Stable Length in Inelastic Design Considering Ductility Requirement

Abstract

The region close to plastic hinge is vulnerable to lateral instability due to the formation of the mechanism. A stable length between lateral restraint at the plastic hinge and adjacent lateral restraint plays a significant role in preventing lateral instability and until after the required hinge rotation has taken place. This phenomenon is studied for beams under uniform moment loading by carrying out an extensive parametric study on a mixed set of I-shaped hybrid and homogeneous plate girders using nonlinear finite element analysis. Geometric dimensions and steel grade of plate girder elements are varied to account for their effect on member slenderness. Effect of overall slenderness is included by varying lateral bracing configuration. Attention is given to the interaction between local and lateral buckling and their influence on inelastic rotation. Regression analysis of the database is carried out to arrive at a prediction equation for stable length to achieve the required rotation capacity. The equation is validated by applying it to thirty-nine selected experiments conducted by others and comparing results with the results of available prediction equations. Statistical analysis of the validation study shows that the proposed equation provides more refined results as compared to available equations. It is observed that use of the new prediction equation when used along with rotation capacity prediction equation suggested by authors in the previous study proves to be a rational solution as available stable length equations that are based only on geometric proportions of sections. At the end of the paper, a flowchart and demonstration examples for use of these equations are presented.

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Correspondence to Abhay Kulkarni.

Appendix

Appendix

Demonstration Example

Design a plate girder of span 12 m subject to static force at 1/3 span from each support. Minimum plastic moment should not be less than 1500 kN m. Work out different alternatives of plate girder to satisfy following criteria:

  1. 1.

    Minimum rotation capacity shall not be less than 4.

  2. 2.

    Calculate different alternatives considering rotation capacity values 4, 5, and 7.

  3. 3.

    Compare solution considering homogeneous and hybrid girders alternative.

Solution

A flow chart is prepared to demonstrate general solution procedure.

Proportioning of section Section shall be proportioned in such a way that:

  1. a.

    The minimum plastic moment capacity of section shall be Mp = 1500 kN m

  2. b.

    As rotation capacity of section is mentioned it is required that the section should qualify as plastic section.

  3. c.

    Hybrid ratio is less than 1.5

Unbraced length Maximum unbraced length for required rotation capacity is calculated using Eq. (7)

Rotation capacity: Rotation capacity of given section is checked using Eq. (8) developed in the previous research by the authors (Kulkarni and Gupta 2018).

$$ R_{u} = 8.125 - 0.55\lambda_{f} - 0.115\lambda_{w} - 6.5\lambda_{LT} + 7.25\left( {\frac{{E_{tf} }}{{E_{tw} }}} \right) + 0.667\left( {\frac{{t_{f} }}{{t_{w} }}} \right) $$
(8)

Modify section or lateral bracing distance if rotation capacity is less than specified. Designer to use his judgment to decide difference between required and actually provided.

In the following tables, the problem is solved using five different combinations for steel grades in plate girders. Three sections are hybrid and other two are homogeneous.

