this study. To my loving parents and siblings, thank you for your patience and encouragement in times of difficulty. Special thanks to my friends who had encouraged and assisted me during this study; especially to Mr. Gan Chun Hou. Thank you.

v ABSTRACT In structural and fabrication technology, new techniques of optimized steel structures design have been developed. One of the developments in steel structure design is the introduction of trapezoidal web beam. One of the phenomena being studied for the trapezoid web beam is the secondary bending moment which is induced in the flanges of section subjected to shear loading in the web, due to the corrugation of the web. A parametric study was carried out to develop a formula for the secondary bending moment. The parametric study involved in this study are depth of web, D, width of flange, B, thickness of flange, T and thickness of web, t. This study was carried out by using finite element method. The formula of secondary bending moment has been successfully derived and can be used for any other sections of trapezoid web beam with same corrugation thickness and corrugation angle. Keywords: trapezoid web beam, secondary bending moment, finite element

vi ABSTRAK Dalam stuktur dan teknologi pembuatan, teknik baru rekabentuk optimum struktur keluli telah dimajukan. Salah satu perkembangan dalam struktur keluli ialah pengenalan rasuk dengan web trapezoid. Salah satu fenomena yang telah dikaji dalam rasuk web trapezoid adalah momen lengkukan kedua yang terhasil dalam bebibir apabila beban rich dikenakan dalam web, disebabkan kerutan web tersebut. Satu kajian parametrik telah dijalankan untuk menghasilkan satu formula bagi momen lengkukan kedua dalam rasuk web trapezoid apabila dikenakan beban ricih. Kajian parametrik ini termasuklah ketigggian web, D, kelebaran web, B, ketebalan bebibir, T dan ketebalan web, t. Kajian ini dijalankan dengan menggunakan kaedah usur terhingga. Formula bagi momen lengkukan kedua telah berjaya dihasilkan dan boleh digunakan untuk apa-apa saja saiz bagi rasuk web trapezoid dengan tebal kerutan dan sudut kerutan yang tetap. Kata kekunci: rasuk web trapezoid, momen lengkukan kedua, usur terhingga

vii TABLE OF CONTENTS CHAPTER TITLE PAGE DECLARATION DEDICATION ACKNOWLEDGEMENTS ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS ii iii iv v vi vii ix xi xiv 1 INTRODUCTION 1 1.1 General 1 1.2 Problem Statement 3 1.3 Objective of Study 3 1.4 Scope of Study 4 2 LITERATURE REVIEWS 5 2.1 Introduction of Trapezoid Web Section 5 2.2 Secondary Bending Moment of Trapezoid Web Beam 6 2.2.1 The Bending Capacity of TWP Section 8 2.3 Finite Element Analysis (FEA) 21 2.4 LUSAS Finite Element Software 22

viii 2.4.1 Introduction 22 2.4.2 LUSAS Finite Element System 23 2.5 Shear Stress Flow 24 2.6 Secondary Bending Moment Capacity of Flange, M cyf 29 2.7 Research Programme 30 3 ANALYSIS OF SECONDARY BENDING MOMENT USING FEM 31 3.1 LUSAS Finite Element Analysis 31 3.2 Preliminary Data Analysis 46 3.3 Comparison Data of FEA and German Table Properties 57 3.4 Parametric Study Analysis 57 3.5 Graph Simulation 66 4 DERIVATION OF FORMULA C o 76 4.1 Manually 76 4.1.1 C o /M sec,o Comparison 88 5 CONCLUSIONS AND SUGGESTIONS 98 5.1 Conclusions 98 5.2 Suggestions 99 REFERENCES 100

ix LIST OF TABLES TABLE NO. TITILE PAGE 2.1 German Table Properties reffered to [1]. 9 2.2 Result of C o from FEA and comparison to C o from existing table of capacities and ratio of C o /M sec,o obtained reffered to [2]. 15 2.3 C o /M sec,o calculated from German table properties with various web depth, fixed flange width and flange thickness. 17 2.4 C o /M sec,o calculated from German table properties with fixed web depth, various flange width and flange thickness. 19 2.5 The reactions in X and Y direction. 25 2.6 Surface stress of the node and element. 25 3.1 Lateral reactions, Fz of each node at top and bottom flange for model with web depth 300mm. 49 3.2 Summary of lateral reaction in each oblique part at top and bottom flange. 50 3.3 Summary of lateral reaction, Q in z direction in each oblique part for each section size 51 3.4 Values of C o from finite element analysis and comparison to existing table capacities. 54 3.5 Values of C o / M cyf and C o / M sec,o from finite element analysis and comparison to existing table. 55 3.6 Summary of lateral reaction, Q in z direction in each oblique part for each section size. 59 3.7 Values of C o and C o /M cyf for various web depth, flange width and flange thickness. 62 3.8 Values of C o and C o /M cyf for various flange width. 65

x 3.9 Values of C o and C o /M cyf for various flange thickness. 65 3.10 Values of C o and C o /M cyf for various web thickness. 66 4.1 Coefficient, P, Y and power, Z of C o and C o /M cyf from graph equation shown in Figure 3.5-3.18 77 4.2 Value of C o calculated based on formula derived and divided by M cyf and compare with FEA. 82 4.3 Value of C o /M cyf calculated based on formula and compare with FEA 85 4.4 Comparison values of C o /M cyf and C o /M sec,o within formula derived, FEA and German table properties. 89

xi LIST OF FIGURES FIGURE NO. TITILE PAGE 2.1 The shear stress flow at the trapezoidal web. 6 2.2 Plan view of lateral force, Q at trapezoidal web subjected to the shear stress flow. 6 2.3 Plan view of secondary bending moment, M sec at trapezoidal web. 7 2.4 The shear stress flow at the flat web. 7 2.5 Plot of C o versus depth of web, d in comparison between finite element analysis to the existing value referred to [2]. 16 2.6 Plot of C o /M sec,o versus depth of web, d in comparison between finite element analysis to the existing value referred to [2]. 16 2.7 Graph C o /M sec,o versus web depth, D from German table properties. 18 2.8 Graph C o /M sec,o versus flange width, B from German table properties. 20 2.9 Graph C o /M sec,o versus flange thickness, T from German table properties. 20 2.10 The FEM model of the web plate subjected to pure shear force 24 2.11 The direction of surface stress SX and reaction FX. 28 3.1 Trapezoidal web beam model with depth web 300mm. 32 3.2 Support condition assigned to the model with depth web 300mm. 33 3.3 Vertical loading with total 20kN assigned to the points to the model with web depth 300mm. 33 3.4 Steps of build up a LUSAS model with web depth 300mm. 34 3.5 Node labels along the connection of the web to the flange which war restrained in z direction for web depth 300mm. 47

