shear capacity of circular concrete...

Shear Capacity of Circular Concrete Sections

Final Year Dissertation

Department of Architecture and Civil Engineering

Accompanying CD notes

John Orr (050308794)

MEng Civil Engineering

University of Bath

20th April 2009

dissertation: john orr i

Contents

1. Introduction .................................................................................................................................................................... 11.1. Outline........................................................................................................................................................................................11.2. Dissertation and Showcase Poster .............................................................................................................................................11.3. Appendices ................................................................................................................................................................................11.4. Spreadsheets .............................................................................................................................................................................11.5. Discussion..................................................................................................................................................................................2

2. Appendices and Website............................................................................................................................................... 32.1. Introduction ................................................................................................................................................................................32.2. Appendix 4.................................................................................................................................................................................32.3. Website ......................................................................................................................................................................................3

3. Spreadsheet details ....................................................................................................................................................... 43.1. Spreadsheet 3.1.........................................................................................................................................................................43.2. Spreadsheet 6.1.........................................................................................................................................................................43.3. Spreadsheet 6.2.........................................................................................................................................................................53.4. Spreadsheet 6.3.........................................................................................................................................................................53.5. Spreadsheet 6.4.........................................................................................................................................................................53.6. Spreadsheet 6.5.........................................................................................................................................................................53.7. Spreadsheet 6.6.........................................................................................................................................................................63.8. Spreadsheet 6.7.........................................................................................................................................................................83.9. Spreadsheet 6.8.........................................................................................................................................................................93.10. Spreadsheet 6.9.......................................................................................................................................................................93.11. Spreadsheet 6.10...................................................................................................................................................................103.12. Spreadsheet 6.11...................................................................................................................................................................103.13. Spreadsheet 6.12...................................................................................................................................................................103.14. Spreadsheet 6.14...................................................................................................................................................................113.15. Spreadsheet 6.15...................................................................................................................................................................113.16. Spreadsheet 6.16...................................................................................................................................................................113.17. Spreadsheet 6.17...................................................................................................................................................................113.18. Spreadsheet 6.18...................................................................................................................................................................123.19. Spreadsheet 6.19...................................................................................................................................................................133.20. Spreadsheet 6.20...................................................................................................................................................................143.21. Spreadsheet 6.21...................................................................................................................................................................143.22. Spreadsheet 6.22...................................................................................................................................................................153.23. Spreadsheet 7.1.....................................................................................................................................................................15

4. Summary....................................................................................................................................................................... 164.1. Conclusions .............................................................................................................................................................................16

5. Bibliography.................................................................................................................................................................. 17

dissertation: john orr 1

1. Introduction

1.1. Outline This document presents notes and supplementary information for the data on the enclosed CD. The following information is provided:

• The full dissertation;

• A1 Showcase Poster;

• Appendices 1 to 4;

• Design Spreadsheets;

• A copy of the accompanying website.

1.2. Dissertation and Showcase Poster The full dissertation is provided as a bookmarked .pdf file. The dissertation showcase poster, which should be printed at 300dpi

on A1 poster paper, is also provided as a .pdf.

1.3. Appendices The four appendices are summarised in Table 1.1. The test data is presented on .xls files, making it easy to use in future work. All data is referenced, and should be checked against the original source before re-use.

Table 1.1: Appendices included on enclosed CD. Number Summary

1 Test data from Capon and De Cossio (1966).

2 Test data from Clarke and Birjandi (1993).

3 Test data from Collins (2002).

4 MCFT analysis files and data.

1.4. Spreadsheets Spreadsheets have been used extensively throughout this project. Some of them have proved useful and are included on this

CD. They are numbered according to the Chapter in which they are principally used. Test data compiled throughout the project is included here also. Table 1.2 presents a summary of all the spreadsheets and their titles.

Table 1.2: Spreadsheets detailed in this document. Number Name (.xls) Summary

3.1 BCA Comparisons A comparison of three design approaches, considering Arup, BCA and BD74 design guidance.

6.1 Variable angle truss Analysis of the proposed variable angle truss model, following work by Feltham (2004). The effect of link

yield strength is considered.

6.2 Shear Enhancement The effect of different shear span limits when using the variable angle truss model. User variables include fy,

cot(θ) and the value of av/d.

6.3 Yield Stress Limits Analysis of taking fy = 250 or 500MPa for data from Clarke (1993), including a comparison between BS5400

and BS EN 1992-1-1.

6.4 Variable angle model (2) Analysis of the second proposed model, after Turmo et al. (2008). Two design cases, including spirally

reinforced sections, are considered.

6.5 Integration A numerical integration of the formulae presented by Turmo et al., 2008 to verify their approach.

6.6 Crushing The proposed crushing method for EN 1992. Full analysis of the method described in the dissertation is

presented.

6.7 Simple Plasticity Simplified method for upper bound plasticity analysis of circular sections. Data presented for experimental

results from Clarke (1993).

6.8 Extended Plasticity The extended method, including the effect of longitudinal steel, and optimisation of the failure plane angle.

6.9 Full UB analysis Rotation method for upper bound analysis of circular sections. Values of rotation and horizontal

displacement are varied for each section to obtain the most accurate results.

6.10 Simple concrete plane Simple analysis of energy dissipated on concrete plane (upper-bound plasticity), for a faceted plane

geometry.

6.11 Extended Concrete Plane Analysis of a curved plane, defined by a cubic equation. Values are varied by the user to obtain the lowest

possible value for EDc.

university of bath: department of architecture and engineering section one: introduction


Number Name (.xls) Summary

6.12 Plastic method comparison Comparison of the four methods presented for upper bound analysis of circular sections. All members from

Clarke (1993) are considered.

6.14 Energy Analysis Analysis of the energy dissipated in the concrete and steel terms (upper bound plasticity approach).

6.15 Unreinforced sections Analysis of unreinforced sections.

6.16 Econ Analy Graphs to accompany economic considerations presented in the dissertation.

6.17 Economics Extend Additional analysis of both BS5400 and BS EN 1992-1-1 approaches to circular section design.

6.18 Force crushing Analysis of the section reinforcement required to force crushing failures in both rectangular and circular

sections.

6.19 ATF Extensive analysis spreadsheet to determine the additional tensile force in circular sections, required to

satisfy equilibrium.

6.20 Comparisons Very simple comparison between BS5400 and BS EN 1992-1-1. The analysis compares an imaginary

section, and considers how much reinforcement is required to resist increasing shear loads.

6.21 ATF Analysis This spreadsheet presents analysis of two methods for determining the additional tensile force in circular

sections.

6.22 Steel Term Analysis Analysis of the contributions of the ‘steel term’ type equations in the upper and lower bound plasticity

methods.

7.1 Final Graphs Final analysis of the proposed truss equations as presented in the Conclusions.

1.5. Discussion Each spreadsheet includes a ‘Notes’ page, where the analysis method used is described in more detail. All spreadsheets have been created on Microsoft Excel for Mac ‘X’. There should, theoretically, be no issues when opening the Spreadsheets on a PC. However, graphs are often given additional trend lines, and Windows tends to alter font sizes and styles. A set of

spreadsheets are provided that have been opened using a ‘Virtual PC’ running Office 2009, and can be found on the enclosed CD if needed (‘PC Versions.zip’).

Compatibility issues should be addressed to the author directly, by emailing [email protected] and every effort will be made to solve any problems with Windows machines.


2. Appendices and Website

2.1. Introduction Appendices 1-3 provide test data only and are therefore self-explanatory (Figure 2.1). Appendix 4 is outlined in more detail in

§2.2.

Figure 2.1: Appendices layout

2.2. Appendix 4 The analysis of circular sections using the MCFT approach was undertaken for the test data provided in Clarke (1993). This was carried out using the ‘Response 2000’ computer program, which is freely available from the University of Toronto. Appendix 4

provides the complete set of ‘.rsp’ files that were made to carry out the analysis. Both reinforced and un-reinforced section data is provided, and the files are numbered according to the Specimen number (Clarke, 1993). The full data is summarised in the

spreadsheets that accompany these files.

2.3. Website The website set up to accompany this dissertation is available from:

h t t p : / / p e o p l e . b a t h . a c . u k / j j o 2 0 / d _ a

The website files are included on the CD in a compressed folder, in case the website server should be unavailable for any

reason.

Appendix 1

Appendix 2

Appendix 3

Appendix 4

!

Test data

Response 2000 MCFT files

http://people.bath.ac.uk/jjo20/d_a


3. Spreadsheet details

3.1. Spreadsheet 3.1 Spreadsheet 3.1 compares three previously used methods to analyse circular sections. This spreadsheet is very simple and

requires little explanation. Results confirm previous analysis by Feltham (2004).

Figure 3.1: Screenshot.

3.2. Spreadsheet 6.1 This spreadsheet analyses the proposed Feltham-variable angle truss model and includes the following data sets:

Sheet Name Details

(1) Analysis for members with flat links only, using fy = 250MPa, Gamma = 1.0; shear enhancement < 2d

(2) Analysis for members with flat links only, using fy = 500MPa Gamma = 1.0; shear enhancement < 2d

(3) Analysis for members with flat links only, using fy = 250MPa, Gamma = 1.0; shear enhancement < 3d

(4) Analysis for members with flat links only, using fy = 500MPa Gamma = 1.0; shear enhancement < 3d

(A) Analysis for members with flat links only, using fy = User Choice; Gamma = User Choice; Shear enhancement < User Choice

Each sheet is essentially the same, and has the layout shown in Figure 3.2.

Figure 3.2: Screenshot of 6.1.

16/3/09

TITLE

Pile information

b 900 mm

BCA 682.37 mm

Recommended 682.37 mm

BD74 680.45 mm

r 450 mm

fcu 30 N/mm2

Ast 3141.592654

As 6283.185307 mm2

sv 240 mm

cover 75 mm

fyv 460 N/mm2

fyv,2 16 N/mm2

gamma mc 1

gamma ms 1

BD74 Xi 0.93 rs 365

vc 0.6716 alpha 0.54 rad vc 0.7104

Xivc 0.62 d 682.37

Av 517509.69 mm^2 As

from BS8110 Table 3.9vc 0.62 Concrete area 517509.69

xi 0.93

vcxi 0.66

V (kN) v Area (mm^2) v Area (mm2) Design shear given by: v define Area (mm2)

0 0.00 Nominal 50.00 0.00 Nominal 50.00 0.00 Nominal 50.00 0.00

















170 0.28 Nominal 50.00 0.33 Minimum 187.83 0.33 Nominal 187.83 129.98

180 0.29 Nominal 50.00 0.35 Minimum 187.83 0.35 Minimum 187.83 137.63

190 0.31 Nominal 50.00 0.37 Minimum 187.83 0.37 Minimum 187.83 145.27

200 0.33 Minimum Links 187.83 0.39 Minimum 187.83 0.39 Minimum 187.83 152.92







BD74 ARUP BCA

d

Shear Capacity of Circular ColumnsFinal Year Dissertation, John Orr

Supporting Spreadsheets

Analysis of three design method for circular sections

!

Asv "2rsv v # vc( )

fyv

Section data and specifications

BD74 method Arup method BCA method

User Variable:

10/12/08

fy(k) 250 N/mm^2

TITLE Gamma s 1

fy 250

Cot Theta 1

Taken from Clarke/Birjandi paper, IStructE 1993

Shear span limit 2

Please read notes.

SpecimenSection

Diameter,

(mm)

Minimum

cover,

(mm)

fcu

(N/mm2) Fi (mm) As (%) FIsv Lsv (mm) Asv/Sv av (mm) Pax (kN)

Failure

ModeVa (kN) Vth (kN)

Va/Vth

(BCA)r rs rsv Alpha Beta As (mm

2) ASV s bw d r(1+sina/1+sinB) (pi/2+b+sinBcosB) Shear

EnhancementVa/Vth Cot Theta 1 1.4 1.8 2.2 2.4 2.5 Va/Vth 1 1.4 1.8 2.2 2.4 2.5

M1/2 152 10 28.00 8.00 2.2 6 Link 100 0.57 310 - S 45.00 35.90 1.25 76 56.00 63 0.49 0.60 399 57 100 152.00 111.65 70.69 71.30 2.64 13.30 1.00 13.30 3.38 13.30 18.62 23.94 29.26 31.92 33.25 3.38 2.42 1.88 1.54 1.41 1.35

M1/3 152 10 28.00 8.00 2.2 6 Link 100 0.57 330 - S 46.00 34.70 1.33 76 56.00 63 0.49 0.60 399 57 100 152.00 111.65 70.69 71.30 2.64 13.30 1.00 13.30 3.46 13.30 18.62 23.94 29.26 31.92 33.25 3.46 2.47 1.92 1.57 1.44 1.38

M1/4 152 10 28.00 8.00 2.2 6 Link 100 0.57 340 - S 38.00 34.60 1.10 76 56.00 63 0.49 0.60 399 57 100 152.00 111.65 70.69 71.30 2.64 13.30 1.00 13.30 2.86 13.30 18.62 23.94 29.26 31.92 33.25 2.86 2.04 1.59 1.30 1.19 1.14

7 300 20 34.40 16.00 2.3 8 Link 150 0.67 330 - S 262.00 175.20 1.50 150 114.00 126 0.51 0.61 1626 101 150 300.00 222.57 83.78 141.23 2.66 31.42 1.82 57.12 4.59 31.42 43.99 56.55 69.12 75.40 78.55 8.34 5.96 4.63 3.79 3.47 3.34

15 300 20 24.30 20.00 3.6 8 Link 150 0.67 660 - S 145.00 107.80 1.35 150 112.00 126 0.50 0.60 2545 101 150 300.00 221.30 83.78 141.33 2.64 31.24 1.00 31.24 4.64 31.24 43.74 56.24 68.74 74.98 78.11 4.64 3.31 2.58 2.11 1.93 1.86

300 20 24.30 20.00 3.6 8 Link 150 0.67 660 - S 148.00 107.80 1.37 150 112.00 126 0.50 0.60 2545 101 150 300.00 221.30 83.78 141.33 2.64 31.24 1.00 31.24 4.74 31.24 43.74 56.24 68.74 74.98 78.11 4.74 3.38 2.63 2.15 1.97 1.89

16 300 20 46.70 20.00 3.6 8 Link 150 0.67 660 - S 185.00 124.90 1.48 150 112.00 126 0.50 0.60 2545 101 150 300.00 221.30 83.78 141.33 2.64 31.24 1.00 31.24 5.92 31.24 43.74 56.24 68.74 74.98 78.11 5.92 4.23 3.29 2.69 2.47 2.37

16b 300 20 46.70 20.00 3.6 8 Link 150 0.67 660 - S 186.00 124.90 1.49 150 112.00 126 0.50 0.60 2545 101 150 300.00 221.30 83.78 141.33 2.64 31.24 1.00 31.24 5.95 31.24 43.74 56.24 68.74 74.98 78.11 5.95 4.25 3.31 2.71 2.48 2.38

11 300 20 24.10 25.00 5.6 8 Link 150 0.67 660 - S 186.00 117.80 1.58 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 6.00 31.02 43.43 55.84 68.25 74.46 77.56 6.00 4.28 3.33 2.73 2.50 2.40

300 20 24.10 25.00 5.6 8 Link 150 0.67 660 - S 188.00 117.80 1.60 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 6.06 31.02 43.43 55.84 68.25 74.46 77.56 6.06 4.33 3.37 2.75 2.53 2.42

12 300 20 23.80 25.00 5.6 8 Link 75 1.34 660 - S 211.00 154.30 1.37 150 109.50 126 0.48 0.59 3958 101 75 300.00 219.71 167.55 141.45 2.62 62.05 1.00 62.05 3.40 62.05 86.86 111.68 136.50 148.91 155.11 3.40 2.43 1.89 1.55 1.42 1.36

300 20 23.80 25.00 5.6 8 Link 75 1.34 660 - S 239.00 154.30 1.55 150 109.50 126 0.48 0.59 3958 101 75 300.00 219.71 167.55 141.45 2.62 62.05 1.00 62.05 3.85 62.05 86.86 111.68 136.50 148.91 155.11 3.85 2.75 2.14 1.75 1.60 1.54

13 300 20 48.40 25.00 5.6 8 Link 150 0.67 660 - S 227.00 139.30 1.63 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 7.32 31.02 43.43 55.84 68.25 74.46 77.56 7.32 5.23 4.07 3.33 3.05 2.93

300 20 48.40 25.00 5.6 8 Link 150 0.67 660 - S 228.00 139.30 1.64 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 7.35 31.02 43.43 55.84 68.25 74.46 77.56 7.35 5.25 4.08 3.34 3.06 2.94

14 300 20 50.50 25.00 5.6 8 Link 75 1.34 660 - S 279.00 177.30 1.57 150 109.50 126 0.48 0.59 3958 101 75 300.00 219.71 167.55 141.45 2.62 62.05 1.00 62.05 4.50 62.05 86.86 111.68 136.50 148.91 155.11 4.50 3.21 2.50 2.04 1.87 1.80

300 20 50.50 25.00 5.6 8 Link 75 1.34 660 - S 288.00 177.30 1.62 150 109.50 126 0.48 0.59 3958 101 75 300.00 219.71 167.55 141.45 2.62 62.05 1.00 62.05 4.64 62.05 86.86 111.68 136.50 148.91 155.11 4.64 3.32 2.58 2.11 1.93 1.86

37 300 20 43.90 25.00 5.6 8 Link 150 0.67 660 270.9 S 232.00 154.70 1.50 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 7.48 31.02 43.43 55.84 68.25 74.46 77.56 7.48 5.34 4.15 3.40 3.12 2.99

300 20 43.90 25.00 5.6 8 Link 150 0.67 660 - S 218.00 135.80 1.61 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 7.03 31.02 43.43 55.84 68.25 74.46 77.56 7.03 5.02 3.90 3.19 2.93 2.81

38 300 20 36.10 25.00 5.6 8 Link 150 0.67 660 270.9 S 209.00 147.30 1.42 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 6.74 31.02 43.43 55.84 68.25 74.46 77.56 6.74 4.81 3.74 3.06 2.81 2.69

300 20 36.10 25.00 5.6 8 Link 150 0.67 660 - S 206.00 129.50 1.59 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 6.64 31.02 43.43 55.84 68.25 74.46 77.56 6.64 4.74 3.69 3.02 2.77 2.66

39 300 20 36.30 25.00 5.6 8 Link 150 0.67 660 270.6 S 217.20 147.50 1.47 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 7.00 31.02 43.43 55.84 68.25 74.46 77.56 7.00 5.00 3.89 3.18 2.92 2.80

300 20 36.30 25.00 5.6 8 Link 150 0.67 660 - S 197.00 129.70 1.52 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 6.35 31.02 43.43 55.84 68.25 74.46 77.56 6.35 4.54 3.53 2.89 2.65 2.54

40 300 20 34.10 25.00 5.6 8 Link 150 0.67 660 274.1 S 225.00 145.40 1.55 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 7.25 31.02 43.43 55.84 68.25 74.46 77.56 7.25 5.18 4.03 3.30 3.02 2.90

300 20 34.10 25.00 5.6 8 Link 150 0.67 660 - S 183.00 127.80 1.43 150 109.50 126 0.48 0.59 3958 101 150 300.00 219.71 83.78 141.45 2.62 31.02 1.00 31.02 5.90 31.02 43.43 55.84 68.25 74.46 77.56 5.90 4.21 3.28 2.68 2.46 2.36

17 300 20 23.70 16.00 2.3 6 Link 150 0.38 660 - S 117.00 81.90 1.43 150 116.00 127 0.51 0.62 1626 57 150 300.00 223.85 47.12 141.54 2.66 17.77 1.00 17.77 6.58 17.77 24.88 31.99 39.10 42.65 44.43 6.58 4.70 3.66 2.99 2.74 2.63

300 20 23.70 16.00 2.3 6 Link 150 0.38 660 - S 115.00 81.90 1.40 150 116.00 127 0.51 0.62 1626 57 150 300.00 223.85 47.12 141.54 2.66 17.77 1.00 17.77 6.47 17.77 24.88 31.99 39.10 42.65 44.43 6.47 4.62 3.59 2.94 2.70 2.59

19 300 20 26.60 20.00 3.6 6 Link 150 0.38 660 - S 113.00 93.70 1.21 150 114.00 127 0.51 0.61 2545 57 150 300.00 222.57 47.12 141.64 2.65 17.67 1.00 17.67 6.39 17.67 24.74 31.81 38.88 42.42 44.19 6.39 4.57 3.55 2.91 2.66 2.56

300 20 26.60 20.00 3.6 6 Link 150 0.38 660 - S 129.00 93.70 1.38 150 114.00 127 0.51 0.61 2545 57 150 300.00 222.57 47.12 141.64 2.65 17.67 1.00 17.67 7.30 17.67 24.74 31.81 38.88 42.42 44.19 7.30 5.21 4.05 3.32 3.04 2.92

20 300 20 49.30 20.00 3.6 6 Link 150 0.38 660 - S 149.00 110.30 1.35 150 114.00 127 0.51 0.61 2545 57 150 300.00 222.57 47.12 141.64 2.65 17.67 1.00 17.67 8.43 17.67 24.74 31.81 38.88 42.42 44.19 8.43 6.02 4.68 3.83 3.51 3.37

300 20 49.30 20.00 3.6 6 Link 150 0.38 660 - S 137.00 110.30 1.24 150 114.00 127 0.51 0.61 2545 57 150 300.00 222.57 47.12 141.64 2.65 17.67 1.00 17.67 7.75 17.67 24.74 31.81 38.88 42.42 44.19 7.75 5.54 4.31 3.52 3.23 3.10

21 300 20 22.20 25.00 5.6 6 Link 150 0.38 660 - S 131.00 99.60 1.32 150 111.50 127 0.49 0.59 3958 57 150 300.00 220.98 47.12 141.75 2.63 17.55 1.00 17.55 7.46 17.55 24.57 31.59 38.61 42.12 43.88 7.46 5.33 4.15 3.39 3.11 2.99

300 20 22.20 25.00 5.6 6 Link 150 0.38 660 - S 151.00 99.60 1.52 150 111.50 127 0.49 0.59 3958 57 150 300.00 220.98 47.12 141.75 2.63 17.55 1.00 17.55 8.60 17.55 24.57 31.59 38.61 42.12 43.88 8.60 6.15 4.78 3.91 3.58 3.44

22 300 20 45.50 25.00 5.6 6 Link 150 0.38 660 - S 163.00 120.90 1.35 150 111.50 127 0.49 0.59 3958 57 150 300.00 220.98 47.12 141.75 2.63 17.55 1.00 17.55 9.29 17.55 24.57 31.59 38.61 42.12 43.88 9.29 6.63 5.16 4.22 3.87 3.71

300 20 45.50 25.00 5.6 6 Link 150 0.38 660 - S 164.00 120.90 1.36 150 111.50 127 0.49 0.59 3958 57 150 300.00 220.98 47.12 141.75 2.63 17.55 1.00 17.55 9.34 17.55 24.57 31.59 38.61 42.12 43.88 9.34 6.67 5.19 4.25 3.89 3.74

43 500 20 37.80 28.30 2.6 8 Link 150 0.72 1200 - S 313.00 250.80 1.25 250 207.85 226 0.56 0.63 5105 101 150 500.00 382.32 83.78 241.14 2.67 53.96 1.00 53.96 5.80 53.96 75.54 97.12 118.70 129.50 134.89 5.80 4.14 3.22 2.64 2.42 2.32

500 20 37.80 28.30 2.6 8 Link 150 0.72 1200 - S 366.00 250.80 1.46 250 207.85 226 0.56 0.63 5105 101 150 500.00 382.32 83.78 241.14 2.67 53.96 1.00 53.96 6.78 53.96 75.54 97.12 118.70 129.50 134.89 6.78 4.85 3.77 3.08 2.83 2.71

44 500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 - S 301.00 242.60 1.24 250 207.85 226 0.56 0.63 5105 101 150 500.00 382.32 83.78 241.14 2.67 53.96 1.00 53.96 5.58 53.96 75.54 97.12 118.70 129.50 134.89 5.58 3.98 3.10 2.54 2.32 2.23

500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 - S 329.00 242.60 1.36 250 207.85 226 0.56 0.63 5105 101 150 500.00 382.32 83.78 241.14 2.67 53.96 1.00 53.96 6.10 53.96 75.54 97.12 118.70 129.50 134.89 6.10 4.36 3.39 2.77 2.54 2.44

1.38 5.76 AVERAGE 5.84 4.17 3.24 2.65 2.43 2.34

0.171 1.8959771 STANDARD DEVIATION 1.92 1.37 1.07 0.87 0.80 0.77

0 0

800 800

SECTION WITH LINKS CHOSEN ONLY. REV 2 OF EQUATIONS

Variations in Cot(Q)



ANALYSIS FOR TRUSS ANGLE COT(Q) BETWEEN 1 AND 2.5

!

