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EC2 Concrete Crack Algorithm

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Dong Li

This document describes an algorithm for analyzing concrete cracks in ANSYS. It defines four stress states for concrete sections: P (partially in compression/tension), T (completely in tension), C (completely in compression), and P* (a special case of P). It provides examples comparing the algorithm's analytic solutions to numerical solutions in ANSYS for each stress state, demonstrating good agreement. The document also shows how the algorithm outputs crack widths and orientations.

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Dong LI Crack March 14, 2017 1 / 34 Concrete Crack Algorithme in ANSYS Structure&Optimazation Dong LI March 14, 2017
Introduction The programme is a post-treatment propramme in ANSYS APDL Language dedicated to ANSYS concrete structural FE model analysis. The programme is aimed to computer concrete shell and beam crack and carry out SLS concrete and rebars stress veri?cation according to European concrete code -Eurocode 2. It can dealt with general case like As = As and give the section stress state noting as P -Section partially in compression/tension, T -Section completely in tension, C -Section completely in compression and P? -Special case for section completely in compression. Dong LI Crack March 14, 2017 2 / 34
Four Con?gurations Four concrete section stress states are de?ned in algorithm. A¨s As x A¨s F¨ As F A¨s As x A¨s As h d x They are noted separately in programme: 1 P 2 T 3 C 4 P* Dong LI Crack March 14, 2017 3 / 34
Special Case - P* De?nition (P*) h ? d ＋ x < h Dong LI Crack March 14, 2017 4 / 34
Special Case - C De?nition (C) | M N | ＋ h 6 if As = As (a) | M N | ＋ f (i, s) if As = As (b) As = As is a special case of (b). Dong LI Crack March 14, 2017 5 / 34
Analytic vs. Numerical Case P A¨s As b h d d¨ Dong LI Crack March 14, 2017 6 / 34
Analytic vs. Numerical Case P A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1MN M ?0.5MNm Dong LI Crack March 14, 2017 6 / 34
Analytic vs. Numerical Case P A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1MN M ?0.5MNm ? Result Analytic Algorithm wk 1.252 1.252 mm x 0.1083 0.1083 m σs 20.02 20.022 MPa σs 365.82 365.818 MPa σc 3.48 3.475 MPa Dong LI Crack March 14, 2017 6 / 34
Analytic vs. Numerical Case P A¨s As b h d d¨ Dong LI Crack March 14, 2017 7 / 34
Analytic vs. Numerical Case P A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N 1MN M 0.25MNm Dong LI Crack March 14, 2017 7 / 34
Analytic vs. Numerical Case P A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N 1MN M 0.25MNm ? Result Analytic Algorithm wk 0.01 0.011 mm x 0.584 0.5839 m σs 13.49 13.489 MPa σs 48.54 48.537 MPa σc 3.01 3.013 MPa Dong LI Crack March 14, 2017 7 / 34
Analytic vs. Numerical Case T A¨s As b h d d¨ Dong LI Crack March 14, 2017 8 / 34
Analytic vs. Numerical Case T A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1MN M 0.15MNm Dong LI Crack March 14, 2017 8 / 34