xii 3.6 The total lateral forces at each node, Q at each oblique sub panel of web depth 300mm. 53 3.7 Graph C o versus depth web,d in comparison between finite element analysis to the existing value. 56 3.8 Graph C o /M sec,o versus depth web, d in comparison between finite element analysis to the existing value. 56 3.9 The total lateral reactions, Q 1, Q 2 and Q 3 of each oblique sub panel in top flange and bottom flange. 58 3.10 Graph C o and C o /M cyf versus web depth, D for fixed flange width, B= 160mm and flange thickness, T = 12mm. 67 3.11 Graph C o and C o /M cyf versus web depth, D for fixed flange width, B = 200mm and flange thickness, T = 15mm. 68 3.12 Graph C o and C o /M cyf versus web depth, D for fixed flange width, B = 240mm and flange thickness, T = 18mm. 69 3.13 Graph C o and C o /M cyf versus web depth, D for fixed flange width, B = 280mm and flange thickness, T = 20mm. 70 3.14 Graph C o and C o /M cyf Versus web depth, D for fixed flange width, B = 300mm and flange thickness, T = 22mm. 71 3.15 Graph C o and C o /M cyf versus web depth, D for fixed flange width, B = 350mm and flange thickness, T = 25mm. 72 3.16 Graph C o and C o /M cyf versus flange width, B for fixed web depth, D = 750mm and flange thickness, T = 10mm. 73 3.17 Graph C o and C o /M cyf versus flange thickness, T for fixed web depth, D = 750mm and flange width, B = 120mm 74 3.18 Graph C o and C o /M cyf versus web thickness, t for fixed web depth, D = 750mm and flange width, B = 120mm, flange thickness, T = 10mm. 75 4.1 Graph P versus Flange Width, B 77 4.2 Graph P versus Flange Thickness, T 78 4.3 Graph Z versus Flange Width, B 78 4.4 Graph Z versus Flange Thickness, T 79 4.5 Graph Y versus Flange Width, B 79 4.6 Graph Y versus Flange Thickness, T 80

xiii 4.7 Graph C o /M cyf versus Web Depth for various flange width and flange thickness. 92 4.8 Graph C o /M cyf versus Flange Width with web depth 750mm, web thickness 3mm. 95 4.9 Graph C o /M cyf versus Flange Thickness with web depth 750mm, web thickness 3mm. 96 4.10 C o /M cyf versus Web Depth from formula result for various flange width and flange thickness 96 4.11 C o /M cyf versus Web Depth from FEM analysis result for various flange width and flange thickness. 97

xiv LIST OF SYMBOLS a - Length of straight part of the web corrugation B - Width of flange C o - Secondary bending moment coefficient D - Depth of web T - Thickness of flange t - Thickness of web M cx - Section bending capacity, which is calculated by neglecting the contribution of web M yf - Secondary bending moment of this study M cyf - Secondary bending moment capacity of this study M sec - Secondary bending moment of German M sec,o - Secondary bending moment capacity of German M x - Applied bending moment V - Shear loading Q - Total lateral reactions at each oblique sub-panel Q avr - Average lateral reactions p y - Design strength Z yf - Elastic modulus of each flange in y axis

CHAPTER 1 INTRODUCTION 1.1 General Demand for steel structures is increasing dramatically especially in construction industry. A number of new structural and fabrication technology has been developed to optimize the efficient use of steel in construction. One of the developments in steel structures is the introduction of trapezoidal web I-beam. Trapezoid web beam is a type of steel section to form an I-section which use corrugated web made in trapezoidal form. The beam with corrugated thin web is continuously welded to the flanges at the top and bottom. Trapezoid web beam is a built up section that able to support vertical loads over long spans. The higher bending capacity is achieved by increasing the depth of the section. Corrugation web in trapezoidal form increases the stability against buckling and can result in very economical designs. Ordinarily, the economic design of steel web I-beam requires thin web. To eliminate the risk of using thicker web of the beam, web stiffeners or by making the web in trapezoidal form is needed for the purpose of strengthening the web. When beams with corrugated webs are compared

2 with those with stiffened flat webs, it can be found that trapezoidal corrugation in the web enables the use of thinner webs and trapezoidal web beams eliminate costly web stiffeners. The trapezoid web beam provides a higher resistance against bending moment about the weak axis, high strength-to-weight ratio, less cost and higher load carrying capacity. Furthermore, it also offers its naturally architectural design element with its own aesthetic quality in the various construction projects. The flange of trapezoid web beam carries the bending moment and the trapezoidal web carries the shear force. Due to the shear force subjected at the trapezoidal web, a lateral bending moment is induced in the flange, which is known as secondary bending moment, M yf. It may cause a minor reduction in the bending moment capacity of the web. This study consists of finite element analysis by using a computer software which is known as LUSAS, to determine the lateral reactions at the flange of the trapezoidal web beam. The lateral reaction depends on the section properties and increase linearly with the applied shear force. Thus, secondary bending moment coefficient, C o is induced. A series of analysis on finite element has been done on various sizes and properties of the beams to determine the value of secondary bending moment coefficient, C o by compared the value between finite element analysis and German existing table properties. In the past, German table properties was developed on the specific size section of trapezoid web beam. From the derivation of formula C o, it is to be used to determine M sec for any size sections. Apart from that, local engineers have widen choice on size section while doing design work.

3 1.2 Problem Statement There is not much work has been done for the secondary bending behavior of trapezoid web beam. From the German table properties of the trapezoid web beam, there is no explanation and formula on how the value of secondary bending moment coefficient, C o and secondary bending moment, M sec is obtained and the information is limited. The behavior of C o can be determined by lateral reaction by using finite element analysis with applied lateral support at the section. The value of secondary bending moment coefficient is acceptable if it is comparable with existing German table properties. Therefore, the study is necessary to determine the clear explanation on the value of C o and derive the formula of C o and extend the knowledge. 1.3 Objective of Study This study has been conducted to address the problem statement mentioned above. This study will give a better understanding on C o and M yf due to various geometric properties by using LUSAS finite element software. The main objectives of this study are: a) To determine the secondary bending moment, M yf in the flange. b) To derive the formula of secondary bending moment coefficient, C o

4 1.4 Scope of Study The scopes in this study consist of: a) Determine the lateral reaction in the flanges of the trapezoidal web beams when subjected to shear loading by using LUSAS finite element software. b) Determine the secondary bending moment, M yf induced at the flanges and the value of C o due to the lateral forces adopted by finite element analysis. c) Carrying out parametric study by varying: i) Flange width ii) Flange thickness iii) Web depth iv) Web thickness v) Aspect ratio of sub-panel vi) Corrugation thickness vii) Corrugation angle and etc. d) Derive the formula of C o from the parametric study using manual calculation. e) Verify the values of M yf and C o by comparing with the German table properties.

