Yielding and Ultimate Deformations of Wide and Deep Reinforced Concrete Beams

Abstract

Current formulations proposed by Eurocode 8 part 3 for the inelastic deformations of existing reinforced concrete members are assessed separately for wide beams (WB) and conventional deep beams (DB). The current approach, based on a large experimental database of members, predicts larger ultimate chord rotation but lower chord rotation ductility for WB rather than for DB despite the similar curvature ductility, due to lower plastic hinge lengths in WB. However, if the data are disaggregated into DB and WB, predicted chord rotations are consistently conservative for DB and not conservative for WB if compared with experimental values, especially at ultimate deformation. Thus, plastic hinge length may be even greater for DB in comparison to WB. Therefore, some feasible corrections of the formulations for chord rotations are proposed, in order to reduce the bias and thus increase the robustness of the model for cross-section shape variability.

1. Introduction

Wide beams are defined by their cross-section aspect ratio in which the width is larger than the depth, and typically also larger than the column dimensions. Their use as an alternative to conventional deep beams in reinforced concrete (RC) frames is quite widespread in seismic regions of the Mediterranean area [1,2,3,4,5]. Traditionally, seismic codes have imposed severe restrictions on the use of wide-beam frames (WBF), such as limitations to their use in high seismicity areas or reduction in behaviour factor (q) [6,7,8]. However, most of current codes dispense similar treatment to WBF as to deep-beam frames (DBF), except for some geometric and mechanical limitations regarding beam–column connections, mainly the ratio between the beam width and the column dimensions. These restrictions have been demonstrated to guarantee proper local performance, as observed in several experimental and analytical tests of wide beam–column sub-assemblages [9,10,11,12,13,14,15,16,17,18,19].

In [7,8], it is shown that WBF may provide similar seismic global performances to DBF when both are designed according to modern performance-based seismic codes such as Eurocode 8 part 1 (EC8-1) [20]. Despite lower local ductility of WB with respect to DB, WBFs show higher effective stiffness and deformation capacity, thanks especially to code provisions regarding the damage limitation limit state. There are some other mechanical benefits of WBF with respect to DBF: higher shear span in first-storey columns, higher ultimate chord rotation of beams, and lower deformability of joints.

Nevertheless, the similar treatment of WBF and DBF in current codes requires an accurate estimation of inelastic deformations of WB under cyclic loads, because the overall performance of the whole structure relies on their member ductility, among other characteristics concerning capacity design. For this aim, some performance-based seismic codes such as the American ASCE-SEI/41-06 (ASCE in the following) [21] or the European Eurocode 8 part 3 (EC8-3) [22] provide different expressions for yielding and ultimate chord rotation capacity of members. The model of EC8-3 is based on a continuous work developed in [23,24,25,26]. These formulations are obtained as a regression of experimental results contained in an increasing database up to 1540 tests. However, only 37 of those elements are WB, so it is not clear whether those formulations can appropriately fit those elements.

The scope of this paper is to evaluate the reliability of the current deformation model adopted by EC8-3 regarding wide beams. Firstly, a comparative numerical analysis of curvatures and chord rotations of a parametric set of eight couples of WB and DB was carried out, in which both current European and American approaches were used in order to understand the different cross-sectional behaviours and the corresponding inelastic member performances. Then, the experimental results of the database on the basis of the EC8-3 approach were disaggregated into DB and WB, and current formulations were applied separately in order to find whether experimental-to-predicted ratios are biased or not for both groups. Finally, some corrections for the current formulations are proposed in order to reduce the bias and thus increase the robustness of the model against cross-sectional shape variations.

2. Numerical Comparison of Deformations of Wide and Deep Beams

Inelastic flexural deformation of members is typically characterised by chord rotation θ at yielding and ultimate deformation (θ_y and θ_u, respectively). Macroscopic value θ is related to local variables, i.e., cross-section curvatures at yielding and ultimate deformation (ϕ_y and ϕ_u, respectively), through shear span (L_V) and plastic hinge length (L_pl). In all the following, b_w and h_b are cross-section width and height, respectively; c_n, concrete cover; d, effective depth; d′, distance from extreme fibres to the axe of reinforcement; z, internal lever arm; x, neutral axis depth; d_bL and d_bt, mean diameter of longitudinal and transverse reinforcement, respectively; A_s1 and A_s2, tensioned and compressed reinforcement areas, respectively; ω, ω′, and ω_tot, bottom, top, and total mechanical reinforcement ratio, respectively; ρ_w, transverse reinforcement ratio; f_c, resistance of concrete; f_y and f_u, yielding and ultimate steel strength, respectively; and M_y and M_u, yielding and ultimate bending moment, respectively. For any parameter A, ratios between values corresponding to WB and DB are indicated as A_W/D (rather than using the heavier notation A_WB/A_DB).

In general, the elastic stiffness of WB is lower than for DB due to h_b,W/D ≤ 1, although b_w,W/D ≥ 1; thus, post-cracked deformability may also be expected to be higher for WB than for DB. In terms of curvature ductility μ_ϕ, traditionally WBs are considered to provide lower values than DBs [4,27], i.e., ϕ_u,W/B ≤ ϕ_y,WB. This statement is based on generic considerations: when h_b is reduced, higher A_s1 is required; thus, a large, compressed concrete area is needed in order to satisfy equilibrium, which sometimes can be only attained by means of higher x, likely causing higher ϕ_y and lower ϕ_u, and thus lower μ_ϕ. However, such an argument does not take into account that b_w,W/D can be quite large, nor that sections designed as high ductility class (DCH) perform as confined ones. On the other hand, it is difficult to find explicit and systematic comparative analyses in the literature for WB and DB regarding θ instead of ϕ. In fact, code provisions guarantee enough member (chord rotation) ductility μ_θ by implicitly regulating μ_ϕ, without any consideration regarding L_pl depending on the cross-sectional aspect ratio.

