M. Vable
Notes for finite element method: Axi-symmetric, plates and shells, 3-D
Polar Coordinates
The small strain-displacement equations in polar coordinates are:
rr = u r r u r 1 v = - + - r r rz = zz = w u r + r z w z 1 w v z = -+ r z
1 u r v v + - r

M. Vable
Notes for finite element method: Intro to FEM 2-D
Storage and Solution Techniques
Banded Matrix Each node has two degrees of freedom (u and v) Element matrix is 6 x 6
2
CST
4
CST CST
6
8
CST CST
Half Band Width =NB
1 1 1 2 3 4 5 6 7 8 9 10 11 12

M. Vable
Notes for finite element method: Intro to FEM 2-D
FEM in two-dimension
1 Strain energy density: U o = - [ xx xx + yy yy + xy xy ] 2 xx ~ Define: cfw_ = yy xy Generalized Hooke's law Plane stress (All stresses with subscript z are zero) Plane str

M. Vable
Notes for finite element method: Review
Review
Linear Strain Energy Density
1 T 1 1 T U o = - [ xx xx + yy yy + zz zz + xy xy + yz yz + zx zx ] = - cfw_ cfw_ = - cfw_ [ E ] cfw_ 2 2 2 xx cfw_ = yy xy
Axial
xx cfw_ = yy = [ B ] cfw_ d xy

1.0 Stress at a Point
REVIEW OF MECHANICS OF MATERIALS
1. 2. 3. 4. 5. Stress is an internal quantity. Stress has units of force per unit area. Stress at a point needs a magnitude and two directions to specify it (i.e. stress is a second-order tensor). The

M. Vable
Notes for finite element method: Modeling and Errors
Modeling
A model is a symbolic representation of the real thing (nature). A model could be experimental, analytical, or numerical or some combination of
the three. Solutions of all models are

EXAM 1
MEEM 4405
Oct. 6th, 2004
Answers to questions 1 and 2 are at the end. _ 1. (a) For the beam shown determine one parameter Rayleigh-Ritz solution using the approximation for the bending displacement given below. (b) For two parameter solution calcul

EXAM 2
MEEM 4405
Nov. 18, 2004
_ 1. A beam has a uniform load and a moment applied to it as shown. Model the beam using two equal beam elements. Assume EI is a constant for the beam. (a) Write the global stiffness matrix before incorporating the loads and

_ 1. The rod shown has a thermal conductivity k. Heat flows into the rod at C and out of the rod at B. Model the rod with two linear elements. Determine the temperature at point B and C and the heat flow at point A assuming the temperature at A is at zero

M. Vable
Notes for finite element method: Thermal Analysis
One-dimensional heat conduction
Axial Member Differential Equation d du EA = p x dx d x EA= Axial Rigidity px= force per unit length Heat Conduction d dT k = qx d x d x k= Thermal conductivity qx=