As fy

2s

1+ sin"

1+ sin#

$

% &

'

( ) rcot*

+

2+ # + sin# cos#

$

% &

'

( ) ,VRd

!

As fy

2s

!

1+ sin"

1+ sin#

$

% &

'

( ) r

!

"

2+ # + sin# cos#

$

% &

'

( )

!

As fy

2s

1+ sin"

1+ sin#

$

% &

'

( ) r cot*

+

2+ # + sin# cos #

$

% &

'

( ) , VRd

Section test data Geometry

Choose fyk

Choose shear span

Results for all truss angles

Analysis to Feltham model

university of bath: department of architecture and engineering section three: spreadsheet details


3.3. Spreadsheet 6.2

Spreadsheet 6.2 is used to analyse the shear enhancement limits for BS EN 1992-1-1. Limits of av = 2d, 2.5d and 3d are

analysed. Each sheet is again laid out as shown in Figure 3.2.

3.4. Spreadsheet 6.3 This set of spreadsheets presents an analysis of both BS5400 and BS EN 1992-1-1 approaches, using different values of link

yield strength. Values of 250MPa and 500MPa are presented. The analysis confirms that Clarke used fy=250MPa in the original

analysis to BS5400-4.

The final spreadsheets consider the effect of varying shear span in BS EN 1992-1-1. The shear span limit may be varied by the

user to determine the effect it has on the subsequent analysis. In all cases the results remain more conservative than the BS5400-4 approach presented by Clarke. Some results are presented in Figure 3.3.

Figure 3.3: Spreadsheet 6.3 analysis for different values of link yield strength to EN1992 (l) and BS5400 (r).

3.5. Spreadsheet 6.4 Spreadsheet 6.4 analyses the second set of variable angle truss equations. Two cases are included (for different values of z), as detailed in the dissertation. The following sheets are included:

Table 3.1: Sheets included in Spreadsheet 6.4: Name Description

Case 1 Analysis using z = 0.9d

Case 2 Analysis using z = 0.8D

Case comparison A comparison of Case 1 and Case 2 design models.

Spirals The design of spirally reinforced sections

3.6. Spreadsheet 6.5 Integration of Turmo et al’s. equation (2008) for link efficiency is undertaken here, and two different methods are presented. The

first provides a direct solution to the equation, as shown below. The second uses Simpson’s rule to solve the equation numerically, verifying the solution to Turmo’s equation. Either approach may be used, as both are too complex to be easily carried out by hand.

• Turmo’s Equation:

€

χ = 1 − z0 − zX′ R

2

dX0

1

∫

• Direct solution:

€

χ =

zX − z0( )′ R 2 − z0 − zX( )

2

′ R 2− ′ R sin−1 z0 − zX

′ R

2z

X=0

X=1

-

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400

Theoretical Failure Load (kN)

Actu

al Failu

re L

oad

(kN

)Clarke, fy = 250MPa

Clarke, fy = 500MPa

FS=1

Linear (Clarke, fy = 250MPa)


-

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400

Theoretical Failure Load (kN)

Actu

al Failu

re L

oad

(kN

)

Clarke, fy = 250MPa

Clarke, fy = 500MPa

FS=1





3.7. Spreadsheet 6.6 Spreadsheet 6.6 considers crushing analysis of circular sections. Two methods are presented for both flat and spirally bound

sections, as summarised below:

• Method 1: Analysis taking z = 0.9d

• Method 2: ‘Accurate analysis’, taking the lever arm to be the distance between the centroid of the more tensile

chord and the centroid of the more compressive chord.

The following worksheets are included:

Name Description

Clarke Data Set Full data from Clarke (1993)

Crushing Analysis 1 See §3.7.1

Crushing Analysis 2 See §3.7.2

Crushing Analysis S1 See §3.7.3

3.7.1. Method 1 – Flat Links

Method 1 assume z = 0.9d. This would appear to be a reasonable assumption if the section is under zero axial load. The

spreadsheet firstly calculates the moments on the section due to the applied loads. The compression chord force is then found by equilibrium, and the required area of the compression chord is determined. This then allows an angle to the compression

chord, ω, to be found by iterating the required concrete area against that provided by a certain value of ω. Microsoft Excel’s

‘goal seek’ function is used to carry out this analysis.

Figure 3.4: Method 1 analysis.

The ‘equivalent rectangle’ is defined as described in the dissertation, and this is then subject to a normal BS EN 1992-1-1

crushing analysis, calculating VRd,max. Results are plotted against the actual failure loads.

3.7.2. Method 2 – Flat Links

The more accurate value for the lever arm is given by:

•

€

z =2rsπ

+4R sin3 0.5ω( )3 ω − sinω( )

This assumes that the layout of the longitudinal bars is unknown, so Feltham’s (2004) definition for the centroid of the tension

steel below the centroidal axis is used (2rs/π). The distance from the centroidal axis to the centre of the compressive chord is

given by the second term in the above equation, and this depends on the value for ω.

Iterate until ! is correct

Final Year Dissertation, John Orr


24/3/09

TITLE

Specimen details and test results theta (degrees) 45.00 theta (degrees) 35.54 theta (degrees) 29.05 theta (degrees) 24.44 theta (degrees) 21.80

Taken from Clarke/Birjandi paper, IStructE 1993 BS EN 1992-1-1 Crushing Equation cot(theta) 1 cot(theta) 1.4 cot(theta) 1.8 cot(theta) 2.2 cot(theta) 2.5

cot(Q) 1 acw 1

Area required ,

Ae (mm^2)

Specimen

Section

Diameter,

(mm)

Minimum

cover,

(mm)

fcu


Failure


Va/Vth

(BCA)r rs rsv Alpha Beta As (mm2) Asv s L Mmax (kNm) Vmax (kN) d z = 0.9d M/z yt/z V/2cot(Q) fcd (N/mm^2) AE

Theta -

Sin(theta) Angle w A (mm^2) c H B Area concrete Area circle

Area - Area

required

Check on

effective

rectangle bw z v1 fcd VRd,max (kN) Va Va/VRd,max VRd,max (kN) Va Va/VRd,max VRd,max (kN) Va Va/VRd,max VRd,max (kN) Va Va/VRd,max VRd,max (kN) Va Va/VRd,max

1 300 20 22.70 10.00 0.9 0 - 0 0.00 660 0 S 65.00 44.40 1.46 150 125.00 130 0.56 0.66 636 0 0 3000 33 51 230 207 161950 0.39 19527 23 10404.88 0.925 1.88 10404.88 61.41 238.59 252.66 60280.95 70685.83 60280.95 0.00 252.66 207 0.6 23 356 65 0.18 336 65 0.19 302 65 0.22 268 65 0.24 245 65 0.27

300 20 22.70 10.00 0.9 0 - 0 0.00 660 0 B 75.00 44.40 1.69 150 125.00 130 0.56 0.66 636 0 0 3000 39 59 230 207 186865 0.39 22531 23 7239.40 0.644 1.64 7239.40 47.72 252.28 251.50 63446.44 70685.83 63446.44 0.00 251.50 207 0.6 23 354 75 0.21 335 75 0.22 300 75 0.25 267 75 0.28 244 75 0.31

2 300 20 49.00 10.00 0.9 0 - 0 0.00 660 0 B 70.00 57.40 1.22 150 125.00 130 0.56 0.66 636 0 0 3000 36 55 230 207 174407 0.39 21029 49 3130.18 0.278 1.22 3130.46 26.88 273.12 247.35 67555.66 70685.83 67555.66 0.00 247.35 207 0.6 49 751 70 0.09 711 70 0.10 638 70 0.11 566 70 0.12 518 70 0.14

300 20 49.00 10.00 0.9 0 - 0 0.00 550 0 B 84.00 68.90 1.22 150 125.00 130 0.56 0.66 636 0 0 3000 38 69 230 207 182606 0.39 26421 49 3187.46 0.283 1.22 3187.46 27.21 272.79 247.44 67498.38 70685.83 67498.38 0.00 247.44 207 0.6 49 752 84 0.11 711 84 0.12 638 84 0.13 566 84 0.15 518 84 0.16

3 300 20 22.80 16.00 2.3 0 - 0 0.00 660 0 S 91.00 60.10 1.51 150 122.00 130 0.54 0.64 1626 0 0 3000 47 71 228 205 228632 0.38 26905 23 8847.66 0.786 1.77 8847.66 54.85 245.15 252.24 61838.18 70685.83 61838.18 0.00 252.24 205 0.6 23 354 91 0.26 334 91 0.27 300 91 0.30 266 91 0.34 244 91 0.37

300 20 22.80 16.00 2.3 0 - 0 0.00 660 0 S 97.00 60.10 1.61 150 122.00 130 0.54 0.64 1626 0 0 3000 50 76 228 205 243706 0.38 28679 23 9431.02 0.838 1.81 9431.02 57.34 242.66 252.43 61254.81 70685.83 61254.81 0.00 252.43 205 0.6 23 354 97 0.27 335 97 0.29 300 97 0.32 267 97 0.36 244 97 0.40

4 300 20 44.00 16.00 2.3 0 - 0 0.00 660 0 S 129.00 74.80 1.72 150 122.00 130 0.54 0.64 1626 0 0 3000 66 101 228 205 324104 0.38 38140 44 6499.18 0.578 1.58 6499.18 44.30 255.70 251.02 64186.65 70685.83 64186.65 0.00 251.02 205 0.6 44 679 129 0.19 642 129 0.20 576 129 0.22 512 129 0.25 468 129 0.28

300 20 44.00 16.00 2.3 0 - 0 0.00 660 0 S 109.00 74.80 1.46 150 122.00 130 0.54 0.64 1626 0 0 3000 56 85 228 205 273855 0.38 32227 44 5491.56 0.488 1.48 5491.56 39.45 260.55 250.22 65194.28 70685.83 65194.28 0.00 250.22 205 0.6 44 677 109 0.16 640 109 0.17 575 109 0.19 510 109 0.21 467 109 0.23

5 300 20 26.70 25.00 5.6 0 - 0 0.00 660 0 S 148.00 83.90 1.76 150 117.50 130 0.52 0.61 3958 0 0 3000 76 115 225 202 376579 0.37 42681 27 12505.56 1.112 2.01 12505.56 69.88 230.12 252.83 58180.28 70685.83 58180.28 0.00 252.83 202 0.6 27 410 148 0.36 388 148 0.38 348 148 0.43 309 148 0.48 283 148 0.52

300 20 26.70 25.00 5.6 0 - 0 0.00 660 0 S 130.00 83.90 1.55 150 117.50 130 0.52 0.61 3958 0 0 3000 67 101 225 202 330779 0.37 37490 27 10984.61 0.976 1.92 10984.61 63.79 236.21 252.75 59701.22 70685.83 59701.22 0.00 252.75 202 0.6 27 410 130 0.32 387 130 0.34 348 130 0.37 309 130 0.42 282 130 0.46

6 300 20 43.60 25.00 5.6 0 - 0 0.00 660 0 S 152.00 98.70 1.54 150 117.50 130 0.52 0.61 3958 0 0 3000 78 119 225 202 386757 0.37 43834 44 7865.20 0.699 1.69 7865.20 50.54 249.46 251.83 62820.64 70685.83 62820.64 0.00 251.83 202 0.6 44 666 152 0.23 630 152 0.24 566 152 0.27 502 152 0.30 460 152 0.33

300 20 43.60 25.00 5.6 0 - 0 0.00 660 0 S 148.00 98.70 1.50 150 117.50 130 0.52 0.61 3958 0 0 3000 76 115 225 202 376579 0.37 42681 44 7658.22 0.681 1.68 7658.22 49.62 250.38 251.72 63027.62 70685.83 63027.62 0.00 251.72 202 0.6 44 666 148 0.22 630 148 0.23 566 148 0.26 502 148 0.29 459 148 0.32

7 300 20 34.40 16.00 2.3 8 Link 150 0.67 330 0 S 262.00 175.20 1.50 150 114.00 126 0.51 0.61 1626 101 150 3000 77 233 223 200 384138 0.36 84481 34 8710.96 0.774 1.76 8710.96 54.26 245.74 252.19 61974.88 70685.83 61974.88 0.00 252.19 200 0.6 34 521 262 0.50 493 262 0.53 443 262 0.59 393 262 0.67 360 262 0.73

300 20 34.40 16.00 2.3 8 Link 150 0.67 330 0 B 261.00 175.20 1.49 150 114.00 126 0.51 0.61 1626 101 150 3000 77 232 223 200 382672 0.36 84158 34 8677.71 0.771 1.75 8677.71 54.11 245.89 252.18 62008.12 70685.83 62008.12 0.00 252.18 200 0.6 34 521 261 0.50 493 261 0.53 443 261 0.59 393 261 0.66 360 261 0.73

300 20 34.40 16.00 2.3 8 Link 150 0.67 660 0 B 126.00 106.20 1.19 150 114.00 126 0.51 0.61 1626 101 150 3000 65 98 223 200 323810 0.36 35607 34 8378.02 0.745 1.73 8378.02 52.81 247.19 252.06 62307.82 70685.83 62307.82 0.00 252.06 200 0.6 34 521 126 0.24 493 126 0.26 442 126 0.28 393 126 0.32 359 126 0.35

8 300 20 38.40 16.00 2.3 8 Link 75 1.34 330 0 B 269.00 217.60 1.24 150 114.00 126 0.51 0.61 1626 101 75 3000 79 239 223 200 394401 0.36 86738 38 8012.06 0.712 1.70 8012.06 51.19 248.81 251.90 62673.78 70685.83 62673.78 0.00 251.90 200 0.6 38 581 269 0.46 550 269 0.49 494 269 0.55 438 269 0.61 401 269 0.67

300 20 38.40 16.00 2.3 8 Link 75 1.34 660 0 B 125.00 146.10 0.86 150 114.00 126 0.51 0.61 1626 101 75 3000 64 98 223 200 321241 0.36 35324 38 7445.74 0.662 1.66 7445.74 48.66 251.34 251.61 63240.09 70685.83 63240.09 0.00 251.61 200 0.6 38 581 125 0.22 549 125 0.23 493 125 0.25 437 125 0.29 400 125 0.31

9 300 20 33.90 16.00 2.3 8 Link 150 0.67 330 0 B 231.00 174.50 1.32 150 114.00 126 0.51 0.61 1626 101 150 3000 68 206 223 200 338686 0.36 74485 34 7793.55 0.693 1.69 7793.55 50.22 249.78 251.79 62892.28 70685.83 62892.28 0.00 251.79 200 0.6 34 513 231 0.45 485 231 0.48 436 231 0.53 386 231 0.60 354 231 0.65

300 20 33.90 16.00 2.3 8 Link 150 0.67 660 0 B 133.00 105.90 1.26 150 114.00 126 0.51 0.61 1626 101 150 3000 68 104 223 200 341800 0.36 37585 34 8973.90 0.798 1.78 8973.90 55.39 244.61 252.29 61711.94 70685.83 61711.94 0.00 252.29 200 0.6 34 514 133 0.26 486 133 0.27 436 133 0.30 387 133 0.34 354 133 0.38

10 300 20 31.30 16.00 2.3 8 Link 75 1.34 330 0 B 259.00 208.20 1.24 150 114.00 126 0.51 0.61 1626 101 75 3000 76 231 223 200 379739 0.36 83513 31 9464.08 0.841 1.81 9464.08 57.48 242.52 252.44 61221.75 70685.83 61221.75 0.00 252.44 200 0.6 31 475 259 0.55 449 259 0.58 403 259 0.64 358 259 0.72 327 259 0.79

300 20 31.30 16.00 2.3 8 Link 75 1.34 660 0 B 128.00 141.40 0.91 150 114.00 126 0.51 0.61 1626 101 75 3000 66 100 223 200 328950 0.36 36172 31 9353.94 0.831 1.80 9353.94 57.01 242.99 252.41 61331.89 70685.83 61331.89 0.00 252.41 200 0.6 31 475 128 0.27 449 128 0.29 403 128 0.32 358 128 0.36 327 128 0.39

11 300 20 24.10 25.00 5.6 8 Link 150 0.67 660 0 S 186.00 117.80 1.58 150 109.50 126 0.48 0.59 3958 101 150 3000 96 145 220 198 484239 0.35 51146 24 17970.66 1.597 2.33 17970.66 90.49 209.51 251.61 52715.18 70685.83 52715.18 0.00 251.61 198 0.6 24 360 186 0.52 340 186 0.55 305 186 0.61 271 186 0.69 248 186 0.75

300 20 24.10 25.00 5.6 8 Link 150 0.67 660 0 S 188.00 117.80 1.60 150 109.50 126 0.48 0.59 3958 101 150 3000 97 147 220 198 489445 0.35 51696 24 18163.89 1.615 2.34 18163.89 91.19 208.81 251.53 52521.94 70685.83 52521.94 0.00 251.53 198 0.6 24 360 188 0.52 340 188 0.55 305 188 0.62 271 188 0.69 248 188 0.76

12 300 20 23.80 25.00 5.6 8 Link 75 1.34 660 0 S 211.00 154.30 1.37 150 109.50 126 0.48 0.59 3958 101 75 3000 109 165 220 198 549324 0.35 58020 24 20643.04 1.835 2.46 20643.04 100.06 199.94 250.29 50042.80 70685.83 50042.80 0.00 250.29 198 0.6 24 353 211 0.60 334 211 0.63 300 211 0.70 266 211 0.79 244 211 0.87

300 20 23.80 25.00 5.6 8 Link 75 1.34 660 0 S 239.00 154.30 1.55 150 109.50 126 0.48 0.59 3958 101 75 3000 123 186 220 198 622221 0.35 65720 24 23382.40 2.078 2.60 23382.40 109.64 190.36 248.49 47303.44 70685.83 47303.44 0.00 248.49 198 0.6 24 351 239 0.68 332 239 0.72 298 239 0.80 264 239 0.90 242 239 0.99

13 300 20 48.40 25.00 5.6 8 Link 150 0.67 660 0 S 227.00 139.30 1.63 150 109.50 126 0.48 0.59 3958 101 150 3000 117 177 220 198 590979 0.35 62420 48 10920.65 0.971 1.91 10920.65 63.53 236.47 252.74 59765.18 70685.83 59765.18 0.00 252.74 198 0.6 48 726 227 0.31 686 227 0.33 616 227 0.37 547 227 0.42 500 227 0.45

300 20 48.40 25.00 5.6 8 Link 150 0.67 660 0 S 228.00 139.30 1.64 150 109.50 126 0.48 0.59 3958 101 150 3000 117 178 220 198 593583 0.35 62695 48 10968.76 0.975 1.92 10968.76 63.73 236.27 252.75 59717.07 70685.83 59717.07 0.00 252.75 198 0.6 48 726 228 0.31 686 228 0.33 616 228 0.37 547 228 0.42 500 228 0.46

14 300 20 50.50 25.00 5.6 8 Link 75 1.34 660 0 S 279.00 177.30 1.57 150 109.50 126 0.48 0.59 3958 101 75 3000 144 218 220 198 726358 0.35 76719 51 12864.14 1.143 2.04 12864.14 71.29 228.71 252.82 57821.69 70685.83 57821.69 0.00 252.82 198 0.6 51 757 279 0.37 716 279 0.39 643 279 0.43 571 279 0.49 522 279 0.53

300 20 50.50 25.00 5.6 8 Link 75 1.34 660 0 S 288.00 177.30 1.62 150 109.50 126 0.48 0.59 3958 101 75 3000 148 225 220 198 749789 0.35 79193 51 13279.12 1.180 2.06 13279.12 72.91 227.09 252.79 57406.72 70685.83 57406.72 0.00 252.79 198 0.6 51 757 288 0.38 716 288 0.40 643 288 0.45 571 288 0.50 522 288 0.55

15 300 20 24.30 20.00 3.6 8 Link 150 0.67 660 0 S 145.00 107.80 1.35 150 112.00 126 0.50 0.60 2545 101 150 3000 75 113 221 199 374783 0.36 40489 24 13756.96 1.223 2.09 13756.96 74.76 225.24 252.75 56928.87 70685.83 56928.87 0.00 252.75 199 0.6 24 367 145 0.40 347 145 0.42 312 145 0.47 276 145 0.52 253 145 0.57

300 20 24.30 20.00 3.6 8 Link 150 0.67 660 0 S 148.00 107.80 1.37 150 112.00 126 0.50 0.60 2545 101 150 3000 76 115 221 199 382537 0.36 41326 24 14041.59 1.248 2.11 14041.59 75.85 224.15 252.71 56644.24 70685.83 56644.24 0.00 252.71 199 0.6 24 367 148 0.40 347 148 0.43 312 148 0.48 276 148 0.54 253 148 0.58

