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26 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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The Fourth International Conference on
Structural Engineering, Mechanics and
Computation: University of Cape Town

A “simple beam” revisited

Roland Prukl

Pr.Eng., Dipl.
-
Ing., FSAICE, AIStructE

Title

1
Dedicated to

Gali

Henry

Katie

Dedication

Contents

Introduction

FEMSA ‘92

SEMC 2001

SEMC 2010
-

A “simple beam” revisited

Conclusions

SEMC 2010

SEMC 2013
-

Axisymmetric models

Conclusions

SEMC 2013

Appeal

Acknowledgements

Contents

Around 1980 Kamel & McCabe stated:

“The finite element method is an
approximate method based on
discretization of both geometry, loading
and boundary conditions, as well as the
use of elements derived using various
assumptions of which the user is often
not quite aware.”

Introduction

At about the same time Bathe
prophesied:

“Within a period of ten years the powerful
tool of finite element analysis will be
available on every analysis engineer’s
desk, but not enough people will be
trained sufficiently by then to correctly
and safely apply this method.”

Introduction

At the
FEMSA 1992 Symposium in Cape
Town,
the author presented:

The CST (Constant Strain Triangle)

An insidious survivor from the infancy of FEA

The results from various finite element
programs, using eight different meshes of
plane stress elements, as well as from one
boundary element mesh, were compared.

FEMSA ‘92

FEMSA ‘92

A concrete beam, 8m long, 2m
high and 0,2m thick is loaded with
a vertical u.d.l. of 100 kN/m at the
top. The material properties are:
Young’s modulus 3,0e7 kPa and
Poisson’s ratio v = 0,2.

The horizontal stresses at the
centre of the span should be

6000 kPa (compression) at the
top and +6000 kPa (tension) at
the bottom.

FEMSA ‘92

Four 4
-
noded quads

Maximum stress
error = 13%

Maximum stress
error =

100%

Eight 3
-
noded triangles

At the SEMC 2001
Symposium in Cape Town,
the
author presented:

Finite Element Analysis (FEA) tests on a simple beam

Important Information for users of FEA software.

For the same problem as used at FEMSA’92, the
results from various finite element programs, using 63
different shell and 23 solid meshes, were compared.

SEMC 2001

SEMC 2001

Using
plane stress elements in program Strand7

for this
mesh, the maximum stress error at the bottom of the beam
at midspan can even be 116 %, i.e. instead of +6000 kPa
tension, we get
-
981 kPa
compression !!!

.

SEMC 2010

Since 1986, the writer has presented a one
-
week course “The Application of the Finite
Element Method in Practice” about every two
months.

In almost every course some of the participants,
when modelling a simply supported deep beam
with shell elements, make the same mistake:
They add
unnecessary restrained freedoms

(translation DZ as well as the rotations RX, RY
and RZ) to all node points of the structure.

SEMC 2010

In the program Strand7, when using 3
-
dimensional
shell elements without any unnecessary restraints
at the nodes, the maximum error in a model with
four 4
-
noded elements is
14 %.

With DZ, RX, RY & RZ restrained at all nodes, the
error increases to
38 %.

This error increase is, in
fact, caused only by the RZ restraint.

SEMC 2010

In the program “Prokon Frame” version W2.72, the
maximum stress error increases from 5 % to 24 %.

In the program Adina, the maximum stress error for
both cases (with and without the additional
restrained freedoms), is 13 %.

SEMC 2010

If one uses 8
-

or 9
-
noded elements in Strand7,
the maximum stress error for both models will
be 3 %.

SEMC 2010

Horizontal stresses at midspan
bottom of beam

Analysis No. 9

Program

% Error

Strand7

-
14

Prokon

-
5

Adina

-
13

Ansys

-
13

Cosmos

-
8

Very coarse mesh !

Conclusions SEMC
2010

Conclusions SEMC 2010

Finite Elements have to be handled with
great care.

Different programs might give different
results and even with the same program
the results from different element types
might differ considerably from each other.

All structures have to be supported
carefully and unnecessary supports must
be avoided.