In Appendix A, preliminary generic considerations aimed at estimating cross-sectional and member ductilities for DB and WB designed in DCH are carried out, taking into account the previous conditions. In that scenario, cross-sectional behaviour can be interpreted as shown in Figure 1: thanks to the confinement, similar ductilities are expected for both types, which is a different conclusion than what is found in the literature. The main reason is that ultimate deformation is ruled by the tensioned steel rather than the failure of compressed concrete; thus, both yielding and ultimate curvature ratios are rather inversely proportional to the depth ratios of the beams.

On the other hand, aimed at an estimation of μ_θ,W/D, two main code-based procedures for obtaining θ can be considered: EC8-3 and ASCE. EC8-3 proposes explicit formulations for θ_y and θ_u. θ_y expression (Equation (1)) depends mainly on ϕ_y, being a_v a zero-one parameter for pre-yielding shear concrete cracking. For θ_u, two approaches are proposed: one with a more fundamental basis (Equation (2)), depending on constant ϕ_pl = ϕ_u − ϕ_y alongside L_pl (calculated as in Equation (3)), and two pure empirical expressions, the first one explicitly for θ_u (Equation (4)) and a second one furnishing values of θ_pl, which is not considered in this work. α is the confinement effectiveness factor, ω_w the transverse mechanical reinforcement ratio, and ρ_d the diagonal reinforcement ratio. Only formulations for members without lap-splices in reinforcement are considered, as lap-splices are recommended to be placed outside critical regions for beams designed for high ductility [28].

$${\mathsf{\theta }}_{y,EC8}={\mathsf{\varphi }}_y\left(\frac{L_V+a_{v}z}{3}+0.13d_{bL}\frac{f_y}{\sqrt{f_c}}\right)+0.0013\left(1+1.5\frac{h_b}{L_V}\right)$$

(1)

$${\mathsf{\theta }}_{u,EC8,fun}={\mathsf{\theta }}_{y,EC8}+\left({\mathsf{\varphi }}_u-{\mathsf{\varphi }}_y\right)L_{pl,EC8}\left(1-\frac{L_{pl,EC8}}{2L_V}\right)$$

(2)

$$L_{pl,EC8}=0.0\stackrel{⌢}{3}{L}_V+0.2h+0.11d_{bL}\frac{f_y}{\sqrt{f_c}}$$

(3)

$${\mathsf{\theta }}_{u,EC8,emp}=0.016{\left(\frac{\max \left\{0.01;{\omega }^{\prime }\right\}}{\max \left\{0.01;\omega \right\}}{f}_c\right)}^{0.225}{\left(\frac{L_V}{h_b}\right)}^{0.35}\cdot {25}^{\alpha {\omega }_w}\cdot {1.25}^{100{\mathsf{\rho }}_d}$$

(4)

Concerning the argument of which type of approach is more suitable, pure empirical or more fundamental, two considerations must be made. Firstly, inelastic behaviour of RC members is a complex phenomenon which is difficult to model satisfactorily from a pure theoretical point of view [27]; in fact, in this specific case empirical model is intended to provide more reliable predictions, showing higher robustness to the variability of single parameters [26,29]. Secondly, and more importantly, is it worth noting that the more fundamental approach is not a pure fundamental one. It adopts an apparent theoretical framework (Equation (2)) but then adds a yielding contribution which contains pure empirical factors to a plastic contribution which depends on a plastic hinge length whose calculation is also purely empirical (Equation (3)). Hence, it is actually another empirical expression.

Conversely, in the ASCE procedure, θ_y (shown in Equation (5) only for flexural deformation) is obtained indirectly as M_y/K_sec (K_sec being the secant-to-yielding member stiffness, obtained as a constant fraction of gross uncracked one taking into account also shear contribution). θ_u is obtained as θ_y + θ_pl, where the plastic contribution θ_pl = θ_u − θ_y is picked from

Table 1. Values of θ_pl,ASCE.

st < d/3 and Vs > 0.75Vpl	$\frac{\mathsf{\rho }-{\mathsf{\rho }}^{\prime }}{{\mathsf{\rho }}_{\mathit{b}\mathit{a}\mathit{l}}}$	$\frac{{\mathit{V}}_{\mathit{p}\mathit{l}}}{{\mathit{b}}_{\mathit{w}}\mathit{d}\sqrt{{\mathit{f}}_{\mathit{c}}}}$	θpl,ASCE
(Y/N)	-	-	(rad)
Y	≤0.0	≤3	0.025
		≥6	0.020
	≥0.5	≤3	0.020
		≥6	0.015
N	≤0.0	≤3	0.020
		≥6	0.010
	≥0.5	≤3	0.010
		≥6	0.005

, being s_t the stirrup spacing, V_s the stirrup shear strength contribution, V_pl the maximum shear corresponding to the attainment of moment resistances, and ρ_bal the reinforcement ratio for balanced strain conditions.

$${\mathsf{\theta }}_{y,ASCE}=\frac{M_y{L}_V}{3\left(0.3E_{c}I\right)}$$

(5)

As shown in Appendix A, fundamental and experimental approaches return different member ductilities for WB and DB. According to the fundamental one, rather similar ductilities to those corresponding to curvatures are expected, while the experimental method furnishes lower ductilities for WB.