Analytic vs. Numerical Case T A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1MN M 0.15MNm ? Result Analytic Algorithm wk 0.26 0.261 mm x m σs ?212.58 ?212.577 MPa σs ?78.46 ?78.459 MPa σc MPa Dong LI Crack March 14, 2017 8 / 34
Analytic vs. Numerical Case C A¨s As b h d d¨ Dong LI Crack March 14, 2017 9 / 34
Analytic vs. Numerical Case C A¨s As b h d d¨ ? Input data h 0.6 As 9.42cm2 Ecm 10.83GPa b 0.3 As 9.42cm2 fctm 3MPa d 0.05 Bar 3HA20 k1 0.8 d 0.05 Es 200GPa k4 0.425 kt 0.4 fyd 435MPa fcd 16.7MPa N 1.14MN M 0.11MNm Dong LI Crack March 14, 2017 9 / 34
Analytic vs. Numerical Case C A¨s As b h d d¨ ? Input data h 0.6 As 9.42cm2 Ecm 10.83GPa b 0.3 As 9.42cm2 fctm 3MPa d 0.05 Bar 3HA20 k1 0.8 d 0.05 Es 200GPa k4 0.425 kt 0.4 fyd 435MPa fcd 16.7MPa N 1.14MN M 0.11MNm ? Result Analytic Algorithm wk mm x m σs 24.5 24.547 MPa σs 139.6 139.67 MPa σc 0.87 0.869 MPa σc 10.07 10.079 MPa Dong LI Crack March 14, 2017 9 / 34
Analytic vs. Numerical Case P* A¨s As b h d d¨ Dong LI Crack March 14, 2017 10 / 34
Analytic vs. Numerical Case P* A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N 0.01836 M ?0.00297 MNm Dong LI Crack March 14, 2017 10 / 34
Analytic vs. Numerical Case P* A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N 0.01836 M ?0.00297 MNm ? Result Analytic Algorithm wk ?0.00 ?0.175E ? 3 mm x 0.77 0.7726 m σs 0.69 0.685 MPa σs ?0.046 ?0.0462 MPa σc 41.0E ? 3 41.03E ? 3 MPa Dong LI Crack March 14, 2017 10 / 34
Analytic vs. Numerical Case As = A s A¨s As b h d d¨ Dong LI Crack March 14, 2017 11 / 34
Analytic vs. Numerical Case As = A s A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 40.21cm2 fctm 3MPa d 0.0745 Bar HA25/32 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1 M ?0.5 MNm Dong LI Crack March 14, 2017 11 / 34
Analytic vs. Numerical Case As = A s A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 40.21cm2 fctm 3MPa d 0.0745 Bar HA25/32 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1 M ?0.5 MNm ? Result Analytic Algorithm wk 1.06 1.0631 mm x 0.12 0.1156 m σs 21.322 21.322 MPa σs 316.71 316.71 MPa σc 3.25 3.249 MPa Dong LI Crack March 14, 2017 11 / 34
ANSYS Model Result Presentation Dong LI Crack March 14, 2017 12 / 34
Input Data A¨s As b h d d¨ h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa Dong LI Crack March 14, 2017 13 / 34
ANSYS Element Forces ELEM Nx Ny Mx My 1 -6507.9 -13141. 2095.8 517.41 2 -12795. -6158.1 1614.0 3306.3 3 -14588. -8323.3 2070.3 8649.2 4 -15088. -12020. 3053.0 15263. 5 -14757. -15870. 4249.9 22181. 6 -13840. -19360 5498.9 28839. 7 -12410. -22363. 6716.6 34957. 8 -10563. -25007. 7874.2 40460. 9 -8580.6 -27858. 8972.3 45411. 10 -6830.2 -32157. 9997.5 49907. 11 -4344.7 -36444. 10192. 53582. 12 2043.2 -2814.5 6140.3 55683. 13 -15683. -15311. 7620.3 1940.1 14 -18209. -9426.2 4712.2 3311.7 15 -18362. -9346.0 2972.1 5977.6 . Dong LI Crack March 14, 2017 14 / 34
ANSYS Model Element Number Figure: Element Number Presentation Dong LI Crack March 14, 2017 15 / 34
