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HATCH Technical Training – Finite Element Analysis

SEMC 2001: FEA Tests on a Simple Beam – Important Information for Users of FEA Software (by R.P. Prukl and P.M. Lopes)

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Finite Element Analysis (FEA) Tests on a Simple Beam – Important Information for Users of FEA Software

by R.P. Prukl, Hatch Africa, Private Bag X20, Gallo Manor, 2052 RSA

P.M. Lopes, 18 Tulp Street, Brackenhurst, Alberton, 1448 RSA Nine years have passed since the FEMSA’92 Symposium (Finite Elements in South Africa) where R.P. Prukl presented a paper with the title “The CST (Constant Strain Triangle) – an insidious survivor from the infancy of FEA” [1]. The information contained in that paper is still very important for all users of finite element software but, unfortunately, the message has not been widely spread. The purpose of this present paper is to bring to the attention of the readers the contents of the paper presented at FEMSA ’92 and the results of further work that has been carried out: • Results from other FEA programs (Strand7 [12], Prokon [13]) were included and

compared with the results from Fesdec [11], • Other meshes and boundary conditions were investigated, • Other elements (plane stress, brick elements) were used. It should be noted that many of the 2-dimensional meshes used in this paper do not contain triangles only, but also triangles and quadrilaterals or quadrilaterals only. Compared with version 9, in this version 10 of this file all Prokon SF runs for shell elements were re-run on version 06/07/2001 of the Prokon SF program.




1. INTRODUCTION

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 (see fig. 0). The material properties are: Young’s modulus 3,0 x 107 kPa and Poisson’s ratio v = 0,2. The horizontal stresses at mid-span can easily be calculated manually using bending theory. They are –6000 kPa (compression) at the top and +6000 kPa (tension) at the bottom. Table 1 shows the results for various meshes from ref. [1]. In the “Element Type” column PS denotes plane stress elements, PN plane strain elements and SH shell elements. The definition of “Error” is as follows: Error = 100 * MAX ( ABS (1 + Sxx,top / 6000), ABS (1 - Sxx,bottom / 6000) ). The Fesdec results are identical for plane stress and shell elements, whereas both Strand7 and Prokon produce different results for different element types. The biggest differences in the “Error” column occur in fig. 1.05. The error for the run using the program Prokon Space Frame is only (!) 29 %, followed by 34% for Fesdec, 64% for Strand7 with shell elements and 119% for Strand7 with plane stress elements, where the program calculates tensile stresses not only at the bottom, but also at the top of the beam, where there is compression. It should be noted, however, that in fig. 1.05 the four-noded elements have been squashed into triangles, with one internal angle being 180 degrees whereas in most of the literature the recommended maximum internal angle is 135 degrees. One of our colleagues at Hatch, Bernard Potgieter, tried to run mesh 1.05 on MSC Nastran. The program however refused to analyse the structure because of the bad shape of the elements. Nevertheless, Prokon SF and Fesdec are very robust and can even handle this problem fairly well. Definitions for aspect ratio, skewness, tapers, convexity etc. as well as other important information can be found in ref. [2] to [10].

2. ADDITIONAL 2D-MESHES

Additional meshes were tested and the results were summarised in table 2. Here are some comments: Fig. 2.01: This skew mesh, consisting of two triangles and three quadrilaterals, surprisingly

yields more accurate results (9% error) than the 4 squares in fig. 1.04 (13% error). Although one can deduce that the skewness of the elements does not affect the accuracy of the results badly, one must not, however, come to the conclusion that skewness is beneficial for accuracy. The Sxx stress contours in the vicinity of the triangular elements do not look very good. Plots of these stresses have not been included in this paper to keep the number of pages as low as possible. Such plots, however, can be obtained from the authors on request.

Fig. 2.02: Again, this skew mesh gives better results (1% error) than the square elements

(13% error) – see fig. 1.08. It must be made clear to the reader that the “maximum error” is based purely on the maximum error in the horizontal stresses at mid-span. No other stresses were investigated, nor were the deflections.