Axisymmetric models

Axisymmetric models

A circular slab with radius of 8 m and a
thickness of 2 m is simply supported

around the outer edge and loaded

with a UDL of 500 kPa.

The material properties are:

Young’s modulus = 3,0e7 kPa and

Poisson’s ratio v = 0,2.

Axisymmetric models

The horizontal stresses at midspan should
be
-

9600 kPa (compression) at the top
and + 9600 kPa (tension) at the bottom.

Centre of slab

Cross section of circular slab:

radius = 8 m, thickness = 2 m

Axisymmetric models

1) 4
-
noded shells, using program Strand7

Maximum stress error at midspan = 0%

Axisymmetric models

Maximum stress error at midspan = 2%

2) 8
-
noded shells, using program Strand7

Axisymmetric models

Maximum stress error at midspan = 5%

3) 8
-
noded solids, using program Strand7

Axisymmetric models

Maximum stress error at midspan = 2%

4) 20
-
noded solids, using program Strand7

Axisymmetric models

Maximum stress error at midspan = 1%

5) 8
-
noded solids, using program Strand7

Axisymmetric models

Maximum stress error at midspan = 1%

6) 20
-
noded solids, using program Strand7

Axisymmetric models

Maximum stress error at midspan =
121%

7) 4
-
noded axisymmetric solids,

with no horizontal supports at the centre,
using program Strand7

Centre of slab

Axisymmetric models

Maximum stress error at midspan =
93%

8) 4
-
noded axisymmetric solids,

with horizontal supports at the centre,

using program Strand7.

Centre of slab

Axisymmetric models

Maximum stress error =
53%

9) 4
-
noded axisymmetric solids,

without horizontal supports at the centre,

using program Strand7.

Centre of slab

Axisymmetric models

Maximum stress error =
44%

10) 4
-
noded axisymmetric solids,

with horizontal supports at the centre,

using program Strand7.

Centre of slab

Axisymmetric models

Maximum stress error = 2%

11) 8
-
noded axisymmetric solids,

without horizontal supports at the centre,

using program Strand7.

Centre of slab

Axisymmetric models

Maximum stress error = 2%

12) 8
-
noded axisymmetric solids,

with horizontal supports at the centre,

using program Strand7.

Centre of slab

Axisymmetric models

Program

Analysis

Stresses at Bottom of Centre

Name

No.

Direction

% Error

Fesdec

7

RR

-
2

TT

-
2

8

RR

2

TT

2

Strand7

7

RR

38

TT

121

8

RR

93

TT

46

Lusas

7

RR

8

TT

8

8

RR

6

TT

6

Conclusions SEMC 2013

Do not use Quad4 elements for axisymmetric
analyses in Strand7 because the program
subdivides each element into four triangles.

Quad8 elements for axisymmetric analyses in
Strand7 are not being subdivided into triangular
elements.

Mr. Stewart Morrison carried out 18 test runs for
this circular plate.

15 of these test runs produced very good
results. The 3 others had errors up to 42 %.

Conclusions SEMC
2013

Appeal

I appeal to all Finite Element users.

Please carry out all or some of my
tests on the software of your choice
and send them to me.

I shall then forward you all my detailed
information on the subject.

Appeal

Final note

The purpose of this presentation is to
warn users of finite element software
about possible pitfalls.

The figures produced should not be
considered as indicative of these
programs for general use.

The author wishes to emphasize that
the aberrations highlighted are specific
to the problems analysed and should
not be regarded as his preference for
one or the other package compared.

Final note

Acknowledgements

Acknowledgements

The author wishes to thank the following persons
who assisted in the preparation of this report:

Mr. Edwin Clarke

for conducting the Strand7 & Prokon analyses

Mr. Steve van Wyk

for conducting the Adina and Ansys analyses

Mr. Jurgen van Wyk

for conducting the Cosmos analyses

Mr. Stewart Morrison

for conducting the Lusas analyses

Mr. Adrian Peirson

for proofreading

End

Thank you for your attention !