Test 2 Review - padtinc.com · Test 2 Review. Test 2 will NOT cover the following 1. All...

Test 2

Review

Test 2 will NOT cover the following

1. All derivations. If any formulas are required, they will be supplied

2. All of lecture 3

The test will focus on material found in lectures 6 through 10 (with the

exception of lecture 7. The test will not cover constraints or

nonlinearities) and the following general themes:

1. Nodal degrees of freedom. Students should be able to be given an

element description and determine the number of nodal degrees of

freedom from the description. They should also understand what this

means for the finite element mesh and assembled system of

equations

2. The distinction between plane stress, plane strain, and axisymmetric

formulations. When to use each and why

3. The distinction between the element types encountered in MAE 323,

when to use each and why.

4. The analysis process. Planning, executing, and validating a FE study

Nodal Degrees of

Freedom

(Review Lecture 6

and Test 1)

•Beams and Shells

x

z

y1

2

3

1 2

v1,y

x

θ2θ1

θy1

θx1θz1

ux1

uy1

uz1

θx2

θy2

θz2 ux2

uy2

uz2

X-X

y

z

Cross

section

X-X

v2

u1u2

3D

2D

•Most commercial FE code will provide

both a 3D and 2D Euler-Bernoulli beam

element

•Dofs per node (3D structural only):

•ux, uy,uz,θx,θy,θz

•DofFs per node (2D structural only):

•ux,uy,θ

•Behavior is governed by small-beam deflection theory (MAE 213), as well

as truss and torsional spring laws (this last usually doesn’t apply in 2D)

•The user must supply cross section definition

•Beam and Shells

z

x

y

2

3

4

rst

θx1

uy1

θy1

ux1

θz1

uz1

θx1

uy1

θy1

ux1

θz

1

uz

1

θx1

uy1

θy1

ux1

θz1

uz1

θx1

uy1

θy1

ux1

θz1

uz1

1

12

3

r

s

t

•These are 3D reduced-continuum elements. They represent a 2D extension

of small beam deflection theory. Their behavior is also primarily dominated

by bending (different assumptions govern whether cross sections remain

parallel or not, as well as how to handle the out-of-plane torsional DoF. See

Chapter 6)

•DoFs per node (structural only): ux, uy,uz,θx,θy,θz

•They are to be used when 2 dimensions are >> than a third and loading is

out-of-plane

•Beam and Shells

•Today, some commercial FE software products offer fully parametric beam and

shell elements like those shown below. The advantage of these types of elements

over traditional small-deflection beams and shells is that they carry transverse

shear components (and so are better at modeling thick beams and shells), and they

more accurately represent large rotations and small curvatures

•These are the elements you get when you mesh lines and surfaces in Workbench

(note: workbench has no truss elements!)

•They may utilize linear, quadratic (default), or cubic polynomials

•DoFs per node (structural only): ux, uy,uz,θx,θy,θz

t

rs

1

2

3

4

2

3

4

rst

1 5

6

7

8

z

x

y

•2D Continuum Elements

rs rs

•All the above element domains (meshes) may be used in plane stress, plane

strain, and axisymmetric analyses in Workbench

•DoFs per node (structural): ux, uy

•Dofs per node (thermal): T

•3D Continuum Elements

r

s

t

r

s

t

•All the above element domains (meshes) may be used in static stuctural, as

well as thermal analyses in workbench

•DoFs per node (structural): ux, uy, uz

•Dofs per node (thermal): T

•Spring Elements

x

y

z

ux1

uy1

uz1

1

2 ux2

Uy2

uz2

1 2

ux1

uy1

uz1

ux2

uy2

uz2

•All the above element domains (meshes) may be used in static structural, as

well as thermal analyses in workbench

•DoFs per node (structural, 3D): ux, uy, uz

•Dofs per node (structural, 2D): ux, uy

•Dofs per node (thermal 2D and 3D): T

•Questions:

•Which of the elements in slide 8 have quadratic (2nd order)

shape functions?

•What is the stiffness matrix size for the linear hexahedral

(cube) in a structural analysis?

•What is the number of shape functions for all of the

elements in a thermal analysis?

Planar Continua

(Review Lecture 4)

•Chapter 4 introduced the planar stress/strain conditions:

1. The cross section normal to a given direction is

constant along that direction

2. The model loads and boundary conditions are constant

along the same direction

•When these conditions are met, we know that we can simplify a

model by going to 2D, using plane stress, plane strain, or

axisymmetry

•However, in order to select of the 2D formulations, more

information is needed. We refer students to the first part of

Chapter 4 for a definition of each of these formulations. Let’s focus

instead on the physical circumstances that imply the selection of

one of these formulations over the others.