Fig. 2.03: This mesh is a refinement of the mesh in fig. 1.05. Compared with that mesh, the

Fesdec results improve from a 34% to a 19% error, the Strand7 shell results from 64% to 37%, the Prokon results from 29% to 13%, but the Strand7 plane stress results become much worse, they change from 119% to 394%. To summarise, Fesdec and Prokon can handle these extreme element shapes relatively well, the




Strand7 shells give much greater errors and the Strand7 plane stress elements are unable to handle such distortions.

Fig. 2.04: It is not recommended to use elements with an aspect ratio greater than 4, but the

example shows that an additional layer of thin elements with an aspect ratio of 10 does not affect the accuracy of the Sxx stress results for Fesdec and Strand7.

Fig. 2.05: The comment for this figure is similar to the comment for fig. 2.04. Fig. 2.06: A big surprise! Who would have thought that the results would be much worse

than for the four square elements (fig. 1.04: 13%)? Fesdec now gives a 24% error, Prokon 25%, Strand7 shells 37% and Strand7 plane stress 43%. Such meshes are being promoted in literature to reduce the total number of nodes in a model, with a finer mesh at areas with higher stresses and a coarser mesh for less important areas. On page 63 in “Analytical modelling of structural systems” by Iain MacLeod [10] there is a very misleading rule: “Keep the number of nodes and elements to a minimum.” Why? There is no explanation given. The author must have been thinking in terms of available computer storage capacities forty years ago.

Take another example of misleading advice. On page 9 of “What every engineer should know about finite element analysis” by John R. Brauer [6] there is a plot of a complicated mesh of a rectangular section consisting of two different materials (see fig. 6.2). A simple rectangular mesh (fig. 6.3) would have been the logical answer, but that would have appeared too easy. How many users of FEA have been mislead by this picture, not knowing that quadrilateral elements are far superior to triangular ones? This is the curse of auto-meshing.

Fig. 2.07: The error of 97% is not unexpected. Fig. 2.08: To add triangular elements at the bottom of the beam is not a good idea. The error

increases from 13 to 23%. Fig. 2.09: In this model plane strain elements were used instead of plane stress elements.

The results are identical. Fig. 2.10: In this example both bottom corners were supported vertically and horizontally.

The difference in the stress results is therefore not surprising. An inexperienced user of FEA could easily make such a mistake (the right hand corner should not be supported horizontally). If the structure had been modelled with beam elements it would not have mattered for this load case whether the right hand support was only vertical or whether it was vertical and horizontal. When using two-dimensional elements, however, the additional horizontal support makes a big difference. Had the beam been loaded with a temperature load, the additional horizontal support would have also made a big difference if beam elements had been used.

Fig. 2.11: Compared with fig. 1.08 the error increases from 3% to 16% in the Fesdec run and

from 3% to 13% in the Strand7 run using shell elements. Fig. 2.12: For this badly distorted mesh the errors are much worse than for fig. 2.11: Fesdec

36% error, Strand7 using shell elements 43% error, Prokon 21% error. Fig. 2.13: If the two elements in fig. 2.12 with internal angles of more than 180 degrees are

“repaired” (angles changed to less than 180 degrees), the errors in Fesdec and Strand7 do not change much, whereas the error in Prokon drops from 21% to 9%.

Fig. 2.14: What a disaster! Compared with fig. 2.06, the Fesdec error doubles (from 24% to

48%), the Strand7 plane stress elements are useless for this example (116% error: compression at the bottom of the beam instead of tension), the error for the




Strand7 shell elements shoots up from 37% to 66% and only the Prokon elements behave relatively well (error increasing from 25% to 28%).

Fig. 2.15: Compared with fig. 2.14, the results are much more accurate. Fig. 2.16: The large errors (53%), caused by the triangular elements, are not unexpected. Fig. 2.17: Compared with fig. 2.16, the errors drop from 53% to 34%. Fig. 2.18: The errors are about the same as for fig. 2.06. Fig. 2.19: Due to the triangular elements the results are worse than for fig. 2.18. Fig. 2.20: As expected for triangular elements, the errors are very high (77%).

