Tài liệu Phân tích đánh giá ổn định động của hệ thống điện khi có tác động của nguồn năng lượng điện gió

ĐẠI HỌC QUỐC GIA TP. HCM TRƯỜNG ĐẠI HỌC BÁCH KHOA -------------------- LỒ SÌU VẪY FRACTURE ANALYSIS IN THIN PLATE USING KIRCHHOFF-LOVE PLATE THEORY AND AN EXTENDED MESHFREE METHOD Chuyên ngành: Cơ kỹ thuật Mã số: 8520101 LUẬN VĂN THẠC SĨ TP. HỒ CHÍ MINH, tháng 7 năm 2021 CÔNG TRÌNH ĐƯỢC HOÀN THÀNH TẠI TRƯỜNG ĐẠI HỌC BÁCH KHOA –ĐHQG -HCM Cán bộ hướng dẫn khoa học : PGS. TS. Trương Tích Thiện Cán bộ chấm nhận xét 1: PGS. TS. Nguyễn Hoài Sơn Cán bộ chấm nhận xét 2: TS. Nguyễn Thanh Nhã Luận văn thạc sĩ được bảo vệ tại Trường Đại học Bách Khoa, ĐHQG Tp. HCM ngày 15 tháng 07 năm 2021 Thành phần Hội đồng đánh giá luận văn thạc sĩ gồm: 1. PGS. TS. Vũ Công Hòa 2. PGS. TS. Trương Tích Thiện 3. PGS. TS. Nguyễn Hoài Sơn 4. TS. Nguyễn Thanh Nhã 5. TS. Trương Quang Tri Xác nhận của Chủ tịch Hội đồng đánh giá LV và Trưởng Khoa quản lý chuyên ngành sau khi luận văn đã được sửa chữa (nếu có). CHỦ TỊCH HỘI ĐỒNG TRƯỞNG KHOA KHOA HỌC ỨNG DỤNG ĐẠI HỌC QUỐC GIA TP.HCM TRƯỜNG ĐẠI HỌC BÁCH KHOA CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM Độc lập - Tự do - Hạnh phúc ________________________________ ____________________________ NHIỆM VỤ LUẬN VĂN THẠC SĨ Họ tên học viên: Lồ Sìu Vẫy MSHV: 1970507 Ngày, tháng, năm sinh: 16/11/1997 Nơi sinh: Đồng Nai Chuyên ngành: Cơ kỹ thuật Mã số: 8520101 I. TÊN ĐỀ TÀI: Phân tích sự rạn nứt trong tấm mỏng sử dụng lý thuyết tấm Kirrchhoff-Love và phương pháp không lưới mở rộng Fracture analysis in thin plate using Kirchhoff-Love plate theory and an extended meshfree method NHIỆM VỤ VÀ NỘI DUNG: 1. Nghiên cứu lý thuyết tấm Kirchhoff-Love 2. Nghiên cứu phương pháp không lưới RPIM 3. Phát triển công thức XRPIM dành cho tấm Kirchhoff-Love bị nứt 4. Phát triển một chương trình giải các bài toán tấm Kirchhoff-Love bằng phương pháp RPIM 5. Phát triển chương trình để phân tích ứng xử dao động tự do của tấm Kirchhoff-Love bị nứt II. NGÀY GIAO NHIỆM VỤ: 22/02/2021 III. NGÀY HOÀN THÀNH NHIỆM VỤ: 05/12/2021 IV.CÁN BỘ HƯỚNG DẪN: PGS. TS. Trương Tích Thiện Tp. HCM, ngày 30 tháng 6 năm 2021 CÁN BỘ HƯỚNG DẪN CHỦ NHIỆM BỘ MÔN ĐÀO TẠO (Họ tên và chữ ký) (Họ tên và chữ ký) TRƯỞNG KHOA KHOA HỌC ỨNG DỤNG (Họ tên và chữ ký) Acknowledgement I would like to express my endless thanks and gratefulness to my supervisor Assoc. Prof. Dr. Thien Tich Truong, Dr. Nha Thanh Nguyen and Dr. Minh Ngoc Nguyen for their supports and advices during the process of completion of my thesis. Without their instructions, the thesis would have been impossible to be done effectively. I really appreciate the lecturers of Department of Engineering Mechanics for their comments and helps while I am doing this thesis, which leads me to the right direction. I also acknowledge the support of time and facilities from Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for this thesis. Last but not least, a special thanks to my parents for their love, care and have most assistances and motivation me for the whole of my life. i Tóm tắt luận văn Đối với các bài toán tấm chịu uốn, sử dụng lý thuyết tấm để mô tả cấu trúc tấm mỏng sẽ ít tốn kém hơn so với sử dụng mô hình 3D. Trong các lý thuyết tấm thì lý thuyết Kirchhoff-Love rất thích hợp để phân tích cấu trúc tấm mỏng. Nếu bỏ qua các bậc tự do trong mặt phẳng của tấm thì mỗi nút chỉ có một bậc tự do - đó là độ võng. Vì lý do đó, các thành phần của trường chuyển vị chỉ tính theo độ võng. Tuy nhiên, phương pháp phần tử hữu hạn cổ điển (FEM) cần các phép biến đổi toán học phức tạp để xây dựng một phần tử mới thỏa mãn các yêu cầu của lý thuyết Kirchhoff-Love. Vì vậy, phương pháp nội suy điểm hướng kính (RPIM) được sử dụng để mô phỏng tấm mỏng Kirchhoff-Love trong luận văn này. Bên cạnh đó, việc phân tích các kết cấu bị nứt rất quan trọng vì nó liên quan đến tuổi thọ của kết cấu. Do đó, luận văn này sử dụng phương pháp nội suy điểm hướng kính mở rộng (XRPIM) để khảo sát dao động tự do của tấm Kirchhoff-Love bị nứt. XRPIM được phát triển dựa trên RPIM nên yêu cầu về đạo hàm cấp hai trong lý thuyết Kirchhoff-Love được xử lý một cách dễ dàng. Kết quả mô phỏng số từ nghiên cứu này được so sánh với các kết quả của các tác giả khác đã công bố để kiểm chứng tính chính xác của phương pháp. ii Abstract For the plate bending problems, using a plate theory for modelling thin plate structure is less computational cost than modelling it in 3D. The Kirchhoff-Love plate theory is appropriate for analysing thin plate structures. If the membrane deformation is ignored in the KirchhoffLove plate, each node has only one degree of freedom – the deflection. For that reason, the components of the displacement