Stable Length in Inelastic Design Considering Ductility Requirement


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  1. AISC-LFRD (2010) Specifications for structural steel buildings. AISC, Chicago

    Google Scholar 

  2. ANSYS Inc (2010) Release 12.0, mechanical user guide, southpointe, vol 275. Technology Drive, Canonsburg

    Google Scholar 

  3. Bayer A, Boissonnade N, Khelil A, Bureau A (2018) Influence of assumed geometric and material imperfections on the numerically determined ultimate resistance of hot-rolled U-shaped steel members. J Constr Steel Res.

    Article  Google Scholar 

  4. Earls CJ (2000a) Geometric factors influencing structural ductility of compact I-shaped beams. J Struct Eng.

    Article  Google Scholar 

  5. Earls CJ (2000b) Influence of material effect on structural ductility of compact I-shaped beams. J Strut Eng.

    Article  Google Scholar 

  6. Earls CJ (2001) Constant moment behavior of high performance steel I-shaped beams. J Constr Steel Res 57:711–728

    Article  Google Scholar 

  7. EN 1993-1-1: Eurocode 3 – Design of steel structures – Part 1-1: General rules and rules for buildings

  8. EN 1993-1-5: Eurocode 3—Design of steel structures—Part 1-5: Plated structural elements

  9. Galambos TV (1967) Summary report on deformation and energy absorption capacity of steel structures in the inelastic range. American Iron and Steel Institute, Washington, DC

    Google Scholar 

  10. Greco N, Earls CJ (2003) Structural ductility in hybrid high performance steel beams. J Struct Eng.

    Article  Google Scholar 

  11. Green PS, Sause R, Ricles JM (2002) Strength and ductility of HPS flexural members. J Constr Steel Res.

    Article  Google Scholar 

  12. Holtz NM, Kulak GL (1973) Web slenderness limit for compact beams. Structural Engineering Report No. 43. Department of Civil Engineering, University of Alberta, Edmonton Alta

    Google Scholar 

  13. Holtz NM, Kulak GL (1975) Web slenderness limit for non-compact beams. Structural Engineering Report No. 51. Department of Civil Engineering, University of Alberta, Edmonton Alta

    Google Scholar 

  14. IS 800-(2007) Indian standard-general construction in steel—code of practice, 3rd edn. Bureau of Indian Standards, New Delhi

    Google Scholar 

  15. Kulkarni AS, Gupta LM (2017) Experimental investigation on flexural response of hybrid steel plate girder. KSCE J Civ Eng.

    Article  Google Scholar 

  16. Kulkarni A, Gupta LM (2018) Evaluation of rotation capacity of I-shaped welded steel plate girders. Arab J Sci Eng.

    Article  Google Scholar 

  17. Lay MG, Adams P F, Galambos TV (1965) Experiments on high strength steel members. Welding research council, bulletin No 110/287

  18. Lee CH, Han KH, Uang CM, Kim DK, Park CH, Kim JH (2013) Flexural strength and rotation capacity of I-shaped beams fabricated from 800-MPa steel. J Struct Eng.

    Article  Google Scholar 

  19. Memon BA, Xiao-Zu S (2004) Arc-length technique for nonlinear finite element analysis. J Zhejiang Univ.

    Article  Google Scholar 

  20. Montgomery DC, Runger GC (2014) Applied statistics and probability for engineers ISV, 6th edn. Wiley, New Delhi

    Google Scholar 

  21. Nakashima M (1994) Variation of ductility capacity of steel beam-column. J Struct Eng.

    Article  Google Scholar 

  22. Ramanan L (2006) Simulation of nonlinear analysis in ANSYS. In: ANSYS India users conference

  23. Shokouhian M, Shi Y (2014a) Investigation of ductility in hybrid and high strength steel beams. Int J Steel Struct.

    Article  Google Scholar 

  24. Shokouhian M, Shi Y (2014b) Classification of I-section flexural members based on member ductility. J Constr Steel Res.

    Article  Google Scholar 

  25. Shokouhian M, Shi Y (2015) Flexural strength of hybrid I-beams based on slenderness. Eng Struct.

    Article  Google Scholar 

  26. SPSS (2015) SPSS—statistical package for social sciences. SPSS, IBM, Armonk

    Google Scholar 

  27. Trahair N, Bradford M, Nethercot D, Gardner L (2007) The behavior and design of steel structures to EN 1993-1-1, 4th edn. Taylor and Francis, New York

    Google Scholar 

  28. Wang CS, Duan L, Chen YF, Wang SC (2016) Flexural behavior and ductility of hybrid high performance steel I-girders. J Constr Steel Res.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Abhay Kulkarni.



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.


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).

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

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
Solution-1: rotation capacity required = 4
Girder dimensionsD594494650590469
Hybrid ratio (Rh) 1.4001.2861.0001.0001.000
Yield stress ratio (ε)Flange0.8450.7451.0000.8450.745
Flange/web slendernessFlange (λf)6.967.905.687.667.90
 Web (λw)34.37533.27837.50040.67335.637
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
 EN 1993-1-139.72636.86555.55737.64738.762
Mcrcritical moment5551.8425586.0025609.3497182.1265579.969
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
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
Hybrid ratio (Rh) 1.4001.2861.0001.0001.000
Yield stress ratio (ε)Flange0.8450.7451.0000.8450.745
Flange/web slendernessFlange (λf)6.967.905.686.967.90
 Web (λw)34.37533.27837.50040.67337.734
NANeutral Axis297247325297247
MpPlastic Mom Cap1513.711568.521531.881634.711649.52
RuRequired rot cap.55555
Appox. Lb/ryPredicted40.9834.9946.8937.5732.75
 EN 1993-1-139.72636.86555.55739.72636.865
Mcrcritical moment7027.4937341.9966902.6368358.9018384.046
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
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
Hybrid ratio (Rh) 1.4001.2861.0001.0001.000
Yield stress ratio (ε)Flange0.8450.7451.0000.8450.745
Flange/web slendernessFlange (λf)5.475.516.105.476.11
 Web (λw)28.75023.66427.50032.53826.833
NANeutral axis258228300248228
MpPlastic mom cap1526.761628.201546.091532.301843.02
RuRequired rot cap.77777
Appox. Lb/ryPredicted41.3242.2440.1439.1537.67
 EN 1993-1-155.17551.57164.25857.44551.169
McrCritical moment6421.5555208.8599597.6876763.1796966.186
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
Check for rot capUsing Eq. (8)7.0577.0577.4637.1507.048
Wt in Kg/m 178.67172.70213.91176.15183.69

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kulkarni, A., Gupta, L.M. Stable Length in Inelastic Design Considering Ductility Requirement. Iran J Sci Technol Trans Civ Eng 44, 825–845 (2020).

Download citation


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