Experimentally investigating the structural capacity of slender web tapered built up plate girder with web opening
AISC 360
AISC 360 specifications primarily address prismatic members, with shear strength predictions based on section-by-section evaluations. However, applying these methods on the girders with opening in the web panel have notable limitations, including the lack of specific provisions for tapered members and a simplified approach to web buckling analysis. The procedures often underestimate the capacity of unstiffened tapered members by ignoring the strength contributions of geometric tapering, leading to overly conservative results by an average of 58% conservation. Which also adds up with the findings of Ibrahim et al.6,7 that tapering of plate girders has a positive effect on their capacities. Therefore, it can be noted that all of the code’s predictions of specimens’ shear capacities are conservative. Equations from Eq. 5 to Eq. 7 show the procedure to determine the shear capacity using this code.
$$\:{V}_{n}=0.6{F}_{y}{A}_{w}{C}_{v}$$
(5)
where Fy is the steel yield stress, Aw is the web cross sectional area and Cv is the shear buckling stress to shear stress ratio. And can be calculated using Eq. 6 to 8. Mentioning that opening in web is taken into consideration by substituting Anet instead of Aw, where Anet is the web cross sectional reduced by the web opening cross sectional area
$$\:{C}_{v}=1.0\:if\:\frac{h}{{t}_{w}}<1.10\sqrt{{k}_{v}E/{F}_{y}}$$
(6)
$$\:{C}_{v}=\frac{1.10\sqrt{{k}_{v}E/{F}_{y}}}{h/{t}_{w}}\:if\:1.10\sqrt{{k}_{v}E/{F}_{y}\:}<\:h/{t}_{w\:}<\:1.37\sqrt{{k}_{v}E/{F}_{y}\:}$$
(7)
$$\:{C}_{v}=\frac{1.51{k}_{v}E}{{F}_{y}{\left(h/{t}_{w}\right)}^{2}}\:if\:1.10\sqrt{{k}_{v}E/{F}_{y}\:}<\:h/{t}_{w\:}<\:1.37\sqrt{{k}_{v}E/{F}_{y}\:}$$
(8)
where h and tw are the web depth and thickness respectively, E is the steel modulus of elasticity and kv is the shear buckling coefficient of simply supported prismatic web that can be determined by Eqs. 9 & 10.
$$\:{k_{v}\:=\:5\:\left(\text{f}\text{o}\text{r}\:\text{w}\text{e}\text{b}\:\text{p}\text{a}\text{n}\text{e}\text{l}\text{s}\:\text{w}\text{i}\text{t}\text{h}\text{o}\text{u}\text{t}\:\text{s}\text{t}\text{i}\text{f}\text{f}\text{e}\text{n}\text{e}\text{r}\text{s}\right)}_{}$$
(9)
$$\:{k}_{v}=5+\frac{5}{{\left(a/h\right)}^{2}}\left(\text{f}\text{o}\text{r}\:\text{w}\text{e}\text{b}\:\text{p}\text{a}\text{n}\text{e}\text{l}\text{s}\:\text{w}\text{i}\text{t}\text{h}\:\text{s}\text{t}\text{i}\text{f}\text{f}\text{e}\text{n}\text{e}\text{r}\text{s}\right)$$
(10)
where α is distance between transverse stiffeners. h is the distance between flanges.
AISC design guide 2
AISC Design Guide 2 provides practical recommendations for the design of plate girders, focusing on prismatic members with web openings. However, when applying its procedure on the tapered plate girders with web opening, the results were found to be overestimating the web shear capacity by 40% compared to the experimental results. This is due to the guide’s lack of consideration of web slenderness and local buckling which are essential to take into account for accurate prediction of shear capacity of such girders. The maximum nominal shear capacity at web opening in this code can be calculated using equations from Eq. 11 and Eq. 12.
$$\:V={V}_{pb}+{V}_{pt}$$
(11)
where Vpb or Vpt is the plastic shear capacity of a tee that can be determined next by Eq. 12.
$$\:{V}_{pb}\:or\:{V}_{pt}=\:\frac{{F}_{y}{t}_{w}s}{\sqrt{3}}$$
(12)
where s is the depth of tee.
