Lect19
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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.
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3) Displacement functions must be capable of representing rigid body displacements, with elements experiencing no strain and zero nodal forces under rigid body motion.
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.
Lect08
The document discusses element aspect ratios in finite element analysis for civil engineering structures. It defines aspect ratio as the ratio of the largest to smallest size of an element. For analysis accuracy, the aspect ratio should be as close to unity as possible. It provides an example of correctly sizing square elements of equal size compared to incorrectly using long rectangular elements.
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.
Lect01
The document discusses the finite element method (FEM), a numerical technique used to solve complex engineering problems. FEM was originally developed for aerospace engineering but is now widely used in civil, mechanical, and electrical engineering. It works by dividing a structure into finite elements and obtaining an approximate solution by analyzing the properties of individual elements and their interaction. The key principles of FEM include using computational methods to solve boundary value problems representing physical structures, with field variables governed by differential equations and boundary conditions specified on the field boundaries.
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- **Computer Networks**: TDM is used in network protocols and systems to manage the transmission of data from multiple sources over a single network medium.
### Advantages of TDM
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Lect19
- 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 Shape functions/Interpolation Function 2 1 1 2 and x x x x N N L L
- 2. Shape functions/Interpolation Function Shape functions • In FEM analysis, the displacement model we assume the variation of displacements within the element since the true variation of displacements are not known. e.g. • But in higher engineering mathematics, analytical solution of some problems is either not known or difficult to find out. • In such cases we replaced that function by another function which is easy to solve mathematically. • That function is called as “Shape function” or “Interpolation Function”. 1 2 u x 1 2 3 u x y 1 1 2 2 u N u N u 1 1 2 2 3 3 u N u N u N u
- 3. 1 2 u x 1 2 3 u x y 1 1 2 2 u N u N u 1 1 2 2 3 3 u N u N u N u 1 2 3 4 u x y xy 1 1 2 2 3 3 4 4 u N u N u N u N u
- 4. Important properties of shape functions • The magnitude of shape function at each node is Unity. • Number of shape functions are equal to number of nodes (Except bending element) • The sum of shape functions is always unity (Except bending element) Element No. Shape Functions Properties of shape function N1 and N2 N1=1 at node 1 and N2=1 at node 2 N1+N2 = 1 N1, N2 and N3 N1=1 at node 1, N2=1 at node 2 and N3=1 at node 3 N1+N2+N3 = 1 N1, N2, N3 and N4 N1=1 at node 1, N2=1 at node 2, N3=1 at node 3 and N4 = 1 at node 4 N1+N2+N3+N4 = 1
- 5. Methods for deriving shape functions • Shape functions using polynomials in Cartesian coordinate system • Shape functions using polynomials in natural coordinates • Shape functions using Lagrange interpolation function in natural coordinates