3. FURTHER 2D-MESHES

Further meshes were tested and the results were summarised in table 3. Here are some comments: Fig. 3.01: Compared with the regular 8x2 square grid (see fig. 1.08) with an error of 3% the

error in Fesdec is unchanged and the error for the Strand7 plane stress elements is remarkably high (55%). The Strand7 shells perform well (6%) and the Prokon shells have an error of 9%.

Fig. 3.02: The shell elements of all three programs give errors of less than 10%. Fig. 3.03: Fesdec cannot handle this type of mesh well (15 square elements, 1 triangular

quadrilateral, 2 triangles) – the error is 37%. The Strand7 shell model has an error of only 8%.

Fig. 3.04: All results have errors between 13 and 16%. Fig. 3.05: Compared with fig. 1.09 (error 72%) this two-layer triangular mesh behaves

similarly (error 70%). Fig. 3.06: Compared with fig. 3.05, this mesh yields more accurate results (43% error). This

is quite surprising. Fig. 3.07: The results from this mesh are similar to the results from fig. 1.04 (4 square

elements, 13% error). Fig. 3.08: The results for this mesh are much more accurate than those for fig. 3.07 (error 1

to 2% only). Fig. 3.09: The results of this model (plane strain) are identical to the results of fig. 1.08

(plane stress), the error is 3%. Fig. 3.10: The errors for this four-layer triangular grid are still two to three times higher than

the errors for one layer of square elements. Fig. 3.11: The errors are 30%, compared with 37% for fig. 3.10. Fig. 3.12: Compared with fig. 3.11, the maximum error drops from 30% to approximately

23%.




4. 3D-MESHES

The meshes are shown in fig. 8 and 9 and the results are summarised in table 4. To create a meaningful test for brick elements, the beam thickness has been increased from 0,200m to 1,000m. Prokon has presently, as solid elements, only 8-noded brick elements; 6-noded wedge elements are in preparation. If Prokon’s 8-noded bricks are shaped into wedges (forming 4-noded triangles on two surfaces), the Prokon program refuses to carry out the analysis. Fig. 4.01: Similar to fig. 1.02. Fig. 4.02: Similar to fig. 1.03. Fig. 4.03: Similar to fig. 1.04. Fig. 4.04: Compared to fig. 1.05, the error increases from 34% to 41% for Fesdec and the

Strand7 elements produce errors of more than 100%. Fig. 4.05: The results are similar to fig. 1.08. Fig. 4.06: The results are similar to fig. 1.09. Fig. 4.07: The results are similar fig. 2.01. Fig. 4.08: The error is only 1%. Fig. 4.09: The Fesdec run has an error of 18%. Strand7 cannot handle such a problem and

produces an error of 185%. Fig. 4.10: The errors for Fesdec and Strand7 are 14%, compared to 13% for fig. 4.03. The

error for Prokon is 10%, twice as much as for fig. 4.03. Fig. 4.11: The results are similar to fig. 4.10. Fig. 4.12: Only 6-noded wedges were used and one layer has aspect ratios of 10. The errors

are close to 100%. Similar to fig. 2.07. Fig. 4.13: Similar to fig. 2.08.

5. ADDITIONAL 3D-MESHES

The meshes in table 4 had only two nodes across the 1.000 m width of the beam. The meshes in table 5 have two to four nodes. For the calculation of the “error”, the most inaccurate of the stresses across the width of the beam was used. Fig. 5.01: The mesh pattern is the same as the mesh pattern in fig. 4.01. The error however,

is reduced dramatically if the elements are loaded on the triangular faces (error 37%) compared to a loading on the quadrilateral faces (100%).

Fig. 5.02: The errors are the same as for fig. 5.01. Fig. 5.03: Compared with fig. 4.04, there is a big improvement (in Fesdec from 41% to 26%,

in Strand7 from 117% to 5%).




Fig. 5.04: Compared with fig. 4.07, the results are worse (the Fesdec error changes from 10% to 21%, the Strand7 error from 16% to 21%).