ANSYS Model Result Presentation Figure: Crack Width in Y direction Top Face -mm Dong LI Crack March 14, 2017 16 / 34
ANSYS Model Result Presentation Figure: Crack Width in X direction Top Face -mm Dong LI Crack March 14, 2017 17 / 34
ANSYS Model Result Presentation Figure: Crack Width in Y direction Bottom Face -mm Dong LI Crack March 14, 2017 18 / 34
ANSYS Model Result Presentation Figure: Crack Width in X direction Bottom Face -mm Dong LI Crack March 14, 2017 19 / 34
Output Data Elem x1 x2 wkx-bot wky-bot wkx-top wky-top xDir yDir 1. 0.49 0.80 0.00 0.00 0.00 0.00 P C 2. 0.80 0.36 0.00 0.00 0.00 0.00 C P 3. 0.80 0.28 0.00 0.00 0.00 0.01 C P 4. 0.68 0.27 0.00 0.00 0.00 0.01 P P 5. 0.53 0.26 0.00 0.00 0.00 0.02 P P 6. 0.42 0.26 0.00 0.00 0.00 0.02 P P 7. 0.36 0.26 0.00 0.00 0.00 0.03 P P 8. 0.31 0.26 0.00 0.00 0.00 0.03 P P 9. 0.28 0.26 0.00 0.00 0.01 0.04 P P 10. 0.26 0.26 0.00 0.00 0.01 0.04 P P 11. 0.24 0.26 0.00 0.00 0.01 0.04 P P 12. 0.20 0.22 0.00 0.00 0.01 0.06 P P 13. 0.38 0.80 0.00 0.00 0.00 0.00 P C 14. 0.57 0.46 0.00 0.00 0.00 0.00 P P 15. 0.77 0.33 0.00 0.00 0.00 0.00 P* P Dong LI Crack March 14, 2017 20 / 34
ANSYS Model Result Presentation - Stress Figure: Concrete Stress Y direction Top Face -Mpa Dong LI Crack March 14, 2017 21 / 34
ANSYS Model Result Presentation - Stress Figure: Concrete Stress X direction Top Face -Mpa Dong LI Crack March 14, 2017 22 / 34
ANSYS Model Result Presentation - Stress Figure: Concrete Stress Y direction Bottom Face -Mpa Dong LI Crack March 14, 2017 23 / 34
ANSYS Model Result Presentation - Stress Figure: Concrete Stress X direction Bottom Face -Mpa Dong LI Crack March 14, 2017 24 / 34
ANSYS Model Result Presentation - Stress Elem x1 x2 CX-b CY-b CX-T CY-T x y 1. 0.49 0.80 0.02 0.02 0.00 0.01 P C 2. 0.80 0.36 0.03 0.04 0.00 0.00 C P 3. 0.80 0.28 0.03 0.10 0.00 0.00 C P 4. 0.68 0.27 0.04 0.17 0.00 0.00 P P 5. 0.53 0.26 0.05 0.24 0.00 0.00 P P 6. 0.42 0.26 0.06 0.31 0.00 0.00 P P 7. 0.36 0.26 0.08 0.38 0.00 0.00 P P 8. 0.31 0.26 0.09 0.44 0.00 0.00 P P 9. 0.28 0.26 0.10 0.49 0.00 0.00 P P 10. 0.26 0.26 0.11 0.54 0.00 0.00 P P 11. 0.24 0.26 0.11 0.58 0.00 0.00 P P 12. 0.20 0.22 0.06 0.57 0.00 0.00 P P 13. 0.38 0.80 0.09 0.03 0.00 0.00 P C 14. 0.57 0.46 0.06 0.04 0.00 0.00 P P 15. 0.77 0.33 0.04 0.07 0.00 0.00 P* P Dong LI Crack March 14, 2017 25 / 34
ANSYS Model Result Presentation - Stress Figure: Bar Stress Y direction Top Face -Mpa Dong LI Crack March 14, 2017 26 / 34
ANSYS Model Result Presentation - Stress Figure: Bar Stress X direction Top Face -Mpa Dong LI Crack March 14, 2017 27 / 34
ANSYS Model Result Presentation - Stress Figure: Bar Stress Y direction Bottom Face -Mpa Dong LI Crack March 14, 2017 28 / 34
ANSYS Model Result Presentation - Stress Figure: Bar Stress X direction Bottom Face -Mpa Dong LI Crack March 14, 2017 29 / 34