•We learned from lecture 4 that the distinction between plane

stress and plane strain is made apparent when tracking the

maximum stress at the “free end” of a 3D model which satisfies

the planar stress/strain conditions:

xy symm

yz symm

L/2What

happens on

this face? 0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

100000

0 5 10 15 20 25 30

max. strain

max. stress

75464.4

5.616e-5

38033.6

0.003

Distance from the free face is measured on the

x-axis

As the distance from the free face grows, the planar solution tends

towards the solution we get when both the end faces are fixed (in the

z-direction). Thus, at distances on or close to the free end, the solution

may be considered as that of a very thin section. Lecture 4 goes on to

show that this solution corresponds to a state of plane stress. At

greater distances, as the solution tends towards to the two-clamped

face solution, we recognize that condition as corresponding to a state

of plane strain. Finally, we learned that the axisymmetric formulation

IS the plane strain formulation in a cylindrical coordinate system

xy symm

yz symm

L/2xy symm2 1.00E-14

1.00E-12

1.00E-10

1.00E-08

1.00E-06

1.00E-04

1.00E-02

1.00E+00

1.00E+02

1.00E+04

0 10 20 30

max. strain

max. stress

75413.4

2.44e-10

•These observations lead to the following general guidelines

for using the plane stress vs. plane strain formulations:

Planar Contuum Problem

Large thickness

One end free

Plane strain

Both ends fixed in z

Plane strain

Small thickness

One end free

Plane stress

Both ends fixed in z

Plane strain

•The last question that must be answered when

contemplating the use of plane stress or plane strain

based on what you know about end effects is this: “What

is the physical mechanism which drives the differences

between the two formulations when modeling parts with

a free end?”

•The answer is: Saint-Venant’s Principle acting in the

plane-normal (z) direction. If there are no singularities,

there are no local peak stresses or strains on the free end

whose magnitude would decay as we traveled down the z-

axis.

Basic Modeling

Guidelines

(Meshing and

Boundary

Conditions)

Rules for Meshing

1. Always construct a coarse mesh first, then refine

selectively as necessary

2. Refine meshes at potential singularities AND small

features*

In a 2D and 3D

continuum, you will

probably need to

refine at these

locations IF looking

at max. Seqv

*Imagine there is a very small fillet at every corner

Rules for Meshing

3. Which potential singularities you must refine depends upon loads,

geometry , BC’s AND which stress component(s) you’re interested in. IF

only displacements are required, the refinement should be more global

(and you don’t need as many elements)

4. Be careful not to allow mesh size transitions to change too abruptly

(this can cause poor element shapes, which in turn may prevent

convergence)

5. If two adjacent bodies share a boundary, then care must be taken to

ensure this happens in most commercial FE code (CAD systems

usually do not allow this. In Workbench, the user can “form new

part” out of the adjacent bodies if they are to receive different

material properties). This may be referred to as maintaining mesh

contiguity.

6. If mixing continuum with non-continuum elements, make sure that

any DoFs on one element type not supported on the other are

constrained

Rules for Meshing

Maintaining Mesh Contiguity

•These two

adjacent

blocks DO

NOT share a

face

•This is what

happens by

default when

you import

an assembly

Rules for Meshing


•Because the blocks don’t

share a face, elements will

not be connected at the

interface (as you can see

when the mesh sizes are

different)

•In Workbench, bonded

contact WILL be

established by default.

But it is not good practice

to rely on this (not all

codes generate bonded

contact automatically)

Rules for Meshing


•Suppose you are

mixing element

types, like solids and

shells. Obviously, this

edge should be

shared between the

solid and shell area.

But this is not

enough! What about

the rotational DoFs?

Rules for Meshing


•A common solution to this

problem is to extend the

shell elements onto one of

the bounding surfaces of

the adjacent solids

•This way all

rotational

DoFs are

constrained

Rules for Meshing


Note: At release 13.0, ANSYS

Workbench does not have the

capability to form contiguous

mixed element type models.

•However, you CAN create “surfaces from

faces” and “surfaces from edges”, which

will become shell elements – they just

won’t be contiguous with the solid

geometry mesh. ANSYS will instead form

bonded contact between the two element

types. In other codes, this might be a step

you would have to perform manually

Rules for Meshing


•This would result in a model like the one below. Note that this

is NOT good modeling practice

Rules for Applying Boundary Conditions

1. Always use symmetry if possible. It reduces model size AND

simplifies BC application generally

2. Always try to establish the “minimum constraint condition”

(and apply it) for problems where the boundary conditions are

poorly defined or obscure. This must be a statically

determinate system

3. Try to avoid BC application at points or vertices UNLESS doing

so satisfies 2. In general, applying BC’s on lines and areas will

result in “milder” singularities

Rules for Applying Boundary Conditions

•The second point is extremely important, but can be somewhat difficult.