Fig. 5.05: The errors are the same as for fig. 4.10. Fig. 5.06: The errors are similar to the errors in fig. 4.11. Fig. 5.07: Compared with fig. 4.12, there is a big improvement (in Fesdec from 97% to 38%,

in Strand7 from 98% to 37%. Fig. 5.08: Compared with fig. 4.13, there is an improvement from 23% to 17%. Fig. 5.09: Compared with fig. 5.08, the error increases from 17% to 30%. This is unexpected.

One would rather expect an improvement because of the lower aspect ratio of the wedge elements.

Fig. 5.10: Compared with fig. 4.03, the errors are almost unchanged.

6. CONCLUSIONS AND COMMENTS 6.1 Triangular 3-noded shells should be avoided if they are loaded in such a way that they

produce mainly midplane stresses (Sxx, Syy, Sxy). If they are loaded such that they produce mainly bending moments (Mxx, Myy, Mxy), these elements perform much better. They also perform much better in heat transfer and similar applications.

It is however, not always easy to predict whether elements will be stressed more in bending or by mid-plane stresses. Changes in the model may drastically change stress patterns and additional load cases might produce high direct stresses, where in previous load cases elements were mainly loaded in bending. To be safe, triangular 3-noded elements should only be used as fillers between quadrilateral elements.

6.2 4-noded quadrilateral shell elements are usually good elements, except in programs

where such elements are invisible to the user, subdivided into four triangular elements internally.

The best performance is achieved by square elements. Try to keep the internal angles close to 90 degrees (maximum 135 degrees) and keep the aspect ratio small (maximum 4).

Certain programs (e.g. Prokon) can even handle quadrilateral elements with internal

angles of more than 180 degrees. However, checks should be built into all programs to restrain models with internal angles greater than 180 degrees from execution.

Different programs behave differently. Shapes, like in fig. 2.01, can be handled well by

Fesdec and Strand7, but not so well by Prokon. For this reason it is advisable that all important analyses are checked by using different software.

Mesh refinement as in figs. 2.06, 2.14, 2.15 and 2.18 causes considerable errors (in some

cases more than 50%). To use a fine model for the whole mesh does not appear as sophisticated, but the results are much more accurate.

6.3 6-noded triangular shell elements are very accurate (more accurate than 4-noded

quadrilaterals) but they also have their disadvantages, especially with regard to their mid-nodes:

• No corner nodes (e.g. from beam elements) should be connected to mid-nodes. • Mid-nodes should be at the edge centres and not off centre. • Mid-nodes should not be loaded by point loads. • If mid-nodes are supported, the reaction forces are wrong.




It is much safer to use 4-noded quadrilaterals instead of 6-noded triangles. 6.4 8-noded quadrilateral shell elements.

The same comments as for 6.3 apply.

6.5 6-noded wedge elements. If loaded by pressure on one of the three 4-noded surfaces, they behave badly (with

errors up to 100%). If loaded by pressure on one of the two 3-noded surfaces, they behave better, but models should be built such that reasonable accuracy will be achieved for all load cases. Therefore, six-noded wedge elements should only be used as fillers between 8-noded brick elements.

6.6 8-noded brick elements. These elements give more accurate results than the 6-noded wedges. The internal angles

on the quadrilateral surfaces should be limited to 135 degrees although certain programs (e.g. Fesdec) can even handle angles of 180 degrees.

It should be noted that the error for the two-dimensional model in fig. 1.05, is 34% and for the same model in three dimensions (see fig. 4.04) the error is 41%.

7. SUMMARY Different programs give different results and often different versions of a program give

different results and often the users are not being informed of these changes. Some programs can handle internal angles of 180 degrees, others fail to do so. To increase the number of nodes and elements in a model is not sufficient to increase its

accuracy. Fig. 1.04, with 10 active nodes gives an error of 13% whereas fig. 2.18 with 39 active nodes yields a 27% error. A mesh consisting of 8x2 square elements and 9x3=27 nodes (see fig 1.08) only creates an error of 3%.

