INTRODUCTION What is finite element analysis, FEM? A Brief history of FEM What is FEM used for? 1D Rod Elements, 2D Trusses
FINITE ELEMENT METHOD –WHAT IS IT? The Finite Element Method (FEM) is a numerical method of solving systems of partial differential equations (PDEs) It reduces a PDE system to a system of algebraic equations that can be solved using traditional linear algebra techniques. In simple terms, FEM is a method for dividing up a very complicated problem into small elements that can be solved in relation to each other.
Overview of the Finite Element Method( S ) ⇔(W ) ≈ ( G ) ⇔( M )Strong Weak Galerkin Matrixform form approx. form
1. Lord John William Strutt Rayleigh (late 1800s), developed a method for predicting the first natural frequency of simple structures. It assumed a deformed shape for a structure and then quantified this shape by minimizing the distributed energy in the structure.2. Ritz then exp Walter ended this into a method, now known as the Rayleigh-Ritz method, for predicting the stress and displacement behavior of structures.
3. Dr. Ray Clough coined the term “finite element” in 1960. The 1960s saw the true beginning of commercial FEA as digital computers replaced analog ones with the capability of thousands of operations per second.4. In the 1950s, a team form Boeing demonstrated that complex surfaces could be analyzed with a matrix of triangular shapes.5. In 1943, Richard Courant proposed breaking a continuous system into triangular segments. (The unveiling of ENIAC at the University of Pennsylvania.)6. In the early 1960s, the MacNeal-Schwendle Corporation (MSC) develop a general purpose FEA code. This original code had a limit of 68,000 degrees of freedom. When the NASA contract was complete, MSC continued development of its own version called MSC/NASTRAN, while the original NASTRAN become available to the public and formed the basis of dozens of the FEA packages available today. Around the time 6 MSC/NASTRAN was released, ANSYS, MARC, and SAP were introduced.
7. By the 1970s, Computer-aided design, or CAD, was introduced later in thedecade.8. standards such as IGES and DXF. Permitted limited geometry transfer between the systems.9. In the 1980s,CAD progressed from a 2D drafting tool to a 3D surfacing tool, and then to a 3D sIn the 1980s, the use of FEA and CAD on the same workstation with developing geometry olid modeling system. Design engineers began to seriously consider incorporating FEA into the general product design process.10. As the 1990s draw to a place, the PC platform has become a major force in high end analysis. The technology has become to accessible that it is actually being “hidden” inside CAD packages. 7
BASIC CONCEPTS Loads f T Pi Equilibrium ~ σ ji , j + fi = 0 Boundary conditions
DEVELOPMENT OF THEORY Rayleigh-Ritz Method Total potential energy equation Galerkin’s Method
1D ROD ELEMENTS To understand and solve 2D and 3D problems we must understand basic of 1D problems. Analysis of 1D rod elements can be done using Rayleigh-Ritz and Galerkin’s method. To solve FEA problems same are modified in the Potential-Energy approach and Galerkin’s approach
1D ROD ELEMENTS Loading consists of three types : body force f , traction force T, point load Pi Body force: distributed force , acting on every elemental volume of body i.e. self weight of body. Traction force: distributed force , acting on surface of body i.e. frictional resistance, viscous drag and surface shear Point load: a force acting on any single point of element
1D ROD ELEMENTS Element -1 Element-2 1 T e Element strain energy U e = q [k ]q 2 Element stiffness matrix E e Ae 1 − 1 [k ] = e − 1 1 le Load vectors Element body load vector Element traction-force vector e Ae l e f 1 f = 2 1 e Tl e 1 T = 2 1
2D TRUSSle = ( x2 − x1 ) 2 + ( y 2 − y1 ) 2 x 2 − x1l = cos θ = le y 2 − y1m = sin θ = le
3D TRUSS STIFFNESS MATRIX 3D Transformation Matrix Direction Cosines l m n 0 0 0 [ L] = 0 0 0 l m n le = ( x 2 − x1 ) 2 + ( y 2 − y1 ) 2 + ( z 2 − z1 ) 2 x 2 − x1 y 2 − y1 z 2 − z1 l = cos θ = m = cos φ = n = cos ϕ = le le le
3D TRUSS STIFFNESS MATRIX 3D Stiffness Matrix l2 lm ln − l 2 − lm − ln lm m2 mn − lm − m 2 − mn E e Ae ln mn n2 − ln − mn − n 2 [k e ] = l e − l 2 − lm − ln l 2 lm ln − lm − m 2 − mn lm m2 mn − ln − mn − n 2 2 ln mn n
EXAMPLE 1D ROD ELEMENTS Example 1 Problem statement: (Problem 3.1 from Chandrupatla and Belegunda’s book) Consider the bar in Fig.1, determine the following by hand calculation: 1) Displacement at point P 2) Strain and stress 3) Element stiffness matrix 4) strain energy in element Given: E = 30 ×106 psi q1 = 0.02in Ae = 1.2 in 2 q2 = 0.025in
Solution:1) Displacement (q) at point PWe haveNow linear shape functions N1( ) and N2( ) are given by And