There are situations in which it is not at all clear what the boundary

conditions should be. Try to use Newton’s Third Law to help with such

situations. The idea is try to identify a statically equivalent system

corresponding to a minimal constraint condition.

Example: Using Newton’s Third Law to Establish BC’s

F

RA RB

Suppose we have a

pinned-pinned

beam with an

applied load at

mid-span


F

RA

•Newton’s Third law tells us that F+RA+RB=0

•It doesn’t make a distinction which is the applied load and which are

the reactions (just as the long as the sum remains the same). This

implies that applied loads and constraints may be swapped without

changing the problem.

•So, this problem is completely equivalent to:

RB

F/2 F/2


•The preceding example might seem a bit obvious and trivial, but the

method really demonstrates its usefulness when dealing with applied

pressures (note: this is rigorously true for reduced-continuum

elements, but only true in a mean sense for continuum elements –

following Saint-Venant’s Principle). Let’s look at an example that’s not

so obvious. Consider this pinned-pinned beam:

RA

p0

p0L/6

L

p0

L/6

p0L/6

RB

L/2

L/3

p0L/6

RB

L/6 L/3


•This methodology is extremely useful for systems in which all

the applied forces and moments are known and balanced, but

no discernable “fixed points” exist. Let’s look at an iconic

example: A partially submerged ship’s hull (let’s forget for a

moment that this problem is symmetric):

mg

A

pdA∫

p ghρ=

h

waterline

x

y


•We can replace all the pressures with equivalent forces at the

centers of pressure…

1

2

3

hg Aρ

2ghAρ

1

2

3

hg Aρ

mg

mg

mg

mg

1

2

3

hg Aρ

•And then

replace some of

the applied

forces with

equivalent

constraints

•Of course,

symmetry can be

applied to this

model in the usual

way. This would

remove the x-

constraint but keep

the pressure along

a side

Symmetry Boundary Conditions

To exploit the symmetry about x in the previous problem, we

could do this:

mg

Mg/21

2

3

hg Aρ

•In general a planar symmetry boundary condition constrains

the degrees of freedom normal to a plane. This constraint

can be used to reduce the size (and complexity) of models

which may be symmetric about such a plane. This is achieved

by exploiting the fact that models respond symmetrically

about either side of such faces, when the geometry, loads,

and boundary conditions are identical on either side


Example: An almost Axi-symmetric pressure

vessel

•Suppose we have a

pressure vessel like

that to the right. It

consists of cylindrical

pressure vessel with a

bolted flange.

Suppose there are 32

bolts and we want to

assess the integrity of

the flange seal by

calculating the

maximum flange

separation•What kind of symmetry can we

exploit?


Example: An almost Axi-symmetric pressure vessel

•Two rather common and obvious choices would be half-

symmetry and quarter symmetry. The quarter-symmetry

model is shown below

•But can we go

further? Remember,

we can’t model this

part as axi-

symmetric because

we still have to

assess the deflection

around bolts (not

axi-symmetric)


Example: An almost Axi-symmetric pressure vessel

Recall from lecture 2,

that we want to

model the bolt

preload pressure

frustra. In other

words, the two

flanges will only be

bonded at the

annular faces

highlighted in green.

The rest of the flange

interface surface will

get have the internal

pressure applied

Internal pressure

applied to all

internal faces AND

flange surfaces

(except for annular

bolt frustra in green)

Modeling Bolted Flanges

•In this course, we strongly emphasized the technique of imprinting

annular faces on bolted flange surfaces. Not just the interface, but also

surfaces underneath bolt heads

•For a refresher, students are encouraged to go back and review lecture 2

on the process. This comes directly from books (such as Shigley and

Mischke) on Mechanical Engineering design, which emphasize that bolts

are not capable of transmitting a clamping force past some critical

distance from bolt head (represented by a swept cone). Evidence for this

assertion is given in the analysis on the next slide in which two plates are

bolted together. The model does NOT use imprinted faces, but uses

pretensioned beam elements to represent the bolts, which are simply

constrained (through constraint equations) to the bolt hole rims

Modeling Bolted Flanges

•The third principle stress is being plotted to show the

compressive stress distribution. It is clear from this model

that the plate compression quickly becomes zero as the

distance from each bolt grows

•Note the stress

contours are

rescaled to

remove the

singularity

associated with

the bolt

constraint

equations (the

grey regions)

Performing a

Structural Analysis

(The Process)

•We learned in lecture 10 that, in addition to all relevant model input data

(materials, geometry, element type, loads and BC’s), an analyst operating

within a typical engineering CAE cycle, must also know the analysis

objectives. Secondly, she must not operate in a vacuum. She must perform

the analysis to determine whether or not these objectives are met, and

must then communicate this with others within the engineering pipeline.