CHAPTER 2 LITERATURE REVIEWS 2.1 Introduction of Trapezoid Web Section The trapezoid web beam is proved to be an alternative to the conventional hot rolled and welded sections with flat web in respects to its strength/weight ratio [1]. Trapezoid web requires no stiffening except at supports. It provides a kind of continuous stiffening in the transverse direction, which in general permits the use of rather thin plates with significant low cost and weight saving. A collaborative research program has been established between TWP manufacturer and the Steel Technology Centre (STC), University Technology Malaysia with the purpose of developing a design guide for section using trapezoid web with slenderness ratio as high as twice the existing sections used in Germany. One of the phenomena being studied for the trapezoidal web beam, which is not the problem in flat web, is the secondary bending moment which is induced at the flanges subjected to the shear force at the web. Analytical works is being carried out to study the problem.

6 2.2 Secondary Bending Moment of Trapezoid Web Beam The shear force subjected to a corrugated web section is designed to be resisted by its web, through the development of a shear stress flow as shown in Figure 2.1. Due to the trapezoidal web, reaction in transverse direction to the horizontal axis will be exerted along the top and bottom edges of the web, which are welded to the top and bottom flanges respectively. Figure 2.1: The shear stress flow at the trapezoidal web. The oblique orientation of alternate web sub panels induces component forces, Q from the shear stress flow, in lateral direction (z direction) of the section as shown in Figure 2.2 in each of the sub panel, these component forces act in opposite direction. This will result into couples and induces secondary bending moment, M yf about the yy axis of the section as illustrated in Figure 2.3. M yf = Q. a Figure 2.2: Plan view of lateral force, Q at trapezoidal web subjected to the shear stress flow.

7 Figure 2.3: Plan view of secondary bending moment, M yf at trapezoidal web. In a flat web plate, the shear stress flow is as shown in Figure 2.4. The plane on which the shear stress flow is acting coincides with the longitudinal neutral axis, therefore no component reaction force is induced in z-direction. Thus, there is no secondary bending moment about the yy axis at the top and bottom flanges [2]. Figure 2.4: The shear stress flow at the flat web. The secondary bending moment, M yf in each flange is given by, M yf = V. C o Where, V is the direct shear loading applied to the web C o is the geometric constant in unit mm.

8 2.2.1 The Bending Capacity of TWP Section The bending capacity of a steel section with trapezoid web is checked by the following interactive equation [5], Where, M x M cx V C o M cyf M x + VC o < 1 M cx M cyf is the applied bending moment is the section bending capacity, which is calculated by neglecting the contribution of web is the applied shear force on the section is the secondary bending moment coefficient is the secondary bending moment capacity of each flange about the minor axis The German table properties for corrugated web section is shown in Table 1. However, there is no clear explanation on how C o and M cyf value is obtained. According to Nina, the only information on C o was that established by Rose in 1985. He made some static calculations of single span girders and frames. The value of secondary bending moment was said to be dependent on the static system and the distribution of the internal forces. He calibrated the influence of the secondary bending moment with these static calculations and creates a simple formula on the safe side, which engineer can apply without complicated calculation.

15 Nina [2] had made some analysis on C o value and C o /M cyf value due to various web depth which is shown in Table 2.2. Graphs of C o and C o / M cyf versus web depth from the analysis and German table properties were plotted as shown in Figure 2.5 and Figure 2.6. She concluded that the values of M cyf and C o from analysis give reasonable comparison with the existing design table. Table 2.2: Result of Co from FEA and comparison to Co from existing table of capacities and ratio of C o / M cyf and Co/M sec,o obtained referred to [2]. D t B T C o C o / M cyf Co/M sec,o (mm) (mm) (mm) (mm) FEA Existing FEA Existing 300 2 120 10 92.50 98.60 10.71 8.47 350 2 120 10 79.27 89.00 9.17 7.19 400 2 120 10 69.39 80.30 8.03 6.24 450 2 120 10 61.69 73.00 7.14 5.52 500 2 120 10 55.54 66.90 6.43 4.94 550 2 120 10 50.48 61.10 5.84 4.47 600 2 120 10 46.30 56.80 5.36 4.09 650 2 120 10 42.74 52.60 4.95 3.76 700 2 120 10 39.69 49.60 4.59 3.51 750 2 120 10 37.04 46.50 4.29 3.25

16 120 100 80 Lusas Co German Co y = 0.0002x 2 0.2942x + 170.96 Co 60 40 y = 27499x -0.9984 20 0 250 350 450 550 650 750 Depth of web, d (mm) Figure 2.5: Plot of C o versus depth of web, d in comparison between finite element analysis to the existing value referred to [2]. y = 3182.7x -0.9984 y = 3240.3x -1.0432 Figure 2.6: Plot of C o /M sec,o versus depth of web, d in comparison between finite element analysis to the existing value reffered to [2].

17 Some graphs were plotted based on German table properties to show the relation between C o /M sec,o with different geometric parameters. The values of C o /M sec,o were calculated in Table 2.3 with various web depth, fixed flange thickness and flange width. The graph was plotted in Figure 2.7. Table 2.4 showed the values of C o /M sec,o with various flange width, flange thickness and fixed web depth. Graphs C o /M sec,o versus flange width and flange thickness were plotted in Figure 2.8 and Figure 2.9. From Figure 2.7-2.9, it is clearly showed that the values of C o /M sec,o decreased with the increased of web depth, flange width and flange thickness. Table 2.3: C o /M sec,o calculated from German table properties with various web depth, fixed flange width and flange thickness. hr = 80mm D B T German (mm) (mm) (mm) C o M sec,o C o /M sec,o 500 160 12 72.4 31.00 2.3355 600 160 12 60.5 31.38 1.9280 700 160 12 52.1 31.71 1.6430 750 160 12 48.9 31.96 1.5300 800 160 12 46.1 32.2 1.4317 900 160 12 41.1 32.45 1.2666 1000 160 12 37.2 32.68 1.1383 600 200 15 64.4 64.53 0.9980 700 200 15 55.1 64.92 0.8487 750 200 15 51.6 65.31 0.7901 800 200 15 48.2 65.29 0.7382 900 200 15 43.1 65.99 0.6531 1000 200 15 38.9 66.29 0.5868 1100 200 15 35.4 66.58 0.5317 1200 200 15 32.5 66.86 0.4861 700 240 18 57.9 116.77 0.4958 750 240 18 54.1 117.35 0.4610 800 240 18 50.5 117.19 0.4309 900 240 18 45 118.23 0.3806 1000 240 18 40.5 118.57 0.3416 1100 240 18 36.6 118.42 0.3091 1200 240 18 33.7 119.26 0.2826