AISC design guide 25
AISC Design Guide 25 focuses on the design of tapered beams specifically. This guide was closest among the comparison between other codes and researches, which is due the guides significant consideration of tapered girders including its slenderness where it accounts for local buckling. However, when its procedure was applied on the specimens in this study considering the reduction of web cross-sectional area for the opening calculating net area, it was found that it underestimated the shear capacity by about 22% conservation. This is due to the neglection of the enhancement caused by the tapering of web on the shear capacity and the stress redistribution that it will lead to according to Ibrahim et al.6,7. Nevertheless, the guide succeeded to predict the shear capacity of specimens S2 and S6, where these specimens had web panel areas relatively smaller than other specimens which reduced the other negative effect occurred by the presence of web opening other than the reduction of the web cross-sectional area. The maximum shear strength of stiffened webs using tension field action can be calculated using equations from Eq. 13 to Eq. 16.
For web panels in which flanges infracting a or b:
$$\:{V}_{n}=0.6{F}_{y}{A}_{w}\:for\:\frac{{h}_{avg}}{{t}_{w}}\le\:1.10\sqrt{\frac{{k}_{v}E\:}{{F}_{y}}}$$
(13)
$$\:{V}_{n}=0.6{F}_{y}{A}_{W}\left({C}_{v}+\frac{1-{C}_{v}}{1.15\left(a/{h}_{min}+\sqrt{1+{\left(a/{h}_{min}\right)}^{2}}\right)}\right)\:for\:\frac{{h}_{avg}}{{t}_{w}}>1.10\sqrt{\frac{{k}_{v}E\:}{{F}_{y}}}$$
(14)
where Aw = havgtw. havg is the average tapered web panel height. hmin = the smallest height in the panel. kv is the web plate buckling coefficient and can be determined using Eq. 12.
$$\:{k}_{v}=5\:\left(\text{f}\text{o}\text{r}\:\text{w}\text{e}\text{b}\:\text{p}\text{a}\text{n}\text{e}\text{l}\text{s}\:\text{w}\text{i}\text{t}\text{h}\text{o}\text{u}\text{t}\:\text{s}\text{t}\text{i}\text{f}\text{f}\text{e}\text{n}\text{e}\text{r}\text{s}\right)$$
(15)
$$\:{k}_{v}=5+\frac{5}{{\left(a/{h}_{avg}\right)}^{2}}\left(\text{f}\text{o}\text{r}\:\text{w}\text{e}\text{b}\:\text{p}\text{a}\text{n}\text{e}\text{l}\text{s}\:\text{w}\text{i}\text{t}\text{h}\:\text{s}\text{t}\text{i}\text{f}\text{f}\text{e}\text{n}\text{e}\text{r}\text{s}\right)$$
(16)
where a is the clear distance between stiffeners. havg is the distance between flanges.
Mentioning that the opening in web is taken into consideration as a reduction in the substitution of Anet instead of Aw. where Anet is Aw reduced by the web opening cross sectional area.
Eurocode 3
Eurocode 3 provides guidelines for designing of steel plated elements and it includes provisions for members with web openings. It specifies to reduce the web cross-sectional area by the area of opening to calculate the net web cross-sectional area to advance and calculate the web shear capacity. However, when applying its procedure on the specimens, the results were overestimating the shear capacity compared to experimental results by an average of 28%. This indicates that the code doesn’t fully consider the local buckling and stress redistribution caused by the presence of web openings in its methodology. Nevertheless, it predicted the shear capacity of S3, S4 and S7, where these specimens relatively have the smallest web opening diameters compared to others which reduced the influence of the presence of web openings in these specimens. The maximum shear strength of stiffened webs using tension field method can be determined using equations from Eq. 17 to Eq. 20.
$$\:V =[(dt_{w} \tau_{bb})+0.9 (g t_{w}\sigma_{bb} sin \Phi )]/\gamma_{M1}$$
(17)
where d is web depth mentioning that it was reduced to the net average depth of web panel by the web opening diameter, tw web thickness, τbb is the initial shear buckling strength, g is the tension field width originated, σbb is its strength and Φ is its inclination and γM1 is a factor of safety.
