Lect12
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The document discusses geometric isotropy or invariance in finite element modeling, which means the displacement shapes of an element do not change with a change in the local coordinate system. This can be achieved in 2D modeling by selecting terms on either side of the Pascal's triangle that correspond to the same order. In 3D modeling, geometric invariance requires selecting terms based on the 3D Pascal's triangle.
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Lect11
The document discusses three criteria for the convergence of displacement/shape functions in the finite element method:
1) Displacement functions must be continuous and compatible within elements, with no discontinuities between elements.
2) Displacement functions must be capable of representing constant strain states within elements.
3) Displacement functions must be capable of representing rigid body displacements, with elements experiencing no strain and zero nodal forces under rigid body motion.
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This document provides an overview of basic equations for the theory of plates and shells. It discusses the state of stress and strain at a point, including defining the six independent stress and strain components. It presents the relationships between strain and displacement, and discusses the equilibrium equations relating stress and body forces. Finally, it provides the equations for both Cartesian and cylindrical coordinate systems. The key concepts covered are the fundamental equations that form the basis of plate and shell theory.
Fem class notes
This document contains lecture notes on the finite element method from Dr. Atteshamuddin S. Sayyad. It covers the fundamentals of elasticity theory including assumptions, basic terms like stresses and strains, equilibrium equations, and strain-displacement relationships. The notes provide definitions and equations for normal and shear stresses and strains, the state of stress and strain at a point, and developing the governing equations of equilibrium in three dimensions from an infinitesimal element.
Lect21
1. The document derives the shape functions for a three-noded constant strain triangular (CST) element.
2. It presents the displacement function in terms of nodal displacements using the element connectivity matrix.
3. The shape functions are obtained by relating the displacement function to the nodal displacement terms, resulting in a set of three linear shape functions that describe the displacement across the element in terms of the displacements at its nodes.
Lect20
1) The document derives the shape functions for a two-noded bar element in finite element analysis.
2) It expresses the displacement function in terms of the nodal displacements using the coordinates of the two nodes.
3) The shape functions are obtained by equating the displacement function to the product of the shape function matrix and the nodal displacement vector.
Lect19
This document discusses shape functions in finite element analysis. It explains that shape functions approximate the variation of displacement within an element since the true variation is unknown. Shape functions, also called interpolation functions, are used to replace difficult analytical solutions with easier mathematical functions. Important properties of shape functions are that the magnitude at each node is unity, the number of functions equals the number of nodes, and the sum of functions is always unity. Common methods for deriving shape functions include using polynomials in Cartesian and natural coordinate systems and Lagrange interpolation in natural coordinates.
Lect18
This document introduces 3D elements that can model structural components in three dimensions. It discusses two families of 3D elements: tetrahedrons generated from triangular elements, and parallelepipeds generated from rectangular elements. Common 3D solid elements for finite element analysis include the 4-noded tetrahedron and 8-noded hexahedron/brick. The tetrahedron has 12 degrees of freedom while the hexahedron has 24 degrees of freedom. Displacement functions are provided for both elements. 3D elements can be used for fully 3D analysis and modeling axisymmetric solids.
Lect17
The document discusses natural coordinates, also called area coordinates, for a three-node triangular element in finite element analysis. It defines the natural coordinates (L1, L2, L3) of any point P within the triangle in terms of the Cartesian coordinates of the three nodes and the fractions of the total triangle area (A1, A2, A3) associated with each node. The natural coordinates represent the point P shifted to each node and satisfy the properties L1 + L2 + L3 = 1 and L1, L2, L3 ≥ 0.
Lect16
The document discusses the natural coordinates of a 1D bar element in finite element analysis. It defines the natural coordinate ξ as the ratio of the distance from any point P to the origin, over the maximum distance from the origin to node 2. This yields expressions for ξ at each node. At node 1, ξ = 0, and at node 2, ξ = 1. Therefore, the natural coordinates of a 1D bar element are ξ, which ranges from 0 at node 1 to 1 at node 2.
Lect15
1) The document discusses natural coordinates for a 1D bar element in finite element analysis.
2) It defines the natural coordinates (L1, L2) of any point P on the bar element in terms of the Cartesian coordinates (x1, x2) of the two nodes and the length L of the element.
3) The natural coordinates at node 1 are derived to be (1, 0) and at node 2 to be (0, 1).
Lect14
This document compares the constant strain triangular (CST) element and the linear strain triangular (LST) element in finite element analysis. It states that the CST element has constant strains inside each element but strains are not continuous across boundaries. In contrast, the LST element has linear strains inside each element and is preferred for stress analysis due to its higher accuracy. The document recommends using LST elements over multiple CST elements for better representation of stresses and displacements.