 Girder designation
HY-1HY-2H-1H-2H-3
Solution-1: rotation capacity required = 4
Girder dimensionsD594494650590469
 dw550450600550425
 h572472625570447
 Bf275275300275275
 tf2222252022
 tw1616161616
Materialfyf350450250350450
 fyw250350250350450
Hybrid ratio (Rh) 1.4001.2861.0001.0001.000
Yield stress ratio (ε)Flange0.8450.7451.0000.8450.745
 Web1.0000.8451.0000.8450.745
Flange/web slendernessFlange (λf)6.967.905.687.667.90
 h/tw34.3828.1337.5034.3826.56
 Web (λw)34.37533.27837.50040.67335.637
ClassFlangePLPLPLPLPL
 WebPLPLPLPLPL
 SectionPlasticPlasticPlasticPlasticPlastic
NANeutral Axis297247325295234.5
MpPlastic Mom Cap1513.711568.521531.881520.751542.08
RuRequired rot cap.44444
Appox. Lb/ryPredicted by Eq. 746.1040.1252.0139.3838.66
 IS800-(2007)37.41135.75653.37034.93637.899
 EN 1993-1-139.72636.86555.55737.64738.762
 AISC42.07237.10449.78042.07237.104
 Galambos34.95930.83141.36434.95930.831
Mcrcritical moment5551.8425586.0025609.3497182.1265579.969
λLTsqrt(Mp/1.156*Mcr)0.48570.49290.48600.42800.4889
Etf/Etw 0.9160.945111
Check for rot capUsing Eq. (8)4.9934.7676.0694.7534.921
Revised Lb/ry 4640524039
McrCritical moment5576.075619.295611.906961.805482.04
λLTsqrt(Mp/1.156*Mcr)0.4850.4910.4860.4350.493
Check for rot capUsing Eq. (8)4.9994.7766.0704.7134.895
Wt in Kg/m 164.07151.51193.11155.43148.37
Solution 2: rotation capacity required = 5
Girder dimensionsD594494650594494
 dw550450600550450
 h572472625572472
 Bf275275300275275
 tf2222252222
 tw1616161616
Materialfyf350450250350450
 fyw250350250350450
Hybrid ratio (Rh) 1.4001.2861.0001.0001.000
Yield stress ratio (ε)Flange0.8450.7451.0000.8450.745
 Web1.0000.8451.0000.8450.745
Flange/web slendernessFlange (λf)6.967.905.686.967.90
 h/tw34.3828.1337.5034.3828.13
 Web (λw)34.37533.27837.50040.67337.734
ClassFlangePLPLPLPLPL
 WebPLPLPLPLPL
 SectionPlasticPlasticPlasticPlasticPlastic
NANeutral Axis297247325297247
MpPlastic Mom Cap1513.711568.521531.881634.711649.52
RuRequired rot cap.55555
Appox. Lb/ryPredicted40.9834.9946.8937.5732.75
 IS800-(2007)37.41135.75653.37037.41135.756
 EN 1993-1-139.72636.86555.55739.72636.865
 AISC42.07237.10449.78042.07237.104
 Galambos34.95930.83141.36434.95930.831
Mcrcritical moment7027.4937341.9966902.6368358.9018384.046
λLTsqrt(Mp/1.156*Mcr)0.43170.42990.43820.41130.4125
Etf/Etw 0.9160.945111
Check for rot capUsing Eq. (8)5.3175.1456.3565.3205.138
Revised Lb/ry 4135473833
Mcrcritical moment7019.027339.486869.448171.038256.07
λLTsqrt(Mp/1.156*Mcr)0.4320.4300.4390.4160.416
Check for rot capUsing Eq. (8)5.3155.1456.3505.2925.119
Wt in Kg/m 164.07151.51193.11164.07151.51
Solution 3: rotation capacity required = 7
Girder dimensionsD516456600496456
 dw460400550440400
 h488428575468428
 Bf275250325275275
 tf2828252828
 tw1620201620
Materialfyf350450250350450
 fyw250350250350450
Hybrid ratio (Rh) 1.4001.2861.0001.0001.000
Yield stress ratio (ε)Flange0.8450.7451.0000.8450.745
 Web1.0000.8451.0000.8450.745
Flange/web slendernessFlange (λf)5.475.516.105.476.11
 h/tw28.7520.0027.5027.5020.00
 Web (λw)28.75023.66427.50032.53826.833
ClassFlangePLPLPLPLPL
 WebPLPLPLPLPL
 SectionPlasticPlasticPlasticPlasticPlastic
NANeutral axis258228300248228
MpPlastic mom cap1526.761628.201546.091532.301843.02
RuRequired rot cap.77777
Appox. Lb/ryPredicted41.3242.2440.1439.1537.67
 IS800-(2007)55.12950.11461.13457.72350.438
 EN 1993-1-155.17551.57164.25857.44551.169
 AISC42.07237.10449.78042.07237.104
 Galambos34.95930.83141.36434.95930.831
McrCritical moment6421.5555208.8599597.6876763.1796966.186
λLTsqrt(Mp/1.156*Mcr)0.45350.52000.37330.44270.4784
Etf/Etw 0.9160.945111
Check for rot capUsing Eq. (8)6.9047.0407.4557.1406.997
Revised Lb/ry 3942403937
McrCritical moment7207.175268.299665.326814.617220.35
λLTsqrt(Mp/1.156*Mcr)0.4280.5170.3720.4410.470
Check for rot capUsing Eq. (8)7.0577.0577.4637.1507.048
Wt in Kg/m 178.67172.70213.91176.15183.69

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Kulkarni, A., Gupta, L.M. Stable Length in Inelastic Design Considering Ductility Requirement. Iran J Sci Technol Trans Civ Eng 44, 825–845 (2020). https://doi.org/10.1007/s40996-019-00258-y

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Keywords

  • Hybrid section
  • Slenderness
  • Stable length
  • Nonlinear analysis
  • Strain hardening
  • Rotation capacity