16 300 20 46.70 20.00 3.6 8 Link 150 0.67 660 0 S 185.00 124.90 1.48 150 112.00 126 0.50 0.60 2545 101 150 3000 95 144 221 199 478171 0.36 51658 47 9133.05 0.812 1.79 9133.05 56.07 243.93 252.34 61552.79 70685.83 61552.79 0.00 252.34 199 0.6 47 704 185 0.26 666 185 0.28 598 185 0.31 531 185 0.35 486 185 0.38

16b 300 20 46.70 20.00 3.6 8 Link 150 0.67 660 0 S 186.00 124.90 1.49 150 112.00 126 0.50 0.60 2545 101 150 3000 96 145 221 199 480756 0.36 51937 47 9182.42 0.816 1.79 9182.42 56.28 243.72 252.35 61503.42 70685.83 61503.42 0.00 252.35 199 0.6 47 704 186 0.26 666 186 0.28 598 186 0.31 531 186 0.35 486 186 0.38

17 300 20 23.70 16.00 2.3 6 Link 150 0.38 660 0 S 117.00 81.90 1.43 150 116.00 127 0.51 0.62 1626 57 150 3000 60 91 224 201 298971 0.37 33452 24 11203.32 0.996 1.93 11203.32 64.68 235.32 252.77 59482.51 70685.83 59482.51 0.00 252.77 201 0.6 24 362 117 0.32 343 117 0.34 307 117 0.38 273 117 0.43 250 117 0.47

300 20 23.70 16.00 2.3 6 Link 150 0.38 660 0 S 115.00 81.90 1.40 150 116.00 127 0.51 0.62 1626 57 150 3000 59 90 224 201 293860 0.37 32880 24 11011.81 0.979 1.92 11011.81 63.90 236.10 252.75 59674.02 70685.83 59674.02 0.00 252.75 201 0.6 24 362 115 0.32 342 115 0.34 307 115 0.37 273 115 0.42 250 115 0.46

18 300 20 49.60 16.00 2.3 6 Link 150 0.38 660 0 B 137.00 98.90 1.39 150 116.00 127 0.51 0.62 1626 57 150 3000 71 107 224 201 350077 0.37 39170 50 6268.28 0.557 1.56 6268.28 43.21 256.79 250.86 64417.56 70685.83 64417.56 0.00 250.86 201 0.6 50 752 137 0.18 711 137 0.19 638 137 0.21 567 137 0.24 519 137 0.26

300 20 49.60 16.00 2.3 6 Link 150 0.38 660 0 B 119.00 98.90 1.20 150 116.00 127 0.51 0.62 1626 57 150 3000 61 93 224 201 304081 0.37 34024 50 5444.71 0.484 1.48 5444.71 39.22 260.78 250.18 65241.13 70685.83 65241.13 0.00 250.18 201 0.6 50 750 119 0.16 709 119 0.17 637 119 0.19 565 119 0.21 517 119 0.23

19 300 20 26.60 20.00 3.6 6 Link 150 0.38 660 0 S 113.00 93.70 1.21 150 114.00 127 0.51 0.61 2545 57 150 3000 58 88 223 200 290401 0.36 31933 27 9716.86 0.864 1.83 9716.86 58.55 241.45 252.51 60968.98 70685.83 60968.98 0.00 252.51 200 0.6 27 404 113 0.28 382 113 0.30 343 113 0.33 304 113 0.37 278 113 0.41

300 20 26.60 20.00 3.6 6 Link 150 0.38 660 0 S 129.00 93.70 1.38 150 114.00 127 0.51 0.61 2545 57 150 3000 66 101 223 200 331520 0.36 36454 27 11092.70 0.986 1.92 11092.70 64.23 235.77 252.76 59593.14 70685.83 59593.14 0.00 252.76 200 0.6 27 404 129 0.32 382 129 0.34 343 129 0.38 304 129 0.42 279 129 0.46

20 300 20 49.30 20.00 3.6 6 Link 150 0.38 660 0 S 149.00 110.30 1.35 150 114.00 127 0.51 0.61 2545 57 150 3000 77 116 223 200 382919 0.36 42106 49 6913.03 0.614 1.61 6913.03 46.23 253.77 251.30 63772.81 70685.83 63772.81 0.00 251.30 200 0.6 49 745 149 0.20 704 149 0.21 632 149 0.24 561 149 0.27 513 149 0.29

300 20 49.30 20.00 3.6 6 Link 150 0.38 660 0 S 137.00 110.30 1.24 150 114.00 127 0.51 0.61 2545 57 150 3000 71 107 223 200 352080 0.36 38715 49 6356.28 0.565 1.56 6356.28 43.63 256.37 250.92 64329.56 70685.83 64329.56 0.00 250.92 200 0.6 49 743 137 0.18 703 137 0.19 631 137 0.22 560 137 0.24 513 137 0.27

21 300 20 22.20 25.00 5.6 6 Link 150 0.38 660 0 S 131.00 99.60 1.32 150 111.50 127 0.49 0.59 3958 57 150 3000 67 102 221 199 339085 0.36 36469 22 13631.36 1.212 2.08 13631.36 74.28 225.72 252.76 57054.48 70685.83 57054.48 0.00 252.76 199 0.6 22 335 131 0.39 317 131 0.41 284 131 0.46 252 131 0.52 231 131 0.57

300 20 22.20 25.00 5.6 6 Link 150 0.38 660 0 S 151.00 99.60 1.52 150 111.50 127 0.49 0.59 3958 57 150 3000 78 118 221 199 390853 0.36 42036 22 15712.48 1.397 2.20 15712.48 82.18 217.82 252.38 54973.35 70685.83 54973.35 0.00 252.38 199 0.6 22 334 151 0.45 316 151 0.48 284 151 0.53 252 151 0.60 231 151 0.65

22 300 20 45.50 25.00 5.6 6 Link 150 0.38 660 0 S 163.00 120.90 1.35 150 111.50 127 0.49 0.59 3958 57 150 3000 84 127 221 199 421915 0.36 45377 46 8275.55 0.736 1.72 8275.55 52.36 247.64 252.02 62410.28 70685.83 62410.28 0.00 252.02 199 0.6 46 684 163 0.24 647 163 0.25 581 163 0.28 515 163 0.32 472 163 0.35

300 20 45.50 25.00 5.6 6 Link 150 0.38 660 0 S 164.00 120.90 1.36 150 111.50 127 0.49 0.59 3958 57 150 3000 84 128 221 199 424503 0.36 45655 46 8326.32 0.740 1.73 8326.32 52.58 247.42 252.04 62359.51 70685.83 62359.51 0.00 252.04 199 0.6 46 684 164 0.24 647 164 0.25 581 164 0.28 516 164 0.32 472 164 0.35

300 20 31.20 20.00 3.6 0 - 0 0.00 660 0 S 86.00 76.80 1.12 150 120.00 130 0.53 0.63 2545 0 0 3000 44 67 226 204 217285 0.37 25150 31 6158.14 0.547 1.55 6158.14 42.69 257.31 250.77 64527.69 70685.83 64527.69 0.00 250.77 204 0.6 31 478 86 0.18 452 86 0.19 406 86 0.21 360 86 0.24 330 86 0.26

300 20 29.70 20.00 3.6 0 - 0 0.00 660 0 S 90.00 75.60 1.19 150 120.00 130 0.53 0.63 2545 0 0 3000 46 70 226 204 227391 0.37 26320 30 6770.05 0.602 1.60 6770.05 45.56 254.44 251.21 63915.78 70685.83 63915.78 0.00 251.21 204 0.6 30 456 90 0.20 431 90 0.21 387 90 0.23 344 90 0.26 315 90 0.29

300 20 20.90 20.00 3.6 0 - 0 0.00 660 0 S 98.00 67.20 1.46 150 120.00 130 0.53 0.63 2545 0 0 3000 50 76 226 204 247603 0.37 28660 21 10475.76 0.931 1.88 10475.76 61.71 238.29 252.67 60210.07 70685.83 60210.07 0.00 252.67 204 0.6 21 323 98 0.30 305 98 0.32 274 98 0.36 243 98 0.40 223 98 0.44

300 20 21.60 25.00 5.6 0 - 0 0.00 660 0 S 116.00 78.10 1.49 150 117.50 130 0.52 0.61 3958 0 0 3000 60 90 225 202 295156 0.37 33452 22 12115.93 1.077 1.99 12115.93 68.34 231.66 252.83 58569.90 70685.83 58569.90 0.00 252.83 202 0.6 22 331 116 0.35 314 116 0.37 281 116 0.41 250 116 0.46 229 116 0.51

300 20 34.80 25.00 5.6 0 - 0 0.00 660 0 S 125.00 91.60 1.36 150 117.50 130 0.52 0.61 3958 0 0 3000 64 98 225 202 318057 0.37 36048 35 8103.70 0.720 1.71 8103.70 51.60 248.40 251.94 62582.14 70685.83 62582.14 0.00 251.94 202 0.6 35 532 125 0.23 503 125 0.25 452 125 0.28 401 125 0.31 367 125 0.34

300 20 37.70 25.00 5.6 0 - 0 0.00 660 0 S 125.00 94.00 1.33 150 117.50 130 0.52 0.61 3958 0 0 3000 64 98 225 202 318057 0.37 36048 38 7480.34 0.665 1.66 7480.34 48.82 251.18 251.63 63205.50 70685.83 63205.50 0.00 251.63 202 0.6 38 576 125 0.22 545 125 0.23 489 125 0.26 434 125 0.29 397 125 0.31

300 20 34.90 25.00 5.6 0 - 0 0.00 660 0 S 136.00 91.70 1.48 150 117.50 130 0.52 0.61 3958 0 0 3000 70 106 225 202 346045 0.37 39220 35 8791.56 0.781 1.76 8791.56 54.60 245.40 252.22 61894.27 70685.83 61894.27 0.00 252.22 202 0.6 35 534 136 0.25 505 136 0.27 454 136 0.30 403 136 0.34 368 136 0.37

300 20 43.90 25.00 5.6 8 Link 150 0.67 660 0 S 218.00 135.80 1.61 150 109.50 126 0.48 0.59 3958 101 150 3000 112 170 220 198 567548 0.35 59945 44 11562.72 1.028 1.95 11562.72 66.13 233.87 252.80 59123.11 70685.83 59123.11 0.00 252.80 198 0.6 44 658 218 0.33 623 218 0.35 559 218 0.39 496 218 0.44 454 218 0.48

300 20 36.10 25.00 5.6 8 Link 150 0.67 660 0 S 206.00 129.50 1.59 150 109.50 126 0.48 0.59 3958 101 150 3000 106 161 220 198 536307 0.35 56645 36 13287.04 1.181 2.06 13287.04 72.94 227.06 252.79 57398.80 70685.83 57398.80 0.00 252.79 198 0.6 36 541 206 0.38 512 206 0.40 460 206 0.45 408 206 0.51 373 206 0.55

300 20 36.30 25.00 5.6 8 Link 150 0.67 660 0 S 197.00 129.70 1.52 150 109.50 126 0.48 0.59 3958 101 150 3000 101 154 220 198 512876 0.35 54171 36 12636.53 1.123 2.02 12636.53 70.40 229.60 252.83 58049.31 70685.83 58049.31 0.00 252.83 198 0.6 36 544 197 0.36 515 197 0.38 462 197 0.43 410 197 0.48 375 197 0.52

300 20 34.10 25.00 5.6 8 Link 150 0.67 660 0 S 183.00 127.80 1.43 150 109.50 126 0.48 0.59 3958 101 150 3000 94 143 220 198 476428 0.35 50321 34 12495.82 1.111 2.01 12495.82 69.85 230.15 252.83 58190.01 70685.83 58190.01 0.00 252.83 198 0.6 34 511 183 0.36 484 183 0.38 434 183 0.42 385 183 0.47 353 183 0.52

41 500 20 34.00 28.30 2.6 0 - 0 0.00 1200 0 S 236.00 175.40 1.35 250 215.85 230 0.58 0.64 5105 0 0 3500 186 155 387 349 533746 0.39 61120 34 13900.76 0.445 1.44 13900.76 61.70 438.30 416.26 182448.78 196349.54 182448.78 0.00 416.26 349 0.6 34 1,480 236 0.16 1,400 236 0.17 1,257 236 0.19 1,115 236 0.21 1,021 236 0.23

42 500 20 33.50 28.30 2.6 0 - 0 0.00 1200 0 S 234.00 174.50 1.34 250 215.85 230 0.58 0.64 5105 0 0 3500 185 154 387 349 529223 0.39 60602 34 13988.67 0.448 1.44 13988.67 61.96 438.04 416.31 182360.87 196349.54 182360.87 0.00 416.31 349 0.6 34 1,459 234 0.16 1,380 234 0.17 1,239 234 0.19 1,099 234 0.21 1,006 234 0.23

500 20 33.50 28.30 2.6 0 - 0 0.00 1200 0 S 222.00 174.50 1.27 250 215.85 230 0.58 0.64 5105 0 0 3500 175 146 387 349 502083 0.39 57495 34 13271.31 0.425 1.41 13271.31 59.77 440.23 415.87 183078.24 196349.54 183078.24 0.00 415.87 349 0.6 34 1,457 222 0.15 1,378 222 0.16 1,237 222 0.18 1,098 222 0.20 1,005 222 0.22

43 500 20 37.80 28.30 2.6 8 Link 150 0.72 1200 0 S 313.00 250.80 1.25 250 207.85 226 0.56 0.63 5105 101 150 3500 247 206 382 344 717322 0.38 79098 38 16884.25 0.540 1.54 16884.25 70.51 429.49 417.85 179465.29 196349.54 179465.29 0.00 417.85 344 0.6 38 1,630 313 0.19 1,542 313 0.20 1,384 313 0.23 1,228 313 0.25 1,124 313 0.28

500 20 37.80 28.30 2.6 8 Link 150 0.72 1200 0 S 366.00 250.80 1.46 250 207.85 226 0.56 0.63 5105 101 150 3500 289 241 382 344 838786 0.38 92491 38 19743.24 0.632 1.63 19743.24 78.54 421.46 419.03 176606.30 196349.54 176606.30 0.00 419.03 344 0.6 38 1,635 366 0.22 1,547 366 0.24 1,388 366 0.26 1,232 366 0.30 1,128 366 0.32

44 500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 0 S 301.00 242.60 1.24 250 207.85 226 0.56 0.63 5105 101 150 3500 237 198 382 344 689821 0.38 76065 33 18655.19 0.597 1.60 18655.19 75.52 424.48 418.62 177694.35 196349.54 177694.35 0.00 418.62 344 0.6 33 1,422 301 0.21 1,345 301 0.22 1,207 301 0.25 1,071 301 0.28 980 301 0.31

500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 0 S 329.00 242.60 1.36 250 207.85 226 0.56 0.63 5105 101 150 3500 259 216 382 344 753990 0.38 83141 33 20390.56 0.652 1.65 20390.56 80.31 419.69 419.26 175958.98 196349.54 175958.98 0.00 419.26 344 0.6 33 1,424 329 0.23 1,347 329 0.24 1,209 329 0.27 1,073 329 0.31 982 329 0.34

45 500 20 29.40 34.60 3.8 0 - 0 0.00 1200 0 - 234.00 191.30 1.22 250 212.70 230 0.57 0.63 7461 0 0 3500 185 154 385 347 531977 0.39 60029 29 16052.65 0.514 1.51 16052.65 68.10 431.90 417.45 180296.89 196349.54 180296.89 0.00 417.45 347 0.6 29 1,277 234 0.18 1,208 234 0.19 1,084 234 0.22 962 234 0.24 881 234 0.27

Shear Capacity of Circular Columns

CRUSHING ANALYSIS FOR CLARKE TEST DATA

INITIAL ANALYSIS REMOVED SPIRAL SECTIONS AND MEMBERS WITH AXIAL LOAD

1: DEFINE SECTION

Define compression zone and equivalent rectangle.

Crushing equations for all values of theta

Loads and Moments Compression chord force Compression area required Find angle w

!

MMAX

=Vaav( ) L " av( )L

!

d = r 1+ sin"( )

!

" # sin" = Ae

2

r2

$

% & &

'

( ) )

!

" + sin" cos" =#

2$A

r2

!

c = r 1" cos#

2

$

% &

'

( )

!

B ="r 2 #

1

2r2 $ # sin$( )

D # c

!

VRd ,max ="cwbwzv1 fcd

cot# + tan#( )

!

Fc =M

z"

yt

z

#

$ %

&

' ( V cot) +

N

2

!

Vmax

=P L " a

v( )L

Section test data Geometry Load and moment calculations

Compression chord force

Concrete area required

Angle !

Area with angle !.

Dimensions of equivalent rectangle

Check on area of circle

EN1992-1-1 crushing equations

Value for "cw



In this analysis, the area of the compression chord, and the value of z now both depend on the value of ω, meaning that two

sets of iteration must be undertaken to determine the correct value for ω. This results in a slightly longer analysis, and multiple

iterations for each section are required. The results are again plotted against the actual failure loads.

Figure 3.5: Method 2 Analysis

3.7.3. Spirally bound sections

Crushing analysis in spirally bound sections is tentatively undertaken. There is much less experimental data for spirally bound

sections. The first method for defining the equivalent rectangle (as detailed in §3.7.1) is used, and two techniques are presented

for calculating VRd,max. The methods are:

• Method A:

€

VRd ,max =αcwbwzv1 fcdcotθ + tanθ

- assuming that the links are flat.

• Method B:

€

VRd ,max =αcwbwzv1 fcd cotθ + cotα( )

1+ cot 2θ - assuming some kind of pitch on the links.

It would seem logical to say that the spirally bound section is simply a special case of the inclined links (Method B). However,

the Method B predicts a significantly larger value for VRd,max than Method A. In the absence of actual crushing failures in spirally

bound sections it is difficult to known which method is more appropriate. Therefore, since Method A appears to be more conservative, this should be used. Analysis is again presented against the actual failure loads.

Iterate until ! is correct

Area with angle !.

Section test data Geometry Load and moment calculations

Compression chord force

Concrete area required

Dimensions of equivalent rectangle

EN1992-1-1 crushing equations

Value for "cw



24/3/09

TITLE

Specimen details and test results theta (degrees) 45.00 theta (degrees) 35.54 theta (degrees) 29.05 theta (degrees) 24.44

Taken from Clarke/Birjandi paper, IStructE 1993 BS EN 1992-1-1 Crushing Equation cot(theta) 1 cot(theta) 1.4 cot(theta) 1.8 cot(theta) 2.2

cot(Q) 1 acw 1

Area required ,

Ae (mm^2)

Specimen

Section

Diameter,

(mm)

Minimum

cover,

(mm)

fcu


Failure


Va/Vth

(BCA)r rs rsv Alpha Beta As (mm2) Asv s L Mmax (kNm) Vmax (kN) d y yg z= y + yg M/z yt/z V/2cot(Q) fcd (N/mm^2) AE

Theta -

Sin(theta) C n C w f(x) f'(x) Angle w A (mm^2)

Area Error

(mm^2) c H B Area concrete Area circle

Area - Area

required

Check on

effective

rectangle bw z v1 fcd VRd,max Analysis 1 !VRd,max Error % VRd,max (kN) Va Va/VRd,max VRd,max (kN) Va Va/VRd,max VRd,max (kN) Va Va/VRd,max VRd,max (kN) Va Va/VRd,max

1 300 20 22.70 10.00 0.9 0 - 0 0.00 660 0 S 65.00 44.40 1.46 150 125.00 130 0.56 0.66 636 0 0 3000 33 51 230 111.65 80 191 174988 0.42 21099 23 11242.56 0.999 0.999 0 0.517 1.90 -0.44 -1.32 1.93 11242.29 -0.27 64.84 235.16 252.78 59443.28 70685.83 59443.28 0.00 252.78 191 0.6 23 356 26 7% 329 65 0.20 311 65 0.21 279 65 0.23 248 65 0.26

300 20 22.70 10.00 0.9 0 - 0 0.00 660 0 B 75.00 44.40 1.69 150 125.00 130 0.56 0.66 636 0 0 3000 39 59 230 121.08 80 201 192413 0.40 23200 23 7454.32 0.663 0.663 1 0.52 1.57 -0.05 -1.00 1.66 7452.44 -1.89 48.69 251.31 251.61 63231.51 70685.83 63231.51 0.00 251.61 201 0.6 23 354 10 3% 344 75 0.22 325 75 0.23 292 75 0.26 259 75 0.29

2 300 20 49.00 10.00 0.9 0 - 0 0.00 660 0 B 70.00 57.40 1.22 150 125.00 130 0.56 0.66 636 0 0 3000 36 55 230 134.33 80 214 168465 0.37 20312 49 3023.53 0.269 0.269 2 0.52 1.52 0.00 -0.95 1.20 3023.89 0.36 26.25 273.75 247.17 67662.30 70685.83 67662.30 0.00 247.17 214 0.6 49 751 -26 -3% 777 70 0.09 735 70 0.10 660 70 0.11 586 70 0.12

300 20 49.00 10.00 0.9 0 - 0 0.00 550 0 B 84.00 68.90 1.22 150 125.00 130 0.56 0.66 636 0 0 3000 38 69 230 134.11 80 214 176566 0.37 25547 49 3082.02 0.274 0.274 3 0.52 1.52 0.00 -0.94 1.21 3086.82 4.80 26.62 273.38 247.29 67603.81 70685.83 67603.81 0.00 247.29 214 0.6 49 752 -25 -3% 777 84 0.11 735 84 0.11 660 84 0.13 585 84 0.14

3 300 20 22.80 16.00 2.3 0 - 0 0.00 660 0 S 91.00 60.10 1.51 150 122.00 130 0.54 0.64 1626 0 0 3000 47 71 228 116.22 78 194 241617 0.40 28433 23 9350.16 0.831 0.831 4 0.52 1.52 0.00 -0.94 1.80 9349.53 -0.64 56.99 243.01 252.40 61335.67 70685.83 61335.67 0.00 252.40 194 0.6 23 354 19 5% 335 91 0.27 317 91 0.29 284 91 0.32 252 91 0.36

300 20 22.80 16.00 2.3 0 - 0 0.00 660 0 S 97.00 60.10 1.61 150 122.00 130 0.54 0.64 1626 0 0 3000 50 76 228 114.48 78 192 259877 0.40 30582 23 10056.82 0.894 0.894 5 0.52 1.52 0.00 -0.94 1.85 10057.08 0.26 59.97 240.03 252.59 60629.01 70685.83 60629.01 0.00 252.59 192 0.6 23 354 22 6% 332 97 0.29 314 97 0.31 282 97 0.34 250 97 0.39

4 300 20 44.00 16.00 2.3 0 - 0 0.00 660 0 S 129.00 74.80 1.72 150 122.00 130 0.54 0.64 1626 0 0 3000 66 101 228 123.35 78 201 330368 0.39 38877 44 6624.80 0.589 0.589 6 0.52 1.52 0.00 -0.94 1.59 6614.38 -10.43 44.84 255.16 251.06 64061.03 70685.83 64061.03 0.00 251.06 201 0.6 44 679 13 2% 666 129 0.19 630 129 0.20 566 129 0.23 502 129 0.26