If results of the empirical approach are interpreted similarly to the fundamental approach, i.e., as the consequence of plastic curvatures alongside plastic hinge length, it is possible to obtain equivalent implicit values of L_pl,eq,W/D ≈ θ_u,W/D/ϕ_u,W/D, again neglecting yielding deformations with respect to ultimate ones. It results in values of 1/(b_w,W/D·d_W/D^1.35) for unconfined beams and d_W/D^0.65 for confined ones, which means that WB may show shorter L_pl,eq than DB for confined sections but higher values in the unconfined case, which is contrary to the trend observed in most of the expressions proposed for plastic hinge length [23,26,27].

The ASCE method may provide different values than EC8-3, as it is not based on curvatures. θ_y,W/D only depends on gross stiffness, which may return large differences between WB and DB. On the other hand, θ_pl for design in DCH may be rather similar in all the cases, as it depends mainly on the stirrup arrangements; however, provided values are independent of geometry, so it may not be possible to predict corresponding ratios for θ_u and μ_θ.

Hence, within the limitations of these preliminary simplified considerations, in general, WBs designed in DCH are expected to provide similar curvature ductilities but chord rotation ductilities lower than or similar to DBs. It is worth noting that such relationships seem to depend mainly on cross-section dimensions.

Aimed at a proper assessment of those preliminary conclusions, a systematic analysis is required. In this section, the set of eight couples of WB and DB already used in [7] is adopted, aimed at a comparative numerical analysis of deformations, but in this case, the contribution of confinement is considered alternatively as null and complete. The actual comparison between DB and WB is based on both magnitudes ϕ and θ, and the corresponding ductilities (μ_ϕ and μ_θ) are also obtained. The characteristics of the set of beams are presented in

Table 2. Characteristics of the analysed set of beams.

Class of Beam	Geometry			Transverse Reinforcement		Longitudinal Reinforcement
	Section Type	bw	hb	Hoops	ρw	Low					High
						ωtot	ω′/ω = 1.5		ω′/ω = 1		ωtot	ω′/ω = 1.5		ω′/ω = 1
							Reinf. Ratio	My	Reinf. Ratio	My		Reinf. Ratio	My	Reinf. Ratio	My
		(mm)	(mm)	(mm)	(%)	(-)	(%)	(kNm)	(%)	(kNm)	(-)	(%)	(kNm)	(%)	(kNm)
DB	A	300	600	2ϕ8/70	0.48	0.10	ρ′ = 0.30	−181	0.25	±152	0.29	ρ′ = 0.90	−524	0.75	±442
							ρ = 0.20	+122				ρ = 0.60	+357
	B	300	500				ρ′ = 0.30	−124	0.25	±104		ρ′ = 0.90	−357	0.75	±301
							ρ = 0.20	+84				ρ = 0.60	+244
WB	A	650	300	4ϕ8/70	0.44	0.19	ρ′ = 0.60	−177	0.50	±149	0.60	ρ′ = 1.89	−513	1.50	±446
							ρ = 0.40	+120				ρ = 1.26	+362
	B	500	300		0.57	0.17	ρ′ = 0.54	−123	0.45	±103	0.53	ρ′ = 1.65	−355	1.38	±301
							ρ = 0.36	+83				ρ = 1.10	+244

, assuming L_V = 2.5 m, c_n = 20 mm, d_bL = 14 mm, d_bt = 8 mm, f_c = 33 MPa, and f_y = 630 MPa.

Five parameters are assumed: (i) class (DB or WB); (ii) cross-sectional aspect ratio (h_b/b_w) for each class (types A and B, providing higher or lower M_y, respectively); (iii) ω′/ω = 1 or 1.5, which in most cases satisfy the requirements of EC8 for DCH; (iv) ω_tot (high and low, which makes top and bottom reinforcement, respectively, correspond to code’s upper and lower limit when ω′/ω = 1.5); and (v) effectiveness of transverse reinforcement on confinement (yes or no). DB and WB show similar M_y for each case, and the high reinforcement case provides approximately three times the flexural strength provided by low reinforcement. Stirrup arrangements satisfy the requirements for DCH of EC8 and also the limitations provided by Eurocode 2 (EC2) [30] regarding the number of transverse legs.

2.1. Curvatures

Full moment–curvature (M-ϕ) relations are obtained through a fibre model for all the cases. Eurocode-based strain–stress models are assumed. For concrete, an EC2 parabolic envelope and confinement model proposed in EC8 are adopted. For steel, a bilinear envelope with hardening is considered, with values of f_u/f_y and ultimate strain ε_su according to those suggested in EC2 for steel type B.

Results for confined cases with asymmetric longitudinal reinforcement are shown in Figure 2. Post-elastic hardening of reinforcement causes that in most cases, M_u > M_y. In almost all the cases, there is spalling of concrete cover before the attainment of ε_su in the tension reinforcement, causing an instantaneous slight drop of M. Hence, ϕ_uof WB reaches larger values than DB, as predicted. It is worth noting that the increment in secant-to-yielding stiffness for high-reinforced sections with respect to low-reinforced ones is rather similar to such an increment in total reinforcement.

In most cases, simplified assumptions made in the preliminary considerations (pre-emptive yielding of steel, concrete and steel failure in unconfined and confined sections, respectively, quasi-linear behaviour of concrete until ϕ_y, negligible values of x with respect to d, similar top reinforcement stresses at ϕ_y, etc.) are confirmed, and estimated values of ϕ and μ_ϕ are predicted with error lower than 10%. In Figure 3, one of the couple DB-WB is studied in detail. It corresponds to a case in which WB presents approximately half the depth and double the width of DB; thus, the cross-sectional area is rather similar. The results confirm the predictions: WB shows double ϕ_y, similar ϕ_u,unconf, and more than double ϕ_u,unconf compared to DB.