18 Table 2.3 (cont ) D B T German (mm) (mm) (mm) C o M sec,o C o /M sec,o 700 280 20 59.7 181.04 0.3298 750 280 20 55.7 181.82 0.3063 800 280 20 52.3 182.57 0.2865 900 280 20 46.4 183.98 0.2522 1000 280 20 41.8 184.32 0.2268 1100 280 20 37.7 183.66 0.2053 1200 280 20 34.6 184.55 0.1875 700 300 22 61.2 232.89 0.2628 750 300 22 57.1 233.84 0.2442 800 300 22 53.5 234.74 0.2279 900 300 22 47.6 236.43 0.2013 1000 300 22 42.6 236.47 0.1801 1100 300 22 38.4 235.58 0.1630 1200 300 22 35.3 236.68 0.1491 Co/Msec,o 2.5000 2.0000 o 1.5000 1.0000 0.5000 0.0000 hr/b = 0.333 hr/b = 0.286 hr/b = 0.267 hr/b = 0.5 hr/b = 0.4 B=160, T=12 B=200, T=15 B=240, T=18 B=280, T=20 B=300, T=22 450 650 850 1050 1250 Web depth, D (mm) Figure 2.7: Graph C o /M sec,o versus web depth, D from German table properties.

19 Table 2.4: C o /M sec,o calculated from German table properties with fixed web depth, various flange width and flange thickness. D B T German (mm) (mm) (mm) C o M sec,o C o /M sec,o 700 120 10 49.40 14.16 3.4887 700 160 12 52.10 31.71 1.6430 700 200 15 55.10 64.92 0.8487 700 240 18 57.90 116.77 0.4958 700 280 20 59.70 181.04 0.3298 700 300 22 61.20 232.89 0.2628 750 120 10 46.50 14.32 3.2472 750 160 12 48.90 31.96 1.5300 750 200 15 51.60 65.31 0.7901 750 240 18 54.10 117.35 0.4610 750 280 20 55.70 181.82 0.3063 750 300 22 57.10 233.84 0.2442 800 160 12 46.10 32.20 1.4317 800 200 15 48.20 65.29 0.7382 800 240 18 50.50 117.19 0.4309 800 280 20 52.30 182.57 0.2865 800 300 22 53.50 234.74 0.2279 800 350 25 54.70 368.00 0.1486 900 160 12 41.10 32.45 1.2666 900 200 15 43.10 65.99 0.6531 900 240 18 45.00 118.23 0.3806 900 280 20 46.40 183.98 0.2522 900 300 22 47.60 236.43 0.2013 900 350 25 48.60 370.65 0.1311 1000 160 12 37.20 32.68 1.1383 1000 200 15 35.40 66.58 0.5317 1000 240 18 40.50 118.57 0.3416 1000 280 20 41.80 184.32 0.2268 1000 300 22 42.60 236.47 0.1801 1000 350 25 43.50 370.67 0.1174

20 3.1000 2.6000 D=700 D=750 D=800 D=900 D=1000 Co/Msec,o 2.1000 1.6000 1.1000 0.6000 0.1000 110 160 210 260 310 Flange width, B (mm) Figure 2.8: Graph C o /M sec,o versus flange width, B from German table properties. 3.1000 2.6000 D=700 D=750 D=800 D=900 D=1000 Co/Msec,o 2.1000 1.6000 1.1000 0.6000 0.1000 8 13 18 23 28 Flange thickness, T (mm) Figure 2.9: Graph C o /M sec,o versus flange thickness, T from German table properties.

21 In order to obtain a clear explanation on how C o value is obtained, finite element method was applied to study the reaction induced in the flanges caused by vertical shear load subjected to the corrugated web. 2.1 Finite Element Analysis (FEA) The finite element method has become a powerful tool for the numerical solution of a wide range of engineering problems. Applications range from deformation and stress analysis of automotive, aircraft, building and bridge structures to field analysis of heat, flux, fluid, flow, magnetic flux, seepage and other flow problems. With the advances in computer technology and CAD systems, complex problems can be modeled with relative ease. Several alternative configurations can be tested on a computer before the first prototype is built. All of this suggests that need to keep pace with these developments by understanding the basic theory, modeling techniques and computational aspects of the finite element method. In this method of analysis, a complex region defining a continuum is discretized into simple geometric shapes called finite elements. The material properties and the governing relationships are considered over these elements and expressed in terms of unknown values at element corners. An assembly process, duly considering the loading and constraints, results in a set of equations. Solution of these equations gives us the approximate behavior of the continuum. Computer use is an essential part of the finite element analysis. Welldeveloped, well-maintained and well-supported computer programs are necessary in solving engineering problems and interpreting results. Many available commercial finite element packages fulfill these needs. It is also the trend in industry that the results are acceptable only when solved using certain standard computer program packages. The commercial packages provide user-friendly data input platforms and elegant and easy to follow display formats. However, the packages do no provide and

22 insight into the formulations and solution methods. Specially developed computer programs with available source codes enhance the learning process [6]. 2.2 LUSAS Finite Element Software 2.2.1 Introduction LUSAS is an associative feature-based Modeller. The model geometry is entered in terms of features which are sub-divided (discretised) into finite elements in order to perform the analysis. Increasing the discretisation of the features will usually result in an increase in accuracy of the solution, but with a corresponding increase in solution time and disk space required. The features in LUSAS form a hierarchy, which is Volumes are comprised of Surfaces, which in turn are made up of Lines or Combined Lines, which are defined by Points. Evaluating a complex engineering design by exact mathematical models, however, is not a simple process. The calculation on the response of a complex shape to any external loading cannot be done. So, the complex shape must be divided up into lots of smaller simpler shapes. These are the finite elements that give the method its name. The shape of each finite element is defined by the coordinates of its nodes. Adjoining elements with common nodes will interact. The real engineering problem responds in an infinite number ways to external forces. The manner in which the finite element model will react is given by the degrees of freedom, which are expressed at the nodes. The response of a single finite element to a known stimulus can be express by built up a model for the whole structure by assembling all of the simple expressions into a set of simultaneous equations with the degrees of freedom at each node as the unknowns. These are then solved using a matrix solution technique [7].