300 20 44.00 16.00 2.3 0 - 0 0.00 660 0 S 109.00 74.80 1.46 150 122.00 130 0.54 0.64 1626 0 0 3000 56 85 228 126.46 78 204 274887 0.38 32348 44 5512.25 0.490 0.490 7 0.52 1.52 0.00 -0.94 1.49 5511.01 -1.23 39.55 260.45 250.23 65173.59 70685.83 65173.59 0.00 250.23 204 0.6 44 677 3 0% 674 109 0.16 638 109 0.17 572 109 0.19 508 109 0.21

5 300 20 26.70 25.00 5.6 0 - 0 0.00 660 0 S 148.00 83.90 1.76 150 117.50 130 0.52 0.61 3958 0 0 3000 76 115 225 105.23 75 180 423194 0.42 47964 27 14053.57 1.249 1.249 8 0.52 1.52 0.00 -0.94 2.11 14053.04 -0.53 75.90 224.10 252.71 56632.27 70685.83 56632.27 0.00 252.71 180 0.6 27 410 45 11% 364 148 0.41 345 148 0.43 309 148 0.48 275 148 0.54

300 20 26.70 25.00 5.6 0 - 0 0.00 660 0 S 130.00 83.90 1.55 150 117.50 130 0.52 0.61 3958 0 0 3000 67 101 225 109.78 75 185 362559 0.41 41092 27 12039.99 1.070 1.070 9 0.52 1.52 0.00 -0.94 1.99 12040.47 0.49 68.04 231.96 252.83 58645.85 70685.83 58645.85 0.00 252.83 185 0.6 27 410 36 9% 374 130 0.35 354 130 0.37 317 130 0.41 282 130 0.46

6 300 20 43.60 25.00 5.6 0 - 0 0.00 660 0 S 152.00 98.70 1.54 150 117.50 130 0.52 0.61 3958 0 0 3000 78 119 225 119.11 75 194 403519 0.39 45734 44 8206.09 0.729 0.729 10 0.52 1.52 0.00 -0.94 1.72 8205.62 -0.47 52.05 247.95 251.98 62479.74 70685.83 62479.74 0.00 251.98 194 0.6 44 666 27 4% 639 152 0.24 605 152 0.25 543 152 0.28 482 152 0.32

300 20 43.60 25.00 5.6 0 - 0 0.00 660 0 S 148.00 98.70 1.50 150 117.50 130 0.52 0.61 3958 0 0 3000 76 115 225 119.74 75 195 391641 0.38 44388 44 7964.54 0.708 0.708 11 0.52 1.52 0.00 -0.94 1.70 7964.96 0.43 50.99 249.01 251.88 62721.30 70685.83 62721.30 0.00 251.88 195 0.6 44 666 25 4% 641 148 0.23 606 148 0.24 544 148 0.27 483 148 0.31

7 300 20 34.40 16.00 2.3 8 Link 150 0.67 330 0 S 262.00 175.20 1.50 150 114.00 126 0.51 0.61 1626 101 150 3000 77 233 223 116.52 73 189 406929 0.38 89493 34 9227.78 0.820 0.820 12 0.52 1.52 0.00 -0.94 1.80 9228.07 0.30 56.48 243.52 252.37 61458.06 70685.83 61458.06 0.00 252.37 189 0.6 34 521 29 6% 492 262 0.53 466 262 0.56 418 262 0.63 371 262 0.71

300 20 34.40 16.00 2.3 8 Link 150 0.67 330 0 B 261.00 175.20 1.49 150 114.00 126 0.51 0.61 1626 101 150 3000 77 232 223 116.62 73 189 405162 0.38 89105 34 9187.73 0.817 0.817 13 0.52 1.52 0.00 -0.94 1.79 9188.23 0.50 56.31 243.69 252.36 61498.11 70685.83 61498.11 0.00 252.36 189 0.6 34 521 29 5% 493 261 0.53 466 261 0.56 418 261 0.62 371 261 0.70

300 20 34.40 16.00 2.3 8 Link 150 0.67 660 0 B 126.00 106.20 1.19 150 114.00 126 0.51 0.61 1626 101 150 3000 65 98 223 117.53 73 190 341213 0.38 37520 34 8828.28 0.785 0.785 14 0.52 1.52 0.00 -0.94 1.77 8828.47 0.19 54.76 245.24 252.24 61857.55 70685.83 61857.55 0.00 252.24 190 0.6 34 521 26 5% 495 126 0.25 468 126 0.27 420 126 0.30 373 126 0.34

8 300 20 38.40 16.00 2.3 8 Link 75 1.34 330 0 B 269.00 217.60 1.24 150 114.00 126 0.51 0.61 1626 101 75 3000 79 239 223 118.63 73 191 413196 0.38 90871 38 8393.86 0.746 0.746 15 0.52 1.52 0.00 -0.94 1.73 8393.92 0.06 52.88 247.12 252.07 62291.97 70685.83 62291.97 0.00 252.07 191 0.6 38 581 26 4% 555 269 0.48 525 269 0.51 471 269 0.57 418 269 0.64

300 20 38.40 16.00 2.3 8 Link 75 1.34 660 0 B 125.00 146.10 0.86 150 114.00 126 0.51 0.61 1626 101 75 3000 64 98 223 120.35 73 193 333549 0.38 36678 38 7731.03 0.687 0.687 16 0.52 1.52 0.00 -0.94 1.68 7730.74 -0.29 49.94 250.06 251.76 62954.81 70685.83 62954.81 0.00 251.76 193 0.6 38 581 21 4% 560 125 0.22 529 125 0.24 475 125 0.26 422 125 0.30

9 300 20 33.90 16.00 2.3 8 Link 150 0.67 330 0 B 231.00 174.50 1.32 150 114.00 126 0.51 0.61 1626 101 150 3000 68 206 223 119.29 73 192 353599 0.38 77765 34 8136.70 0.723 0.723 17 0.52 1.52 0.00 -0.94 1.71 8136.09 -0.61 51.74 248.26 251.95 62549.13 70685.83 62549.13 0.00 251.95 192 0.6 34 513 21 4% 492 231 0.47 465 231 0.50 417 231 0.55 370 231 0.62

300 20 33.90 16.00 2.3 8 Link 150 0.67 660 0 B 133.00 105.90 1.26 150 114.00 126 0.51 0.61 1626 101 150 3000 68 104 223 115.74 73 188 363594 0.39 39981 34 9546.11 0.849 0.849 18 0.52 1.52 0.00 -0.94 1.82 9545.90 -0.21 57.83 242.17 252.46 61139.73 70685.83 61139.73 0.00 252.46 188 0.6 34 514 30 6% 483 133 0.28 457 133 0.29 411 133 0.32 364 133 0.37

10 300 20 31.30 16.00 2.3 8 Link 75 1.34 330 0 B 259.00 208.20 1.24 150 114.00 126 0.51 0.61 1626 101 75 3000 76 231 223 114.27 73 187 407130 0.39 89537 31 10146.73 0.902 0.902 19 0.52 1.52 0.00 -0.94 1.86 10146.54 -0.19 60.34 239.66 252.61 60539.11 70685.83 60539.11 0.00 252.61 187 0.6 31 475 32 7% 443 259 0.58 419 259 0.62 376 259 0.69 334 259 0.78

300 20 31.30 16.00 2.3 8 Link 75 1.34 660 0 B 128.00 141.40 0.91 150 114.00 126 0.51 0.61 1626 101 75 3000 66 100 223 114.59 73 187 352057 0.39 38713 31 10011.01 0.890 0.890 20 0.52 1.52 0.00 -0.94 1.85 10011.07 0.07 59.78 240.22 252.58 60674.83 70685.83 60674.83 0.00 252.58 187 0.6 31 475 31 6% 444 128 0.29 420 128 0.30 377 128 0.34 334 128 0.38

11 300 20 24.10 25.00 5.6 8 Link 150 0.67 660 0 S 186.00 117.80 1.58 150 109.50 126 0.48 0.59 3958 101 150 3000 96 145 220 87.49 70 157 609119 0.44 64336 24 22605.10 2.009 2.009 2.56 22605.30 0.20 106.94 193.06 249.05 48080.74 70685.83 48080.74 0.00 249.05 157 0.6 24 360 77 21% 283 186 0.66 268 186 0.69 240 186 0.77 213 186 0.87

300 20 24.10 25.00 5.6 8 Link 150 0.67 660 0 S 188.00 117.80 1.60 150 109.50 126 0.48 0.59 3958 101 150 3000 97 147 220 86.82 70 157 618288 0.45 65304 24 22945.38 2.040 2.040 Value for w 1.5153 2.58 22944.72 -0.66 108.12 191.88 248.81 47740.46 70685.83 47740.46 0.00 248.81 157 0.6 24 360 78 22% 282 188 0.67 266 188 0.71 239 188 0.79 212 188 0.89

12 300 20 23.80 25.00 5.6 8 Link 75 1.34 660 0 S 211.00 154.30 1.37 150 109.50 126 0.48 0.59 3958 101 75 3000 109 165 220 77.74 70 147 736698 0.47 77811 24 27684.35 2.461 2.461 2.80 27683.32 -1.03 124.34 175.66 244.80 43001.49 70685.83 43001.49 0.00 244.80 147 0.6 24 353 96 27% 258 211 0.82 244 211 0.87 219 211 0.96 194 211 1.09

300 20 23.80 25.00 5.6 8 Link 75 1.34 660 0 S 239.00 154.30 1.55 150 109.50 126 0.48 0.59 3958 101 75 3000 123 186 220 65.93 70 136 907068 0.51 95805 24 34086.67 3.030 3.030 3.09 34085.66 -1.01 145.81 154.19 237.36 36599.17 70685.83 36599.17 0.00 237.36 136 0.6 24 351 121 34% 230 239 1.04 217 239 1.10 195 239 1.22 173 239 1.38

13 300 20 48.40 25.00 5.6 8 Link 150 0.67 660 0 S 227.00 139.30 1.63 150 109.50 126 0.48 0.59 3958 101 150 3000 117 177 220 109.81 70 180 650941 0.39 68753 48 12028.69 1.069 1.069 1.98 12027.85 -0.84 67.99 232.01 252.82 58657.15 70685.83 58657.15 0.00 252.82 180 0.6 48 726 67 9% 659 227 0.34 623 227 0.36 560 227 0.41 497 227 0.46

300 20 48.40 25.00 5.6 8 Link 150 0.67 660 0 S 228.00 139.30 1.64 150 109.50 126 0.48 0.59 3958 101 150 3000 117 178 220 109.67 70 179 654341 0.39 69112 48 12091.51 1.075 1.075 1.99 12091.03 -0.47 68.24 231.76 252.83 58594.33 70685.83 58594.33 0.00 252.83 179 0.6 48 726 67 9% 659 228 0.35 623 228 0.37 559 228 0.41 496 228 0.46

14 300 20 50.50 25.00 5.6 8 Link 75 1.34 660 0 S 279.00 177.30 1.57 150 109.50 126 0.48 0.59 3958 101 75 3000 144 218 220 103.92 70 174 827229 0.40 87373 51 14650.62 1.302 1.302 2.14 14650.77 0.14 78.18 221.82 252.61 56035.21 70685.83 56035.21 0.00 252.61 174 0.6 51 757 93 12% 664 279 0.42 629 279 0.44 564 279 0.49 501 279 0.56

300 20 50.50 25.00 5.6 8 Link 75 1.34 660 0 S 288.00 177.30 1.62 150 109.50 126 0.48 0.59 3958 101 75 3000 148 225 220 102.64 70 172 860224 0.40 90858 51 15234.98 1.354 1.354 2.18 15235.22 0.23 80.39 219.61 252.49 55450.85 70685.83 55450.85 0.00 252.49 172 0.6 51 757 98 13% 659 288 0.44 624 288 0.46 560 288 0.51 497 288 0.58

15 300 20 24.30 20.00 3.6 8 Link 150 0.67 660 0 S 145.00 107.80 1.35 150 112.00 126 0.50 0.60 2545 101 150 3000 75 113 221 101.26 71 173 432586 0.41 46733 24 15878.71 1.411 1.411 2.21 15878.51 -0.20 82.80 217.20 252.33 54807.13 70685.83 54807.13 0.00 252.33 173 0.6 24 367 50 14% 317 145 0.46 300 145 0.48 270 145 0.54 239 145 0.61

300 20 24.30 20.00 3.6 8 Link 150 0.67 660 0 S 148.00 107.80 1.37 150 112.00 126 0.50 0.60 2545 101 150 3000 76 115 221 100.37 71 172 443803 0.42 47945 24 16290.44 1.448 1.448 2.24 16290.87 0.43 84.33 215.67 252.22 54395.39 70685.83 54395.39 0.00 252.22 172 0.6 24 367 51 14% 316 148 0.47 299 148 0.50 268 148 0.55 238 148 0.62

16 300 20 46.70 20.00 3.6 8 Link 150 0.67 660 0 S 185.00 124.90 1.48 150 112.00 126 0.50 0.60 2545 101 150 3000 95 144 221 115.23 71 187 510576 0.38 55159 47 9751.97 0.867 0.867 1.83 9751.51 -0.46 58.69 241.31 252.51 60933.87 70685.83 60933.87 0.00 252.51 187 0.6 47 704 44 6% 660 185 0.28 624 185 0.30 560 185 0.33 497 185 0.37

16b 300 20 46.70 20.00 3.6 8 Link 150 0.67 660 0 S 186.00 124.90 1.49 150 112.00 126 0.50 0.60 2545 101 150 3000 96 145 221 115.08 71 186 513747 0.38 55501 47 9812.55 0.872 0.872 1.84 9812.51 -0.03 58.95 241.05 252.53 60873.29 70685.83 60873.29 0.00 252.53 186 0.6 47 704 45 6% 659 186 0.28 624 186 0.30 560 186 0.33 497 186 0.37

17 300 20 23.70 16.00 2.3 6 Link 150 0.38 660 0 S 117.00 81.90 1.43 150 116.00 127 0.51 0.62 1626 57 150 3000 60 91 224 109.10 74 183 329226 0.40 36837 24 12337.09 1.097 1.097 2.00 12337.41 0.32 69.22 230.78 252.83 58348.75 70685.83 58348.75 0.00 252.83 183 0.6 24 362 33 9% 329 117 0.36 311 117 0.38 279 117 0.42 248 117 0.47

300 20 23.70 16.00 2.3 6 Link 150 0.38 660 0 S 115.00 81.90 1.40 150 116.00 127 0.51 0.62 1626 57 150 3000 59 90 224 109.68 74 184 322586 0.40 36094 24 12088.24 1.075 1.075 1.99 12087.87 -0.37 68.23 231.77 252.83 58597.59 70685.83 58597.59 0.00 252.83 184 0.6 24 362 32 9% 330 115 0.35 312 115 0.37 280 115 0.41 249 115 0.46

18 300 20 49.60 16.00 2.3 6 Link 150 0.38 660 0 B 137.00 98.90 1.39 150 116.00 127 0.51 0.62 1626 57 150 3000 71 107 224 123.99 74 198 356498 0.37 39889 50 6383.25 0.567 0.567 1.57 6383.31 0.07 43.75 256.25 250.94 64302.59 70685.83 64302.59 0.00 250.94 198 0.6 50 752 13 2% 739 137 0.19 699 137 0.20 627 137 0.22 557 137 0.25

300 20 49.60 16.00 2.3 6 Link 150 0.38 660 0 B 119.00 98.90 1.20 150 116.00 127 0.51 0.62 1626 57 150 3000 61 93 224 126.57 74 200 305660 0.37 34201 50 5472.97 0.486 0.486 1.48 5472.98 0.01 39.36 260.64 250.20 65212.86 70685.83 65212.86 0.00 250.20 200 0.6 50 750 4 1% 746 119 0.16 706 119 0.17 634 119 0.19 562 119 0.21

19 300 20 26.60 20.00 3.6 6 Link 150 0.38 660 0 S 113.00 93.70 1.21 150 114.00 127 0.51 0.61 2545 57 150 3000 58 88 223 113.51 73 186 312614 0.39 34376 27 10460.11 0.930 0.930 1.88 10459.95 -0.16 61.64 238.36 252.67 60225.73 70685.83 60225.73 0.00 252.67 186 0.6 27 404 28 7% 375 113 0.30 355 113 0.32 319 113 0.35 283 113 0.40

300 20 26.60 20.00 3.6 6 Link 150 0.38 660 0 S 129.00 93.70 1.38 150 114.00 127 0.51 0.61 2545 57 150 3000 66 101 223 109.39 73 182 364956 0.40 40131 27 12211.45 1.085 1.085 2.00 12211.53 0.07 68.72 231.28 252.83 58474.38 70685.83 58474.38 0.00 252.83 182 0.6 27 404 37 9% 367 129 0.35 347 129 0.37 312 129 0.41 277 129 0.47

20 300 20 49.30 20.00 3.6 6 Link 150 0.38 660 0 S 149.00 110.30 1.35 150 114.00 127 0.51 0.61 2545 57 150 3000 77 116 223 121.98 73 195 394263 0.37 43354 49 7117.84 0.633 0.633 1.63 7117.94 0.10 47.17 252.83 251.42 63568.00 70685.83 63568.00 0.00 251.42 195 0.6 49 745 21 3% 723 149 0.21 684 149 0.22 614 149 0.24 545 149 0.27

300 20 49.30 20.00 3.6 6 Link 150 0.38 660 0 S 137.00 110.30 1.24 150 114.00 127 0.51 0.61 2545 57 150 3000 71 107 223 123.70 73 196 359332 0.37 39513 49 6487.21 0.577 0.577 1.58 6486.94 -0.27 44.24 255.76 251.01 64198.62 70685.83 64198.62 0.00 251.01 196 0.6 49 743 15 2% 729 137 0.19 689 137 0.20 619 137 0.22 549 137 0.25

21 300 20 22.20 25.00 5.6 6 Link 150 0.38 660 0 S 131.00 99.60 1.32 150 111.50 127 0.49 0.59 3958 57 150 3000 67 102 221 101.63 71 173 390702 0.41 42020 22 15706.40 1.396 1.396 2.20 15706.28 -0.12 82.15 217.85 252.38 54979.43 70685.83 54979.43 0.00 252.38 173 0.6 22 335 45 13% 290 131 0.45 274 131 0.48 246 131 0.53 219 131 0.60

300 20 22.20 25.00 5.6 6 Link 150 0.38 660 0 S 151.00 99.60 1.52 150 111.50 127 0.49 0.59 3958 57 150 3000 78 118 221 95.10 71 166 468061 0.43 50340 22 18816.25 1.673 1.673 2.37 18816.19 -0.06 93.55 206.45 251.24 51869.59 70685.83 51869.59 0.00 251.24 166 0.6 22 334 56 17% 278 151 0.54 263 151 0.57 236 151 0.64 209 151 0.72

22 300 20 45.50 25.00 5.6 6 Link 150 0.38 660 0 S 163.00 120.90 1.35 150 111.50 127 0.49 0.59 3958 57 150 3000 84 127 221 117.80 71 189 444483 0.38 47804 46 8718.22 0.775 0.775 1.76 8718.72 0.50 54.29 245.71 252.20 61967.62 70685.83 61967.62 0.00 252.20 189 0.6 46 684 34 5% 650 163 0.25 615 163 0.27 552 163 0.30 490 163 0.33

300 20 45.50 25.00 5.6 6 Link 150 0.38 660 0 S 164.00 120.90 1.36 150 111.50 127 0.49 0.59 3958 57 150 3000 84 128 221 117.65 71 189 447571 0.38 48136 46 8778.78 0.780 0.780 1.76 8778.86 0.07 54.55 245.45 252.22 61907.05 70685.83 61907.05 0.00 252.22 189 0.6 46 684 35 5% 649 164 0.25 614 164 0.27 551 164 0.30 489 164 0.34

300 20 31.20 20.00 3.6 0 - 0 0.00 660 0 S 86.00 76.80 1.12 150 120.00 130 0.53 0.63 2545 0 0 3000 44 67 226 124.36 76 201 220536 0.38 25527 31 6250.30 0.556 0.556 1.56 6250.69 0.39 43.13 256.87 250.84 64435.54 70685.83 64435.54 0.00 250.84 201 0.6 31 478 7 1% 471 86 0.18 446 86 0.19 400 86 0.21 355 86 0.24

300 20 29.70 20.00 3.6 0 - 0 0.00 660 0 S 90.00 75.60 1.19 150 120.00 130 0.53 0.63 2545 0 0 3000 46 70 226 122.47 76 199 232986 0.38 26968 30 6936.62 0.617 0.617 1.62 6936.79 0.17 46.34 253.66 251.31 63749.21 70685.83 63749.21 0.00 251.31 199 0.6 30 456 11 2% 445 90 0.20 421 90 0.21 378 90 0.24 335 90 0.27

300 20 20.90 20.00 3.6 0 - 0 0.00 660 0 S 98.00 67.20 1.46 150 120.00 130 0.53 0.63 2545 0 0 3000 50 76 226 111.35 76 188 268718 0.41 31104 21 11369.08 1.011 1.011 1.94 11368.32 -0.76 65.35 234.65 252.79 59316.76 70685.83 59316.76 0.00 252.79 188 0.6 21 323 25 8% 298 98 0.33 281 98 0.35 253 98 0.39 224 98 0.44

300 20 21.60 25.00 5.6 0 - 0 0.00 660 0 S 116.00 78.10 1.49 150 117.50 130 0.52 0.61 3958 0 0 3000 60 90 225 106.40 75 181 329549 0.41 37350 22 13527.72 1.202 1.202 2.08 13527.10 -0.62 73.87 226.13 252.77 57158.12 70685.83 57158.12 0.00 252.77 181 0.6 22 331 35 10% 297 116 0.39 281 116 0.41 252 116 0.46 224 116 0.52

300 20 34.80 25.00 5.6 0 - 0 0.00 660 0 S 125.00 91.60 1.36 150 117.50 130 0.52 0.61 3958 0 0 3000 64 98 225 118.39 75 193 333082 0.39 37751 35 8486.53 0.754 0.754 1.74 8486.97 0.44 53.28 246.72 252.11 62199.30 70685.83 62199.30 0.00 252.11 193 0.6 35 532 24 4% 508 125 0.25 481 125 0.26 432 125 0.29 383 125 0.33

300 20 37.70 25.00 5.6 0 - 0 0.00 660 0 S 125.00 94.00 1.33 150 117.50 130 0.52 0.61 3958 0 0 3000 64 98 225 120.28 75 195 329863 0.38 37386 38 7758.01 0.690 0.690 1.68 7758.25 0.24 50.07 249.93 251.78 62927.83 70685.83 62927.83 0.00 251.78 195 0.6 38 576 20 4% 556 125 0.23 525 125 0.24 472 125 0.27 419 125 0.30