ANSYS Model Result Presentation - Stress Elem x1 x2 SX-b SY-b SX-T SY-T x y 1. 0.49 0.80 0.38 0.32 0.22 0.21 P C 2. 0.80 0.36 0.43 0.55 0.08 0.72 C P 3. 0.80 0.28 0.51 1.29 0.07 2.77 C P 4. 0.68 0.27 0.64 2.22 0.05 5.22 P P 5. 0.53 0.26 0.80 3.19 0.35 7.80 P P 6. 0.42 0.26 0.97 4.11 0.84 10.30 P P 7. 0.36 0.26 1.12 4.96 1.47 12.62 P P 8. 0.31 0.26 1.24 5.72 2.16 14.72 P P 9. 0.28 0.26 1.34 6.41 2.88 16.55 P P 10. 0.26 0.26 1.43 7.08 3.55 17.99 P P 11. 0.24 0.26 1.39 7.65 3.96 19.07 P P 12. 0.20 0.22 0.69 6.99 3.02 24.44 P P 13. 0.38 0.80 1.29 0.51 1.50 0.10 P C 14. 0.57 0.46 0.91 0.60 0.29 0.42 P P 15. 0.77 0.33 0.68 0.97 -0.05 1.49 P* P Dong LI Crack March 14, 2017 30 / 34
Analytic Solution-Element 15 A¨s As h xInput data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa Elem 15 Nx 18.36KN Mx ?2.97KNm Ny 9.35KN My ?5.08 Elem 3 Nx 14.588KN Mx ?2.07KNm Dong LI Crack March 14, 2017 31 / 34
Analytic Solution-Element 15 - Direction X A¨s As h 0.77 ? Section stress statu ?★ P? ? y3 + p ， y + q = 0 ?★ x = 0.77 p = ?0.469E ? 01 q = ?0.128 ? σb = N，x b，x2 2 ?Es /Ec ，As ，(d?x)?Es /Ec ，As ，(x?d ) ?★ σ = 41.03Kpa ? σs = Es/Ec ， σb ， d?x x ?★ σs = 0.685Mpa ? σs = Es/Ec ， σb ， x?d x ?★ σs = ?0.05Mpa Dong LI Crack March 14, 2017 32 / 34
Analytic Solution-Element 15 - Direction Y A¨s As x ? Section stress statu ?★ P ? y3 + p ， y + q = 0 ?★ x = 0.3304 p = 0.3148 q = ?0.3647 ? σb = N，x b，x2 2 ?Es /Ec ，As ，(d?x)?Es /Ec ，As ，(x?d ) ?★ σ = 0.067Kpa ? σs = Es/Ec ， σb ， d?x x ?★ σs = 0.965Mpa ? σs = Es/Ec ， σb ， x?d x ?★ σs = 1.49Mpa ? wk = 0.273e ? 2mm Dong LI Crack March 14, 2017 33 / 34
Analytic Solution-Element 3 - Direction X A¨s As x ? Section stress statu ?★ C ? S = 0.9269m2 v = 0.4m I = 0.0561m4 ? σc = 0.98Kpa ? σc = 30.5Kpa ? σs = 68.9Kpa ? σs = 512.4Kpa ? wk = 0mm Dong LI Crack March 14, 2017 34 / 34

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EC2 Concrete Crack Algorithm

  • 1. Dong LI Crack March 14, 2017 1 / 34 Concrete Crack Algorithme in ANSYS Structure&Optimazation Dong LI March 14, 2017
  • 2. Introduction The programme is a post-treatment propramme in ANSYS APDL Language dedicated to ANSYS concrete structural FE model analysis. The programme is aimed to computer concrete shell and beam crack and carry out SLS concrete and rebars stress veri?cation according to European concrete code -Eurocode 2. It can dealt with general case like As = As and give the section stress state noting as P -Section partially in compression/tension, T -Section completely in tension, C -Section completely in compression and P? -Special case for section completely in compression. Dong LI Crack March 14, 2017 2 / 34
  • 3. Four Con?gurations Four concrete section stress states are de?ned in algorithm. A¨s As x A¨s F¨ As F A¨s As x A¨s As h d x They are noted separately in programme: 1 P 2 T 3 C 4 P* Dong LI Crack March 14, 2017 3 / 34
  • 4. Special Case - P* De?nition (P*) h ? d ＋ x < h Dong LI Crack March 14, 2017 4 / 34
  • 5. Special Case - C De?nition (C) | M N | ＋ h 6 if As = As (a) | M N | ＋ f (i, s) if As = As (b) As = As is a special case of (b). Dong LI Crack March 14, 2017 5 / 34
  • 6. Analytic vs. Numerical Case P A¨s As b h d d¨ Dong LI Crack March 14, 2017 6 / 34