field are calculated only in terms of deflection. The classical finite element method (FEM), however, needs complex mathematical transformations to formulate a new element that satisfies the Kirchhoff-Love theory. For this reason, the radial point interpolation method (RPIM) is used for modelling thin Kirchhoff-Love plate in this thesis. Besides, the analysis of cracked structures is important because it is related to the lifetime of the structures. Therefore, this thesis employs the extended radial point interpolation method (XRPIM) to investigate the free vibration of the cracked Kirchhoff-Love plate. The XRPIM is based on RPIM so the requirement for calculating the second-order derivative in the Kirchhoff-Love theory is easily done. The numerical results from this study are compared with other published results to verify the accuracy of the method. iii Declarations I hereby declare that this master thesis represents my own work which has been done after registration for the degree of Master in Engineering Mechanics at Ho Chi Minh city University of Technology, VNU – HCM, Viet Nam and has not been previously included in a thesis or dissertation submitted to this or any other institution for a degree, diploma or other qualifications. Author iv Contents List of Figures vii List of Tables ix List of Abbreviations and Nomenclatures x 1 Introduction 1 1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Scope of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Author’s contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Methodology 4 2.1 The Kirchhoff-Love plate theory . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Equilibrium equation . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Constitutive equation . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.3 Governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.4 Finite element approximation . . . . . . . . . . . . . . . . . . . . 10 The Radial Point Interpolation Method . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Brief introduction to the RPIM . . . . . . . . . . . . . . . . . . . . 14 2.2.2 RPIM shape functions construction . . . . . . . . . . . . . . . . . 14 The extended RPIM for the cracked Kirchhoff-Love plate . . . . . . . . . . 19 2.2 2.3 3 Implementation 23 3.1 Compute stiffness matrix and mass matrix . . . . . . . . . . . . . . . . . . 23 3.2 Compute strain computing matrix . . . . . . . . . . . . . . . . . . . . . . 25 v 4 Numerical Results 27 4.1 Square plate under uniform pressure . . . . . . . . . . . . . . . . . . . . . 27 4.2 Plate with a central crack . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2.1 Simply supported square plate . . . . . . . . . . . . . . . . . . . . 29 4.2.2 Simply supported rectangular plate . . . . . . . . . . . . . . . . . 31 4.2.3 Clamped circular plate . . . . . . . . . . . . . . . . . . . . . . . . 34 Plate containing a side crack . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3.1 Simply supported square plate . . . . . . . . . . . . . . . . . . . . 38 4.3.2 Simply supported rectangular plate . . . . . . . . . . . . . . . . . 38 4.3.3 Clamped annular plate . . . . . . . . . . . . . . . . . . . . . . . . 42 Square plate with an oblique crack . . . . . . . . . . . . . . . . . . . . . . 42 4.4.1 Central crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.4.2 Side crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 4.4 5 Conclusion and outlook 51 5.1 Conlusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Bibliography 54 vi List of Figures 2.1 Internal and external forces on the plate . . . . . . . . . . . . . . . . . . . 5 2.2 7 11 2.4 Plate before and after deformation . . . . . . . . . . . . . . . . . . . . . . ∂w ∂w A rectangular element with three degrees of freedom w, , per node ∂x1 ∂x2 Discrete nodes (gray dots) and support domains . . . . . . . . . . . . . . . 2.5 Pascal triangle of monomials for 2D case . . . . . . . . . . . . . . . . . . 16 2.6 Split nodes and tip nodes around the crack curve . . . . . . . . . . . . . . . 20 2.7 Illustration of tangential and normal direction for a crack . . . . . . . . . . 21 2.8 Global coordinate system and local coordinate systems . . . . . . . . . . . 21 3.1 Algorithm for computing K and M matrices. . . . . . . . . . . . . . . . . 24 3.2 Algorithm for computing B and N matrices. . . . . . . . . . . . . . . . . . 26 4.1 Geometry of the square plate and the coordinate system. . . . . . . . . . . 28 4.2 Convergence of the result. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 Deflection of the simply supported square plate. . . . . . . . . . . . . . . . 