This paper is not a complete entity. It can be improved and extended (including 15-noded

wedges, 20-noded bricks and 4- and 10-noded tetrahedrons). It should, however, give the reader, the inexperienced as well as the experienced, valuable information and guidance. Hopefully it will trigger further research at universities and technikons. Without such research Finite Element Analysis is a dangerous tool even in the hands of experts.

8. ACKNOWLEDGEMENTS

A great thank you to all our colleagues whose keen interest in the subject has inspired this paper.




9. REFERENCES

[1] The CST (Constant Strain Triangle) – An insidious survivor from the infancy of FEA, by R.P. Prukl, FEMSA92 Symposium, 11th Symposium on Finite Element Methods in South Africa, 15-17 January 1992, organised by FRD/UCT Centre for Research in Computational and Applied Mechanics, University of Cape Town

[2] A new look at element shape parameters, by John Robinson, April 1985, NAFEMS [3] Basic and shape sensitivity tests for membrane and plate bending finite elements,

by John Robinson, April 1985, NAFEMS [4] An introduction to hierarchical displacement elements and the adaptive technique,

by John Robinson, November 1986, NAFEMS [5] Distortion measures for quadrilaterals with curved boundaries, by John Robinson,

August 1987, NAFEMS [6] What every engineer should know about finite element analysis, edited by John R.

Brauer, CAD Comp. Inc., Milwaukee, Wisconsin, Marcel Dekker Inc., New York and Basel

[7] Distortion effects in isoparametric elements, an analytic approach, by John Barlow,

Proceedings of NAFEMS AGM / Technical Session, 1987 [8] The standard NAFEMS benchmarks, October 1989, NAFEMS [9] Shell finite element evaluation tests, by, A.J. Morris, August 1985, NAFEMS [10] Analytical modelling of structural systems, by Iain A. MacLeod, 1990, Ellis

Horwood [11] Fesdec, General Finite Element Program from Cape Engineering UK Limited,

Birmingham Road,Warwick, England CV34 4TX [12] Strand7, General Finite Element Program from G+D Computing, Suite1, Level 7,

541 Kent Street, Sydney NSW 2000, Australia [13] Prokon, Program Module Space Frame (SF) from Prokon Software Consultants,

P.O Box 17295, Groenkloof 0027, South Africa.







Sxx Stress (kPa) Fig. No.

Elevation

Program

Name

Element

Type Mid-Span

Top Mid-Span Bottom

Error

%

1.01 Bending Theory

- 6000

+ 6000

0

1.02

FESDEC STRAND7 PROKON

PS SH SH

0 0 0

+ 1347 + 1347 + 2080

100 100 100

1.03

FESDEC

PS - 426 + 424 93

1.04

FESDEC STRAND7 STRAND7 PROKON

PS PS SH SH

- 5250 - 5250 - 5323 - 5410

+ 5250 + 5250 + 5177 + 5530

13 13 14 10

1.05

FESDEC STRAND7 STRAND7 PROKON

PS PS SH SH

- 3940 + 1144 - 2147 - 4260

+ 6772 + 4306 + 4211 + 5290

34 119 64 29

1.06

FESDEC STRAND7

PS PS

- 5962 - 5764

+ 5978 + 5959

1 4

1.07

FESDEC STRAND7

PS PS

- 6022 - 6187

+ 6275 + 6187

5 3

1.08

FESDEC FESDEC

STRAND7 STRAND7 PROKON PROKON

PS SH PS SH PS SH

- 5802 - 5802 - 5802 - 5842 - 6093 - 5990

+ 5795 + 5795 + 5795 + 5856 + 6086 + 5950

3 3 3 3 1 1

1.09

FESDEC STRAND7 STRAND7

PS PS SH

- 1683 - 1683 - 1683

+ 3289 + 3289 + 3289

72 72 72

Table 1

• element corner nodes

element mid-nodes Table 1 is a modified and shortened copy of table 1 in reference [1]. See references [11], [12] and [13].




Sxx Stress (kPa)

Fig. No.