Lect13
This document discusses displacement functions or polynomial functions for various finite elements. It provides examples of the displacement functions for:
1) A two-noded bar element with two degrees of freedom.
2) A three-noded bar element with three degrees of freedom.
3) A two-noded beam element with four degrees of freedom.
4) A constant strain triangular element with six degrees of freedom.
It explains that the displacement functions are selected from Pascal's triangle and include terms for x- and y-coordinates for two-dimensional elements.
Lect10
The document discusses the different types of finite elements used in civil engineering. It outlines three main types: 1) 1D elements which are used to analyze trusses, beams, pipes and tubes; 2) 2D elements which are used for plane stress, plane strain and plate bending as well as shell structures; and 3) 3D elements which are used for three-dimensional analysis and axisymmetric solids.
Lect09
1. The document outlines the step-by-step procedure for finite element analysis, which includes: selecting suitable field variables and elements, discretizing the continuum, selecting interpolation functions, finding element properties, assembling global properties, imposing boundary conditions, and making additional calculations.
2. Key steps include discretizing the continuum by dividing it into sub-domains called elements, expressing unknown field variables within each element using interpolation functions, assembling element properties like stiffness matrices, and assembling global properties from the element properties to obtain system equations.
3. The procedure allows reducing infinite unknowns to a finite number and solving the system of equations imposed with boundary conditions to find nodal values and make additional calculations for required outputs.
Lect07
This document discusses effective node numbering schemes for finite element analysis. It explains that effective node numbering should minimize the bandwidth of the stiffness matrix. The bandwidth is calculated as B = (1+D)f, where D is the maximum difference in node numbers between any two connected nodes, and f is the degrees of freedom per node. Several examples are provided to demonstrate calculating the bandwidth for different node numbering approaches applied to structures like rectangular slabs and trusses. Effective node numbering reduces computation time by producing a banded, sparse stiffness matrix.
Lect06
Nodes are points where basic unknowns are determined in finite element analysis and where elements connect. There are two main types of nodes: external nodes located at element edges or surfaces and shared by multiple elements, and internal nodes located inside a single element. External nodes are further divided into primary nodes at corners or ends, and secondary nodes along edges but not at corners. Internal nodes are specific to each individual element.
Lect05
Three coordinate systems are used in finite element analysis: the local coordinate system defines each element's properties, the global coordinate system defines points in the entire structure, and the natural coordinate system specifies points within an element using dimensionless numbers between -1 and 1 with the origin at the element's center.
Lect04
The document discusses applications of the finite element method (FEM) in various engineering fields. It lists applications of FEM in civil engineering such as analyzing structures like bars, trusses, beams and frames; earthquake analysis; and bending, buckling and vibration analysis of beams and plates. It also lists applications of FEM in mechanical engineering, aerospace engineering, medical engineering, and electrical engineering.
Lect03
The document discusses the advantages and disadvantages of the finite element method (FEM) in civil engineering. It states that FEM can effectively handle problems with irregular shapes, boundary conditions, and loading conditions. It can also solve problems involving anisotropic and non-linear material and geometric properties that classical methods cannot. However, FEM provides approximate solutions rather than exact ones, requires skill in element selection and modeling, and demands significant computer resources.
Lect02
Discretization is the process of dividing a continuous structure into smaller, finite pieces or elements to simulate the structure on a computer for finite element analysis. The structure is divided into a number of elements that are connected at nodes. The number, shape, size and configuration of elements must closely simulate the real structure, and the discretization must allow results to converge to the true solution. For example, a rectangular slab can be discretized by dividing it into n subdomains, with each subdomain being an element connected to others at nodes to form a finite element mesh.
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Lect12
- 1. Dr. A. S. Sayyad Professor & Head Department of Structural Engineering Sanjivani College of Engineering, Kopargaon 423603. (An Autonomous Institute, Affiliated to Savitribai Phule Pune University, Pune) Finite Element Method In Civil Engineering 2D and 3D Pascal’s Triangle Geometric Isotropy or Geometric Invariance
- 2. Geometric Isotropy or Geometric Invariance. Besides the convergence and compatibility requirement, one of the important considerations in choosing proper terms in the polynomial expansion is that the element should have no preferred direction. That is displacement shapes will not change with change in local coordinate system. This property is known as Geometric Isotropy or Geometric Invariance.
- 3. 2D Pascal’s triangle Geometric Invariance can be achieved by selecting the corresponding order of terms on either side of triangle.