  • 7. Analytic vs. Numerical Case P A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1MN M ?0.5MNm Dong LI Crack March 14, 2017 6 / 34
  • 8. Analytic vs. Numerical Case P A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1MN M ?0.5MNm ? Result Analytic Algorithm wk 1.252 1.252 mm x 0.1083 0.1083 m σs 20.02 20.022 MPa σs 365.82 365.818 MPa σc 3.48 3.475 MPa Dong LI Crack March 14, 2017 6 / 34
  • 9. Analytic vs. Numerical Case P A¨s As b h d d¨ Dong LI Crack March 14, 2017 7 / 34
  • 10. Analytic vs. Numerical Case P A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N 1MN M 0.25MNm Dong LI Crack March 14, 2017 7 / 34
  • 11. Analytic vs. Numerical Case P A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N 1MN M 0.25MNm ? Result Analytic Algorithm wk 0.01 0.011 mm x 0.584 0.5839 m σs 13.49 13.489 MPa σs 48.54 48.537 MPa σc 3.01 3.013 MPa Dong LI Crack March 14, 2017 7 / 34
  • 12. Analytic vs. Numerical Case T A¨s As b h d d¨ Dong LI Crack March 14, 2017 8 / 34
  • 13. Analytic vs. Numerical Case T A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1MN M 0.15MNm Dong LI Crack March 14, 2017 8 / 34
  • 14. Analytic vs. Numerical Case T A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1MN M 0.15MNm ? Result Analytic Algorithm wk 0.26 0.261 mm x m σs ?212.58 ?212.577 MPa σs ?78.46 ?78.459 MPa σc MPa Dong LI Crack March 14, 2017 8 / 34
  • 15. Analytic vs. Numerical Case C A¨s As b h d d¨ Dong LI Crack March 14, 2017 9 / 34
  • 16. Analytic vs. Numerical Case C A¨s As b h d d¨ ? Input data h 0.6 As 9.42cm2 Ecm 10.83GPa b 0.3 As 9.42cm2 fctm 3MPa d 0.05 Bar 3HA20 k1 0.8 d 0.05 Es 200GPa k4 0.425 kt 0.4 fyd 435MPa fcd 16.7MPa N 1.14MN M 0.11MNm Dong LI Crack March 14, 2017 9 / 34
  • 17. Analytic vs. Numerical Case C A¨s As b h d d¨ ? Input data h 0.6 As 9.42cm2 Ecm 10.83GPa b 0.3 As 9.42cm2 fctm 3MPa d 0.05 Bar 3HA20 k1 0.8 d 0.05 Es 200GPa k4 0.425 kt 0.4 fyd 435MPa fcd 16.7MPa N 1.14MN M 0.11MNm ? Result Analytic Algorithm wk mm x m σs 24.5 24.547 MPa σs 139.6 139.67 MPa σc 0.87 0.869 MPa σc 10.07 10.079 MPa Dong LI Crack March 14, 2017 9 / 34
  • 18. Analytic vs. Numerical Case P* A¨s As b h d d¨ Dong LI Crack March 14, 2017 10 / 34
  • 19. Analytic vs. Numerical Case P* A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N 0.01836 M ?0.00297 MNm Dong LI Crack March 14, 2017 10 / 34
  • 20. Analytic vs. Numerical Case P* A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N 0.01836 M ?0.00297 MNm ? Result Analytic Algorithm wk ?0.00 ?0.175E ? 3 mm x 0.77 0.7726 m σs 0.69 0.685 MPa σs ?0.046 ?0.0462 MPa σc 41.0E ? 3 41.03E ? 3 MPa Dong LI Crack March 14, 2017 10 / 34
  • 21. Analytic vs. Numerical Case As = A s A¨s As b h d d¨ Dong LI Crack March 14, 2017 11 / 34
  • 22. Analytic vs. Numerical Case As = A s A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 40.21cm2 fctm 3MPa d 0.0745 Bar HA25/32 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1 M ?0.5 MNm Dong LI Crack March 14, 2017 11 / 34