30 4.4 Deflection at the center line y = 0. . . . . . . . . . . . . . . . . . . . . . . 30 4.5 Geometries of a square plate and a rectangular plate containing a central crack. 31 4.6 Mode shapes of five lowest modes of a square plate containing a central crack. 33 4.7 Mode shapes of five lowest modes of a rectangular plate containing a central 2.3 15 crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.8 A circular plate containing a central crack. . . . . . . . . . . . . . . . . . . 35 4.9 Discrete model of the circular plate. . . . . . . . . . . . . . . . . . . . . . 36 4.10 Mode shapes of five lowest modes of a circular plate containing a central crack. 37 4.11 Geometry of square plate and rectangular plate with a side crack. . . . . . . 38 4.12 Mode shapes of five lowest modes of a square plate with a side crack. . . . 40 4.13 Mode shapes of five lowest modes of a rectangular plate with a side crack. . 40 4.14 A annular plate containing two symmetric side cracks. . . . . . . . . . . . 42 vii 4.15 Mode shapes of five lowest modes of a rectangular plate containing a central crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.16 Geometry of square plate with a central oblique crack and a side oblique crack. 44 4.17 Mode shapes of five lowest modes of a square plate containing a central oblique crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.18 Node distribution. Left: Not aligned with the crack, Right: Aligned with the crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.19 Mode shapes of five lowest mode of a square plate containing a side oblique crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 50 List of Tables 2.1 Some radial basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.1 Convergence of the non-dimensional deflection of different methods . . . . 28 4.2 Non-dimensional frequency parameter ω̄ of a simply supported square plate containing a central crack. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Non-dimensional frequency parameter ω̄ of a simply supported rectangular plate containing a central crack. . . . . . . . . . . . . . . . . . . . . . . . 4.4 43 Non-dimensional frequency parameter ω̄ of a simply supported square plate with a central oblique crack. . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 41 Non-dimensional frequency parameter ω̄ of the clamped annular plate with two side cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 39 Non-dimensional frequency parameter ω̄ of a simply supported rectangular plate with a side crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 37 Non-dimensional frequency parameter ω̄ of a simply supported square plate with a side crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 34 Non-dimensional frequency parameter ω̄ of the clamped circular plate with a central crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 32 45 Non-dimensional frequency parameter ω̄ of a simply supported square plate with a central oblique crack. . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.10 Non-dimensional frequency parameter ω̄ of a simply supported square plate with a central oblique crack. . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.11 Non-dimensional frequency parameter ω̄ of a simply supported square plate with a central crack. The c/a ratio is equal 0.6 and α = 45o . . . . . . . . . 48 4.12 Non-dimensional frequency parameter ω̄ of a simply supported square plate with a side oblique crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.13 Non-dimensional frequency parameter ω̄ of a simply supported square plate with a side oblique crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 50 List of Abbreviations and Nomenclatures Abbreviation 2D Two dimensional 3D Three dimensional CS-DSG3 Cell-based smoothed discrete shear gap method using triangular element DDM Domain Decomposition Method DOF Degree Of Freedom FEM Finite Element Method FSDT First-order Shear Deformation Theory HSDT Higher-order Shear Deformation Theory MQ Multiquadrics PIM Point Interpolation Method RBF Radial Basis Function RPIM Radial Point Interpolation Method SBTP4 Strain-based triangular plate with four nodes TPS Thin Plate Spline XCS-DSG3 Extended cell-based smoothed discrete shear gap method using triangular element XFEM Extended Finite Element Method XRPIM Extended Radial Point Interpolation Method x Nomenclature 1 , 2 , γ12 Strain components p Pseudo-strain κ1 , κ2 , χ Plate curvature components ∇ Laplacian operator ν Poisson ratio ω Free vibation frequency ∂x1 , ∂x2 , ∂x3 Partial derivative in x1 , x2 , x3 directions ρ Mass density σ1 , σ2 , σ12 Stress components σp Pseudo-stress B Strain computing matrix D Material matrix fb Body force