Elevation

Program

Name

Element

Type Mid-Span

Top Mid-Span Bottom

Error

%

2.01

FESDEC STRAND 7 PROKON

PS PS SH

- 5436 - 5436 - 4650

+ 5486 + 5486 + 4920

9 9

23

2.02


PS PS SH

- 6043 - 6043 - 5440

+ 6066 + 6066 + 5450

1 1 9

2.03

FESDEC STRAND 7 STRAND 7 PROKON

PS PS SH SH

- 4836 + 14274

- 3776 - 5230

+ 6301 + 29639

+ 5851 + 6030

19 394 37 13

2.04


PS PS SH

- 5232 - 5232 - 5710

+ 5331 + 5331 + 4860

13 13 19

2.05

FESDEC STRAND 7

PS PS

- 5401 - 5401

+ 5314 + 5314

11 11

2.06


PS PS SH SH

- 4531 - 3639 - 3801 - 4520

+ 5293 + 3419 + 5607 + 5820

24 43 37 25

2.07

FESDEC STRAND 7

PS PS

- 167 - 166

+ 1670 + 1670

97 97

2.08

FESDEC STRAND 7

PS PS

- 4664 - 4664

+ 4643 + 4643

23 23

2.09

FESDEC STRAND 7

PN PN

- 5250 - 5250

+ 5250 + 5250

13 13

2.10

FESDEC STRAND 7

PS PS

- 3325 - 3325

+ 1400 + 1400

77 77

Table 2.A

Comments: Fig. 2.09: Plane strain elements. Fig. 2.10: Both bottom corners supported vertically and horizontally. Note: The thicknesses of the thin layers in fig. 2.04, fig. 2.05, fig. 2.07 and fig. 2.08 are

0.2m. For better clarity, these thicknesses have been increased in these graphs.




Sxx Stress (kPa) Fig. No.

Elevation

Program

Name

Element

Type Mid-Span

Top Mid-Span Bottom

Error

%

2.11


PS PS SH SH

- 5046 - 5532 - 5237 - 5880

+ 5729 + 5348 + 5640 + 5960

16 11 13

2 2.12


PS PS SH SH

- 3445 - 4604 - 3850 - 4720

+ 4958 + 4562 + 4860 + 4950

43 32 36 21

2.13


PS PS SH SH

- 3425 - 4198 - 3807 - 5470

+ 4990 + 4458 + 4844 + 5840

43 30 37

9 2.14


PS PS SH SH

- 3109 - 1605 - 2020

- 4320

+ 3570 - 981

+ 4851 + 6090

48 116 66 28

2.15


PS PS SH SH

- 6289 - 5959 - 5359 - 5510

+ 4444 + 4525 + 4676 + 5480

26 25 22

9 2.16

FESDEC STRAND 7

PS PS

- 3112 - 3112

+ 2849 + 2849

53 53

2.17

FESDEC STRAND 7

PS PS

- 4096 - 4096

+ 3955 + 3955

34 34

2.18


PS SH SH

- 4384 - 3792 - 4480

+ 5192 + 5175 + 5720

27 37 25

2.19

FESDEC STRAND 7

PS PS

- 3730 - 3748

+ 5181 + 5197

38 38

2.20

FESDEC STRAND 7

PS PS

- 1387 - 1388

+ 1413 + 1412

77 77

Table 2.B




Sxx Stress (kPa)

Fig. No.