$$\:{\tau\:}_{bb}={f}_{yw}/\sqrt{3}\:\text{f}\text{o}\text{r}\:{\stackrel{-}{\lambda\:}}_{w}\le\:0.8$$
(18)
$$\:{\tau\:}_{bb}=\:\left.\left[1\:-\:0.8\left({\stackrel{-}{\lambda\:}}_{w}-\:0.8\right)\right.\right]\left({f}_{yw}/\sqrt{3}\right)\:\text{f}\text{o}\text{r}\:{0.8<\stackrel{-}{\lambda\:}}_{w}<1.25$$
(19)
$$\:{\tau\:}_{bb}=\:\left.\left[1/{{\stackrel{-}{\lambda\:}}_{w}}^{2}\right.\right]\left({f}_{yw}/\sqrt{3}\right)\:\text{f}\text{o}\text{r}\:{\stackrel{-}{\lambda\:}}_{w}\ge\:1.25$$
(20)
where fyw is the web steel yield stress. \(\:{\stackrel{-}{\lambda\:}}_{w}=\frac{d/{t}_{w}}{37.5\epsilon\:\sqrt{{k}_{T}}}\), ε is strain coefficient = (235/fy)0.5 (fy is the steel yield stress in N/mm2). kT is shear buckling factor = 5.34 (for vertical stiffeners at supports and no intermediate stiffeners).
Egyptian code of practice (ECP)
The Egyptian Code of Practice provides general guidelines for the design of steel girders. When its procedures were applied to the specimens, the results obtained were very conservative by an average of 58% compared to the experimental results. This is due to the code’s simplified approach, which does not fully address the effect of tapering and presence of web openings. It focuses on prismatic members and assumes uniform stress distributions. The maximum shear strength of stiffened webs can be determined using equations from Eq. 21 to Eq. 26.
$$\:V={A}_{w}{q}_{b}$$
(21)
where Aw is the web cross sectional area while taking the web opening into consideration in the reduction of web cross sectional area by the area of web opening. qb is the allowable shear stress and can be calculated from Eqs. 25 & 26
$$\:{q}_{b}={0.35F}_{y}\:\text{f}\text{o}\text{r}\:{\lambda\:}_{q}\le\:0.8$$
(22)
$$\:{q}_{b}=\:\left(1.5\:-\:0.625{\lambda\:}_{q}\right)\left({0.35F}_{y}\right)\:\text{f}\text{o}\text{r}\:0.8<{\lambda\:}_{q}<1.2$$
(23)
$$\:{q}_{b}=\:\left(\frac{0.9}{{\lambda\:}_{q}}\right)\left({0.35F}_{y}\right)\:\text{f}\text{o}\text{r}\:{\lambda\:}_{q}\ge\:1.2$$
(24)
where Fy is the steel yield stress. λq is the web slenderness parameter \(\:=\frac{d/{t}_{w}}{57}\sqrt{\frac{Fy}{{k}_{q}}}\). kq is shear buckling factor shear.
$$\:{k}_{q}=4+\frac{5.34}{{\alpha\:}^{2}}\:if\:\alpha\:\le\:1.0$$
(25)
$$\:{k}_{q}=5.34+\frac{4}{{\alpha\:}^{2}}\:if\:\alpha\:>1.0\:\alpha\:=\frac{\text{s}\text{p}\text{a}\text{c}\text{i}\text{n}\text{g}\:\text{o}\text{f}\:\text{t}\text{r}\text{a}\text{n}\text{s}\text{v}\text{e}\text{r}\text{s}\:\text{s}\text{t}\text{i}\text{f}\text{f}\text{e}\text{n}\text{e}\text{r}\text{s}}{web\:depth}$$
(26)
Serror et al.2
The study proposed a methodology for predicting the shear capacity of tapered girders, considering the effects of local buckling, slenderness, and stress redistribution. However, when applying its procedure on the specimens, it overestimated the shear capacity by an average of 45%. This indicated that the approach may not fully capture the actual shear behavior of girders with web openings where their presence significantly reduces the shear capacity. While the study provides valuable insights, its methodology appears to overestimate the effect of certain parameters. This indicates the need for refining the approach considering the presence of web opening. They proposed a shear buckling coefficient for predicting the shear capacity of tapered girders accurately when applying on Eq. 13 from Eurocode 3, The maximum shear capacity from this procedure can be calculated using Eq. 27
$$V=K_{n}[(dt_{w} \tau_{bb})+0.9\:(g\:t_{w} \sigma_{bb} sin \Phi )]/1.05$$
(27)
where, Kn is the nominal shear buckling coefficient, d is web depth, tw web thickness, τbb is the initial shear buckling strength and g is the tension field width originated, σbb is its strength and Φ is its inclination.