Results for all the cases are shown in Figure 4, in which mean ratios between WB and DB are indicated as W/B in the bottom of the graphics. In general, more satisfactory values are obtained for asymmetric reinforced sections in positive bending than in the rest. For unconfined cases, high-reinforced sections show much poorer performances in terms of μ_ϕ than low-reinforced sections (almost half values), while for confined sections, the bias is much lower.

It is worth noting that provisions of EC2 regarding the distribution of stirrup legs within the width of the section causes quite a higher contribution of confinement in WB rather than in DB: in terms of μϕ, DB is multiplied by 1.5 on average, while for WB, the factor is almost 3.0. Even in the cases of asymmetric high-reinforced WB to negative bending, which does not satisfy DCH provisions on longitudinal reinforcement, high confinement causes similar values of μϕ than in the rest of the cases.

2.2. Chord Rotations

EC8-3 and ASCE procedures are carried out for all the cases. Regarding the first approach, θ_y(Figure 5a) shows a similar relative distribution of values to ϕ_y, although θ_y,W/D is 15% lower than ϕ_y,W/D on average due to the different shear contribution at yielding, independent of curvature. Mean secant-to-yielding stiffness values are on average 9% and 23% of the uncracked gross stiffness for low- and high-reinforced DB, respectively, and 15% and 38% for WB, respectively, due to the higher reinforcement ratios in WB than in DB. The mean global value is 21%, consistently with [23].

In Figure 5b,c, θ_u values for unconfined and confined cases, respectively, obtained following the EC8-3 fundamental approach, are shown. L_pl of WB is 0.86 times that of DB, on average. This is exactly the ratio between mean values of θ_u,W/D/ϕ_u,W/D for confined beams (see Figure 4c and Figure 5c); however, for unconfined beams, still larger θ_u values for WB rather than for DB are shown, notwithstanding the lower L_pl for WB, because in this case, large yielding deformations are not negligible with respect to ultimate ones. Consequently, μ_θ is 40% lower for WB rather than for DB for the unconfined section, while similar ductilities are expected for confined beams (see Figure 5d and Figure 5e, respectively).

Figure 6a,b corresponds to θ_u for unconfined and confined cases, respectively, obtained following the EC8-3 empirical approach. The relative positive influence of confinement on WB is quite lower than for the fundamental approach: the mean increment in θ_u is only 16% instead of 125%. For unconfined beams, notwithstanding the similar ϕ_u for WB and DB, higher values of θ_u are observed for WB rather than for DB; in fact, the implicit equivalent plastic hinge length is 32% higher for WB, on average. Instead, for confined cases, mean L_pl,eq,W/D = 0.62, which is more consistent with explicit values within the fundamental approach. The lower influence of confinement on WB causes that, even on confined beams, μ_θ is 25% lower for WB than for DB.

Finally, in Figure 7b,c, θ_uvalues for unconfined and confined beams, respectively, corresponding to the ASCE approach, are presented. Values of θ_y (Figure 7a) correspond by definition to a constant degradation of 30% from the uncracked gross stiffness; thus, much larger differences between high- and low-reinforced sections are observed. Values are much lower (about half times) than in both EC8-3 approaches for confined cases, because within this method, increment due to confinement is very low. The differences in θ_u between the different cases are only due to the contribution of θ_y, because values of θ_pl are rather constant for all the beams. It is worth noting that similar mean ratios between θ_u,W/D for confined sections are obtained with ASCE and EC8-3 empirical approaches: about 1.37. According to the ASCE approach, WB shows less than half the μ_θ of DB (see Figure 7d,e).

3. Disaggregation of Experimental Database

Affirming the reliability of the results obtained in the previous section requires the EC8-3 method to be appropriate, aimed at predicting deformations of WB. In this section, those formulations are assessed at this scope.

The EC8-3 approach has almost fully adopted the formulations corresponding to members under cyclic loading, with proper seismic design and with potential slippage of longitudinal bars, proposed in [25,26] for deformations at yielding and ultimate, respectively. All those expressions are obtained as a regression of experimental results contained in a large database of about 1540 tests [24]; for more detailed information on the composition of the database, see Appendix B.

The current paper only considers beams with a full rectangular cross-section and ribbed bars, with neither lap-splices nor precompression or retrofitting, whose failure is governed by uniaxial flexure. Hence, 277 DB and 37 WB are selected. Regarding wide cross-sections, only 5% of the total amount of specimens are tested in the parallel direction to the cross-section axe of minimum stiffness (members oriented as “wide” sections). Hence, the reliability of the models based on such databases for this minority might be under discussion.

For this aim, the model by Biskinis and Fardis [25,26]—B&F in the following—and also the preliminary one by Panagiotakos and Fardis [23]—P&F in the following—are applied separately to the sub-databases of DB and WB, in order to obtained disaggregated values of experimental-to-predicted ratios and thus assess the possible bias of results within the two groups. The ASCE model is also employed in order to compare the accuracy regarding WB and DB, although it is actually regressed from another database [31]. The results of the disaggregated application of deformation models should be carefully considered considering that the sub-database of WB is quite reduced and also unbalanced regarding the previous items.

All the graphics presented in the following show (i) the median value—which is intended to be more representative than the mean in the case of large dispersion [23]—of single experimental-to-predicted ratios, which is indicated as “Median exp/pred” and which corresponds to the slope of the plotted thick line; (ii) the 16th and 84th percentiles (associated with standard deviation in a normal distribution), corresponding to the slope of the dashed lines; and (iii) the coefficient of variation (CoV).. It is also indicated in each case which half of the graphic corresponds to conservative results (i.e., when formulations provide overestimation at yielding and underestimation at ultimate deformation).