  • 23. Analytic vs. Numerical Case As = A s A¨s As b h d d¨ ? Input data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 40.21cm2 fctm 3MPa d 0.0745 Bar HA25/32 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa N ?1 M ?0.5 MNm ? Result Analytic Algorithm wk 1.06 1.0631 mm x 0.12 0.1156 m σs 21.322 21.322 MPa σs 316.71 316.71 MPa σc 3.25 3.249 MPa Dong LI Crack March 14, 2017 11 / 34
  • 24. ANSYS Model Result Presentation Dong LI Crack March 14, 2017 12 / 34
  • 25. Input Data A¨s As b h d d¨ h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa Dong LI Crack March 14, 2017 13 / 34
  • 26. ANSYS Element Forces ELEM Nx Ny Mx My 1 -6507.9 -13141. 2095.8 517.41 2 -12795. -6158.1 1614.0 3306.3 3 -14588. -8323.3 2070.3 8649.2 4 -15088. -12020. 3053.0 15263. 5 -14757. -15870. 4249.9 22181. 6 -13840. -19360 5498.9 28839. 7 -12410. -22363. 6716.6 34957. 8 -10563. -25007. 7874.2 40460. 9 -8580.6 -27858. 8972.3 45411. 10 -6830.2 -32157. 9997.5 49907. 11 -4344.7 -36444. 10192. 53582. 12 2043.2 -2814.5 6140.3 55683. 13 -15683. -15311. 7620.3 1940.1 14 -18209. -9426.2 4712.2 3311.7 15 -18362. -9346.0 2972.1 5977.6 . Dong LI Crack March 14, 2017 14 / 34
  • 27. ANSYS Model Element Number Figure: Element Number Presentation Dong LI Crack March 14, 2017 15 / 34
  • 28. ANSYS Model Result Presentation Figure: Crack Width in Y direction Top Face -mm Dong LI Crack March 14, 2017 16 / 34
  • 29. ANSYS Model Result Presentation Figure: Crack Width in X direction Top Face -mm Dong LI Crack March 14, 2017 17 / 34
  • 30. ANSYS Model Result Presentation Figure: Crack Width in Y direction Bottom Face -mm Dong LI Crack March 14, 2017 18 / 34
  • 31. ANSYS Model Result Presentation Figure: Crack Width in X direction Bottom Face -mm Dong LI Crack March 14, 2017 19 / 34
  • 32. Output Data Elem x1 x2 wkx-bot wky-bot wkx-top wky-top xDir yDir 1. 0.49 0.80 0.00 0.00 0.00 0.00 P C 2. 0.80 0.36 0.00 0.00 0.00 0.00 C P 3. 0.80 0.28 0.00 0.00 0.00 0.01 C P 4. 0.68 0.27 0.00 0.00 0.00 0.01 P P 5. 0.53 0.26 0.00 0.00 0.00 0.02 P P 6. 0.42 0.26 0.00 0.00 0.00 0.02 P P 7. 0.36 0.26 0.00 0.00 0.00 0.03 P P 8. 0.31 0.26 0.00 0.00 0.00 0.03 P P 9. 0.28 0.26 0.00 0.00 0.01 0.04 P P 10. 0.26 0.26 0.00 0.00 0.01 0.04 P P 11. 0.24 0.26 0.00 0.00 0.01 0.04 P P 12. 0.20 0.22 0.00 0.00 0.01 0.06 P P 13. 0.38 0.80 0.00 0.00 0.00 0.00 P C 14. 0.57 0.46 0.00 0.00 0.00 0.00 P P 15. 0.77 0.33 0.00 0.00 0.00 0.00 P* P Dong LI Crack March 14, 2017 20 / 34
  • 33. ANSYS Model Result Presentation - Stress Figure: Concrete Stress Y direction Top Face -Mpa Dong LI Crack March 14, 2017 21 / 34
  • 34. ANSYS Model Result Presentation - Stress Figure: Concrete Stress X direction Top Face -Mpa Dong LI Crack March 14, 2017 22 / 34
  • 35. ANSYS Model Result Presentation - Stress Figure: Concrete Stress Y direction Bottom Face -Mpa Dong LI Crack March 14, 2017 23 / 34
  • 36. ANSYS Model Result Presentation - Stress Figure: Concrete Stress X direction Bottom Face -Mpa Dong LI Crack March 14, 2017 24 / 34