300 20 34.90 25.00 5.6 0 - 0 0.00 660 0 S 136.00 91.70 1.48 150 117.50 130 0.52 0.61 3958 0 0 3000 70 106 225 116.33 75 191 366309 0.39 41517 35 9306.36 0.827 0.827 1.80 9306.64 0.28 56.81 243.19 252.39 61379.47 70685.83 61379.47 0.00 252.39 191 0.6 35 534 29 5% 505 136 0.27 478 136 0.28 429 136 0.32 381 136 0.36

300 20 43.90 25.00 5.6 8 Link 150 0.67 660 0 S 218.00 135.80 1.61 150 109.50 126 0.48 0.59 3958 101 150 3000 112 170 220 107.87 70 178 631962 0.39 66748 44 12875.02 1.144 1.144 2.04 12874.99 -0.03 71.34 228.66 252.82 57810.81 70685.83 57810.81 0.00 252.82 178 0.6 44 658 67 10% 591 218 0.37 559 218 0.39 502 218 0.43 445 218 0.49

300 20 36.10 25.00 5.6 8 Link 150 0.67 660 0 S 206.00 129.50 1.59 150 109.50 126 0.48 0.59 3958 101 150 3000 106 161 220 102.62 70 172 615381 0.40 64997 36 15246.10 1.355 1.355 2.18 15245.81 -0.29 80.43 219.57 252.49 55439.74 70685.83 55439.74 0.00 252.49 172 0.6 36 541 70 13% 471 206 0.44 446 206 0.46 400 206 0.51 355 206 0.58

300 20 36.30 25.00 5.6 8 Link 150 0.67 660 0 S 197.00 129.70 1.52 150 109.50 126 0.48 0.59 3958 101 150 3000 101 154 220 104.61 70 174 581764 0.40 61446 36 14333.81 1.274 1.274 2.12 14333.28 -0.52 76.97 223.03 252.67 56352.03 70685.83 56352.03 0.00 252.67 174 0.6 36 544 65 12% 480 197 0.41 454 197 0.43 407 197 0.48 361 197 0.55

300 20 34.10 25.00 5.6 8 Link 150 0.67 660 0 S 183.00 127.80 1.43 150 109.50 126 0.48 0.59 3958 101 150 3000 94 143 220 105.04 70 175 539100 0.40 56940 34 14139.58 1.257 1.257 2.11 14139.91 0.34 76.23 223.77 252.70 56546.26 70685.83 56546.26 0.00 252.70 175 0.6 34 511 60 12% 452 183 0.41 427 183 0.43 384 183 0.48 340 183 0.54

41 500 20 34.00 28.30 2.6 0 - 0 0.00 1200 0 S 236.00 175.40 1.35 250 215.85 230 0.58 0.64 5105 0 0 3500 186 155 387 213.42 137 351 530458 0.39 60744 34 13815.13 0.442 0.442 1.43 13813.99 -1.14 61.43 438.57 416.21 182534.41 196349.54 182534.41 0.00 416.21 351 0.6 34 1480 -9 -1% 1,489 236 0.16 1,409 236 0.17 1,265 236 0.19 1,122 236 0.21

42 500 20 33.50 28.30 2.6 0 - 0 0.00 1200 0 S 234.00 174.50 1.34 250 215.85 230 0.58 0.64 5105 0 0 3500 185 154 387 213.25 137 351 526218 0.39 60258 34 13909.24 0.445 0.445 1.44 13908.48 -0.76 61.72 438.28 416.26 182440.30 196349.54 182440.30 0.00 416.26 351 0.6 34 1459 -8 -1% 1,467 234 0.16 1,388 234 0.17 1,246 234 0.19 1,105 234 0.21

500 20 33.50 28.30 2.6 0 - 0 0.00 1200 0 S 222.00 174.50 1.27 250 215.85 230 0.58 0.64 5105 0 0 3500 175 146 387 214.63 137 352 497271 0.39 56944 34 13144.11 0.421 0.421 1.41 13144.87 0.75 59.38 440.62 415.79 183205.43 196349.54 183205.43 0.00 415.79 352 0.6 34 1457 -14 -1% 1,471 222 0.15 1,392 222 0.16 1,249 222 0.18 1,108 222 0.20

43 500 20 37.80 28.30 2.6 8 Link 150 0.72 1200 0 S 313.00 250.80 1.25 250 207.85 226 0.56 0.63 5105 101 150 3500 247 206 382 207.73 132 340 725841 0.39 80037 38 17084.76 0.547 0.547 1.55 17084.30 -0.46 71.08 428.92 417.94 179264.78 196349.54 179264.78 0.00 417.94 340 0.6 38 1630 19 1% 1,612 313 0.19 1,525 313 0.21 1,368 313 0.23 1,214 313 0.26

500 20 37.80 28.30 2.6 8 Link 150 0.72 1200 0 S 366.00 250.80 1.46 250 207.85 226 0.56 0.63 5105 101 150 3500 289 241 382 202.46 132 335 862105 0.40 95062 38 20292.12 0.649 0.649 1.65 20292.63 0.51 80.04 419.96 419.22 176057.42 196349.54 176057.42 0.00 419.22 335 0.6 38 1635 43 3% 1,592 366 0.23 1,506 366 0.24 1,351 366 0.27 1,199 366 0.31

44 500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 0 S 301.00 242.60 1.24 250 207.85 226 0.56 0.63 5105 101 150 3500 237 198 382 204.45 132 337 704806 0.39 77717 33 19060.43 0.610 0.610 1.61 19060.54 0.11 76.65 423.35 418.78 177289.11 196349.54 177289.11 0.00 418.78 337 0.6 33 1422 30 2% 1,392 301 0.22 1,317 301 0.23 1,182 301 0.25 1,049 301 0.29

500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 0 S 329.00 242.60 1.36 250 207.85 226 0.56 0.63 5105 101 150 3500 259 216 382 201.28 132 334 777691 0.40 85754 33 21031.51 0.673 0.673 1.67 21032.80 1.29 82.05 417.95 419.47 175318.03 196349.54 175318.03 0.00 419.47 334 0.6 33 1424 43 3% 1,381 329 0.24 1,307 329 0.25 1,173 329 0.28 1,041 329 0.32

45 500 20 29.40 34.60 3.8 0 - 0 0.00 1200 0 - 234.00 191.30 1.22 250 212.70 230 0.57 0.63 7461 0 0 3500 185 154 385 209.32 135 345 535283 0.39 60402 29 16152.43 0.517 0.517 1.52 16151.24 -1.19 68.39 431.61 417.50 180197.11 196349.54 180197.11 0.00 417.50 345 0.6 29 1277 8 1% 1,269 234 0.18 1,201 234 0.19 1,078 234 0.22 956 234 0.24

Find angle w

This value will change when it is input to column 'AX'. Therefore,

you will need to repeatedly enter the value into column AX until it

has converged, i.e. that the lever arm and area values are using the

same angle!

This method (Newton Raphson) avoids circular references. Macros could be used also.

See comments. Error when compared to Analysis 1


CRUSHING ANALYSIS FOR CLARKE TEST DATA

INITIAL ANALYSIS REMOVED SPIRAL SECTIONS AND MEMBERS WITH AXIAL LOAD

1: DEFINE SECTION

Define compression zone and equivalent rectangle.

Crushing equations for all values of theta

Loads and Moments Compression chord force Compression area required

!

MMAX

=Vaav( ) L " av( )L

!

d = r 1+ sin"( )

!

" # sin" = Ae

2

r2

$

% & &

'

( ) )

!

" + sin" cos" =#

2$A

r2

!

c = r 1" cos#

2

$

% &

'

( )

!

B ="r 2 #

1

2r2 $ # sin$( )

D # c

!


cot# + tan#( )

!

Fc =M

z"

yt

z

#

$ %

&

' ( V cot) +

N

2

!

Vmax

=P L " a

v( )L

!

y =4R sin

30.5"( )

3 " # sin"( )

Angle !

Lever arm between centroids of comp. and tension

Calculate by Newton-Raphson iteration to find z, then to find !. Two iterations required.



3.8. Spreadsheet 6.7 Spreadsheet 6.7 presents a simple upper bound analysis of the circular sections, where the angle of the failure plane can be

varied for each member. This provides a reliable method of analysis, although it is not as accurate as some of the other methods.

Figure 3.6: Spreadsheet 6.7.

Similarly to all the upper bound plastic analysis spreadsheets, there are a number of worksheets, labeled, for example as:

• d300 s150 av660

• d300 s75 av660

• etc…

These sheets calculate the link geometries to become part of the steel contribution. The output from each worksheet depends

on the failure plane angle (β), and the geometry of the section (radius, stirrup radius, cover etc…). This is all calculated

automatically as the failure plane angle is varied using the sliders (Figure 3.6).

The energy dissipated in the steel may be summarised by Eq.3.1. The output from the geometry calculation is therefore the

value for cosφ. The area (Asi) depends on the number of links intersected, which is easily calculated (and depends on the failure

plane angle). The procedure for finding the value of cosφ is described in Figure 3.7.

€

EDs = Asi fy cosφ∑ Eq. 3.1

Figure 3.7: Link intersection calculations

Steel contributionConcrete contribution

Section test data Geometry Links to worksheets to calculate link intersections etc...

Predicted failure load

Optimise results using slider to vary Beta



24/3/09

TITLE 0 0

5000 500053

92

93

Specimen details and test results BETA 69


CONCRETE CONTRIBUTION STEEL CONTRIBUTION

Specimen

Section

Diameter,

(mm)

Minimum

cover,

(mm)

fcu


Failure


Va/Vth

(BCA)r rs rsv As (mm2) Asv s

Ideal

Beta

Vary Beta

(rads)Delta fcu v fc Concrete Area EDC Number of links Asv (two legs) fy Sum Cos(phi) EDS Vth Va Va/Vth Angle B Degrees

12 300 20 23.80 25.00 5.6 8 Link 75 1.34 660 0 S 211.00 154.30 1.37 150 109.50 126 3958 101 75 0.43 0.43 1.00 23.80 0.68 16.21 170,820 124,080 124928.61 8.00 100.53 200 4.84 97,257 221 211 0.95 0.53 30

300 20 23.80 25.00 5.6 8 Link 75 1.34 660 0 S 239.00 154.30 1.55 150 109.50 126 3958 101 75 0.43 0.43 1.00 23.80 0.68 16.21 170,820 124,080 124928.61 8.00 100.53 200 4.84 97,257 221 239 1.08

14 300 20 50.50 25.00 5.6 8 Link 75 1.34 660 0 S 279.00 177.30 1.57 150 109.50 126 3958 101 75 0.43 0.43 1.00 50.50 0.55 27.65 170,820 211,668 213114.66 8.00 100.53 200 4.84 97,257 309 279 0.90

300 20 50.50 25.00 5.6 8 Link 75 1.34 660 0 S 288.00 177.30 1.62 150 109.50 126 3958 101 75 0.43 0.43 1.00 50.50 0.55 27.65 170,820 211,668 213114.66 8.00 100.53 200 4.84 97,257 309 288 0.93

7 300 20 34.40 16.00 2.3 8 Link 150 0.67 330 0 S 262.00 175.20 1.50 150 114.00 126 1626 101 150 0.74 0.74 1.00 34.40 0.63 21.60 105,082 295,182 295735.72 2.00 100.53 200 1.84 37,018 332 262 0.79 0.92 53

11 300 20 24.10 25.00 5.6 8 Link 150 0.67 660 0 S 186.00 117.80 1.58 150 109.50 126 3958 101 150 0.43 0.43 1.00 24.10 0.68 16.38 170,820 125,368 126224.70 4.00 100.53 200 3.14 63,056 188 186 0.99

300 20 24.10 25.00 5.6 8 Link 150 0.67 660 0 S 188.00 117.80 1.60 150 109.50 126 3958 101 150 0.43 0.43 1.00 24.10 0.68 16.38 170,820 125,368 126224.70 4.00 100.53 200 3.14 63,056 188 188 1.00

13 300 20 48.40 25.00 5.6 8 Link 150 0.67 660 0 S 227.00 139.30 1.63 150 109.50 126 3958 101 150 0.43 0.43 1.00 48.40 0.56 27.01 170,820 206,756 208169.64 4.00 100.53 200 3.14 63,056 270 227 0.84

300 20 48.40 25.00 5.6 8 Link 150 0.67 660 0 S 228.00 139.30 1.64 150 109.50 126 3958 101 150 0.43 0.43 1.00 48.40 0.56 27.01 170,820 206,756 208169.64 4.00 100.53 200 3.14 63,056 270 228 0.85

15 300 20 24.30 20.00 3.6 8 Link 150 0.67 660 0 S 145.00 107.80 1.35 150 112.00 126 2545 101 150 0.43 0.43 1.00 24.30 0.68 16.49 170,820 126,222 127084.90 4.00 100.53 200 3.14 63,056 189 145 0.77

300 20 24.30 20.00 3.6 8 Link 150 0.67 660 0 S 148.00 107.80 1.37 150 112.00 126 2545 101 150 0.43 0.43 1.00 24.30 0.68 16.49 170,820 126,222 127084.90 4.00 100.53 200 3.14 63,056 189 148 0.78

16 300 20 46.70 20.00 3.6 8 Link 150 0.67 660 0 S 185.00 124.90 1.48 150 112.00 126 2545 101 150 0.43 0.43 1.00 46.70 0.57 26.46 170,820 202,533 203917.56 4.00 100.53 200 3.14 63,056 266 185 0.70

16b 300 20 46.70 20.00 3.6 8 Link 150 0.67 660 0 S 186.00 124.90 1.49 150 112.00 126 2545 101 150 0.43 0.43 1.00 46.70 0.57 26.46 170,820 202,533 203917.56 4.00 100.53 200 3.14 63,056 266 186 0.70

17 300 20 23.70 16.00 2.3 6 Link 150 0.38 660 0 S 117.00 81.90 1.43 150 116.00 127 1626 57 150 0.43 0.43 1.00 23.70 0.68 16.15 170,820 123,650 124495.04 4.00 56.55 200 3.14 35,469 159 117 0.74

300 20 23.70 16.00 2.3 6 Link 150 0.38 660 0 S 115.00 81.90 1.40 150 116.00 127 1626 57 150 0.43 0.43 1.00 23.70 0.68 16.15 170,820 123,650 124495.04 4.00 56.55 200 3.14 35,469 159 115 0.72

19 300 20 26.60 20.00 3.6 6 Link 150 0.38 660 0 S 113.00 93.70 1.21 150 114.00 127 2545 57 150 0.43 0.43 1.00 26.60 0.67 17.74 170,820 135,827 136755.66 4.00 56.55 200 3.14 35,469 171 113 0.66 0.93 53

300 20 26.60 20.00 3.6 6 Link 150 0.38 660 0 S 129.00 93.70 1.38 150 114.00 127 2545 57 150 0.43 0.43 1.00 26.60 0.67 17.74 170,820 135,827 136755.66 4.00 56.55 200 3.14 35,469 171 129 0.75

20 300 20 49.30 20.00 3.6 6 Link 150 0.38 660 0 S 149.00 110.30 1.35 150 114.00 127 2545 57 150 0.43 0.43 1.00 49.30 0.55 27.29 170,820 208,903 210330.56 4.00 56.55 200 3.14 35,469 244 149 0.61

300 20 49.30 20.00 3.6 6 Link 150 0.38 660 0 S 137.00 110.30 1.24 150 114.00 127 2545 57 150 0.43 0.43 1.00 49.30 0.55 27.29 170,820 208,903 210330.56 4.00 56.55 200 3.14 35,469 244 137 0.56

21 300 20 22.20 25.00 5.6 6 Link 150 0.38 660 0 S 131.00 99.60 1.32 150 111.50 127 3958 57 150 0.43 0.43 1.00 22.20 0.69 15.30 170,820 117,099 117898.97 4.00 56.55 200 3.14 35,469 153 131 0.86

300 20 22.20 25.00 5.6 6 Link 150 0.38 660 0 S 151.00 99.60 1.52 150 111.50 127 3958 57 150 0.43 0.43 1.00 22.20 0.69 15.30 170,820 117,099 117898.97 4.00 56.55 200 3.14 35,469 153 151 0.99

22 300 20 45.50 25.00 5.6 6 Link 150 0.38 660 0 S 163.00 120.90 1.35 150 111.50 127 3958 57 150 0.43 0.43 1.00 45.50 0.57 26.05 170,820 199,419 200781.97 4.00 56.55 200 3.14 35,469 235 163 0.69

300 20 45.50 25.00 5.6 6 Link 150 0.38 660 0 S 164.00 120.90 1.36 150 111.50 127 3958 57 150 0.43 0.43 1.00 45.50 0.57 26.05 170,820 199,419 200781.97 4.00 56.55 200 3.14 35,469 235 164 0.70

37 300 20 43.90 25.00 5.6 8 Link 150 0.67 660 270.9 S 232.00 154.70 1.50 150 109.50 126 3958 101 150 0.43 0.43 1.00 43.90 0.58 25.48 170,820 195,095 196428.53 4.00 100.53 200 3.14 63,056 258 232 0.90

300 20 43.90 25.00 5.6 8 Link 150 0.67 660 0 S 218.00 135.80 1.61 150 109.50 126 3958 101 150 0.43 0.43 1.00 43.90 0.58 25.48 170,820 195,095 196428.53 4.00 100.53 200 3.14 63,056 258 218 0.84

38 300 20 36.10 25.00 5.6 8 Link 150 0.67 660 270.9 S 209.00 147.30 1.42 150 109.50 126 3958 101 150 0.43 0.43 1.00 36.10 0.62 22.36 170,820 171,209 172379.79 4.00 100.53 200 3.14 63,056 234 209 0.89

300 20 36.10 25.00 5.6 8 Link 150 0.67 660 0 S 206.00 129.50 1.59 150 109.50 126 3958 101 150 0.43 0.43 1.00 36.10 0.62 22.36 170,820 171,209 172379.79 4.00 100.53 200 3.14 63,056 234 206 0.88

39 300 20 36.30 25.00 5.6 8 Link 150 0.67 660 270.6 S 217.20 147.50 1.47 150 109.50 126 3958 101 150 0.43 0.43 1.00 36.30 0.62 22.45 170,820 171,880 173055.00 4.00 100.53 200 3.14 63,056 235 217 0.92

300 20 36.30 25.00 5.6 8 Link 150 0.67 660 0 S 197.00 129.70 1.52 150 109.50 126 3958 101 150 0.43 0.43 1.00 36.30 0.62 22.45 170,820 171,880 173055.00 4.00 100.53 200 3.14 63,056 235 197 0.84

40 300 20 34.10 25.00 5.6 8 Link 150 0.67 660 274.1 S 225.00 145.40 1.55 150 109.50 126 3958 101 150 0.43 0.43 1.00 34.10 0.63 21.47 170,820 164,335 165458.07 4.00 100.53 200 3.14 63,056 227 225 0.99

300 20 34.10 25.00 5.6 8 Link 150 0.67 660 0 S 183.00 127.80 1.43 150 109.50 126 3958 101 150 0.43 0.43 1.00 34.10 0.63 21.47 170,820 164,335 165458.07 4.00 100.53 200 3.14 63,056 227 183 0.80

43 500 20 37.80 28.30 2.6 8 Link 150 0.72 1200 0 S 313.00 250.80 1.25 250 207.85 226 5105 101 150 0.39 0.39 1.00 37.80 0.61 23.10 510,509 453,485 446630.94 8.00 100.53 200 5.75 115,557 569 313 0.55

500 20 37.80 28.30 2.6 8 Link 150 0.72 1200 0 S 366.00 250.80 1.46 250 207.85 226 5105 101 150 0.39 0.39 1.00 37.80 0.61 23.10 510,509 453,485 446630.94 8.00 100.53 200 5.75 115,557 569 366 0.64 0.69 40

44 500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 0 S 301.00 242.60 1.24 250 207.85 226 5105 101 150 0.39 0.39 1.00 32.90 0.64 20.91 510,509 410,527 404321.88 8.00 100.53 200 5.75 115,557 526 301 0.57

500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 0 S 329.00 242.60 1.36 250 207.85 226 5105 101 150 0.39 0.39 1.00 32.90 0.64 20.91 510,509 410,527 404321.88 8.00 100.53 200 5.75 115,557 526 329 0.63

Mean 0.80

SD 0.14

COV 17%


ANLAYSIS FOR TAN BETA = D/AV. CIRCULAR SECTION, UPPER BOUND PLASTICITY

See 2nd G for more accurate analysis

Vary Beta with slider

!

EDs = Asi fy cos"( )#

!

10/12/08

TITLE

SECTION VARIABLES

r 150 mm

rsv 122 mm

s 75 mm

beta 0.43 rads 0.43

edge to centre rsv 28

Beta Diameter Av

24 300 660

Number of links crossing the inclined plane 42 300 330

n 8.00 Rounded down to nearest integer 23 500 1200

n x y dy/dx Phi (rads) Cos Phi

8.00 #NUM! 122.73

7.00 -83.8307524 88.64 0.95 0.81 0.69

6.00 -109.127418 54.55 2.00 0.46 0.89

5.00 -120.273071 20.45 5.88 0.17 0.99

4.00 -121.235513 -13.64 -8.89 -0.11 0.99

3.00 -112.276923 -47.73 -2.35 -0.40 0.92

2.00 -90.4974316 -81.82 -1.11 -0.74 0.74

1.00 -38.0668182 -115.91 -0.33 -1.25 0.31

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

0.00 #NUM! -150.00

Sum 5.54



SECTION ANALYSIS USING EQUATION 6.69. DETERMINATION OF LINK FORCES, AND GRADIENT OF LINKS AT THE INTERSECTION WITH THE INCLINED FAILURE PLANE

!

dy

dx=a " x

y " b=

rsv " ns tan# " r( )2( )0.5

ns tan# " rInput section variables

Number of links crossing plane

Coordinates of the link at the point of intersection

Gradient of the link at the point of intersection of the failure plane

Tangential angle of the link (phi)

Cos of the angle

Sum of the values for cos(phi). This value goes into the main spreadsheet



3.9. Spreadsheet 6.8 Spreadsheet 6.8 works in the same manner as Spreadsheet 6.7, but here longitudinal steel is added to the steel contribution.

This makes only a small difference to accuracy, but does increase the complexity of the calculations somewhat. The longitudinal steel contributions are calculated as shown in Figure 3.8.

Figure 3.8: Longitudinal steel contributions

3.10. Spreadsheet 6.9 Spreadsheet 6.9 presents the most complete upper bound analysis of circular sections. The analysis method is fully explained in the dissertation. Essentially, the method properly accounts for the asymmetric loading condition, by allowing the section to

rotate. The rotation is coupled with horizontal displacement of the section, and the angle of the failure plane itself can also be varied. Each of these three variables is controlled with a simple slider, and the analysis is undertaken for every member in the Clarke database. Calculations of link intersections and longitudinal steel contributions are also undertaken for each design case.