23 2.2.2 LUSAS Finite Element System A complete finite element analysis involves three stages which include preprocessing, finite element solver and results-processing. The LUSAS finite element system consists of two parts to perform a full analysis: a) LUSAS Modeller is a fully interactive pre and post-processing graphical user interface. b) LUSAS Solver performs the finite element analysis. Pre-processing involves creating a geometric representation of the structure, then assigning properties, then outputting the information as a formatted data file suitable for processing by LUSAS. A model is a graphical representation consisting of Geometry (Points, Lines, Combined Lines, Surfaces and Volumes) and Attributes (Materials, Loading, Supports, Mesh, etc.). Each part of the model is created in two steps: define the Feature or Attribute and assign the Feature or Attribute. Features can be defined by entering coordinates, selecting Points on the screen or by using utilities such as transformations. An attribute is first defined by creating an attribute dataset. The dataset is then assigned to chosen features. To complete a model it may be necessary to define additional utilities called control datasets. These are used to control the progress of advanced analysis. Once a model has been created click on the solve button to begin the solution stage. LUSAS create a data file from the model, solves the stiffness matrix and produces a result file. Result-processing involves using a selection of tools for viewing and analyzing the results file produced by the Solver [7].

24 2.5 Shear Stress Flow A finite element analysis was carried out on a plate to study the surface stress and the shear stress flow in the plate when subjected to a pure shear force. The plate has the dimension 250x250mm and 10mm thickness. A vertical load of 2kN was assigned to each point at the right side of the web. The supports were fixed in translation in X and Y direction. The result of the reaction in X and Y direction and surface stress is shown in Table 2.5 and Table 2.6 Figure 2.10: The FEM model of the web plate subjected to pure shear force.

25 Table 2.5: The reactions in X and Y direction. Node FX FY FZ MX MY MZ RSLT 1 7.27 2.25 0 0 0 0 7.61 2 7.21 1.8 0 0 0 0 7.43 9 2.04 1.94 0 0 0 0 2.82 14-2.04 1.94 0 0 0 0 2.82 19-7.21 1.8 0 0 0 0 7.43 24-7.27 2.25 0 0 0 0 7.61 Table 2.6: Surface stress of the node and element. Element Node SX SY SZ SXY SYZ SZX 1 1-0.0304-0.00648 0-0.00454 0 0 1 2-0.0129-0.00648 0-0.00454 0 0 1 4-0.0129-0.00044 0-0.00454 0 0 1 6-0.0304-0.00044 0-0.00454 0 0 2 2-0.0124-0.0025 0-0.00498 0 0 2 9-0.00426-0.0025 0-0.00498 0 0 2 11-0.00426-0.00148 0-0.00498 0 0 2 4-0.0124-0.00148 0-0.00498 0 0 3 9-0.00366-8.67E-19 0-0.00495 0 0 3 14 0.00366-2.43E-18 0-0.00495 0 0 3 16 0.00366 9.71E-18 0-0.00495 0 0 3 11-0.00366 9.19E-18 0-0.00495 0 0 4 14 0.00426 0.0025 0-0.00498 0 0 4 19 0.0124 0.0025 0-0.00498 0 0 4 21 0.0124 0.00148 0-0.00498 0 0 4 16 0.00426 0.00148 0-0.00498 0 0 5 19 0.0129 0.00648 0-0.00454 0 0 5 24 0.0304 0.00648 0-0.00454 0 0 5 26 0.0304 0.000442 0-0.00454 0 0 5 21 0.0129 0.000442 0-0.00454 0 0 6 6-0.02 0.0012 0-0.00255 0 0 6 4-0.0123 0.0012 0-0.00255 0 0 6 29-0.0123-0.00101 0-0.00255 0 0 6 31-0.02-0.00101 0-0.00255 0 0 7 4-0.0125-0.0014 0-0.00612 0 0 7 11-0.0036-0.0014 0-0.00612 0 0 7 34-0.0036-0.00026 0-0.00612 0 0 7 29-0.0125-0.00026 0-0.00612 0 0 8 11-0.00335 8.50E-18 0-0.00666 0 0 8 16 0.00335 6.77E-18 0-0.00666 0 0 8 37 0.00335 1.18E-17 0-0.00666 0 0 8 34-0.00335 8.50E-18 0-0.00666 0 0 9 16 0.0036 0.0014 0-0.00612 0 0 9 21 0.0125 0.0014 0-0.00612 0 0 9 40 0.0125 0.000255 0-0.00612 0 0 9 37 0.0036 0.000255 0-0.00612 0 0

26 Table 2.6 (cont ) Element Node SX SY SZ SXY SYZ SZX 10 21 0.0123-0.0012 0-0.00255 0 0 10 26 0.02-0.0012 0-0.00255 0 0 10 43 0.02 0.00101 0-0.00255 0 0 10 40 0.0123 0.00101 0-0.00255 0 0 11 31-0.0145 0.000368 0-0.00233 0 0 11 29-0.00858 0.000368 0-0.00233 0 0 11 46-0.00858-0.00061 0-0.00233 0 0 11 48-0.0145-0.00061 0-0.00233 0 0 12 29-0.00862 0.00046 0-0.00593 0 0 12 34-0.00275 0.00046 0-0.00593 0 0 12 51-0.00275-0.00094 0-0.00593 0 0 12 46-0.00862-0.00094 0-0.00593 0 0 13 34-0.00268 9.02E-18 0-0.00749 0 0 13 37 0.00268 1.20E-17 0-0.00749 0 0 13 54 0.00268 3.87E-17 0-0.00749 0 0 13 51-0.00268 3.78E-17 0-0.00749 0 0 14 37 0.00275-0.00046 0-0.00593 0 0 14 40 0.00862-0.00046 0-0.00593 0 0 14 57 0.00862 0.000939 0-0.00593 0 0 14 54 0.00275 0.000939 0-0.00593 0 0 15 40 0.00858-0.00037 0-0.00233 0 0 15 43 0.0145-0.00037 0-0.00233 0 0 15 60 0.0145 0.000608 0-0.00233 0 0 15 57 0.00858 0.000608 0-0.00233 0 0 16 48-0.0104 0.000677 0-0.00191 0 0 16 46-0.00417 0.000677 0-0.00191 0 0 16 63-0.00417-0.00175 0-0.00191 0 0 16 65-0.0104-0.00175 0-0.00191 0 0 17 46-0.00403-1.51E-05 0-0.00622 0 0 17 51-0.00118-1.51E-05 0-0.00622 0 0 17 68-0.00118-0.00019 0-0.00622 0 0 17 63-0.00403-0.00019 0-0.00622 0 0 18 51-0.00115 3.55E-17 0-0.00775 0 0 18 54 0.00115 4.44E-17 0-0.00775 0 0 18 71 0.00115-3.11E-17 0-0.00775 0 0 18 68-0.00115-2.29E-17 0-0.00775 0 0

