A Cost Effective Method to Retrofit Steel GirdersOmar Y. El Masri

Department of Civil & Environmental EngineeringSyracuse University Advisor: Eric M. Lui

1. IntroductionIn steel structures, I-sections are commonly used for beamsand columns. These cross-sections often lack lateral rigidityand torsional stiffness. In this research, an effective methodto improve their lateral rigidity and overall flexuralresistance by welding two inclined rectangular plates to thecompression flange and the compression portion of the webof hot-rolled or welded I-section is explored. The resultingsection, referred to as a Delta girder can greatly increase theload-carrying capacity of the beam with appreciable cost andtime savings.

ℎ

𝑏𝑏𝑐𝑐

𝑑𝑑

𝑏𝑏𝑑𝑑

ℎ𝑑𝑑

2. Research Objectives and Practical ApplicationsThe primary objectives of this research are to investigate theelastic and inelastic behavior of the proposed Delta girdersand to develop guidelines and equations for the design ofthese girders. The research involves analytical and finiteelement formulations and verifications against availabletheoretical and experimental results.Anticipated applications for Delta girders include the designof new beams and the retrofit of existing beams.

3. Major Advantages of Delta Girdersa) Improve flexural capacity.b) Increase shear resistance.c) Avoid local buckling by reducing the width-to-thickness

ratio of both the compression flange and the web.d) Reduce incidences for lateral-torsional buckling (LTB)

during transportation and erection.e) Provide a simple and cost effective means to strengthen

existing beams.f) Reduce the need for longitudinal and intermediate

stiffeners.g) Enhance torsional resistance for carrying eccentric loads.h) Improve the appearance of the girder and provide an

obstruction free path for the wheels of a crane trolley

4. Finite Element ModelingThe high fidelity finite element model used in thesimulation study takes into account bothgeometric and material nonlinearities includingthe effects of initial imperfections and residualstresses.

5. Analysis for Flexure

450

400

350

) 300

m-kN 250

(cr

M 200

150

100

50

00 4 8 12 16 20

Lb (m)

Welded IPE 360

Delta girder 1

Delta girder 2

Delta girder 3

Delta girder 4

6. Analysis for Shear

2.602.402.202.001.80

,InV 1.60

/M 1.40

FEV 1.20

1.000.800.600.400.200.00

0.00 0.02 0.04 0.06 0.08 0.10γ

7. Design RecommendationsFlexural capacity:The following equations are proposed:

𝑀𝑀𝑛𝑛 =1

Φ𝐿𝐿𝐿𝐿 + Φ𝐿𝐿𝐿𝐿2 −

3𝑍𝑍𝑥𝑥𝐹𝐹𝑦𝑦4𝑀𝑀𝑐𝑐𝑐𝑐

⁄1 2 𝑍𝑍𝑥𝑥𝐹𝐹𝑦𝑦

in which Φ𝐿𝐿𝐿𝐿 = 0.5 1 + 𝛼𝛼𝐿𝐿𝐿𝐿𝑍𝑍𝑥𝑥𝐹𝐹𝑦𝑦𝑀𝑀𝑐𝑐𝑐𝑐

− 0.4 + 3𝑍𝑍𝑥𝑥𝐹𝐹𝑦𝑦4𝑀𝑀𝑐𝑐𝑐𝑐

Shear capacity:The following equation is proposed:

𝑉𝑉𝑛𝑛 = 𝐹𝐹𝑦𝑦3𝐴𝐴𝑤𝑤𝑤𝑤 + 0.5𝐴𝐴𝑑𝑑 1 + 0.5𝐴𝐴𝑑𝑑

𝐴𝐴𝑤𝑤𝑤𝑤+0.5𝐴𝐴𝑑𝑑

Guidelines for delta stiffener configurations:

𝑤𝑤𝑑𝑑𝑡𝑡𝑑𝑑

≤ 72 ⁄235 𝐹𝐹𝑦𝑦

Limits Length ℎ𝑑𝑑 𝑏𝑏𝑑𝑑

⁄𝑑𝑑 𝑏𝑏𝑐𝑐 ≤ 2𝐿𝐿𝑏𝑏 ≤ 𝐿𝐿𝑐𝑐𝑟𝑟∗ ⁄ℎ 5 to ⁄ℎ 4 ⁄2𝑏𝑏𝑐𝑐 5 to ⁄𝑏𝑏𝑐𝑐 2

𝐿𝐿𝑏𝑏 > 𝐿𝐿𝑐𝑐𝑟𝑟∗ ⁄ℎ 5 to ⁄ℎ 3 ⁄2𝑏𝑏𝑐𝑐 3 to ⁄3𝑏𝑏𝑐𝑐 4

⁄𝑑𝑑 𝑏𝑏𝑐𝑐 > 2 Any ⁄ℎ 5 to ⁄ℎ 4 ⁄2𝑏𝑏𝑐𝑐 3 to ⁄3𝑏𝑏𝑐𝑐 4* 𝐿𝐿𝑐𝑐𝑟𝑟 is 𝐿𝐿𝑐𝑐 of the base I-section

Cross-section Limits 𝛼𝛼𝐿𝐿𝐿𝐿

Delta Girders ⁄𝑑𝑑 𝑏𝑏𝑐𝑐 ≤ 2⁄𝑑𝑑 𝑏𝑏𝑐𝑐 > 2

0.210.34

Notations:𝑍𝑍𝑥𝑥 is the plastic section modulus about the strong axis, 𝐹𝐹𝑦𝑦 is the yield strength, 𝑀𝑀𝑐𝑐𝑐𝑐 is the elastic LTB moment, 𝐴𝐴𝑤𝑤𝑤𝑤 is the area of the web below the delta region, 𝐴𝐴𝑑𝑑 is the area of the delta region, 𝐿𝐿𝑏𝑏 is the unbraced length, 𝐿𝐿𝑐𝑐 is the limiting length for inelastic LTB, 𝑤𝑤𝑑𝑑 and 𝑡𝑡𝑑𝑑 are the width and thickness of the delta stiffener, respectively.

8. SummaryUsing theoretical analysis and high fidelity finiteelement modeling, the load carrying capacity ofDelta girders were determined. Design guidelinesand equations that engineers can use to designthese Delta girders were derived. Based on acomparative study between Delta girders and I-sections, it has been shown that Delta girders canprovide up to 394% and 89% increase in theflexural and shear resistance capacities,respectively. Moreover, comparisons that includethe weight and fabrication cost show that costsavings greater than 35% can be achieved.