This method provides the most accurate analysis results.

The link intersections are complicated by the axial displacement component, but in principal are the same as those sheets used

in Spreadsheets 6.7 and 6.8 (Figure 3.10). Some extra columns and calculation stages are required in to get consistent results. It is with this sort of analysis that a full C++ approach could be useful.

Figure 3.9: Spreadsheet 6.9

Figure 3.10: Link geometry calculation

!

10/12/08

TITLE

8

67

45

r150 mm 23

rsv126 mm 1

rs109.5

s75 mm

beta 0.43 rads


Long. bars 8 mm

Diameter 25 mm L 3000

Beta Diameter Av

Cover 20 mm 24 300 660

Stirrups 8 mm 42 300 330

Effective depth 219.71 mm

No of spaces 4.00

Height/space 54.75

Longitudinal bars, no.1 at base then pairs and last at top

Bar Number x y (even) y (odd) y z Length

Neta (both even and odd) li P Q sin(Q) ONLY TAKING LONGITUDINAL BARS IN THE TENSION ZONE

8.00 259.50 259.50 565.83 3000.00 0.11 2447.97 1.46 0.11

7.00 204.75 204.75 446.45 3000.00 0.08 2561.75 1.49 0.08

6.00 204.75 204.75 446.45 3000.00 0.08 2561.75 1.49 0.08

5.00 150.00 150.00 327.07 3000.00 0.06 2677.14 1.51 0.06

4.00 150.00 150.00 327.07 3000.00 0.06 2677.14 1.51 0.06

3.00 95.25 95.25 207.69 3000.00 0.03 2793.94 1.54 0.03 0.03

2.00 95.25 95.25 207.69 3000.00 0.03 2793.94 1.54 0.03 0.03

1.00 40.50 40.50 88.31 3000.00 0.01 2911.97 1.56 0.01 0.01

3000.00

3000.00

3000.00

3000.00

3000.00

3000.00

3000.00

3000.00

3000.00

3000.00

3000.00

3000.00

3000.00

3000.00

3000.00

SUM 0.08

SECTION VARIABLES




!

dy

dx=a " x

y " b=

rsv " ns tan# " r( )2( )0.5

ns tan# " r

!

tan" =yi

L # zi

!

li =yi

sin"

!

cosP =yi

li

Input section variables (from main spreadsheet)

Coordinates of the longitudinal bar on the cross section.

Bar number

Angle to the bar

Length of the plane to the bar

Angle ‘P’ (see diagrams)

Angle ‘Q’ (see diagrams)

Value of sin(Q)

Sum of the sin(Q) values

--> goes to main spreadsheet

1

2

3

4

1 2 3 4

Steel contributionConcrete contribution

Uh 2100 81

Final Year Dissertation, John Orr Neta 910

Supporting Spreadsheets Beta 15

24/3/09FY 200

TITLE 0 0

5000 35

Va/Vth 0.94

DELTA 0.8193



STEEL CONTRIBUTION

LINKS Longitudinal Steel

Specimen

Section

Diameter,

(mm)

Minimum

cover,

(mm)

fcu


Failure


Va/Vth


No of

Long Bars Diameter L Ideal Beta Beta Neta Uh zi Angle R

NETA

CONCRETE

LIMITED FOR

CRACK l1 Delta fcu v fc Concrete Area EDC Number of links Asv (two legs)

Sum

Cos(phi)*cos(Q) As fy Delta Asv (per bar)

SUM

RESULTANT

DELTA As fy delta EDS Edc + Eds Va/Vth

44 500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 0 S 301.00 242.60 1.24 250 207.85 226 5105 101 150 16 20 3500 0.15 0.910 21.000 1654.1 0.13 0.0050 1862.71 1695.06 32.90 0.64 20.91 1,313,919 261,440,861 22.00 100.53 31641.88 636,197,797 319.07 -3329 -212,415,157 423,782,640 685,224 0.94


ANLAYSIS FOR TAN BETA = D/AV. CIRCULAR SECTION, UPPER BOUND PLASTICITY

See Circle Plastic

!

"#

!

EDc =1

21" cos#( ) fc $

%r 2

sin#

&

' ( (

)

* + +

Section test data Geometry Vary Beta

Vary Neta

Vary horizontal displacementLinks to worksheets to calculate link intersections etc...

Predicted failure load

!

10/12/08

TITLE

av 660

157

Range of rotation

x = 0 0

x=1 0.17

r 250 mm Rotation, x 1.57 89.95437384

rsv 226 mm

s 150 mm

beta 0.15 rads


L 3500

Beta Diameter Av

24 300 660


n 22.00 Rounded down to nearest integer 23 500 1200 Take sin for vertical steel

neta x length GRADIENT OF LINK RESOLVE INTO LINK Uh 90-angle

n x y z y (abs) R Length to

link

DELTA dy/dx Phi (rads) Cos Phi R Delta - cos R Delta sin R delta cos R

times cos

Phi22.00 #NUM! 248.75 3300.00 498.75 1.19 537.35 488.99 1.19 182.0000 21

21.00 #NUM! 226.08 3150.00 476.08 0.94 590.89 537.71 0.94 318.5000 21

20.00 -98.4994409 203.41 3000.00 453.41 0.74 674.96 614.22 0.4843 1.1198 0.44 0.74 455.0000 412.60 198.31 21 -1.10204064 2.67 457.78 277.49

19.00 -135.686129 180.74 2850.00 430.74 0.59 779.76 709.59 0.7507 0.9268 0.60 0.59 591.5000 391.97 355.13 21 -0.80721523 2.38 528.92 490.69

18.00 -161.528411 158.07 2700.00 408.07 0.47 898.06 817.24 1.0219 0.7746 0.71 0.47 728.0000 371.34 520.32 21 -0.59258971 2.16 639.24 677.90

17.00 -180.953714 135.39 2550.00 385.39 0.39 1025.20 932.93 1.3365 0.6424 0.80 0.39 864.5000 350.71 692.19 21 -0.44453219 2.02 775.96 842.26

16.00 -195.88053 112.72 2400.00 362.72 0.32 1158.26 1054.02 1.7377 0.5222 0.87 0.32 1001.0000 330.08 867.59 21 -0.34223052 1.91 928.26 992.89

15.00 -207.28298 90.05 2250.00 340.05 0.27 1295.43 1178.84 2.3018 0.4098 0.92 0.27 1137.5000 309.45 1043.29 21 -0.26974119 1.84 1088.22 1136.21

14.00 -215.720658 67.38 2100.00 317.38 0.22 1435.53 1306.33 3.2014 0.3028 0.95 0.22 1274.0000 288.82 1216.05 21 -0.21677588 1.79 1249.88 1275.75

13.00 -221.532587 44.71 1950.00 294.71 0.19 1577.77 1435.77 4.9545 0.1992 0.98 0.19 1410.5000 268.19 1382.62 21 -0.17691445 1.75 1408.39 1413.36

12.00 -224.922406 22.04 1800.00 272.04 0.16 1721.63 1566.68 10.2036 0.0977 1.00 0.16 1547.0000 247.56 1539.62 21 -0.14610395 1.72 1559.40 1549.99

11.00 -225.999131 -0.63 1650.00 249.37 0.13 1866.73 1698.73 -360.51 0.00 1.00 0.13 1683.5000 226.93 1683.49 21 -0.12171805 1.69 1698.72 1686.16

10.00 -224.796 -23.30 1500.00 226.70 0.11 2012.81 1831.65 -9.65 -0.10 0.99 0.11 1820.0000 206.30 1810.30 21 -0.10200301 1.67 1822.02 1822.13

9.00 -221.27583 -45.97 1350.00 204.03 0.09 2159.66 1965.29 -4.81 -0.20 0.98 0.09 1956.5000 185.67 1915.60 21 -0.0857515 1.66 1924.58 1958.07

8.00 -215.325012 -68.64 1200.00 181.36 0.08 2307.14 2099.50 -3.14 -0.31 0.95 0.08 2093.0000 165.04 1994.14 21 -0.0721063 1.64 2000.96 2094.04

7.00 -206.733755 -91.31 1050.00 158.69 0.06 2455.13 2234.17 -2.26 -0.42 0.91 0.06 2229.5000 144.41 2039.44 21 -0.06043794 1.63 2044.54 2230.09

6.00 -195.153648 -113.98 900.00 136.02 0.05 2603.56 2369.24 -1.71 -0.53 0.86 0.05 2366.0000 123.78 2043.07 21 -0.05026418 1.62 2046.81 2366.24

5.00 -180.008788 -136.65 750.00 113.35 0.04 2752.34 2504.62 -1.32 -0.65 0.80 0.04 2502.5000 103.15 1993.24 21 -0.04119091 1.61 1995.91 2502.50

4.00 -160.291915 -159.32 600.00 90.68 0.03 2901.42 2640.29 -1.01 -0.78 0.71 0.03 2639.0000 82.52 1871.73 21 -0.03285612 1.60 1873.55 2638.86

3.00 -133.999808 -181.99 450.00 68.01 0.02 3050.76 2776.19 -0.74 -0.94 0.59 0.02 2775.5000 61.89 1645.65 21 -0.02484216 1.60 1646.81 2775.33

2.00 -95.8671781 -204.66 300.00 45.34 0.01 3200.32 2912.29 -0.47 -1.13 0.42 0.01 2912.0000 41.26 1235.24 21 -0.01640007 1.59 1235.93 2911.90

1.00 #NUM! -227.33 150.00 22.67 0.01 3350.08 3048.57 0.01 3048.5000 21 #VALUE!

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

36722.85 sum 31641.88

SECTION VARIABLES




!

dy

dx=a " x

y " b=

rsv " ns tan# " r( )2( )0.5

ns tan# " r

!

tan" =#$

isinR +U

h

$icosR

!

"

Input section variables (from main spreadsheet)

Geometry calculations

Length of plane to the link

Gradient of the link at the intersection

Resolve into the link direction

Add components from horizontal movement

Resultant components (i.e. Rotation x length)

Sum of components --> goes to the main spreadsheet

10/12/08

TITLE

av 660

157

Range of rotation

x = 0 0

x=1 0.17

r 250 mm Rotation, x 1.57 89.95437384

rsv 226 mm

s 150 mm

beta 0.15 rads


L 3500

Beta Diameter Av

24 300 660


n 22.00 Rounded down to nearest integer 23 500 1200 Take sin for vertical steel

neta x length GRADIENT OF LINK RESOLVE INTO LINK Uh 90-angle

n x y z y (abs) R Length to

link

DELTA dy/dx Phi (rads) Cos Phi R Delta - cos R Delta sin R delta cos R

times cos

Phi22.00 #NUM! 248.75 3300.00 498.75 1.19 537.35 488.99 1.19 182.0000 21

21.00 #NUM! 226.08 3150.00 476.08 0.94 590.89 537.71 0.94 318.5000 21

20.00 -98.4994409 203.41 3000.00 453.41 0.74 674.96 614.22 0.4843 1.1198 0.44 0.74 455.0000 412.60 198.31 21 -1.10204064 2.67 457.78 277.49

19.00 -135.686129 180.74 2850.00 430.74 0.59 779.76 709.59 0.7507 0.9268 0.60 0.59 591.5000 391.97 355.13 21 -0.80721523 2.38 528.92 490.69

18.00 -161.528411 158.07 2700.00 408.07 0.47 898.06 817.24 1.0219 0.7746 0.71 0.47 728.0000 371.34 520.32 21 -0.59258971 2.16 639.24 677.90

17.00 -180.953714 135.39 2550.00 385.39 0.39 1025.20 932.93 1.3365 0.6424 0.80 0.39 864.5000 350.71 692.19 21 -0.44453219 2.02 775.96 842.26

16.00 -195.88053 112.72 2400.00 362.72 0.32 1158.26 1054.02 1.7377 0.5222 0.87 0.32 1001.0000 330.08 867.59 21 -0.34223052 1.91 928.26 992.89

15.00 -207.28298 90.05 2250.00 340.05 0.27 1295.43 1178.84 2.3018 0.4098 0.92 0.27 1137.5000 309.45 1043.29 21 -0.26974119 1.84 1088.22 1136.21

14.00 -215.720658 67.38 2100.00 317.38 0.22 1435.53 1306.33 3.2014 0.3028 0.95 0.22 1274.0000 288.82 1216.05 21 -0.21677588 1.79 1249.88 1275.75

13.00 -221.532587 44.71 1950.00 294.71 0.19 1577.77 1435.77 4.9545 0.1992 0.98 0.19 1410.5000 268.19 1382.62 21 -0.17691445 1.75 1408.39 1413.36

12.00 -224.922406 22.04 1800.00 272.04 0.16 1721.63 1566.68 10.2036 0.0977 1.00 0.16 1547.0000 247.56 1539.62 21 -0.14610395 1.72 1559.40 1549.99

11.00 -225.999131 -0.63 1650.00 249.37 0.13 1866.73 1698.73 -360.51 0.00 1.00 0.13 1683.5000 226.93 1683.49 21 -0.12171805 1.69 1698.72 1686.16

10.00 -224.796 -23.30 1500.00 226.70 0.11 2012.81 1831.65 -9.65 -0.10 0.99 0.11 1820.0000 206.30 1810.30 21 -0.10200301 1.67 1822.02 1822.13

9.00 -221.27583 -45.97 1350.00 204.03 0.09 2159.66 1965.29 -4.81 -0.20 0.98 0.09 1956.5000 185.67 1915.60 21 -0.0857515 1.66 1924.58 1958.07

8.00 -215.325012 -68.64 1200.00 181.36 0.08 2307.14 2099.50 -3.14 -0.31 0.95 0.08 2093.0000 165.04 1994.14 21 -0.0721063 1.64 2000.96 2094.04

7.00 -206.733755 -91.31 1050.00 158.69 0.06 2455.13 2234.17 -2.26 -0.42 0.91 0.06 2229.5000 144.41 2039.44 21 -0.06043794 1.63 2044.54 2230.09

6.00 -195.153648 -113.98 900.00 136.02 0.05 2603.56 2369.24 -1.71 -0.53 0.86 0.05 2366.0000 123.78 2043.07 21 -0.05026418 1.62 2046.81 2366.24

5.00 -180.008788 -136.65 750.00 113.35 0.04 2752.34 2504.62 -1.32 -0.65 0.80 0.04 2502.5000 103.15 1993.24 21 -0.04119091 1.61 1995.91 2502.50

4.00 -160.291915 -159.32 600.00 90.68 0.03 2901.42 2640.29 -1.01 -0.78 0.71 0.03 2639.0000 82.52 1871.73 21 -0.03285612 1.60 1873.55 2638.86

3.00 -133.999808 -181.99 450.00 68.01 0.02 3050.76 2776.19 -0.74 -0.94 0.59 0.02 2775.5000 61.89 1645.65 21 -0.02484216 1.60 1646.81 2775.33

2.00 -95.8671781 -204.66 300.00 45.34 0.01 3200.32 2912.29 -0.47 -1.13 0.42 0.01 2912.0000 41.26 1235.24 21 -0.01640007 1.59 1235.93 2911.90

1.00 #NUM! -227.33 150.00 22.67 0.01 3350.08 3048.57 0.01 3048.5000 21 #VALUE!

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

0.00 #NUM! -250.00 0.00 21

36722.85 sum 31641.88

SECTION VARIABLES




!

dy

dx=a " x

y " b=

rsv " ns tan# " r( )2( )0.5

ns tan# " r

!

tan" =#$

isinR +U

h

$icosR

!

"



3.11. Spreadsheet 6.10 This spreadsheet presents a simple concrete plane calculation method, using facets rather than a smooth curve. The angle

between each facet is defined, and the resulting energy dissipation is determined once the user has entered an effective concrete strength.


3.12. Spreadsheet 6.11 Spreadsheet 6.11 considers two approaches to defining the energy dissipated on the concrete plane. The first is similar to that found in Spreadsheet 6.10. The second defines the plane by a single equation (here a cubic is used, it could be altered to be

any other equation), and the values of x, x2 and x3 are variables, allowing different failure plane shapes to be investigated.

Using an equation to define the concrete plane has the great advantage that one method can define all intersection points along

the failure plane. However, intersections with links and longitudinal steel remain tricky, and every section would require its own worksheet. The spreadsheet therefore only considers the concrete plane, but an extension to include all the steel would be a useful area of future work. Developing a C++ approach for this analysis could also be fruitful.


3.13. Spreadsheet 6.12 Spreadsheet 6.12 presents a comparison between all of the upper bound plasticity approaches used in this dissertation.

10/12/08

TITLE

Process:

1 Define section variables, including effective concrete strength Output

2 Choose 'n' number of segments on the inclined plane

3 Input each segment angle

4 Check energy dissipated value

5 Iterate for minimum value

Note Only cells highlighted in Blue are for user input values

ENERGY DISSIPATED

808,085

r 250 mm

rsv 225 mm

s 150 mm


D 500

fc 23.10 Column length 3500 EDC 808,085

Beta Diameter Av

24 300 660

Total length of failure plane 42 300 330

n 10.00 23 500 1200

Height of each 50 l(total) 1124.17 mm

OK

Coordinate calculation Circle width at mid height of slice a 0

b 0

Segment Number

Segment Angle Segment x xi yi

Height from center of circle Plane length x1 x2 width Area Neta l1 P Q Delta Edc

1 10 141.78 141.78 25 -225 287.94 108.97 -108.97 217.94 62754.7464 0.01 3358.31 1.56 0.01 1.00 11011.30

2 30 43.30 326.87 75 -175 100.00 178.54 -178.54 357.07 35707.1421 0.02 3174.02 1.55 0.02 1.00 55238.04

3 40 29.79 399.96 125 -125 77.79 216.51 -216.51 433.01 33682.4089 0.04 3102.56 1.53 0.04 1.00 90942.26

4 50 20.98 450.73 175 -75 65.27 238.48 -238.48 476.97 31131.9797 0.06 3054.29 1.51 0.06 1.00 128233.41

5 60 14.43 486.14 225 -25 57.74 248.75 -248.75 497.49 28722.8132 0.07 3022.24 1.4963 0.0745 1.00 165413.93

6 50 20.98 521.55 275 25 65.27 248.75 -248.75 497.49 32471.5963 0.09 2991.11 1.48 0.09 1.00 133404.00

7 40 29.79 572.33 325 75 77.79 238.48 -238.48 476.97 37101.6486 0.11 2945.66 1.46 0.11 0.99 99643.50

8 30 43.30 645.42 375 125 100.00 216.51 -216.51 433.01 43301.2702 0.13 2879.11 1.44 0.13 0.99 66433.88

9 20 68.69 757.41 425 175 146.19 178.54 -178.54 357.07 52200.3497 0.15 2775.33 1.42 0.15 0.99 35931.31

10 20 68.69 894.78 475 225 146.19 108.97 -108.97 217.94 31861.4198 0.18 2648.17 1.39 0.18 0.98 21833.15

0 #VALUE!

#VALUE!

Number of segments to inclined plane

Example is input for a 500mm diameter section with fc = 23.10N/mm2

SECTION VARIABLES



EXAMPLE OF A SIMPLIFIED CONCRETE PLANE ANALYSIS. SECTION DOES NOT INCLUDE LONGITUDINAL STEEL CONTRIBUTION, THIS IS SIMPLY TO DEMONSTRATE HOW THE ANGLE OF EACH SEGMENT CAN BE VARIED

0

100

200

300

400

500

0 500 1000 1500 2000 2500 3000 3500

!

dy

dx=a " x

y " b=

rsv " ns tan# " r( )2( )0.5

ns tan# " r

!

ltotal

=D

n sin"

#

$ %

&

' ( )

!

x " a( )2

+ y " b( )2

= r 2

!

x2" 2ax + a 2 = r 2 " y " b( )

2

a = 0

x2 = r 2 " y " b( )

2

x = r2" y " b( )

2

Section test data

Number of segments

Angle of each segment Concrete contribution...

Energy dissipated For this failure plane shape

Effective concrete strength

10/12/08

TITLE

Process

1 Choose section diameters

2 Determine effective concrete strength, fc

3 Vary a, b and c to redefine the concrete failure plane.

4 Iterate until lowest Edc is found.

Note Cells highlighted in blue are for user input

r 250 mm

rsv 225 mm

s 150 mm


D 500

Set Equation for failure plane:

a 20

b 19

c 41

EDC 1,005,772,783 Unitless (comparisons only)

Concrete strength

NuFc 23.1

VARY THE 'SEGMENT ANGLES' TO REDRAW THE GRAPH AND PRODUCE A DIFFERENT FAILURE PLANE. THIS ANALYSIS IS USED IN SPREADSHEET 6.6 ALSO

SECTION VARIABLES




Column elevation (L=3500)

-250

-150

-50

50

150

250

0 500 1000 1500 2000 2500 3000 3500

close up of failure plane

-250

-150

-50

50

150

250

0 200 400 600 800 1000 1200 1400

!

dy

dx=a " x

y " b=

rsv " ns tan# " r( )2( )0.5

ns tan# " r

!

x = ay3

+ by2

+ cy

Section test data

Energy dissipated

Effective concrete strength

Change variables a b c to redefine the equation that describes the concrete failure plane

Section elevation



3.14. Spreadsheet 6.14 Spreadsheet 6.14 provides an analysis of the proportion of energy dissipated in both concrete and steel terms in the upper

bound plasticity approach. The spreadsheet is described in Figure 3.13 for the optimised analysis (varying concrete plane angle); the spreadsheet also includes a fixed angle plane analysis.


3.15. Spreadsheet 6.15 Spreadsheet 6.15 presents analysis undertaken of unreinforced sections. The method is detailed in the dissertation, as are the

results. Worksheets are presented both with and without partial safety factors, to both BS5400 and BS EN 1992-1-1 methods.

3.16. Spreadsheet 6.16 The brief economic analysis undertaken at the end of Chapter 6 is taken partly from this spreadsheet. The following aspects were investigated:

Worksheet name Description

Non-reinforced,

€

cotθ = 2.5 Comparison of reinforced and unreinforced sections for

€

cotθ = 2.5

Non-reinforced,

€

cotθ = 1 Comparison of reinforced and unreinforced sections for

€

cotθ = 1

Concrete Shear Capacity Variation in VRd,c with diameter

VRd,c graph with Vrd,max Comparison between crushing limit and VRd,c for different section diameters

VRd,c graph VRd,c for different longitudinal steel percentages

VRd,c graph + axial Effect of axial load on VRd,c

VRd,c graph ALL Comparison of all VRd,c analyses presented above

3.17. Spreadsheet 6.17 More detailed economic analysis is discussed in this Spreadsheet. The initial worksheets analyse the effect of varying section diameter to both BS EN 1992-1-1 and BS5400 and how this interacts with crushing limits and unreinforced section capacity. These sheets are presented in Figure 3.14 and Figure 3.15. All sections are analysed to take a constant 565kN shear force

applied across them. This was chosen arbitrarily and may be changed if desired.