27 Table 2.6 (cont ) Element Node SX SY SZ SXY SYZ SZX 19 54 0.00118 1.51E-05 0-0.00622 0 0 19 57 0.00403 1.51E-05 0-0.00622 0 0 19 74 0.00403 0.000186 0-0.00622 0 0 19 71 0.00118 0.000186 0-0.00622 0 0 20 57 0.00417-0.000677 0-0.00191 0 0 20 60 0.0104-0.000677 0-0.00191 0 0 20 77 0.0104 0.00175 0-0.00191 0 0 20 74 0.00417 0.00175 0-0.00191 0 0 21 65-0.00443-0.000344 0-0.0032 0 0 21 63-0.000726-0.000344 0-0.0032 0 0 21 80-0.000726 0.00738 0-0.0032 0 0 21 82-0.00443 0.00738 0-0.0032 0 0 22 63-0.000867 0.000481 0-0.0057 0 0 22 68 0.000102 0.000481 0-0.0057 0 0 22 85 0.000102 0.00561 0-0.0057 0 0 22 80-0.000867 0.00561 0-0.0057 0 0 23 68-0.000812-3.17E-17 0-0.0062 0 0 23 71 0.000812-1.24E-17 0-0.0062 0 0 23 88 0.000812-2.09E-17 0-0.0062 0 0 23 85-0.000812 1.89E-17 0-0.0062 0 0 24 71-0.000102-0.000481 0-0.0057 0 0 24 74 0.000867-0.000481 0-0.0057 0 0 24 91 0.000867-0.00561 0-0.0057 0 0 24 88-0.000102-0.00561 0-0.0057 0 0 25 74 0.000726 0.000344 0-0.0032 0 0 25 77 0.00443 0.000344 0-0.0032 0 0 25 94 0.00443-0.00738 0-0.0032 0 0 25 91 0.000726-0.00738 0-0.0032 0 0

28 From Table 2.6, it is clearly showed that the values of SXY have negative sign. In this case, the flow of SXY are in the same direction either anticlockwise or clockwise. Figure 2.11 shows the direction of surface stress, SX is in opposite direction with the reaction FX. Meanwhile the flow of surface stress, SXY is acting in opposite direction with the vertical loading. Refer to [8] for further information about shear stress flow. SX 24 SX 19 SX 14 SX 9 SX 2 SX 1 Figure 2.11: The direction of surface stress SX and reaction FX.

29 2.6 Secondary Bending Moment Capacity of Flange, M cyf In the past, existing German table properties was not defined in detail how the value of M sec,o and C o obtained. The information about C o and M sec,o is limited. Therefore, in this study, M sec,o is defined as the secondary bending moment capacity of each flange about the minor axis of the section, represented by M cyf. The formula of M cyf is given as, M cyf = p y. Z yf Where, p y Z yf is the design strength of the flange is the elastic modulus of each flange in y axis. Thus, in this study secondary bending moment capacity of flange, M cyf = p y. Z yf.

30 2.7 Research Programme The overall research programme can be summarized into a flow chart as shown below: Determine objective and scope of study. Do literature review. Identify the formula of secondary bending moment, M sec and C o and static calculation on trapezoidal web beam. Analysis the lateral reaction at the flange of trapezoidal web beam which subjected to shear loading at web by using the LUSAS finite element software, with various geometric properties. Calculate and determine the secondary bending moment, M sec and C o due to the lateral reactions that obtained from finite element analysis. Compare the value of M sec and C o between finite element analysis and German table properties by plotting the graph. Derive the formula of C o. Compare the values of Co/Msec from formula, FEA and German table properties.

CHAPTER 3 ANALYSIS OF SECONDARY BENDING MOMENT USING FEM 3.1 LUSAS Finite Element Analysis To study the lateral reaction and secondary bending moment induced by the trapezoidal web profile, numerical study was carried out by using finite element analysis. LUSAS finite element software was adopted. In order to build up a LUSAS model, points and lines are defined to produce surfaces and has to be inter connected to give actual situation of the model. A finite element model which was made up of two cycles web panel was studied. The number of cycles in trapezoidal web was found to be of no effect to the result. The model consists of top and bottom flanges, trapezoidal web and right and left side stiffeners plates as shown in Figure 3.1.

32 Y X Z Figure 3.1: Trapezoidal web beam model with web depth 300mm. Each surface is formed form 4 nodal lines. Semiloof curved thin shell element (QSL8) which is a family of shell element in 3D dimension was chosen to represent the element type of model in this study. Thin shell is selected as the generic element type, Quadrilateral as the element shape with Quadratic as the interpolation order. Regular mesh was assigned by allowed transition pattern with 1 local x and y division. Surface geometric are assigned to the web with the thickness 2mm, however, flange and side plates with the thickness are depended to the section properties of the model. Isotopic material properties are assigned to the model with Young s modulus of 205x10 6 kn/m 2 and Poisson ratio of 0.3. The support at one of the side plates were restrained in translation x, y, z direction. The other side are pinned in translation x and z direction. Nodes connected between web and flanges are restrained in z direction as shown in Figure 3.2. The total concentrated load in vertical direction, 20kN is assigned to the points along the pinned side of the plate which allowing deformation in vertical direction (y direction). Figure 3.3 shows the loading arrangement assigned to the model in the analysis.

33 Y X Z Figure 3.2: Support condition assigned to the model with web depth 300mm Y X Z Figure 3.3: Vertical loading with total 20kN assigned to the points to the model with web depth 300mm. The steps of build up a LUSAS model with 300mm web depth were shown in Figure 3.4.