J Jacobian matrix K Stiffness matrix M Mass matrix m Inertia matrix N Shape function W The set containing all the nodes Ws The set containing the nodes in the support domain which is cut by the crack Wt The set containing the crack tip θ Shape parameter D Bending stiffness xi E Young’s modulus f (x) Sign distance funtion G(x) Tip enrichment funtion h Plate’s thickness H(x) Heaviside funtion Mij Moment resultant mij Moment per unit length n Number of nodes inside the support domain p(x) Polynomial basis funtion Px3 Lateral force Qi Shear force resultant qi Shear force per unit length R(x) Radial basis funtion ui Displacement field w Transverse deflection x1 , x2 , x3 Cartesian coordinate xii Chapter 1 Introduction 1.1 State of the Art Plate and shell structures are very common in practice, its application can be mentioned as: ship’s hull, fluid container, aircraft fuselage and so on. Therefore, investigating the mechanical behavior of thin-walled structures is really important. In addition to this, modelling plate and shell structure by using an appropriate plate theory can be more efficient than considering it as a 3D model. There are many types of plate theories can be employed to reduce a 3D model to a 2D plate, such as the Kirchhoff-Love plate theory [1, 2, 3, 4, 5, 6, 7, 8], the Reissner-Mindlin plate theory (also known as the First-order Shear Deformation Theory) [9, 10, 11, 12, 13, 14, 15, 16, 17, 18], the Higher-order Shear Deformation Theory [19, 20, 21, 22, 23]. In the scope of this thesis, only thin plate structures are examined. Hence, the Kirchhoff-Love plate theory which is widely used in the analysis of thin plate structure, is appropriate for this study. Besides, fracture analysis is also an important task because it is related to the lifetime of the structures. However, the number of researches on cracked plate is still limited, especially, thin plate using Kirchhoff-Love theory. Therefore, more investigations on the modelling of cracked thin plates are necessary. In fracture analysis, a powerful method to model crack discontinuity without remeshing is the eXtened Finite Element Method (XFEM) proposed by [24] and has been widely used [25, 26, 27, 28, 29]. In XFEM formulation, the discontinuity (jump) in displacement fields and the singularity in the stress fields are described by using enrichment functions. XFEM has been developed to examined the behavior of the thick cracked plates using the Reissner-Mindlin theory [30, 31, 32] and thin cracked plates using the Kirchhoff-Love theory [33, 34, 35]. However, in the finite element analysis of KirchhoffLove plate, it is necessary to construct higher-order shape function for the requirement of 1 the second-order derivatives in the Kirchhoff-Love formulation [36]. Therefore, it brings computational difficulties which lead to the need for an alternative numerical method, take the XIGA [37, 38] for example. In this study, the alternative numerical method employed is the Meshfree method - the Radial Point Interpolation Method (RPIM) [39, 40, 41]. The shape function in the RPIM is formulated from the radial basis and the polynomial basis so that the second-order derivative of the shape function is easily obtained. Hence, RPIM is easier to model the Kirchhoff-Love plate than FEM. Similar to the formulation of XFEM, Nguyen et al. introduced the extended RPIM by combining RPIM and enrichment functions [42, 43, 44] to investigate the fracture mechanics. They used this method for 2D fracture analysis. The XRPIM has also been employed to examined the cracked Reissner-Mindlin plate [45]. Nevertheless, the number of report on fracture analysis in Kirchhoff-Love plate by using XRPIM is still limited. This thesis investigates the fracture behavior of the cracked Kirchhoff-Love plate using the XRPIM. In the scope of this study, only the free vibration behavior of cracked plate is investigated. This thesis analyzes the free vibration behavior of thin plates with throughthickness crack. The method used for computing is the XRPIM and the Kirchhoff-Love plate theory is used for modelling thin plate behavior. The validity of the proposed method is examined through numerous numerical examples, demonstrating the accuracy of the approach. 1.2 Scope of study In this study, the author concentrates on the following aspects: • The Kirchhoff-Love plate theory: only thin plate structures are consider in this thesis and the Kirchhoff-Love theory is employed for modelling thin plate. • The eXtended Radial Point Interpolation Method: the Radial Point Interpolation Method is used as the main numerical method in this thesis, the XRPIM is based on RPIM and used for fracture analysis. • The free vibration behavior of the cracked structure: the free vibration frequencies are considered, while the evaluation of the stress intensity factors is not mentioned in this study. Other issues not mentioned above are beyond the scope of this study and will not be discussed in this thesis. 