3.1. Curvatures

Stress–strain models are similar to those adopted in the original approach (see Appendix B). Experimental and predicted ϕ_y (through the fibre model) are compared in Figure 8a,b for DB and WB, respectively. In both cases, adequate fitting is shown, although the fibre model slightly underestimates values. Rather similar trends are obtained if the simplified procedures in [23,25] are performed.

Conversely, quite poorer fitting is shown for ϕ_u. If stirrups with 90° closed hooks are assumed to not provide any confinement at all, corresponding beams show large underestimation of ϕ_u than the rest (see Figure 9a). In fact, this assumption is intended to be feasible for design purposes, given that it furnishes conservative results. However, the real influence of hooks on stress–strain models is not clearly quantified; a modification of Mander’s confinement model for columns with 90° closed hooks is proposed in [32]. If full confinement is assumed for beams with 90° closed hooks in which some confinement would be expected if 135° closed hooks were used (i.e., in beams with α > 0), the error reduces largely (see Figure 9b), even when only 56 out of 277 beams belong to this group. Regarding the application of fundamental approaches for the estimation of θ_u (based on ϕ values), the last assumption is adopted herein. For empirical approaches, it is not relevant because the influence of confinement is significantly lower (see Section 1) and also because, in the particular case of the present database, beams with 90° closed hooks show lower density of stirrups, thus providing low values of α.

In Figure 10, experimental and predicted ϕ_u are compared. High underestimation and very large dispersion of results are shown especially for DB. It is worth noting that the adopted confinement model has been obtained as a regression of the whole original database, including columns. It should be necessary to apply the same procedure to the columns belonging to the database in order to know whether the generalised bias in beams is balanced by columns or not. In the last case, the difference of results may rely on the different approach on curvature calculation, even when similar models are adopted. It emphasises the higher sensitivity of fundamental procedures for θ with respect to empirical ones regarding steel type or seismic detailing.

3.2. Chord Rotations

Regarding θ, different formulations proposed in [23,25,26] are applied separately to the disaggregated sub-databases DB and WB. In Equations C1 to C8 of Appendix B, all the expressions are presented in a homogenised form. In the following, subscripts “emp” and “fun” denote empirical and fundamental approaches, respectively. Formulations from ASCE (Equation (5) and Table 1) are also performed.

Median values and dispersion magnitudes of all the cases are shown in

Table 3. Fitting of different expressions for predicted θ with respect to experimental disaggregated data.

Experimental-to-Predicted Ratio		Expression for Prediction	DB		WB
			Median	CoV	Median	CoV
θy,exp/	/θy,P&F	Equation (A18)	1.02	35%	1.06	21%
	/θy,B&F	Equation (A19)	1.07	34%	1.14	21%
	/θy,ASCE	Equation (5)	1.30	68%	4.24	44%
θu,exp/	/θu,P&F,emp1	Equation (A20)	1.14	47%	0.88	40%
	/θu,P&F,emp2	Equation (A21)	1.00	52%	0.95	44%
	/θu,B&F,emp1	Equation (A22)	1.19	46%	0.84	38%
	/θu,B&F,emp2	Equation (A23)	1.21	47%	0.90	37%
	/θu,B&F,fun	Equations (A24) and (A25)	2.06	68%	0.88	46%
	/θu,ASCE	Table 1 + Equation (5)	2.19	74%	1.60	51%

. As expected, ASCE formulations show much poorer fitting than the rest, as they largely underestimate chord rotations. Both P&F and B&F approaches slightly underestimate θ_y both for DB and WB, which is not conservative; conversely, they underestimate θ_u for DB (which is conservative) and overestimate θ_u for WB (not conservative). Empirical approaches show better fitting than fundamental ones: in the first case, median experimental-to-predicted ratios are always within ± 20% with respect to perfect fitting, while in the second case, it can reach 100%, due to the high uncertainty regarding the calculation of curvatures.

In general, the P&F model shows better fitting than B&F, as the original database from which it comes out as a regression is more similar to the sub-databases used herein (e.g., it does not include sections different from rectangular shape). However, the present work focuses mainly on B&F because it is the basis of current EC8-3 formulations. Except for the fundamental approach, dispersion levels in all the cases are rather similar to those observed in the original works.

In Figure 11, experimental and predicted θ_y for the B&F model are compared. Larger underestimation but lower dispersion is shown for WB rather than for DB. The median experimental-to-predicted ratio for all the beams (DB + WB) is 1.11 if WB values are weighted in order to provide a similar contribution to the median despite their lower number of tests, or 1.08 otherwise. No particular bias is shown for the different sub-groups (i.e., steel class, type of loading, possibility of slippage, or hook closure angle).

Regarding θ_u, in Figure 12, experimental and predicted values for the B&F first empirical model are compared. Almost exactly symmetric bias is shown for DB and WB: median experimental-to-predicted ratios for both cases are inverse (1.19 for DB and 0.84 for WB); better fitting is shown by P&F’s second model (1.00 and 0.95, respectively). If both sub-databases are merged, weighted median values of 0.94 are obtained, which is not conservative. It is worth noting that the bias is more important for the sub-groups that likely represent current seismic-designed buildings: hot-rolled ductile steel, cyclic loading, slippage of longitudinal reinforcement, and seismic detailing of stirrup hooks (see Figure 13). In fact, EC8-3 assumes by default the formulations corresponding to cyclic loading and slippage.