Thus, it is extremely helpful when these objectives are formal (written down

as product specification). Unfortunately, it is rare that a product’s various

functional specs are detailed enough to directly provide load and BC input to

an analysis. In these situations, the analyst must improvise, by “filling in the

gaps”. This should effectively produce what we might call an “analysis

specification”.

•In general, the analysis objectives fall under three main

categories:

1. Requirements for burst/material rupture. This consists of a

set of maximal loads and boundary conditions for which the

part(s) must not “break” (cease to function –irreversibly)

2. Proof requirements. This consists of a set of loads which are

less severe than 1. The desired part or assembly performance

under this set of loads is usually that no irreversible

performance degradation has occurred

3. Operating requirements. This consists of a set of loads

representative of repetitive use. The part or assembly must

demonstrate a minimum acceptable life under these conditions

•In MAE 323, students may do well to go over previous lab

assignments and ask: “which type of requirement would this

loading environment represent”?

•In lecture 10, we reviewed two hypothetical examples in which we

deduced a simulated loading environment from existing functional

specifications (even though these were not explicitly contained within

those specifications)

•Once all the analysis objectives are clear (loads to be applied and metrics

to assess the response), assuming the analyst has all other model inputs,

she is ready to perform the analysis. In lecture 10, we reviewed a

procedure first introduced in lecture 2 (and we explained how the analysis

objective guide this process as well):

1. Formulation of Physical Problem

2. Analysis Type Selection

3. Physical Reduction (Idealization)

4. Element Type Selection

5. Model Creation

a) Loads and Boundary Conditions

b) Material Properties

c) Mesh

6. Solution

7. Post-processing

8. Validation

Example: The Analysis Process

•Consider the

following pressure

vessel. It is made of a

mild steel (so we

know E and ν). It

contains a gas under

pressure, p. All the

dimensions of the

part are given (ri, ro, L,

etc) and we build the

solid model at left.

•Let’s assume we

want to assess the

integrity of the weld

joints under this

pressure

Weld

beads


•Let us further assume that we know that the pressure, p is a proof load

operating at 120 degrees C. We will use this last bit of information to

look up the material’s yield strength at that temperature –that is what

we’ll use as a reference to calculate the proof margin. We also need an

appropriate stress-based failure metric, like von Mises stress or first

principle stress (if the weld is brittle). In any case, we are ready to

begin. We start with the:

1. Formulation of the Physical Problem

•Load Type: Static pressure

•Energy Dissipation Mechanism: Mechanical

strain

•Material Response: Linear

•Time Dependence: no

So, this will be a linear static problem


2. Analysis Type Selection

In Workbench: Static Structural

3. Physical Reduction (Idealization)

•Does the problem require reduced continuum

elements (springs, beams, or shells)? NO

•Can symmetry be exploited. Yes. In fact, it is

completely axisymmetric

4. Element Type Selection

•In Workbench, the answers to 3 make the decision for us. We

only have to decide whether to use quadratic (mid-side node)

elements or linear ones. We might as well stick with the more

accurate default.


5. Model Creation

•Use your available modeling (sometimes referred to as pre-

processing) tools to retain only the quarter-cross-sectional

areas shown below


5. Model Creation

Pressure, p

gets applied to

these edges

Pressure loading mesh

Weld bead


5. Model Creation

Frictionless

support on this

edge (uy=0)

Boundary Conditions

x

y


6. Solution

•Since this is a purely linear system, solution is

straightforward and completely automated. We accept all

the default options

7. PostProcessing

•We are only interested in stresses in the weld, so this is all

we will report. We want to compare Seq (von Mises) with

our yield stength at 120 C. So, the first thing we have to do

is find a reliable value. We will perform a convergence

study


7. PostProcessing …we seem to be converging to a value

around 2700-2800. This is by no means

guaranteed, however. It is quite likely

that we might encounter a singularity

in situations like this


8. Validation

•The first thing we’ll want to do is validate

the reaction forces. We expect zero in the

radial direction. In the axial direction, we

should get a reaction force at the

fricitonless support equal to 2

0p rπ

•In workbench, we

place a force

reaction probe at

the frictionless

support to verify

this


8. Validation

•Next, we want to verify that the mean

stress fields are equal to those predicted

by theory (thick-walled cylinders):

( )( )

( )

( )

2 22 2

0 00 0

2 2 2 2 2

0 0

i ii i

i i

r r p pp r p r

r r r r rθσ

−−= −

− −

( )( )

( )

( )

2 22 2

0 00 0

2 2 2 2 2

0 0

i ii i

r

i i

r r p pp r p r

r r r r rσ

−−= +

− −

( )( )

2 2

0 0

2 2

0

i i

a

i

p r p r

r rσ

−=

−

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