[...]... facts that element 1 is not connected to node 3 and element 2 is not connected to node 1) :  (1)  f 1  U1 k1 −k1 0   U2 = f (1) −k1 k1 0 (2 .11 )  2    0 0 0 0 0    0  0 0 0 0   (2) 0 k2 −k2 U2 = f 2 (2 .12 )    (2)  0 −k2 k2 U3 f 3 The addition of Equations 2 .11 and 2 .12 yields k1 −k1 0 −k1 k1 + k2 −k2 0 −k2 k2 U1 U2 U3 =      f (1) 1 f (1) + f (2) 2 2 f (2) 3      (2 .13 ) Next,... Bar Elements CHAPTER 2 Text © The McGraw−Hill Companies, 2004 Stiffness Matrices, Spring and Bar Elements (1) u1 u (1) 2 (2) u1 1 (1) f1 1 u(2) 2 2 2 k1 f (1) 2 f (2) 2 2 3 k2 (a) f (2) 3 (b) (1) f1 F1 f (1) ϩ f (2) 2 2 1 2 f (2) 3 F2 (d) (c) F3 (e) Figure 2.3 Free-body diagrams of elements and nodes for the two -element system of Figure 2.2 forces are F1, F2, and F3 Assuming the system of two spring elements... Figure 1. 8 (a) Deployable truss module showing details of folding joints (b) A sample vibration-mode shape of a five-module truss as obtained via finite element analysis (Courtesy: AIAA) 14 Hutton: Fundamentals of Finite Element Analysis 1 Basic Concepts of the Finite Element Method Text © The McGraw−Hill Companies, 2004 1. 5 Examples of Finite Element Analysis 0.0 019 7Љ Z X 0.488Љ 0.25Љ Figure 1. 9 Finite element. .. total of 41 elements is shown (b) Refined finite element mesh showing reduction of the area not included in the model A total of 19 2 elements is shown Hutton: Fundamentals of Finite Element Analysis 1 Basic Concepts of the Finite Element Method Text © The McGraw−Hill Companies, 2004 1. 2 How Does the Finite Element Method Work? of the cylinder (Figure 1. 3b) The uniform bar is a link or bar finite element. .. terms of the range of weight loading anticipated, so as to properly size the component and select the material Figure 1. 10b shows a prototype of the completed hand design 15 Hutton: Fundamentals of Finite Element Analysis 16 1 Basic Concepts of the Finite Element Method CHAPTER 1 Text © The McGraw−Hill Companies, 2004 Basic Concepts of the Finite Element Method (a) (b) Figure 1. 10 (a) A finite element. .. Element Analysis of Fluid Flows.” Proceedings of the Second Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Kilborn, Ohio, October 19 68 13 Wilson, E L., and R E Nickell “Application of the Finite Element Method to Heat Conduction Analysis. ” Nuclear Engineering Design 4 (19 66) 17 Hutton: Fundamentals of Finite Element Analysis 18 1 Basic Concepts of the Finite Element. .. Hutton: Fundamentals of Finite Element Analysis 6 1 Basic Concepts of the Finite Element Method CHAPTER 1 Text © The McGraw−Hill Companies, 2004 Basic Concepts of the Finite Element Method Exact Four elements ␦ x (L) ␦(x ϭ L) Exact 1 2 3 Number of elements (a) 4 0.25 0.5 0.75 1. 0 x L (b) Figure 1. 4 (a) Displacement at x ϭ L for tapered cylinder in tension of Figure 1. 3 (b) Comparison of the exact solution... Application of Finite Element Methods in Civil Engineering, Vanderbilt University, Nashville, 19 69 10 Wilson, E L “Structural Analysis of Axisymmetric Solids.” Journal of the American Institute of Aeronautics and Astronautics 3, (19 65) 11 Melosh, R J “Structural Analysis of Solids.” Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, August 19 63 12 Martin, H C Finite Element. .. characteristic of the finite element method The formulation of the finite element method for a given problem is such that the primary field variable is continuous from element to element but Hutton: Fundamentals of Finite Element Analysis 1 Basic Concepts of the Finite Element Method Text © The McGraw−Hill Companies, 2004 1. 2 How Does the Finite Element Method Work? 4.5 4.0 Exact Two elements Four elements... 2.8, we obtain k1 −k 1 −k 1 k1 U1 U2 = f k2 −k 2 −k 2 k2 U2 U3 = f f (1) 1 (1) 2 (2 .10 a) (2) 2 (2) 3 (2 .10 b) and Here, we use the notation f node i ( j) i f to represent the force exerted on element j at Hutton: Fundamentals of Finite Element Analysis 2 Stiffness Matrices, Spring and Bar Elements Text © The McGraw−Hill Companies, 2004 2.2 Linear Spring as a Finite Element Equation 2 .10 is the equilibrium . exact CHAPTER 1 Hutton: Fundamentals of Finite Element Analysis 1. Basic Concepts of the Finite Element Method Text © The McGraw−Hill Companies, 2004 2 CHAPTER 1 Basic Concepts of the Finite Element. showing reduction of the area not included in the model. A total of 19 2 elements is shown. Hutton: Fundamentals of Finite Element Analysis 1. Basic Concepts of the Finite Element Method Text. the domain. (c) Additional elements showing a partial finite element mesh of the domain. Hutton: Fundamentals of Finite Element Analysis 1. Basic Concepts of the Finite Element Method Text © The