Reference no: EM13859390
Objectives:
• To understand the procedure of discretization and interpolation.
• To learn the algorithm of Gauss quadrature.
1. 1D element shape functions
Consider a 4-node cubic element in 1D. The length of the element is 3 with the first node located at xi = -2. The remaining nodes are equally spaced.
1) Construct the Lagrange shape functions for this element.
2) Interpolate the function ue(x) (e.g. the displacement along the axis of a uniform bar) in the element when the nodal values are given as
with the same unit as the coordinates of the nodes. (Writing out the expression for u°(x) would suffice).
3) Evaluate the derivatives of the shape functions, i.e. the Be matrix, and use it to find the expression for the derivative of u°(x) (i.e. the axial strain e(x) = due /dx).
4) Plot ue(x) and a°(x) obtained in (2) and (3).
5) What are the corresponding displacement field and strain field when the nodal displacements are given by
Why is this result expected?
2. Gauss quadrature
1) Write a short Matlab (or whatever language you prefer) program to perform 2 point Gauss integration of a function f over a given interval (a, b). Inputs: a, b, and the functional values of f at the two Gauss points x1 and x2, f 1) and f (x2). (The program should calculate the actual coordinates of the two Gauss points). Output: the (approximate) value of the integral ∫fdx. (If you are able, write a program with adjustable order of integration.)
2) Check your program with the following integrals by comparing to the exact values:
0∫4 (x3+1)dx, -1∫2 (x4 + 3x2)dx, -1∫1 cos2Πx dx
|
What is a job analysis
: What is a job analysis? What data is included in a job analysis? What are three key considerations in job analysis?
|
|
Describe some practical guidelines for hr managers
: Describe some practical guidelines for HR managers to ensure that recruitment and selection practices are nondiscriminatory
|
|
How successful was reconstruction
: How Successful was Reconstruction? Reconstruction was truly a critical time in U.S. history, but what really came out if it? Were the newly freed slaves' new lives much better than under slavery? Were the federal government's programs successful or f..
|
|
The views and opinions within culture began evolving
: The views and opinions within culture began evolving during the 60s and 70s. Takingthis into consideration, choose an individual subject, title, or artist from the list of eligible media genres below. Next,assess the role that your topic played as a ..
|
|
What are corresponding displacement field and strain field
: Write a short Matlab (or whatever language you prefer) program to perform 2 point Gauss integration of a function f over a given interval (a, b). Inputs: a, b, and the functional values of f at the two Gauss points x1 and x2, f 1) and f (x2). (The..
|
|
Provide a detailed description of the work
: Provide a detailed description of the work
|
|
Evaluate the importance of the material in understanding
: Evaluate the importance of the material in understanding how these ancient cultures developed
|
|
What does epicurus think is the end of all human action
: What does Epicurus think is the end of all human action? What would a successful human life look like, on his view? Does he present a compelling picture of what it is to live a good life? Defend your answer.
|
|
What problems inhibited the smooth flow of the treaty
: What problems inhibited the smooth flow of the treaty and how were these problems handled?
|