Figure 3.14: BS5400 analysis



24/3/09

TITLE Steep Flat 0 0

Steel 18.09% 19 47 5000 500078

24.63718519 Concrete 81.91% 81 6376

76



CONCRETE CONTRIBUTION STEEL CONTRIBUTION

Specimen

Section

Diameter,

(mm)

Minimum

cover,

(mm)

fcu


Failure


Va/Vth


Ideal

Beta

Vary Beta

(rads)Delta fcu v fc Concrete Area EDC Number of links Asv (two legs) fy Sum Cos(phi) EDS Vth Va Average Vth Average Eds Average Edc % Edc % Eds Va/Vth Angle B Degrees

12 300 20 23.80 25.00 5.6 8 Link 75 1.34 660 0 S 211.00 154.30 1.37 150 109.50 126 3958 101 75 0.43 0.78 1.00 23.80 0.68 16.21 100,509 235,465 4.00 100.53 200 2.59 52,019 287 211 370980 52409 318571 86% 14% 0.73 0.78 45

300 20 23.80 25.00 5.6 8 Link 75 1.34 660 0 S 239.00 154.30 1.55 150 109.50 126 3958 101 75 0.43 0.78 1.00 23.80 0.68 16.21 100,509 235,465 4.00 100.53 201 2.59 52,279 288 239 0.83

14 300 20 50.50 25.00 5.6 8 Link 75 1.34 660 0 S 279.00 177.30 1.57 150 109.50 126 3958 101 75 0.43 0.78 1.00 50.50 0.55 27.65 100,509 401,678 4.00 100.53 202 2.59 52,539 454 279 0.61

300 20 50.50 25.00 5.6 8 Link 75 1.34 660 0 S 288.00 177.30 1.62 150 109.50 126 3958 101 75 0.43 0.78 1.00 50.50 0.55 27.65 100,509 401,678 4.00 100.53 203 2.59 52,799 454 288 0.63

7 300 20 34.40 16.00 2.3 8 Link 150 0.67 330 0 S 262.00 175.20 1.50 150 114.00 126 1626 101 150 0.74 0.76 1.00 34.40 0.63 21.60 102,604 304,960 2.00 100.53 200 1.00 20,071 325 262 325031 20071 304960 94% 6% 0.81 0.76 44

11 300 20 24.10 25.00 5.6 8 Link 150 0.67 660 0 S 186.00 117.80 1.58 150 109.50 126 3958 101 150 0.43 0.76 1.00 24.10 0.68 16.38 102,604 231,170 2.00 100.53 200 1.00 20,071 251 186 322453 17737 304716 94% 6% 0.74

300 20 24.10 25.00 5.6 8 Link 150 0.67 660 0 S 188.00 117.80 1.60 150 109.50 126 3958 101 150 0.43 0.76 1.00 24.10 0.68 16.38 102,604 231,170 2.00 100.53 201 1.00 20,172 251 188 0.75

13 300 20 48.40 25.00 5.6 8 Link 150 0.67 660 0 S 227.00 139.30 1.63 150 109.50 126 3958 101 150 0.43 0.76 1.00 48.40 0.56 27.01 102,604 381,245 2.00 100.53 202 1.00 20,272 402 227 0.57

300 20 48.40 25.00 5.6 8 Link 150 0.67 660 0 S 228.00 139.30 1.64 150 109.50 126 3958 101 150 0.43 0.76 1.00 48.40 0.56 27.01 102,604 381,245 2.00 100.53 203 1.00 20,372 402 228 0.57

15 300 20 24.30 20.00 3.6 8 Link 150 0.67 660 0 S 145.00 107.80 1.35 150 112.00 126 2545 101 150 0.43 0.76 1.00 24.30 0.68 16.49 102,604 232,745 2.00 100.53 204 1.00 20,473 253 145 0.57

300 20 24.30 20.00 3.6 8 Link 150 0.67 660 0 S 148.00 107.80 1.37 150 112.00 126 2545 101 150 0.43 0.76 1.00 24.30 0.68 16.49 102,604 232,745 2.00 100.53 205 1.00 20,573 253 148 0.58

16 300 20 46.70 20.00 3.6 8 Link 150 0.67 660 0 S 185.00 124.90 1.48 150 112.00 126 2545 101 150 0.43 0.76 1.00 46.70 0.57 26.46 102,604 373,457 2.00 100.53 206 1.00 20,673 394 185 0.47 0

16b 300 20 46.70 20.00 3.6 8 Link 150 0.67 660 0 S 186.00 124.90 1.49 150 112.00 126 2545 101 150 0.43 0.76 1.00 46.70 0.57 26.46 102,604 373,457 2.00 100.53 207 1.00 20,774 394 186 0.47

17 300 20 23.70 16.00 2.3 6 Link 150 0.38 660 0 S 117.00 81.90 1.43 150 116.00 127 1626 57 150 0.43 0.76 1.00 23.70 0.68 16.15 102,604 228,002 2.00 56.55 208 1.00 11,742 240 117 0.49

300 20 23.70 16.00 2.3 6 Link 150 0.38 660 0 S 115.00 81.90 1.40 150 116.00 127 1626 57 150 0.43 0.76 1.00 23.70 0.68 16.15 102,604 228,002 2.00 56.55 209 1.00 11,798 240 115 0.48

19 300 20 26.60 20.00 3.6 6 Link 150 0.38 660 0 S 113.00 93.70 1.21 150 114.00 127 2545 57 150 0.43 0.76 1.00 26.60 0.67 17.74 102,604 250,456 2.00 56.55 210 1.00 11,855 262 113 0.43 0.76 44

300 20 26.60 20.00 3.6 6 Link 150 0.38 660 0 S 129.00 93.70 1.38 150 114.00 127 2545 57 150 0.43 0.76 1.00 26.60 0.67 17.74 102,604 250,456 2.00 56.55 211 1.00 11,911 262 129 0.49

20 300 20 49.30 20.00 3.6 6 Link 150 0.38 660 0 S 149.00 110.30 1.35 150 114.00 127 2545 57 150 0.43 0.76 1.00 49.30 0.55 27.29 102,604 385,202 2.00 56.55 212 1.00 11,967 397 149 0.38

300 20 49.30 20.00 3.6 6 Link 150 0.38 660 0 S 137.00 110.30 1.24 150 114.00 127 2545 57 150 0.43 0.76 1.00 49.30 0.55 27.29 102,604 385,202 2.00 56.55 213 1.00 12,024 397 137 0.34

21 300 20 22.20 25.00 5.6 6 Link 150 0.38 660 0 S 131.00 99.60 1.32 150 111.50 127 3958 57 150 0.43 0.76 1.00 22.20 0.69 15.30 102,604 215,922 2.00 56.55 214 1.00 12,080 228 131 0.57

300 20 22.20 25.00 5.6 6 Link 150 0.38 660 0 S 151.00 99.60 1.52 150 111.50 127 3958 57 150 0.43 0.76 1.00 22.20 0.69 15.30 102,604 215,922 2.00 56.55 215 1.00 12,137 228 151 0.66

22 300 20 45.50 25.00 5.6 6 Link 150 0.38 660 0 S 163.00 120.90 1.35 150 111.50 127 3958 57 150 0.43 0.76 1.00 45.50 0.57 26.05 102,604 367,715 2.00 56.55 216 1.00 12,193 380 163 0.43

300 20 45.50 25.00 5.6 6 Link 150 0.38 660 0 S 164.00 120.90 1.36 150 111.50 127 3958 57 150 0.43 0.76 1.00 45.50 0.57 26.05 102,604 367,715 2.00 56.55 217 1.00 12,250 380 164 0.43

37 300 20 43.90 25.00 5.6 8 Link 150 0.67 660 270.9 S 232.00 154.70 1.50 150 109.50 126 3958 101 150 0.43 0.76 1.00 43.90 0.58 25.48 102,604 359,742 2.00 100.53 218 1.00 21,878 382 232 0.61

300 20 43.90 25.00 5.6 8 Link 150 0.67 660 0 S 218.00 135.80 1.61 150 109.50 126 3958 101 150 0.43 0.76 1.00 43.90 0.58 25.48 102,604 359,742 2.00 100.53 219 1.00 21,978 382 218 0.57

38 300 20 36.10 25.00 5.6 8 Link 150 0.67 660 270.9 S 209.00 147.30 1.42 150 109.50 126 3958 101 150 0.43 0.76 1.00 36.10 0.62 22.36 102,604 315,699 2.00 100.53 220 1.00 22,078 338 209 0.62

300 20 36.10 25.00 5.6 8 Link 150 0.67 660 0 S 206.00 129.50 1.59 150 109.50 126 3958 101 150 0.43 0.76 1.00 36.10 0.62 22.36 102,604 315,699 2.00 100.53 221 1.00 22,179 338 206 0.61

39 300 20 36.30 25.00 5.6 8 Link 150 0.67 660 270.6 S 217.20 147.50 1.47 150 109.50 126 3958 101 150 0.43 0.76 1.00 36.30 0.62 22.45 102,604 316,935 2.00 100.53 222 1.00 22,279 339 217 0.64

300 20 36.30 25.00 5.6 8 Link 150 0.67 660 0 S 197.00 129.70 1.52 150 109.50 126 3958 101 150 0.43 0.76 1.00 36.30 0.62 22.45 102,604 316,935 2.00 100.53 223 1.00 22,379 339 197 0.58

40 300 20 34.10 25.00 5.6 8 Link 150 0.67 660 274.1 S 225.00 145.40 1.55 150 109.50 126 3958 101 150 0.43 0.76 1.00 34.10 0.63 21.47 102,604 303,022 2.00 100.53 224 1.00 22,480 326 225 0.69

300 20 34.10 25.00 5.6 8 Link 150 0.67 660 0 S 183.00 127.80 1.43 150 109.50 126 3958 101 150 0.43 0.76 1.00 34.10 0.63 21.47 102,604 303,022 2.00 100.53 225 1.00 22,580 326 183 0.56

43 500 20 37.80 28.30 2.6 8 Link 150 0.72 1200 0 S 313.00 250.80 1.25 250 207.85 226 5105 101 150 0.39 0.76 1.00 37.80 0.61 23.10 285,010 905,638 3.00 100.53 200 2.49 49,974 956 313 912717 49974 862743 95% 5% 0.33

500 20 37.80 28.30 2.6 8 Link 150 0.72 1200 0 S 366.00 250.80 1.46 250 207.85 226 5105 101 150 0.39 0.76 1.00 37.80 0.61 23.10 285,010 905,638 3.00 100.53 200 2.49 49,974 956 366 0.38 0.76 44

44 500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 0 S 301.00 242.60 1.24 250 207.85 226 5105 101 150 0.39 0.76 1.00 32.90 0.64 20.91 285,010 819,848 3.00 100.53 200 2.49 49,974 870 301 0.35

500 20 32.90 28.30 2.6 8 Link 150 0.72 1200 0 S 329.00 242.60 1.36 250 207.85 226 5105 101 150 0.39 0.76 1.00 32.90 0.64 20.91 285,010 819,848 3.00 100.53 200 2.49 49,974 870 329 0.38

Vary Beta with slider


BS8110 ANALYSIS OF CLARKE TEST RESULTS

Energy Analysis

!

EDs = Asi fy cos"( )#

Section geometry Concrete contribution for failure plane of angle !

Steel contribution for angle !

Energy analysis. % energy on concrete

and steel terms.

Vary ! with sliders for each section

10/12/08

TITLE

Designation Abbreviation Asw/s Designation Abbreviation Asw/s

VRd,s(1) H12-50 4.52 VRd,s(4) H12-175 1.293

fyv 500 N/mm2 VRd,s(2) H12-100 2.26 VRd,s(5) H12-200 1.131

VRd,s(3) H12-150 1.51

BS5400

Cover 10 mm

Section Diameter

% Long Steel

Longitudinal Steel

(mm2)

r rs alpha d z Av mm2 fcu b

Concrete term

Force in concrete (kN)

Asv/Sv required

vsForce to be carried in links (kN)

LimitsSum (check)

Plot

0 2 2500 0 -20 #DIV/0! #DIV/0! #DIV/0! #DIV/0! 40 0 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Design Shear Force 2 N/mm2

#DIV/0! #DIV/0! #DIV/0! #DIV/0!

50 2 2500 25 5 0.13 28.18 25.36 1140.47 40 50 5.57 6.35 49.03 490.28 559.15 Shear Force 565.5 kN 495.85 5.42 565.50 5.41723998

100 2 2500 50 30 0.39 69.10 62.19 5789.33 40 100 3.24 18.75 18.89 94.44 546.75 97.68 27.50 565.50 27.4993387

150 2 2500 75 55 0.49 110.01 99.01 13890.27 40 150 2.42 33.61 11.49 38.29 531.89 40.71 65.98 565.50 65.978806

200 2 2500 100 80 0.53 150.93 135.84 25434.58 40 200 1.98 50.31 8.10 20.26 515.19 22.23 120.81 565.50 120.814271

250 2 2500 125 105 0.56 191.85 172.66 40420.24 40 250 1.69 68.51 6.15 12.30 496.99 13.99 192.00 565.50 191.996142

300 2 2500 150 130 0.58 232.76 209.48 58846.52 40 300 1.50 88.01 4.87 8.11 477.49 9.61 279.52 565.50 279.520952

350 2 2500 175 155 0.60 273.68 246.31 80713.08 40 350 1.35 108.64 3.96 5.66 456.86 7.01 383.39 565.50 383.387135

400 2 2500 200 180 0.61 314.59 283.13 106019.76 40 400 1.23 130.30 3.28 4.10 435.20 5.33 503.59 565.50 503.593876

450 2 2500 225 205 0.62 355.51 319.96 134766.47 40 450 1.13 152.90 2.76 3.06 412.60 4.20 640.14 565.50 565.5

500 2 2500 250 230 0.63 396.42 356.78 166953.13 40 500 1.06 176.37 2.33 2.33 389.13 3.39 793.03 565.50 565.5

550 2 2500 275 255 0.63 437.34 393.60 202579.71 40 550 0.99 200.64 1.98 1.80 364.86 2.79 962.25 565.50 565.5

600 2 2500 300 280 0.64 478.25 430.43 241646.19 40 600 0.93 225.68 1.69 1.41 339.82 2.34 1147.82 565.50 565.5

650 2 2500 325 305 0.64 519.17 467.25 284152.55 40 650 0.88 251.42 1.44 1.11 314.08 1.99 1349.72 565.50 565.5

700 2 2500 350 330 0.64 560.08 504.08 330098.76 40 700 0.84 277.84 1.22 0.87 287.66 1.71 1567.97 565.50 565.5

750 2 2500 375 355 0.65 601.00 540.90 379484.84 40 750 0.80 304.90 1.03 0.69 260.60 1.49 1802.55 565.50 565.5

800 2 2500 400 380 0.65 641.92 577.72 432310.75 40 800 0.77 332.58 0.86 0.54 232.92 1.31 2053.48 565.50 565.5

850 2 2500 425 405 0.65 682.83 614.55 488576.51 40 850 0.74 360.84 0.71 0.42 204.66 1.16 2320.74 565.50 565.5

900 2 2500 450 430 0.65 723.75 651.37 548282.11 41 900 0.72 392.89 0.57 0.31 172.61 1.03 2604.34 565.50 565.5

950 2 2500 475 455 0.66 764.66 688.20 611427.54 42 950 0.70 425.92 0.43 0.23 139.58 0.92 2904.28 565.50 565.5

1000 2 2500 500 480 0.66 805.58 725.02 678012.79 43 1000 0.68 459.89 0.31 0.16 105.61 0.83 3220.56 565.50 565.5

1050 2 2500 525 505 0.66 846.49 761.84 748037.88 44 1050 0.66 494.82 0.20 0.09 70.68 0.76 3553.18 565.50 565.5

1100 2 2500 550 530 0.66 887.41 798.67 821502.79 45 1100 0.65 530.67 0.09 0.04 34.83 0.69 3902.14 565.50 565.5

1150 2 2500 575 555 0.66 928.32 835.49 898407.52 46 1150 0.63 567.43 0.00 0.00 -1.93 0.63 4267.44 565.50 565.5

VRd,max = concrete crushing capacity



COMPARISON OF ECONOMIC DESIGN APPROACHES, FOR BS 5400 AND BS EN 1992-1-1. CONSIDERING A CIRCULAR SECTION

GENERAL FORMULAE

!

v = vc

+ vs

!

v = 0.27100As

bwd

"

# $

%

& '

1 3

fcu( )1 3

"

#

$ $

%

&

' ' +

Asv fyv

bsv

"

# $ $

%

& ' '

!

z = 0.9d

!

d = r 1+ sin"( )

!

Av

= r2 "

2+# + sin# cos#

$

% &

'

( )

Section geometry

Forces carried by concrete and steel

Limits analysed

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000 1200

Section diameter (mm)

Co

ncre

te a

nd

Ste

el C

ontr

ibutio

n (kN

)

BS5400, Concrete

Contribution

BS5400, Steel

Contribution

Shear limit

Capacity (Vc + Vs)



Figure 3.15: BS EN 1992-1-1 Analysis

3.18. Spreadsheet 6.18 Spreadsheet 6.18 considers the crushing limits of BS EN 1992-1-1, and how much reinforcement is required to force crushing failures in a circular section. Graphs are presented for different values of truss angle, as well as for rectangular sections. The

method simply equates the stirrup capacity to concrete strut capacity. The result is graphs such as that shown in Figure 3.16 for circular sections of varying diameter.

Figure 3.16: Results from Spreadsheet 6.18 for circular sections.

The rectangular analysis is slightly different. Here, section geometry (width and height) and loads are input by the user, and the

reinforcement ratio (Asv/sv) is varied using a slider to determine the required percentage to force a crushing failure. The truss

angle may also be varied from

€

1≤ cotθ ≤ 2.5, as in all other Eurocode analysis.

Figure 3.17: Results for rectangular sections.

0

200

400

600

800

1000

1200

0 200 400 600 800 1000 1200

Section diameter (mm)

Co

ncre

te a

nd

Ste

el C

ontr

ibutio

n (kN

)

EC2, Unreinforced

capacity

EN1992, Section

capacity

EN1992, Crushing

Limit

Section geometry Crushing limits analysedFull force carried by links up to point at which unreinforced section capacity (large diameter) is sufficient

10/12/08

TITLE Crdc 0.18

k1 0.15

Designation Abbreviation Asw/s Designation Abbreviation Asw/s

VRd,s(1) H12-50 4.52 VRd,s(4) H12-175 1.293

fyv 500 N/mm2 VRd,s(2) H12-100 2.26 VRd,s(5) H12-200 1.131

EC2

VRd,s(3) H12-150 1.51

Cover 10 mm

Section Diameter

% Long Steel

Longitudinal Steel

(mm2)

r rs alpha d z Av mm2 fcu b Concrete

Asv/Sv required

vs Crdc k Vrd,c Vrd,max

0 2 2500 0 -20 #DIV/0! #DIV/0! #DIV/0! #DIV/0! 40 0 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Design Shear Force 2 N/mm2

#DIV/0! #DIV/0! 565.50 #DIV/0! #DIV/0!

50 2 2500 25 5 0.13 28.18 25.36 1140.47 40 50 5.57 6.35 49.03 490.28 559.15 Shear Force 565.5 kN 3.24 3.66 565.50 15.22 15.22

100 2 2500 50 30 0.39 69.10 62.19 5789.33 40 100 3.24 18.75 18.89 94.44 546.75 12.13 2.70 565.50 74.63 74.63

150 2 2500 75 55 0.49 110.01 99.01 13890.27 40 150 2.42 33.61 11.49 38.29 531.89 25.30 2.35 565.50 178.22 178.22

200 2 2500 100 80 0.53 150.93 135.84 25434.58 40 200 1.98 50.31 8.10 20.26 515.19 42.44 2.15 565.50 326.01 326.01

250 2 2500 125 105 0.56 191.85 172.66 40420.24 40 250 1.69 68.51 6.15 12.30 496.99 63.36 2.02 565.50 517.98 517.98

300 2 2500 150 130 0.58 232.76 209.48 58846.52 40 300 1.50 88.01 4.87 8.11 477.49 87.95 1.93 565.50 565.50 754.14

350 2 2500 175 155 0.60 273.68 246.31 80713.08 40 350 1.35 108.64 3.96 5.66 456.86 116.12 1.85 565.50 565.50 1034.50

400 2 2500 200 180 0.61 314.59 283.13 106019.76 40 400 1.23 130.30 3.28 4.10 435.20 147.79 1.80 565.50 565.50 1359.04

450 2 2500 225 205 0.62 355.51 319.96 134766.47 40 450 1.13 152.90 2.76 3.06 412.60 182.92 1.75 565.50 565.50 1727.76

500 2 2500 250 230 0.63 396.42 356.78 166953.13 40 500 1.06 176.37 2.33 2.33 389.13 221.46 1.71 565.50 565.50 2140.68

550 2 2500 275 255 0.63 437.34 393.60 202579.71 40 550 0.99 200.64 1.98 1.80 364.86 263.37 1.68 565.50 565.50 2597.79

600 2 2500 300 280 0.64 478.25 430.43 241646.19 40 600 0.93 225.68 1.69 1.41 339.82 308.62 1.65 565.50 565.50 3099.08

650 2 2500 325 305 0.64 519.17 467.25 284152.55 40 650 0.88 251.42 1.44 1.11 314.08 357.18 1.62 565.50 565.50 3644.57

700 2 2500 350 330 0.64 560.08 504.08 330098.76 40 700 0.84 277.84 1.22 0.87 287.66 409.02 1.60 565.50 565.50 4234.24

750 2 2500 375 355 0.65 601.00 540.90 379484.84 40 750 0.80 304.90 1.03 0.69 260.60 464.12 1.58 565.50 565.50 4868.10

800 2 2500 400 380 0.65 641.92 577.72 432310.75 40 800 0.77 332.58 0.86 0.54 232.92 522.46 1.56 565.50 565.50 5546.15

850 2 2500 425 405 0.65 682.83 614.55 488576.51 40 850 0.74 360.84 0.71 0.42 204.66 584.02 1.54 565.50 584.02 6268.39

900 2 2500 450 430 0.65 723.75 651.37 548282.11 40 900 0.71 389.67 0.58 0.32 175.83 648.79 1.53 565.50 648.79 7034.82

950 2 2500 475 455 0.66 764.66 688.20 611427.54 40 950 0.69 419.04 0.46 0.24 146.46 716.75 1.51 565.50 716.75 7845.43

1000 2 2500 500 480 0.66 805.58 725.02 678012.79 40 1000 0.66 448.94 0.34 0.17 116.56 787.88 1.50 565.50 787.88 8700.24

1050 2 2500 525 505 0.66 846.49 761.84 748037.88 40 1050 0.64 479.34 0.24 0.12 86.16 862.18 1.49 565.50 862.18 9599.23

1100 2 2500 550 530 0.66 887.41 798.67 821502.79 40 1100 0.62 510.23 0.15 0.07 55.27 939.64 1.47 565.50 939.64 10542.41

1150 2 2500 575 555 0.66 928.32 835.49 898407.52 40 1150 0.60 541.60 0.06 0.03 23.90 1020.23 1.46 565.50 1020.23 11529.78

BS EN 1992-1-1 Analysis

GENERAL FORMULAE

VRd,max = concrete crushing capacity



COMPARISON OF ECONOMIC DESIGN APPROACHES, FOR BS 5400 AND BS EN 1992-1-1. CONSIDERING A CIRCULAR SECTION

!

v = vc

+ vs

!

v = 0.27100As

bwd

"

# $

%

& '

1 3

fcu( )1 3

"

#

$ $

%

&

' '

+Asv fyv

bsv

"

# $ $

%

& ' '

!

z = 0.9d

!

d = r 1+ sin"( )

!

