ME 305 INTRODUCTION TO FINITE ELEMENTS
IN MECHANICAL ENGINEERING
Winter 2014
Homework 1 Solution
Problem 1. a
Similarly, for all the nodes,
Node ID
G
F
E
D
C
B
A
Weight-equivalent force (pointing downwards)
1/2W1
1/2W1+1/2W2
1/2W2+1/2W3
1/2W3+1/2W4
1/2W4+
ME305 INTRODUCTION TO FINITE ELEMENTS
IN MECHANICAL ENGINEERING
Displacement accuracy and convergence for 3 node triangles
To obtain a clear picture of the issues when using triangles for solving elasticity problems
(in contrast to fluid dynamics), we wil
ME305 W14
March 10
Modeling Symmetry
HW3 handout
HW3 due Friday 03/21
1
Symmetry/Antisymmetry
Geometric planes of symmetry may
Reduce model size = less analysis time
Enable larger models = more accurate solutions
Symmetry/Antisymmetry modeling requires
Ge
ME 305 INTRODUCTION TO FINITE ELEMENTS
IN MECHANICAL ENGINEERING
Introduction to ME305 FE Software
Part 1
In what follows, unless stated otherwise, click means to click only once(!) with the leftmouse button. Phrases in bold indicates the menu choices, bu
ME305 INTRODUCTION TO FINITE ELEMENTS
IN MECHANICAL ENGINEERING
March 14 exercise
Stress accuracy and convergence
1. Open Hyperworks and choose the NASTRAN user profile.
2. Import your 4 x 2 quad element NASTRAN input file you created last class
period, u
ME305 W14
March 26
2D Model Cleanup
HW4: Due Friday April 4
1
Move nodes
Use the place node option in the Quality Index panel
Left click and hold mouse on an element node
Drag node and watch element quality color vary
2
Delete and add elements
Use the
ME305 W14
March 12
2D Elements: Stress Analysis
1
Stress Analysis
1D Elements
Axial stress, Bending stress, Combined stress
,
g
,
2D Elements
Normal stresses
Shear stress
Principal stresses Principal stress angle
stresses,
Von Mises stress
3D Elem
ME305 W14
March 17
Constraints
Equivalent Nodal Loads
1
Stress Analysis
Interpreting Stresses: Stress concentrations
Stress concentrations
S
Stress concentrations very dependent on
Mesh refinement around concentration
Whether or not stress is unbounded
ME305 W14
March 31
3D (Solid) Elements
HW4: Due Friday 04/04
1
3D Elements
Geometry of the element defined by a volume
Both the simplest and most complex elements
Greater usage over the past 20 years
No geometric parameters
Some meshing difficulty
Large m
ME305 W14
March 10
Modeling Symmetry
HW3 handout
HW3 due Friday 03/21
1
Symmetry/Antisymmetry
Geometric planes of symmetry may
Reduce model size = less analysis time
Enable larger models = more accurate solutions
Symmetry/Antisymmetry modeling requires
Ge
% Let outside radius= ro, inside radius=ri
%Let Node A location = ya, Node B location = yb
%
syms ro ri ya yb
%
% Here the area over an element is assumed constant
% Want to make area a function of ro, ri, ya, yb
% Even though area here doesn't vary with
ME305 INTRODUCTION TO FINITE ELEMENTS
IN MECHANICAL ENGINEERING
March 12 exercise
Stress accuracy and convergence
We return to the quadrilateral meshes we looked at (for displacement accuracy), but now
want to examine the accuracy and convergence trends f
ME 305 INTRODUCTION TO FINITE ELEMENTS
IN MECHANICAL ENGINEERING
Creating Truss (CROD) Element Model Using Hyperworks
In this exercise, we will use Hyperworks to build a 2 element truss (CROD) finite element model of the
turbine tower.
1. Start Hyperworks
Foundations of Finite Element Method
1
2
Principal of minimum potential energy:
p
p
gy
The displacements that satisfy equilibrium are distinguished
by a stationary value of the total potential energy
Foundations of Finite Element Method
Principal of minim
Two Truss Element Problem
Two Truss Element Problem
Two Truss Element Problem
Two Truss Element Problem
Two Truss Element Problem
Two Truss Element Problem
Two Truss Element Problem
ME 305 INTRODUCTION TO FINITE ELEMENTS
IN MECHANICAL ENGINEERING
Introduction to ME305 FE Software
Part 2
In this second part of our FE software introduction, you will create from scratch a simplified
model of the wind turbine tower, conduct linear static
Foundations of Finite Element Method
For structural problems:
Principal of minimum potential energy
Equivalent spring network
More generally methods of variational
generally,
calculus
(principal of virtual displacements )
displacements, )
Foundations of F
Foundations of Finite Element Method
1
2
Principal of minimum potential energy:
p
p
gy
The displacements that satisfy equilibrium are distinguished
by a stationary value of the total potential energy
Foundations of Finite Element Method
Principal of minim
ME 305 INTRODUCTION TO FINITE ELEMENTS IN
MECHANICAL ENGINEERING
Optimization with Optistruct
A design concept for an automotive control arm required to meet performance specifications will
be created in this tutorial. The finite element mesh containing d
ME305 INTRODUCTION TO FINITE ELEMENTS
IN MECHANICAL ENGINEERING
Displacement accuracy and convergence for Second-order triangles
CTRIA6 elements
Last week we investigated the use of the standard 3 node, linear (first-order) triangle
(CTRIA3) for structura
ME305 W14
March 12
2D Elements: Stress Analysis
1
Stress Analysis
1D Elements
Axial stress, Bending stress, Combined stress
,
g
,
2D Elements
Normal stresses
Shear stress
Principal stresses Principal stress angle
stresses,
Von Mises stress
3D Elem
ME305 W14
March 17
Constraints
Equivalent Nodal Loads
1
Stress Analysis
Interpreting Stresses: Stress concentrations
Stress concentrations
S
Stress concentrations very dependent on
Mesh refinement around concentration
Whether or not stress is unbounded
ME305 F13
March 14
2D Elements: Stress Analysis
1
Stress Analysis
Nodal stresses
Nodal stresses from each element most inaccurate
of all stress locations
Centroidal stresses
Compromise between accuracy and efficiency
x
x = centroidal location
x
2
Stre
ME305 INTRODUCTION TO FINITE ELEMENTS
IN MECHANICAL ENGINEERING
March 12 exercise
Stress accuracy and convergence
We return to the quadrilateral meshes we looked at (for displacement accuracy), but now
want to examine the accuracy and convergence trends f
ME305 W14
March 17
Constraints
Equivalent Nodal Loads
1
Stress Analysis
Interpreting Stresses: Stress concentrations
Stress concentrations
S
Stress concentrations very dependent on
Mesh refinement around concentration
Whether or not stress is unbounded
ME305 F13
October 11
2D Elements
1
2D Elements
Still most widely used class of finite elements
Key attributes
Characterized by one geometric parameter:
Thickness, T
Visually imaged as (flat) surface
T
2
2D Elements
Complexity of modeling
Easier than
ME305 W14
March 21
Stress convergence
Element quality and repair
1
Stress Convergence
Convergence formulas are not practical
Compare maximum values between nested meshes
Start with consistent load model from last class (the first
(
split model)
2
Stres