34 (a) Give the file name and title. The units knmtcs chosen. (b) Click geometry point coordinates

35 (c) Enter coordinates x, y, z. (d) The points as shown above.

36 (e) Select all points and click copy (f) Copy points in translation Y with 0.05m, number of copies 1.

37 Surface icon (g) Select four points and click on surface icon. Same step repeat for all the points. (h) Select the surfaces and copy in translation Y with 0.05m and 5 number of copies.

38 (i) Copy the surfaces in translation X with 0.5m. (j) The web with 2 cycles is shown above.

39 (k) Steps (c) (i) repeated for the flange at top, bottom and both sides. (l) Click Attributes Geometric Surface.

40 (m) Input the thickness of the flange and web and assigned. (n) Click Attributes Material Isotropic.

41 (n) Input the value of Young s modulus, Poisson s ratio and Mass density. (o) Click Atrributes Mesh Surface.

42 (p) For Generic element type, choose Thin shell. For Element shape, choose Quadrilateral. For Interpolation order, choose Quadratic. Then click Regular mesh and Allow transition pattern with 1 Local X and Y divisions. Assigned to the model. (q) Click Atrributes Support Strutural.

43 (r) Fixed the supports translation in X, Y, Z for the left side of the model and assigned. (r) Fixed the supports Translation in X and Z, and Free in Translation Y for the right side of the model and assigned.

44 (r) Free the supports Translation in X and Y, and Fixed in Translation Z for the connection between web and flange at top and bottom of the model and assigned. (s) The supports assigned to the model.

45 Click Atrributes Loading Strutural. (t) Input the value of Concentrated load in Y Dir and assigned to right side of the model. Run analysis icon (u) The supports condtion and loadings were assigned to the model. Finally click run analysis icon to get the result. Figure 3.4: Steps of build up a LUSAS model with web depth 300mm.

46 3.2 Preliminary Data Analysis The finite element analysis was carried out to determine the reaction at lateral direction at the top and bottom flanges. For the preliminary data analysis, a parametric study was carried out on 10 finite element models with the web depth ranging from 300mm to 750mm, flange width of 120mm and flange thickness of 10mm. Figure 3.5 show the labels of each node along the connection of the web to the flange at top and bottom, which were restrained in z direction for model with web depth 300mm. The lateral reactions for each node were given in Table 3.1. The summary of lateral reactions in each oblique part of the section is shown in Table 3.2. The lateral reactions of each section size, in each oblique part for the trapezoidal web are summarized into Table 3.3 and the phenomena of one section with 300mm depth web is shown in Figure 3.6.

47

48

49 Table 3.1: Lateral reactions, Fz of each node at top and bottom flange for model with web depth 300mm. Top Flange Bottom Flange Node Fz Node Fz 215 2.37 499-2.37 216 0.00539 500-0.00539 213 0.664 1-0.664 220-0.332 3 0.332 218-0.165 2 0.165 223-0.021 10 0.021 221-0.00665 9 0.00665 226-0.236 15 0.236 224-0.159 14 0.159 229-0.636 20 0.636 227-2.360 19 2.360 232-0.004 25 0.004 230-3.750 24 3.750 235-0.003 30 0.003 233-1.080 29 1.080 238-0.27 35 0.27 236-0.149 34 0.149 241-0.106 40 0.106 239-0.00924 39 0.00924 244 0.134 45-0.134 242 0.143 44-0.143 247 0.447 50-0.447 245 1.727 49-1.727 250 0.004 55-0.004 248 3.748 54-3.748 457 0.004 252-0.004 455 1.727 251-1.727 460 0.447 257-0.447 458 0.143 256-0.143 463 0.134 262-0.134 461-0.00922 261 0.00922 466-0.106 267 0.106 464-0.149 266 0.149 469-0.27 272 0.27 467-1.079 271 1.079

50 Table 3.1 (cont ) Top Flange Bottom Flange Node Fz Node Fz 472-0.003 277 0.003 470-3.752 276 3.752 475-0.004 282 0.004 473-2.357 281 2.357 478-0.636 287 0.636 476-0.159 286 0.159 481-0.236 292 0.236 479-0.0071 291 0.0071 484-0.0206 297 0.0206 482-0.161 296 0.161 487-0.325 302 0.325 485 0.646 301-0.646 490 0.0054 307-0.0054 488 2.48 306-2.48 Table 3.2: Summary of lateral reaction in each oblique part at top and bottom flange. Top Flange Bottom Flange Node Fz Node Fz 467-1.079 271 1.079 472-0.003 277 0.003 470-3.752 276 3.752 475-0.004 282 0.004 473-2.357 281 2.357 Total: -7.195 Total: 7.195 245 1.727 49-1.727 250 0.004 55-0.004 248 3.748 54-3.748 457 0.004 252-0.004 455 1.727 251-1.727 Total: 7.210 Total: -7.210 227-2.360 19 2.360 232-0.004 25 0.004 230-3.750 24 3.750 235-0.003 30 0.003 233-1.080 29 1.080 Total: -7.197 Total: 7.197

51 Table 3.3: Summary of lateral reaction, Q in z direction in each oblique part for each section size. Section Size Top Flange Bottom Flange Q avr 300 x 120 x 10-2.357 1.727-1.080 2.357-1.727 1.080-0.004 0.004-0.003 0.004-0.004 0.003-3.752 3.748-3.752 3.752-3.748 3.752-0.003 0.004-0.004 0.003-0.004 0.004-1.079 1.727-2.358 1.079-1.727 2.358-7.195 7.210-7.197 7.195-7.210 7.197 7.201 350 x 120 x 10-2.022 1.481-0.927 2.022-1.481 0.927-0.003 0.003-0.003 0.003-0.003 0.003-3.218 3.213-3.218 3.218-3.213 3.218-0.003 0.003-0.003 0.003-0.003 0.003-0.926 1.481-2.023 0.926-1.481 2.023-6.172 6.182-6.173 6.172-6.182 6.173 6.176 400 x 120 x 10-1.771 1.297-0.811 1.771-1.297 0.811-0.003 0.003-0.002 0.003-0.003 0.002-2.817 2.812-2.817 2.817-2.812 2.817-0.002 0.003-0.003 0.002-0.003 0.003-0.811 1.297-1.771 0.811-1.297 1.771-5.404 5.411-5.404 5.404-5.411 5.404 5.406 450 x 120 x 10-1.575 1.153-0.722 1.575-1.153 0.722-0.003 0.003-0.002 0.003-0.003 0.002-2.504 2.499-2.504 2.504-2.499 2.504-0.002 0.002-0.003 0.002-0.002 0.003-0.721 1.153-1.575 0.721-1.153 1.575-4.806 4.812-4.806 4.806-4.812 4.806 4.808 500 x 120 x 10-1.418 1.039-0.650 1.418-1.039 0.650-0.002 0.002-0.002 0.002-0.002 0.002-2.254 2.250-2.254 2.254-2.250 2.254-0.002 0.002-0.002 0.002-0.002 0.002-0.650 1.038-1.419 0.650-1.038 1.419-4.326 4.331-4.327 4.326-4.331 4.327 4.328