2 1.3 Research objectives The target of this research is to examine the free vibration behavior of the cracked KirchhoffLove plate using the XRPIM. To achieve this goal, the following tasks must be fulfilled: • Develop a RPIM-based program to analyze behavior of Kirchhoff-Love plate • Construct XRPIM formulation for the Kirchhoff-Love plate • Develop a program to analyze the behavior of the cracked plate The achievement of this thesis will be the premise for further study in related topics. 1.4 Author’s contributions The scientific aspects that this thesis contributes to: • Formulating the Kirchhoff-Love theory in RPIM. • The analysis of cracked Kirchhoff-Love plate using XRPIM. • A MATLAB program for the analysis. 1.5 Thesis outline The structure of this thesis is following this outline. Chapter 2 introduces the related theories employed in this thesis. First is the Kirchhoff-Love theory for thin plate structure. Then, a brief summary of the Radial Point Interpolation Method is provided. In the last of this chapter, the eXtended Radial Point Interpolation Method is presented which provides mathematical formulation of XRPIM for thin-cracked plates in Kirchhoff-Love theory. The implementation of the XRPIM is shown in Chapter 3. Chapter 4 brings various numerical examples, demonstrating the accuracy of the approach. Finally, some concluding remarks and outlooks are given in Chapter 5. 3 Chapter 2 Methodology 2.1 The Kirchhoff-Love plate theory This section is written based on Publication B. A three-dimensional structure is considered to be a plate structure when it has one dimension much smaller than the other two. In other words, a plate is a flat structure that has a small thickness compared to the in-plane dimensions. In practice, analyzing this structure by using a 3D model is not absolutely necessary and the problem can be reduced to a 2D analysis. However, when considering it as a 2D analysis, an appropriate plate theory is required so that the 3D behavior of the structure is reconstructed. For thin plate structure, the Kirchhoff-Love plate theory is widely used in the analysis. Besides the Kirchhoff-Love theory, there are many other plate theories, for instance, the First-order Shear Deformation Theory (FSDT), the Higher-order Shear Deformation Theory (HSDT) and so on. Each plate theory has its own assumptions for the purpose of simplifying the computational work while preserving the accuracy of the solutions. Particularly, the Kirchhoff-Love plate is based on the assumption that the transverse shear deformation of the plate is neglected. The presumption above can be explicitly written in two separated assumptions [1]: 1. Kinematic assumption: The cross-section remains straight and perpendicular to the middle surface of the plate. 2. Static assumption: The transverse normal stresses are negligible. In this section, the formulation of the Kirchhoff-Love plate is introduced through the equilibrium equations, the constitutive equations and the governing equations. The construction of finite element analysis for the Kirchhoff-Love plate element is performed in the latter of this section as well. 4 2.1.1 Equilibrium equation Giving consideration to a piece of cut out of the plate as shown in Figure 2.1. This differential element has the thickness h in x3 direction and the lengths of dx1 and dx2 in x1 and x2 directions. On the top surface, the plate is subjected to lateral loads Px3 only. Also shown in the figure, Qi and Mij (i, j = 1, 2) are the shear force resultant and the moment resultant, respectively. All the internal and external forces acting on the plate must satisfies the following equilibrium equations X Mx1 = 0, X Mx2 = 0 and X Px3 = 0 (2.1) The meaning of the first two equations is that the summation of the moments around x1 and x2 axis must be zeros. While the last equation means that the sum of all lateral forces in the x3 direction must equal zero. Figure 2.1: Internal and external forces on the plate Taking all the moments into the first equation of Eq (2.1), this gives the following expression     ∂M1 ∂M21 dx1 − M1 + M21 + dx2 M1 + ∂x1 ∂x2   dx1 ∂Q1 dx1 − Q1 − Q1 + dx1 = 0 (2.2) 2 ∂x1 2 To ensure the consistency of using resultant terms in this thesis (see Section 2.1.3), the moment is turned into moment per unit length, i.e. m1 = M1 /dx2 , and the transverse shear 5