The B&F’s fundamental approach (see Figure 14) shows rather similar underestimation of θ_u for WB (median values of 0.88), but the overestimation for DB is huge (2.06), even when curvatures corresponding to perfect confinement also for 90° closed hooks are assumed. This may suggest that values of L_pl are highly underestimated for DB and overestimated for WB, even when they are higher for DB rather than for WB as they increase with h_b. Conversely to the empirical approach, experimental-to-predicted ratios do not show significant bias when disaggregated for the different sub-groups (i.e., loading, steel, etc.). For the merge of sub-databases, a weighted median ratio of 1.16 is shown.

Undoubtedly, the reliability of the results obtained in this section may be under discussion, considering the limited number of tests belonging to sub-database WB and also the reduced variability of cases. However, in almost all the cases, the median experimental-to-predicted ratios for the merge of both sub-databases DB and WB are roughly near to 1.0, which means that those few results of WB provide a kind of balance to DB ones, whose reliability is higher. Additionally, lower bias is observed for WB than for DB.

4. Proposal of Corrected Expressions

In the previous section, the application of formulations on the basis of the current procedure in EC8-3 separately to DB and WB shows that experimental θ_u is lower than that predicted for WB and higher than that predicted for DB, while experimental θ_y is slightly higher than that predicted mainly for WB.

In this section, some corrections for the formulations of [25,26] are proposed, in order to reduce the bias and thus increase the robustness of the deformation model against cross-sectional shape variations. This proposal should be understood purely as an available simple alternative for the assessment of buildings with WB, or to be used for compared analysis of WB and DB, for instance. The current approach in EC8-3 makes no explicit distinction between columns and beams aimed at the estimation of θ. Hence, any alternative set of formulations able to account for the cross-sectional aspect ratio should also be checked for columns, which is not possible to be carried out with the existing database because cross-sectional orientation is always similar in most cases.

The proposals are intended as slight modifications within the framework of the formulations, which is not altered. In some cases, independent contribution factors are added, while in other cases, new parameters are placed within other contributions. In all the cases, corrections are carried out only on the part of the body of formulations which is purely empirical, aimed at best-fitting.

However, some premises according to previous results could be followed aimed at the definition of the correction parameters. Firstly, they must refer to the geometry of the section (h_b and/or b_w), which are also responsible for different performances of DB and WB regarding curvatures (see Section 1). In order to be consistent with the disaggregation of the original database that allowed determining the bias (see Section 2), maybe the most feasible factor to be used would be the cross-sectional aspect ratio (h_b/b_w), which is on the basis of the definition of DB and WB consistent with a “corner” value of 1.0. Still, all the expressions already contain terms depending on h_b; thus, different attempts aimed at avoiding such duplicity are carried out.

Regarding θ_u, the influence of aspect ratio can be intended as being divided into two contributions, as it influences both ϕ and L_pl (see Section 1). The fundamental approach already takes into account the important influence on ϕ; thus, any further influence of aspect ratio should concern only L_pl, in such a way that it increases for DB and decreases for WB. Conversely, in an empirical approach, both implicit contributions of aspect ratio on ϕ and L_pl are concentrated mainly in the factor h_b^−0.35 and to a lesser extent on the confinement factor 25^α·ωw; the latter is not modified in the proposal.

Firstly, the form of the expressions is chosen. Four different forms for the corrected empirical formulations for θ_u are proposed; they are shown in Equations (6)–(9), in which parameters C₁ and C₂ are defined in each formula for best fitting, i.e., experimental-to-predicted ratio equal to 1.0 and the lowest dispersion. Aimed at easing the awareness of the differences between formulations, some of their members are condensed with respect to the original expression in Equation (A22) (see Appendix B): a = a_st[1 − a_old·a_cy(1 + 0.25a_pl)/6](1 − 0.43a_cy)(1 + a_sl/2); k_α = 25^α·ωw; k_ρ = 1.25^100ρd; k_ω = (max{0.01;ω}/max{0.01;ω′}·f_c)^0.225.

$${\mathsf{\theta }}_{u,emp1,\mathrm{mod}1}=a\cdot k_{\omega }\cdot k_{\alpha }\cdot k_{\mathsf{\rho }d}{\left(\min \left\{\frac{L_V}{h_b};9\right\}\right)}^{0.35}{\left(\frac{h_b}{b_w}\right)}^{C_1}$$

(6)

$${\mathsf{\theta }}_{u,emp1,\mathrm{mod}2}=a\cdot k_{\alpha }\cdot k_{\mathsf{\rho }d}\cdot k_{\omega }{\left(\min \left\{\frac{L_V}{h_b};9\right\}\right)}^{0.35}{\left(\frac{C_2}{b_w}\right)}^{C_1}$$

(7)

$${\mathsf{\theta }}_{u,emp1,\mathrm{mod}3}=a\cdot k_{\alpha }\cdot k_{\mathsf{\rho }d}\cdot k_{\omega }{\left(\min \left\{\frac{L_V}{\sqrt{b_w{h}_b}};9\right\}\right)}^{C_1}$$

(8)

$${\mathsf{\theta }}_{u,emp1,\mathrm{mod}4}=a\cdot k_{\alpha }\cdot k_{\mathsf{\rho }d}\cdot k_{\omega }{\left(\min \left\{\frac{L_V}{\sqrt[3]{b_w{h}_b{}^2}};9\right\}\right)}^{C_1}$$

(9)

The first proposal is to multiply the original formulation by a power of aspect ratio (Equation (6)), which increases θ_u of DB and decreases θ_u of WB. In order to avoid the duplicity of terms depending on h_b, a second option, based on a factor only depending on b_w, is proposed (Equation (7)). However, this option needs to be given a “corner” value for b_w in order to define the threshold for the increase or decrease in θ_u, which is actually kind of a definition of DB and WB regarding only b_w, being in some cases insufficient. On the other hand, the third and fourth options are essentially based on the combined influence of h_b and b_w on relative ultimate curvatures (see Section 1). In the third option (Equation (8)), the original denominator h_b is replaced by the geometric mean of h_b and b_w (in order to keep the dimensionless character of the shear span). In the fourth option (Equation (9)), a similar approach is proposed, but more importance is given to h_b.