They proposed the nominal shear buckling coefficient Kn that’s function of parameters and coefficients with their values included in their research.
Ibrahim et al.6
The study proposed a methodology for predicting the shear capacity of tapered girders. When applying its methodology on the specimens, which include web openings, the predictions were found to be conservative compared to the experimental results by an average of 40%. This conservation is due to the methodology being developed for girders without openings, where the shear resistance is primarily governed by web slenderness and tapering effects. Hence, the methodology does not fully account for the influence of openings on stress redistribution and shear capacity reduction. While the study provides valuable insights, its direct application to girders with web openings may require further improvements. They proposed shear buckling factor for shear for predicting the shear capacity of tapered girders when applying on Eq. 6 from AISC 360. The maximum shear capacity from this procedure can be calculated using equations from Eq. 28.
$$\:{V}_{n}=0.6{F}_{y}{A}_{w}{C}_{vt}$$
(28)
where Fy is the steel yield stress, Aw is the web cross sectional area and Cvt is the shear buckling stress to shear stress ratio, it takes web tapering and aspect ratio effects into consideration by using their proposed coefficient of shear buckling kvtFR and correction factor Cr. They can be calculated using equations from Eq. 29 to Eq. 35.
$$\:{k}_{vtFR}={k}_{vtSS}+{\beta\:}_{vt}\left({k}_{vtFF}-{k}_{vtSS}\right)$$
(29)
$$\:{k}_{vtSS}=5.907-\frac{0.604}{{R}^{2}}+\frac{8.202}{\alpha\:}-\frac{6.748}{\left(\alpha\:R\right)}$$
(30)
$$\:{k}_{vtFF}=9.775-\frac{0.558}{{R}^{2}}+\frac{13.558}{\alpha\:}-\frac{12.358}{\left(\alpha\:R\right)}$$
(31)
$$\:{\beta\:}_{vt}=\left({t}_{f}/{t}_{w}\right)-0.64-0.16\:{\left({t}_{f}/{t}_{w}\right)}^{2}\le\:1.00$$
(32)
$$\:{C}_{r}=0.87-\frac{0.635}{R}-\frac{65.68}{R\left(h/{t}_{w}\right)}$$
(33)
$$\:{C}_{vt}=1.0\:if\:\:h/{t}_{w\:}\le\:1.10\sqrt{{k}_{vtFR}E/{F}_{y}}$$
(34)
$$\:{C}_{vt}={C}_{r}\frac{\sqrt{{k}_{vtFR}E/{F}_{y}}}{h/{t}_{w}}\le\:\frac{1}{R}\:if\:\:h/{t}_{w\:}>1.10\sqrt{{k}_{vtFR}E/{F}_{y}\:}$$
(35)
R.I. Shain et al.14
This study proposed a methodology for determining the coefficient of elastic local critical buckling. Applying this method by calculating the buckling coefficient by their method and advancing to calculating the shear capacity, it was found that it was conservative by an average of 48% compared to the experimental results. Although they considered the tapering ratio and slenderness ratio, the web opening presence had additional effects that need to be considered in the study to determine the shear capacity of such girders. Their shear buckling coefficient can be calculated from Eq. 36 to 38.
$$\:k\:=\:{\gamma\:}_{k}*\:\left(20-\frac{7.78}{{\alpha\:}^{1.34}}+\frac{17.19}{{R}^{0.27}}-\frac{0.186}{\left({{\alpha\:}^{3.97}R}^{1.82}\right)}\right)$$
(36)
$$\:{\gamma\:}_{k}=\left\{\begin{array}{c}1,\:\:R=1\\\:{{\lambda\:}_{n}}^{0.09},\:\:R\ne\:1\end{array}\right.$$
(37)
where k is the shear buckling coefficient, γk is a correction factor and λn is the normalized slenderness ratio.
$$\:{\lambda\:}_{n}=\frac{h}{100{t}_{w}}$$
(38)
Considering that, the maximum shear capacity (V) was calculated using the rest of AISC procedure in Eq. 5 and to calculate (Cv)in advance using Eq. 6 to 8.
Table 4; Fig. 16 show comparison between experimental results of this research and the calculated capacities using previously mentioned design codes and earlier researches.
Comparison between results from experimental and specifications results.
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