Finite Element Analysis

Consider the patch test shown below, modeled after the figure in the Cook

Consider the plane quadrilateral element shown below. (a) Numerically evalute the strains

Sketch a quadrilateral with corners properly letterd and r,s axes properly oriented if shape functions

Q3. Sketch a four-node isoparametric element for which the determinant J is a function of r but not of s.

Q4.

Consider a 4 node rectangular finite element given earlier in the class notes with x dimension a=3 cm and y dimension b-4 cm, and a thickness of 1 cm. For steel which has a mass density of 8 g/cm³, find the first row of the consistent mass matrix (8 terms) for the element and the first diagonal entry of the lumped mass matrix using the row-sum technique.

Q2. Using Hermite cubic shape functions derive the element force vector for a distributed load of constant intensity fo (namely, a constant transverse applied load, in units of force/length). The solution (take h² = l) is:

Q1. Using Hermite cubic shape functions in the strain-displacement matrix, show that for constant flexural rigidity EI, the element stiffness matrix is given as below. Hint: the functions are given on the next page, for an element spanning Ω = (0,l), so set h=l.

Problem 2: Determine the displacements and rotations in the beam shown in the following figure. The beam is fixed at both ends and was discretized into four elements as shown below. (EI = (101º) N-cm² throughout the beam length).

(10 points) Using the nondimensionalized shape functions listed in equation (10.46) on the next page, only calculate