Av

= r2 "

2+# + sin# cos#

$

% &

'

( )

Equating the two equations for different section diameters:

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

020

040

060

080

0

Section Diameter (mm)

VR

d,m

ax (kN

)

0

1.5

3

4.5

6

7.5

9

10.5

12

13.5

15

Asv/S

v

Theta = 45º

H8-150

H8-100

H12-100

H10-100

H12-50

Asv/s (45º truss)

H16-50

Linear (H12-100)

Linear (H12-50)

Linear (H10-100)

Linear (H8-100)

Linear (H8-150)

Linear (H16-50)

Linear (Asv/s (45º truss))

Capacity provided by concrete strut

Link capacity (right hand scale)

Horizontal lines represent capacity provided by certain link percentages

!

Vrd ,s =Asw

szfywd cot"

!


cot# + tan#

10/12/08

TITLE

Define Section Properties

b 250 mm fck 40 MPa

h 500 mm fyk 500 MPa

cover 20 mm Gamma s 1.15

ø long. steel 20 mm Gamma c 1.5

ø links 8 mm

d 462

z 415.8

Asv 101 mm2

Asv/Sv provided

2.6 Required Spacing

39mm

Applied Loads

Shear 500 kN

cot(Q) 1

Failure mode

Notes

Section has insufficient shear capacity for the applied loads

Link spacing for proposed section is less than 75mm

Analysis

Vrd,s 470 kN

VRd,max 832 kN

STIRRUPS

Truss angle - vary using slider

Reinforcement - vary using slider



Crushing analysis for rectangular section

Section geometry defined Material properties

Applied loads (shear)

Vary the reinforcement ratio using slider

Vary truss angle using slider

Output = failure mode of section.



3.19. Spreadsheet 6.19 The additional tensile force, required for equilibrium of the truss model, is analysed in this spreadsheet. The method is best

investigated by reading Chapter 6 of the dissertation, and trying out a few designs with the spreadsheet. The spreadsheet includes two method examples, taking moments either about the centroidal axis or the centre of the compression chord. The best results are found when moments are taken about the compression chord. A brief summary of the analysis is shown in

Figure 3.18.

Figure 3.18: ATF analysis


Supporting Spreadsheets10/12/08

TITLE

Additional Tensile Force

1 2

Bar Forces

Diameter 500 mm Position under consideration: 0.66 m from the LHSø long. bar 20 mm At this location V = 173.600 kNNo. long bars 14 M = 114.576 kNmø links 8 mm N = 0 kNcover 20 mm If the section considered is closer than 2.5d from the support, no !F is applied.

r 250 mm Here: NO !F APPLIEDrs 212 mmrsv 226 mm Total force in the tension chord:

d 405 mmz 364 mm

Section length 5 m

Axial load

P(axial) 0 kN Chord Forces EQUIL

Fcd 314.49 kN Fcd - !F 240.72 0.00

Fct 314.49 kN Fct + !F 414.32

!F(t) 99.84 kN Vcot(Q) 173.60

!F(c ) 73.76 kN cotQ 1

Point Load 200.00 kN Axial Load

at av(m) from LHS 0.66 m N 0.00

N (to tension chord) 0.00

UDL 0.00 kN/m N (to compression chord) 0.00

M at centre 0.00 From w

66.00 From PL

66.00 Moment equilibrium is taken about the compression chord

At PL 114.58 From PL364 mm

0.00 From w

114.58

M(Fct) 114.58 kNm

Maximum moment 114.58

The distance x is the position under consideration for additional tensile force This should be equal to the applied loads if equilibrium is correct

x 0.66 m OK

Reaction from w 0.00 kNM(w) 0.00 kNm Now determine bar forces by equilibrium and by moments about the centroid

This will need to be repeated for each new section analysed.RHS reaction

from PL 26.40 kN

LHS reaction 173.60 kN6

M(x), PL 114.58 kNm

Centroid of chords

TOTAL MOMENT 114.576 kNm Zone x y

Compression 0 209.52

Tension 0 -154.81

Reaction from w 0.00 kNV(x) 0.00 Distance from centroid of compression zone to longitudinal bars

V(x) 173.60 kN Bar Number x y! distance from effective depth

Distance from comp (mm) Force (kN)

Distance as a percentage of 'z'

Sum of ratios Make <1

6 -191.01 -91.98 62.82 301.50 57.15 0.83 6.00 0.14

5 -132.18 -165.75 -10.94 375.27 71.13 1.03 0.17TOTAL SHEAR 173.600 kN 4 -47.17 -206.68 -51.88 416.20 78.89 1.14 0.19

3 47.17 -206.68 -51.88 416.20 78.89 1.14 0.19

2 132.18 -165.75 -10.94 375.27 71.13 1.03 0.17

1 191.01 -91.98 62.82 301.50 57.15 0.83 0.14

0 #N/A #N/A #N/A 0.00

0 #N/A #N/A #N/A 0.00

0 #N/A #N/A #N/A 0.00

0 #N/A #N/A #N/A 0.00

0 #N/A #N/A #N/A 0.00

F 414.32 kN

M 117.14 kNm

Horizontal equilibrium OVERALL EQUILIBRIUM CHECK

!F + Ftd 414.3 Horizontal 0.00 OK

!H 0.0 Vertical N/A

Moment 2.56 OK

Moment Equilibrium

M 114.6

!M -2.6

Moment about compression chord

Conclusion

Bar Force (including !F)

6 57.15

5 71.13

4 78.89

3 78.89

2 71.13

1 57.15

0 0.00

0 0.00

0 0.00

0 0.00

0 0.00

Scale Factor 1

Section Definitions

Calculate shear at a distance 'x' from LHS

VALUES TO CHECK EQUILIBRIUM FROM

Calculate moment at a distance 'x' from LHS

Number of bars in the tension zone

Section Loads

Distance of tension chord from compression chord

SECTION AND LOAD DEFINITIONS BAR FORCE CALCULATIONS

Axial load will alter the lever arm, therefore initial analysis should avoid using this function.

!"#$

%&'($

)*#&+*,*#

!"#$

%&'($

)*#&+*,*#

-*'$&.'/0*',#.&1

0.+2#*$$&.'

0*',#.&1

!"#/3.#4*$

0.+2#*$$&.'/3.#4*

%*5*#/6#+

!

Ftd

= Mz

+ "F

Section geometry defined

Loads applied, UDL and point load may be applied along the section.

Choose position at which to determine the value of !F

Chord forces are determined. Choose a value for truss angle.

Moment equilibrium checked

Force in each longitudinal bar is analysed. Total tensile force distributed between the bars.

Longitudinal and moment equilibrium is checked.

Forces in the compression and tension zones are plotted automatically. Scale of plot can be altered.



3.20. Spreadsheet 6.20 This spreadsheet presents a comparison between BS5400 and BS EN 1992-1-1. The analysis compares an imaginary section,

and considers how much reinforcement is required to resist increasing shear loads. The BS5400 analysis uses fcu = 40MPa, 1%

longitudinal steel, and partial safety factors set to 1.00. Thus, the required steel area begins at 0, since the concrete provides

full capacity. The 1% longitudinal steel ratio may be varied by the user.

The EN1992-1-1 approach is presented for cot(θ) = 1 and cot(θ) = 2.5. At cot(θ) =2.5, there is a point at which the EN1992-1-1

approach requires less steel than BS5400, although in all cases the EN1992-1-1 approach requires more steel initially, as it has no concrete term.

By increasing the longitudinal steel to 3%, the BS5400 and BS EN 1992-1-1 approaches are more similar. BS EN 1992-1-1

remains more economical at high shear loads (up until crushing governs, which occurs at about the same load in both methods), and BS5400 is more economical at low shear loads (due to the concrete term). Analysis may be undertaken for any

section diameter, concrete strength and longitudinal steel percentage.


3.21. Spreadsheet 6.21 This spreadsheet presents analysis of two methods for determining the additional tensile force. The detailed analysis is explained

in Spreadsheet 6.19, and apportions the full ∆F between the top and bottom chords in proportion to their distance from the centroidal axis. The simple analysis assumes that the full ∆F is divided equally between the chords (I.e. ∆F = 0.5Vcot(θ)).


!

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0 500 1000 1500 2000

Applied Shear (kN)

Asv/S

v (re

quired

)

BS5400

BSEN1992-1-1

EN1992 (cotQ = 1)

10/12/08

TITLE

Section Data

Section ø 500.00 rs 198 mm

Long steel 1.00 % r 250 mm

Section

Diameter As/Sv Asv/sv

0 0.00 0.00

vc 0.99 Gamma m 1 50 0.28 1.01

d 376.05 mm Gamma ms 1 100 0.56 2.02

fcu 40.00 N/mm2

150 0.83 3.02

alpha 0.53 200 1.11 4.03

Av 158417.32 mm2

250 1.39 5.04

300 1.67 6.05

Vc 157.08 kN 350 1.95 7.06

400 2.22 8.06

fywd 500 N/mm2

450 2.50 9.07

500 2.78 10.08

BS5400: cot Q 2.50 cot Q 1.00 550 3.06 11.09

Limiting stress in BS54004.75 N/mm2

Max As/s, EC2 2.78

Max As/s,

EC2 10.08 600 3.34 12.10Equivalent

shear 752.48 kN zfywdcotQ 423.06 kN zfywdcotQ 169.22 kN 650 3.61 13.10

Equivalent steel

area (As/s) 3.17 z 0.9d z 0.9d 700 3.89 14.11

750 4.17 15.12

800 4.45 16.13

COTQ = 2.5 COTQ = 1 850 4.73 17.14

Shear (kN)

BS5400, As/s

(required) EN1992, As/s

Check for

crushing EN1992, As/s

Check for

crushing 900 5.00 18.14

0 0.00 0.00 0.00 0.00 0.00 950 5.28 19.15

100 0.00 0.24 0.24 0.59 0.59 1000 5.56 20.16

200 0.23 0.47 0.47 1.18 1.18 1050 5.84 21.17

300 0.76 0.71 0.71 1.77 1.77 1100 6.12 22.18

400 1.29 0.95 0.95 2.36 2.36 1150 6.40 23.18

500 1.82 1.18 1.18 2.95 2.95 1200 6.67 24.19

600 2.36 1.42 1.42 3.55 3.55 1250 6.95 25.20

700 2.89 1.65 1.65 4.14 4.14 1300 7.23 26.21

800 3.17 1.89 1.89 4.73 4.73 1350 7.51 27.22

900 3.17 2.13 2.13 5.32 5.32 1400 7.79 28.22

1000 3.17 2.36 2.36 5.91 5.91 1450 8.06 29.23

1100 3.17 2.60 2.60 6.50 6.50

1200 3.17 2.84 2.78 7.09 7.09

1300 3.17 3.07 2.78 7.68 7.68

1400 3.17 3.31 2.78 8.27 8.27

1500 3.17 3.55 2.78 8.86 8.86

1600 3.17 3.78 2.78 9.45 9.45

1700 3.17 4.02 2.78 10.05 10.05

1800 3.17 4.25 2.78 10.64 10.08

1900 3.17 4.49 2.78 11.23 10.08

2000 3.17 4.73 2.78 11.82 10.08

2100 3.17 4.96 2.78 12.41 10.08

2200 3.17 5.20 2.78 13.00 10.08

2300 3.17 5.44 2.78 13.59 10.08

2400 3.17 5.67 2.78 14.18 10.08

2500 3.17 5.91 2.78 14.77 10.08

2600 3.17 6.15 2.78 15.36 10.08

2700 3.17 6.38 2.78 15.96 10.08

2800 3.17 6.62 2.78 16.55 10.08

2900 3.17 6.85 2.78 17.14 10.08

3000 3.17 7.09 2.78 17.73 10.08

3100 3.17 7.33 2.78 18.32 10.08

3200 3.17 7.56 2.78 18.91 10.08

3300 3.17 7.80 2.78 19.50 10.08

3400 3.17 8.04 2.78 20.09 10.08

3500 3.17 8.27 2.78 20.68 10.08

3600 3.17 8.51 2.78 21.27 10.08

3700 3.17 8.75 2.78 21.86 10.08

3800 3.17 8.98 2.78 22.46 10.08

3900 3.17 9.22 2.78 23.05 10.08

4000 3.17 9.45 2.78 23.64 10.08

4100 3.17 9.69 2.78 24.23 10.08

4200 3.17 9.93 2.78 24.82 10.08

4300 3.17 10.16 2.78 25.41 10.08

4400 3.17 10.40 2.78 26.00 10.08

4500 3.17 10.64 2.78 26.59 10.08

4600 3.17 10.87 2.78 27.18 10.08

4700 3.17 11.11 2.78 27.77 10.08

4800 3.17 11.35 2.78 28.36 10.08

4900 3.17 11.58 2.78 28.96 10.08

5000 3.17 11.82 2.78 29.55 10.08

5100 3.17 12.06 2.78 30.14 10.08

5200 3.17 12.29 2.78 30.73 10.08

5300 3.17 12.53 2.78 31.32 10.08

5400 3.17 12.76 2.78 31.91 10.08

5500 3.17 13.00 2.78 32.50 10.08

5600 3.17 13.24 2.78 33.09 10.08

5700 3.17 13.47 2.78 33.68 10.08

5800 3.17 13.71 2.78 34.27 10.08

5900 3.17 13.95 2.78 34.87 10.08

6000 3.17 14.18 2.78 35.46 10.08



Economics of design, comparing BS5400 and BS EN 1992-1-1

Limiting values of As/sv to

ensure crushing for cot(Q)

= 2.5


ensure crushing for cot(Q) =

1

BS EN 1992-1-1

10/12/08

TITLE

Section Data

Section ø 500.00 rs 198 mm

Long steel 1.00 % r 250 mm

Section

Diameter As/Sv Asv/sv

0 0.00 0.00

vc 0.99 Gamma m 1 50 0.28 1.01

d 376.05 mm Gamma ms 1 100 0.56 2.02

fcu 40.00 N/mm2

150 0.83 3.02

alpha 0.53 200 1.11 4.03

Av 158417.32 mm2

250 1.39 5.04

300 1.67 6.05

Vc 157.08 kN 350 1.95 7.06

400 2.22 8.06

fywd 500 N/mm2

450 2.50 9.07

500 2.78 10.08

BS5400: cot Q 2.50 cot Q 1.00 550 3.06 11.09

Limiting stress in BS54004.75 N/mm2

Max As/s, EC2 2.78

Max As/s,

EC2 10.08 600 3.34 12.10Equivalent

shear 752.48 kN zfywdcotQ 423.06 kN zfywdcotQ 169.22 kN 650 3.61 13.10

Equivalent steel

area (As/s) 3.17 z 0.9d z 0.9d 700 3.89 14.11

750 4.17 15.12

800 4.45 16.13

COTQ = 2.5 COTQ = 1 850 4.73 17.14

Shear (kN)

BS5400, As/s

(required) EN1992, As/s

Check for

crushing EN1992, As/s

Check for

crushing 900 5.00 18.14

0 0.00 0.00 0.00 0.00 0.00 950 5.28 19.15

100 0.00 0.24 0.24 0.59 0.59 1000 5.56 20.16

200 0.23 0.47 0.47 1.18 1.18 1050 5.84 21.17

300 0.76 0.71 0.71 1.77 1.77 1100 6.12 22.18

400 1.29 0.95 0.95 2.36 2.36 1150 6.40 23.18

500 1.82 1.18 1.18 2.95 2.95 1200 6.67 24.19

600 2.36 1.42 1.42 3.55 3.55 1250 6.95 25.20

700 2.89 1.65 1.65 4.14 4.14 1300 7.23 26.21

800 3.17 1.89 1.89 4.73 4.73 1350 7.51 27.22

900 3.17 2.13 2.13 5.32 5.32 1400 7.79 28.22

1000 3.17 2.36 2.36 5.91 5.91 1450 8.06 29.23

1100 3.17 2.60 2.60 6.50 6.50

1200 3.17 2.84 2.78 7.09 7.09

1300 3.17 3.07 2.78 7.68 7.68

1400 3.17 3.31 2.78 8.27 8.27

1500 3.17 3.55 2.78 8.86 8.86

1600 3.17 3.78 2.78 9.45 9.45

1700 3.17 4.02 2.78 10.05 10.05

1800 3.17 4.25 2.78 10.64 10.08

1900 3.17 4.49 2.78 11.23 10.08

2000 3.17 4.73 2.78 11.82 10.08

2100 3.17 4.96 2.78 12.41 10.08

2200 3.17 5.20 2.78 13.00 10.08

2300 3.17 5.44 2.78 13.59 10.08

2400 3.17 5.67 2.78 14.18 10.08

2500 3.17 5.91 2.78 14.77 10.08

2600 3.17 6.15 2.78 15.36 10.08

2700 3.17 6.38 2.78 15.96 10.08

2800 3.17 6.62 2.78 16.55 10.08

2900 3.17 6.85 2.78 17.14 10.08

3000 3.17 7.09 2.78 17.73 10.08

3100 3.17 7.33 2.78 18.32 10.08

3200 3.17 7.56 2.78 18.91 10.08

3300 3.17 7.80 2.78 19.50 10.08

3400 3.17 8.04 2.78 20.09 10.08

3500 3.17 8.27 2.78 20.68 10.08

3600 3.17 8.51 2.78 21.27 10.08

3700 3.17 8.75 2.78 21.86 10.08

3800 3.17 8.98 2.78 22.46 10.08

3900 3.17 9.22 2.78 23.05 10.08

4000 3.17 9.45 2.78 23.64 10.08

4100 3.17 9.69 2.78 24.23 10.08

4200 3.17 9.93 2.78 24.82 10.08

4300 3.17 10.16 2.78 25.41 10.08

4400 3.17 10.40 2.78 26.00 10.08

4500 3.17 10.64 2.78 26.59 10.08

4600 3.17 10.87 2.78 27.18 10.08

4700 3.17 11.11 2.78 27.77 10.08

4800 3.17 11.35 2.78 28.36 10.08

4900 3.17 11.58 2.78 28.96 10.08

5000 3.17 11.82 2.78 29.55 10.08

5100 3.17 12.06 2.78 30.14 10.08

5200 3.17 12.29 2.78 30.73 10.08

5300 3.17 12.53 2.78 31.32 10.08

5400 3.17 12.76 2.78 31.91 10.08

5500 3.17 13.00 2.78 32.50 10.08

5600 3.17 13.24 2.78 33.09 10.08

5700 3.17 13.47 2.78 33.68 10.08

5800 3.17 13.71 2.78 34.27 10.08

5900 3.17 13.95 2.78 34.87 10.08

6000 3.17 14.18 2.78 35.46 10.08



Economics of design, comparing BS5400 and BS EN 1992-1-1


ensure crushing for cot(Q)

= 2.5


ensure crushing for cot(Q) =

1

BS EN 1992-1-1

Section geometry defined

Output

Data from crushing analysis (Spreadsheet 6.19)

Analysis to determine required As/Sv, and ensure its within the crushing limits

Applied shear

!

10/12/08

TITLE

Diameter % !Ft % !Fc % !Ft % !Fc ø long. bar 20 mm

200 65 35 50 50 No. long bars 12

300 60 40 50 50 ø links 8 mm

400 58 42 50 50 cover 20 mm

500 57 43 50 50

600 56 44 50 50 r = d/2 mm

700 56 44 50 50 rs = r - cover - øbar/2mm

800 55 45 50 50 rsv =r-cover-øbar-ølink/2mm

900 55 45 50 50 d = r + 2rs/pi mm

1000 55 45 50 50 z =0.9d mm

2000 54 46 50 50

3000 53 47 50 50 Section length = 10 x diameterm

4000 53 47 50 50

Section Loads

Point Load 400 kNat av(m) from

LHS =2.5d m

Analysis is done using spreadsheet 6.19

ANALYSIS - detailed method SECTION DETAILS (constants)ANALYSIS - simple method



Proportion of ATF given in top and bottom chords.

Detailed analysis of !F

Output

30

35

40

45

50

55

60

65

70

0 1000 2000 3000 4000 5000

Diameter

% V

cot(Q

)

% !Ft

% !Fc

Simple method

Section geometry



3.22. Spreadsheet 6.22 Spreadsheet 6.22 compares the upper and lower bound steel terms. The two methods provide different values for the steel

contribution, and this disparity appears to stem from different considerations of cracking in the section prior to failure. The analysis is explained in the Spreadsheet, and in the attached file 6.22P.

3.23. Spreadsheet 7.1 The final spreadsheet essentially presents the same analysis as Spreadsheet 6.4, providing a last analysis of the proposed truss equations. This is then used in the Conclusion section.


4. Summary

4.1. Conclusions Using spreadsheets is a simple way to analyse circular sections, and has been used to good effect throughout this dissertation.

All of the procedures developed in the Spreadsheets (especially upper bound plastic analysis) are well suited to further analysis, for example through the use of C++ computer programming. The development of computer programs was felt to be outside

the bounds of this dissertation.

Any further questions, queries or usage problems should again be directed to the Author ([email protected]).


5. Bibliography

CAPON, M. J. F., DE COSSIO, R. D., 1966 Diagonal tension in concrete members of circular section. Foreign Literature Study No.466, Portland Cement Association, Illinois. Originally published in Ingenieria, April 1965.

CLARKE, J., BIRJANDI, F., 1993 The behaviour of reinforced concrete columns in shear. The Structural Engineer, 71(5), pp73-81.

COLLINS, M. P., BENTZ, E. C., 2002. Shear Strength of Circular Reinforced Concrete Columns. ACI Special Publication SP197-03.

FELTHAM, I., 2004 Shear in reinforced concrete piles and circular columns. The Structural Engineer. 84(11) pp.27-31

TURMO J., RAMOSB, I., APARICIO, A. C., 2008 Shear truss analogy for concrete members of solid and hollow circular cross section. Engineering Structures doi:10.1016/j.engstruct.2008.09.002

shear capacity of circular concrete...

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