Elevation

Program

Name

Element

Type Mid-Span

Top Mid-Span Bottom

Error

%

3.01


PS PS SH SH

- 5818 - 5159 - 5801 - 5990

+ 5982 + 9311 + 5606 + 5460

3 55

6 9

3.02


PS PS SH SH

- 5628 - 5627 - 5621 - 6410

+ 5581 + 5446 + 5693 + 5950

7 9 6 7

3.03


PS PS SH SH

- 5549 - 5806 - 5756 - 6850

+ 3772 + 4425 + 5545 + 4690

37 26

8 22

3.04


PS PS SH SH

- 5578 - 5579 - 5589 - 6770

+ 5030 + 5030 + 5018 + 5340

16 16 16 13

3.05

FESDEC STRAND 7

PS SH

- 1779 - 1779

+ 1779 + 1779

70 70

3.06

FESDEC STRAND 7

PS SH

- 3441 - 3441

+ 3467 + 3466

43 43

3.07

FESDEC STRAND 7

PS SH

- 5305 - 5310

+ 5304 + 5190

12 14

3.08

FESDEC STRAND 7

PS SH

- 5953 - 5859

+ 5953 + 5859

1 2

3.09

FESDEC STRAND 7

PN PN

- 5798 - 5798

+ 5791 + 5791

3 3

3.10


PS PS SH SH

- 3792 - 3792 - 3792 - 4490

+ 4938 + 4938 + 4937 + 5150

37 37 37 25

3.11

FESDEC STRAND 7 STRAND 7

PS PS SH

- 4418 - 4418 - 4389

+ 4196 + 4196 + 4121

30 30 31

3.12

FESDEC STRAND 7 STRAND 7

PS PS SH

- 4470 - 4770 -4770

+ 4667 + 4667 + 4588

22 22 24

Table 3




Sxx Stress (kPa)

Fig. No.

Elevation

Program

Name

Element

Type Mid-Span

Top Mid-Span Bottom

Error

%

4.01


WEDGE6 WEDGE6 WEDGE6

- 25 + 44 NYA

+ 1355 + 1280

100 100

4.02



- 462 - 40

NYA

+ 456 + 367

92 99

4.03


BRICK8 BRICK8 BRICK8

- 5192 - 5192 - 5730

+ 5192 + 5192 + 5730

13 13

5

4.04



- 3995 + 1035

DNR

+ 3564 + 4432

41 117

4.05



- 5796 - 5796 - 6090

+ 5783 + 5783 + 6070

4 4 1

4.06



- 1712 - 1080

NYA

+ 3307 + 3222

71 82

4.07


BR8+WE6 BR8+WE6 BR8+WE6

- 5389 - 5489

NYA

+ 5437 + 5015

10 16

4.08



- 6036 - 6036

NYA

+ 6057 + 6038

1 1

4.09



- 5087 + 5085

DNR

+ 4918 + 13119

18 185

4.10



- 5179 - 5179 - 6340

+ 5328 + 5328 + 5400

14 14 10

4.11



- 5401 - 5401 - 5840

+ 5310 + 5310 + 6010

12 12

3

4.12



- 174 - 112 NYA

+ 1650 + 1648

97 98

4.13



- 4608 - 4608

NYA

+ 4638 + 4632

23 23

Table 4

Note: The thickness of the thin layers in fig. 4.10 - fig. 4.13 are 0.2m. For better clarity, these thicknesses have been increased in these graphs.




Sxx Stress (kPa)

Fig. No.

Plan

Program

Name

Element

Type Mid-Span

Top Mid-Span Bottom

Error

%

5.01



- 3844 - 3860

NYA

+ 3785 + 3809

37 37

5.02



- 3828 - 3827

NYA

+ 3754 + 3754

37 37

5.03

FESDEC STRAND 7

BRICK8 BRICK8

- 4551 - 6152

+ 4468 + 5689

26 5

5.04



- 4859 - 4859

NYA

+ 4762 + 4762

21 21

5.05



- 5182 - 5182 - 5750

+ 5184 + 5184 + 5750

14 14

4

5.06



- 5182 - 5182 - 5625

+ 5184 + 5184 + 5625

14 14

6

5.07



- 3805 - 3807

NYA

+ 3720 + 3755

38 37

5.08



- 4991 - 4991

NYA

+ 4976 + 4975

17 17

5.09



- 4261 - 4255

NYA

+ 4230 + 4229

30 30

5.10



- 5181 - 5181 - 5720

+ 5181 + 5185 + 5720

14 14

5

Table 5 Note: The thicknesses of the thin layers in fig. 5.05 – fig. 5.08 are 0.1m. For better clarity, these thicknesses have been increased in these graphs. NYA – Not yet available DNR – Did not run