Regarding the fundamental approach for θ_u, corrections should be performed on the value of L_pl (Equation (A25), see Appendix B). The theoretical relation between L_pl and cross-sectional geometry (h_b and b_w) is not clearly defined in the literature. The expression proposed in [27] considers L_pl as a constant ratio (8%) of L_V plus an increment due to slippage, thus independent of the cross-sectional geometry. In [23], a similar form of the expression is assumed. Conversely, in [26], a term depending on h_b is added, whose weight may be comparable to that of L_V. In any of those approaches b_w is proposed as a relevant variable.

Two options for modifying Equation (A25) are proposed, in order to obtain larger values for DB and shorter ones for WB. Parameters C₁ to C₄ are analogously defined in each formula for best fitting with experimental data. In the first one (Equation (10)), h_b is multiplied by the aspect ratio; conversely, in the second proposal (Equation (11)), the higher difference between DB and WB is intended to be reached by emphasising the relative contribution of h_b at the expense of the term dependent on L_V, without any contribution of b_w. However, several attempts aimed at conducting an optimisation of the last expression show values of C₃ = C₄ = 0 and still very large dispersion. Hence, only the first option (Equation (10)) is developed.

$$L_{pl,\mathrm{mod}1}=\{\begin{array}{l}0.04\min \left\{L_V;9h_b\right\}+C_1\cdot h_b\left(\frac{h_b}{b_w}\right)\hspace{1em}\left(\mathrm{monotonic}\right)\\ 0.0\stackrel{⌢}{6}\min \left\{L_V;9h_b\right\}+C_2\cdot h_b\left(\frac{h_b}{b_w}\right)\hspace{1em}(\mathrm{cyclic},\mathrm{seismic} \mathrm{design})\end{array}$$

(10)

$$L_{pl,\mathrm{mod}2}=\{\begin{array}{l}{C}_3\cdot \min \left\{L_V;9h_b\right\}+C_1\cdot h_b\hspace{1em}\left(\mathrm{monotonic}\right)\\ C_4\cdot \min \left\{L_V;9h_b\right\}+C_2\cdot h_b\hspace{1em}(\mathrm{cyclic}, \mathrm{seismic} \mathrm{design})\end{array}$$

(11)

Finally, a proposal for a correction of the formulation for θ_y (Equation (12)) is also made. Only the shear contribution (last term) should be modified. Analogously to Equation (8), h_b is replaced by the geometric mean of h_b and b_w.

$${\mathsf{\theta }}_{y,\mathrm{mod}}={\mathsf{\varphi }}_y\left(\frac{L_V+a_{v}z}{3}+a_{sl}\cdot 0.125d_{bL}\frac{f_y}{\sqrt{f_c}}\right)+0.0014\left(1+C_1\frac{\sqrt{b_w{h}_b}}{L_V}\right)$$

(12)

Proposed parameters for all the formulations are shown in

Table 4. Selected values of parameters providing best fitting of proposed corrected expressions for θ within merged experimental database.

Corrected Expression	Equation	Proposed Parameters		Experimental-to-Predicted Ratios
				All Cases		DB		WB
		C1	C2	Median	CoV	Median	CoV	Median	CoV
θy,mod	(12)	4.64	-	1.00	28%	0.99	34%	1.00	21%
θu,emp1,mod1	(6)	0.20	-	1.00	43%	1.04	48%	0.98	38%
θu,emp1,mod2	(7)	0.40	262 mm	1.00	45%	1.00	47%	1.00	43%
θu,emp1,mod3	(8)	0.35	-	1.00	43%	1.06	47%	0.96	38%
θu,emp1,mod4	(9)	0.33	-	1.00	43%	1.14	47%	0.95	38%

. Rather satisfactory solutions are found when compared to Table 3: similar dispersion levels to the original formulations are shown, except for the corrected fundamental approach (Equation (10)). Perfect fitting is shown for corrected θ_y (see Figure 15) and for the second option of corrected θ_u (see Figure 16). In the rest of the expressions, the bias of results is rather symmetric and much more reduced than in the original ones: mean ratios are approximately within ±5% with respect to perfect fitting, except for the fourth option for corrected θ_u in DB (+14%).

Regarding bias corresponding to different disaggregations of sub-databases (steel type, slippage, loading type, and hook closure; see

Table 5. Disaggregated experimental-to-predicted ratios for proposed corrected expressions on different subsets of DB from the experimental database.

Corrected Expression	Median Experimental-to-Predicted Ratios
	Steel Type			Slippage		Loading		Hooks
	Hot-Rolled	Tempcore	Cold	Yes	No	Monotonic	Cyclic	135°	90°
θy,mod	1.00	0.94	1.01	1.01	0.98	0.97	1.06	1.05	0.97
θu,emp1,mod1	1.03	1.15	0.83	1.15	0.89	0.93	1.18	1.17	0.92
θu,emp1,mod2	0.99	1.08	0.79	1.13	0.89	0.91	1.19	1.18	0.90
θu,emp1,mod3	1.05	1.16	0.84	1.17	0.92	0.95	1.19	1.19	0.94
θu,emp1,mod4	1.13	1.22	0.91	1.23	1.00	1.02	1.26	1.25	1.01

), corrected θ_y values show quite good balance, in accordance with the reduced global CoV. The first two proposals for corrected empirical θ_u also show rather good balance, but quite large bias is shown by the third and fourth proposals and especially by the fundamental approach, whose reliability is actually lower.