  • 37. ANSYS Model Result Presentation - Stress Elem x1 x2 CX-b CY-b CX-T CY-T x y 1. 0.49 0.80 0.02 0.02 0.00 0.01 P C 2. 0.80 0.36 0.03 0.04 0.00 0.00 C P 3. 0.80 0.28 0.03 0.10 0.00 0.00 C P 4. 0.68 0.27 0.04 0.17 0.00 0.00 P P 5. 0.53 0.26 0.05 0.24 0.00 0.00 P P 6. 0.42 0.26 0.06 0.31 0.00 0.00 P P 7. 0.36 0.26 0.08 0.38 0.00 0.00 P P 8. 0.31 0.26 0.09 0.44 0.00 0.00 P P 9. 0.28 0.26 0.10 0.49 0.00 0.00 P P 10. 0.26 0.26 0.11 0.54 0.00 0.00 P P 11. 0.24 0.26 0.11 0.58 0.00 0.00 P P 12. 0.20 0.22 0.06 0.57 0.00 0.00 P P 13. 0.38 0.80 0.09 0.03 0.00 0.00 P C 14. 0.57 0.46 0.06 0.04 0.00 0.00 P P 15. 0.77 0.33 0.04 0.07 0.00 0.00 P* P Dong LI Crack March 14, 2017 25 / 34
  • 38. ANSYS Model Result Presentation - Stress Figure: Bar Stress Y direction Top Face -Mpa Dong LI Crack March 14, 2017 26 / 34
  • 39. ANSYS Model Result Presentation - Stress Figure: Bar Stress X direction Top Face -Mpa Dong LI Crack March 14, 2017 27 / 34
  • 40. ANSYS Model Result Presentation - Stress Figure: Bar Stress Y direction Bottom Face -Mpa Dong LI Crack March 14, 2017 28 / 34
  • 41. ANSYS Model Result Presentation - Stress Figure: Bar Stress X direction Bottom Face -Mpa Dong LI Crack March 14, 2017 29 / 34
  • 42. ANSYS Model Result Presentation - Stress Elem x1 x2 SX-b SY-b SX-T SY-T x y 1. 0.49 0.80 0.38 0.32 0.22 0.21 P C 2. 0.80 0.36 0.43 0.55 0.08 0.72 C P 3. 0.80 0.28 0.51 1.29 0.07 2.77 C P 4. 0.68 0.27 0.64 2.22 0.05 5.22 P P 5. 0.53 0.26 0.80 3.19 0.35 7.80 P P 6. 0.42 0.26 0.97 4.11 0.84 10.30 P P 7. 0.36 0.26 1.12 4.96 1.47 12.62 P P 8. 0.31 0.26 1.24 5.72 2.16 14.72 P P 9. 0.28 0.26 1.34 6.41 2.88 16.55 P P 10. 0.26 0.26 1.43 7.08 3.55 17.99 P P 11. 0.24 0.26 1.39 7.65 3.96 19.07 P P 12. 0.20 0.22 0.69 6.99 3.02 24.44 P P 13. 0.38 0.80 1.29 0.51 1.50 0.10 P C 14. 0.57 0.46 0.91 0.60 0.29 0.42 P P 15. 0.77 0.33 0.68 0.97 -0.05 1.49 P* P Dong LI Crack March 14, 2017 30 / 34
  • 43. Analytic Solution-Element 15 A¨s As h xInput data h 0.8 As 34.36cm2 Ecm 10.83GPa b 1 As 34.36cm2 fctm 3MPa d 0.0745 Bar 7HA25 k1 0.8 d 0.0745 Es 200GPa k4 0.425 kt 0.4 fyd 250MPa fcd 21MPa Elem 15 Nx 18.36KN Mx ?2.97KNm Ny 9.35KN My ?5.08 Elem 3 Nx 14.588KN Mx ?2.07KNm Dong LI Crack March 14, 2017 31 / 34
  • 44. Analytic Solution-Element 15 - Direction X A¨s As h 0.77 ? Section stress statu ?★ P? ? y3 + p ， y + q = 0 ?★ x = 0.77 p = ?0.469E ? 01 q = ?0.128 ? σb = N，x b，x2 2 ?Es /Ec ，As ，(d?x)?Es /Ec ，As ，(x?d ) ?★ σ = 41.03Kpa ? σs = Es/Ec ， σb ， d?x x ?★ σs = 0.685Mpa ? σs = Es/Ec ， σb ， x?d x ?★ σs = ?0.05Mpa Dong LI Crack March 14, 2017 32 / 34
  • 45. Analytic Solution-Element 15 - Direction Y A¨s As x ? Section stress statu ?★ P ? y3 + p ， y + q = 0 ?★ x = 0.3304 p = 0.3148 q = ?0.3647 ? σb = N，x b，x2 2 ?Es /Ec ，As ，(d?x)?Es /Ec ，As ，(x?d ) ?★ σ = 0.067Kpa ? σs = Es/Ec ， σb ， d?x x ?★ σs = 0.965Mpa ? σs = Es/Ec ， σb ， x?d x ?★ σs = 1.49Mpa ? wk = 0.273e ? 2mm Dong LI Crack March 14, 2017 33 / 34
  • 46. Analytic Solution-Element 3 - Direction X A¨s As x ? Section stress statu ?★ C ? S = 0.9269m2 v = 0.4m I = 0.0561m4 ? σc = 0.98Kpa ? σc = 30.5Kpa ? σs = 68.9Kpa ? σs = 512.4Kpa ? wk = 0mm Dong LI Crack March 14, 2017 34 / 34