52 Table 3.3 (cont ) Size Section Top Flange Bottom Flange Q avr 550 x 120 x 10-1.290 0.944-0.591 1.290-0.944 0.591-0.002 0.002-0.002 0.002-0.002 0.002-2.050 2.045-2.050 2.050-2.045 2.050-0.002 0.002-0.002 0.002-0.002 0.002-0.591 0.944-1.290 0.591-0.944 1.290-3.934 3.938-3.935 3.934-3.938 3.935 3.935 600 x 120 x 10-1.183 0.866-0.542 1.183-0.866 0.542-0.002 0.002-0.002 0.002-0.002 0.002-1.879 1.875-1.879 1.879-1.875 1.879-0.002 0.002-0.002 0.002-0.002 0.002-0.542 0.866-1.183 0.542-0.866 1.183-3.607 3.610-3.607 3.607-3.610 3.607 3.608 650 x 120 x 10-1.092 0.800-0.500 1.092-0.800 0.500-0.002 0.002-0.001 0.002-0.002 0.001-1.735 1.730-1.735 1.735-1.730 1.735-0.001 0.002-0.002 0.001-0.002 0.002-0.500 0.800-1.093 0.500-0.800 1.093-3.330 3.333-3.331 3.330-3.333 3.331 3.331 700 x 120 x 10-1.015 0.743-0.464 1.015-0.743 0.464-0.002 0.002-0.001 0.002-0.002 0.001-1.611 1.607-1.611 1.611-1.607 1.611-0.001 0.002-0.002 0.001-0.002 0.002-0.464 0.743-1.015 0.464-0.743 1.015-3.093 3.095-3.093 3.093-3.095 3.093 3.094 750 x 120 x 10-0.947 0.693-0.434 0.947-0.693 0.434-0.002 0.001-0.001 0.002-0.001 0.001-1.504 1.500-1.504 1.504-1.500 1.504-0.001 0.001-0.002 0.001-0.001 0.002-0.433 0.693-0.947 0.433-0.693 0.947-2.887 2.889-2.887 2.887-2.889 2.887 2.888

53 7.197kN 7.210kN 7.195kN 7.195kN 7.210kN 7.197kN Figure 3.6: The total lateral forces at each node, Q at each oblique sub panel of web depth 300mm. The total lateral forces at the upper flange is equal to the total forces of lower flange but in opposite direction. At the oblique part of the web, the total forces at the top flange are balanced to the total forces at the oblique part of bottom flange. These opposite forces forms a couple, which is known as secondary bending moment, M yf due to the lever arm between them, a = 250mm for each couple along the section. M yf = Q x 250mm Q varies linearly with V. Therefore M yf varies linearly with V and giving M yf = V.C o. C o is a coefficient. C o is obtained by calibrating between the applied shear loading and the secondary bending moment resulted from the couples. M yf = V.C o C o = Q x 250 mm V

54 The values of C o from finite element analysis are calculated based on the equation above as shown in Table 3.4. The values are compared to C o values from German existing table capacities. Table 3.5 shows the ratio of C o /M cyf from calculation compared to C o /M sec,o ratio from the existing capacities table. The relations of C o and C o /M sec,o versus web depth are illustrated in Figure 3.7 and 3.8 respectively. Table 3.4: Values of C o from finite element analysis and comparison to existing table capacities. D B T t Q avr a V C o (mm) (mm) (mm) (mm) (mm) (kn) (mm) (kn) FEA Existing 300 120 10 2 7.201 250 20 90.01 98.60 350 120 10 2 6.176 250 20 77.20 89.00 400 120 10 2 5.406 250 20 67.58 80.30 450 120 10 2 4.808 250 20 60.10 73.00 500 120 10 2 4.328 250 20 54.10 66.90 550 120 10 2 3.935 250 20 49.19 61.10 600 120 10 2 3.608 250 20 45.10 56.80 650 120 10 2 3.310 250 20 41.38 52.60 700 120 10 2 3.094 250 20 38.68 49.60 750 120 10 2 2.888 250 20 36.10 46.50

55

56 120 100 80 60 Co 40 Co Vs Depth 11722x -0.8336 26977x -0.9997 Lusas Co German Co B=120mm T= 10mm 20 0 200 300 400 500 600 700 800 Depth, D (mm) Figure 3.7: Graph C o versus depth web,d in comparison between finite element analysis to the existing value. 3166.3x -0.9997 B=120mm T= 10mm 3252.4x -1.0439 Figure 3.8: Graph C o /M sec,o versus depth web, d in comparison between finite element analysis to the existing value.

57 3.3 Comparison Data of FEA and German Table Properties The results from finite element analysis show that the values of C o and ratio C o /M cyf decrease with the increase of the web depth. From the Table 3.4, values of C o from finite element analysis are lower than C o given in the existing German table properties. However, values of ratio C o /M cyf from the analysis are higher than the existing C o /M sec,o as shown in Figure 3.8. From both of the figure, the pattern of graphs plotted from finite element analysis is similar with the German existing table capacities. Hence, the derivation of secondary bending moment, M yf and secondary bending moment, C o from the finite element analysis give reasonable comparison with the existing table capacities. 3.4 Parametric Studies Analysis There were several sets of parametric studies carried out by using LUSAS Finite Element Analysis to determine the reaction at lateral direction at the top and bottom flanges in order to calculate C o and C o / M cyf. There are six set of parametric studies cover different ranges of web depth but with a fixed size of flange width and flange thickness. However, there are three sets of parametric studies contain different ranges of flange widths, flange thicknesses, web thicknesses with fixed size of web depth 750mm. The surface meshing, material properties, types of supports of the section are the same as mentioned above. The concentrated load in vertical direction, 0.25kN is assigned to each point along the loaded side which allowing deformation in vertical direction. The total lateral reactions, Q 1, Q 2 and Q 3 of each oblique sub panel in top flange and bottom flange is shown in Figure 3.9.