Finally, some of those proposals are applied to the set of DB and WB analysed in Section 1 (see Table 2), in order to obtain more realistic values of deformations and thus a more representative comparison of ductilities between both types. In most cases, it is possible to apply the same corrections, previously proposed for expressions in [25,26], to the EC8-3 formulations (Equations (1) and (4)), because they correspond to a particular case of those ones. The last is not possible only for the fundamental approach, given that the two approaches show different confinement models, different contributions of fixed-end rotation at ultimate deformation, and also different expressions of L_pl. Hence, the final proposal of corrected expressions for EC8-3 is presented in Equation (13) for θ_y and Equations (14) and (15) for θ_u, since they show better fitting than the rest.

$${\mathsf{\theta }}_{y,EC8,\mathrm{mod}}={\mathsf{\varphi }}_y\left(\frac{L_V+a_{v}z}{3}+0.13d_{bL}\frac{f_y}{\sqrt{f_c}}\right)+0.0013\left(1+4.64\frac{\sqrt{b_w{h}_b}}{L_V}\right)$$

(13)

$${\mathsf{\theta }}_{u,EC8,emp,\mathrm{mod}1}=0.016{\left(\frac{\max \left\{0.01;{\omega }^{\prime }\right\}}{\max \left\{0.01;\omega \right\}}{f}_c\right)}^{0.225}{\left(\frac{L_V}{h_b}\right)}^{0.35}{\left(\frac{h_b}{b_w}\right)}^{0.2}\cdot {25}^{\alpha {\omega }_w}\cdot {1.25}^{100{\mathsf{\rho }}_d}$$

(14)

$${\mathsf{\theta }}_{u,EC8,emp,\mathrm{mod}2}=0.016{\left(\frac{\max \left\{0.01;{\omega }^{\prime }\right\}}{\max \left\{0.01;\omega \right\}}{f}_c\right)}^{0.225}{\left(\frac{L_V}{h_b}\right)}^{0.35}{\left(\frac{262mm}{b_w}\right)}^{0.4}\cdot {25}^{\alpha {\omega }_w}\cdot {1.25}^{100{\mathsf{\rho }}_d}$$

(15)

The results are shown in Figure 17 for Equations (13) and (14). Equation (15) causes a reduction in values also for DB, given that those beams have b_w = 300 mm, which is common for this kind of beam but higher than the “corner value” of 262 mm. In fact, such a value is obtained for best fitting with the sub-database of DB, which contains a high number of scaled specimens (255 out of 272 with b_w < 262 mm). This is not an issue for the original expressions and for the rest of the proposed corrected formulations, in which cross-section geometry measures are always rated to L_V. Hence, Equation (14) may be more robust than Equation (15).

Mean θ_u (Figure 17a,b) shows an increase of 13% for DB and a decrease of 12% for WB, which are slightly lower than the bias of median experimental-to-predicted ratios for the corresponding sub-databases (+19% for DB and −16% for WB, see Table 3 and Figure 12). These modifications result in rather similar θ_u values for WB and DB for both confined and unconfined cases (θ_u,W/D,unconf = 0.96 and θ_u,W/D,conf = 1.08, on average). On the other hand, equivalent implicit L_pl according to these values becomes always lower for WB than for DB (L_pl,et,W/D = 0.95 and 0.45 for unconfined and confined sections, respectively), which is consistent with the explicit expression of L_pl in the fundamental approach.

Consequently, according to the corrected model, even lower local ductilities are expected for WB than for DB (μ_θ,W/D = 0.53 and 0.59 for unconfined and confined cases instead of 0.67 and 0.75, respectively; see Figure 17c,d).

Nevertheless, it cannot be the cause of the imposition of lower behaviour factors (q) for WBF in past codes, because even the original deformation models are more recent than those prescriptions. Moreover, such a lack of local ductility of WB with respect to DB should not become a reason for any further prescription in current seismic codes regarding a limit of q for wide-beam frames (WBF). In fact, the local ductility of beams appears to be only one of the parameters governing the global capacity of WBF when the damage limitation limit state is the most critical condition of design, as in most EC8-designed buildings [7,8]. First of all, local ductility of columns is able to balance the global ductility of the frame; the rest of the modern code’s provisions result in favourable design results for WBF rather than for DBF (e.g., larger L_V of first storey columns or higher stiffness of joints).

5. Conclusions

The current model proposed by Eurocode 8 part 3 predicts that the chord rotation ductility of wide beams is lower than in conventional deep beams despite the similar curvature ductilities, due to lower plastic hinge lengths in WB.

However, those formulations have been proved to show some bias when applied separately to the wide and deep beams belonging to the original database from which that model was derived. Predicted chord rotations compared with experimental values are consistently conservative for DB and not conservative for WB, especially at ultimate deformation. Thus, the current model is still underestimating the difference in plastic hinge between both beam types of beams: they should be even larger for DB.

Therefore, some feasible correction factors have been proposed in order to improve the prediction capacity of the current model, for both the pure empirical and fundamental formulations. Factors considering cross-sectional geometry are included in the original formulations, and parameters aimed at best fitting with experimental data are searched. Rather satisfactory solutions with similar dispersion and no bias have been obtained; thus, the correction may increase the robustness of the model against cross-sectional shape variability. Given the reduced amount of WB in the original database, further experimental research would be required in order to increase the reliability of the corrections.