Methods to Determine Ductility of Structural Members: A Review

Ductility plays a crucial role in ensuring the safety of a structure, as its inadequacy can lead to sudden and brittle failure. Despite its significance, there is no explicit method for determining, leading to inconsistency and confusion in selecting appropriate techniques. Misjudging a structure’s ductile behaviour can have catastrophic consequences. Therefore, this study examined several preliminary studies and identified twenty-one methods for computing ductility indices. These indices were categorized into three types, namely conventional, displacement-based, and energy-based. The conventional ductility indices are commonly applied to steel-reinforced members, deformation-based ductility indices to FRP-reinforced members, and energy-based ductility indices to earthquake-resistant and static-load structures. Conventional ductility indices are specific to ductile reinforcements, while displacement-based and energy-based ductility indices apply to both ductile and non-ductile reinforcements. However, different calculation methods can lead to significant variations in the computed ductility, particularly for those involving the first crack, and load factor, thereby leading to different ductility requirements for ensuring structural safety. Additionally, not all methods are explicit, and it is crucial to avoid indiscriminately applying requirements from one method to another.


INTRODUCTION
The importance of ductility cannot be overstated when it comes to structural safety due to its ability to warn of impending failure (Wang and Belarbi, 2011).According to El Zareef and El Madawy (2018), a structure that exhibits ductility can deform significantly prior to failure.Additionally, ductility allows a structure to absorb and dissipate substantial amounts of energy before failure (Park, 1988), which is crucial for structures in areas with a high risk of earthquakes (Muralidhara Rao et al., 2015).The quantification of ductility is currently not standardized due to the unavailability of explicit method to determine the process (Nogueira and Rodrigues, 2017).Preliminary studies employed a range of methods to assess ductility, with similar parameters but varying computational approaches.The lack of consistency between methods can result in a structure being evaluated with differing degrees of ductility, depending on the method utilized.
In practice, most studies employed one method to determine ductility because it was deemed suitable.Research by Barrera et al. (2012) and Spadea et al. (2001) used two or more methods.Several types of ductility methods have been demonstrated in previous studies (Tann et al., 2004;Ghallab, 2014;Zou, 2003;El Zareef and El Madawy, 2018;Oudah and El-Hacha, 2012).However, only a few common methods have been identified, and many previously used ductility methods remain unidentified.This study aims to provide a comprehensive summary of the different ductility methods available in the literature, along with their computations and applications.Additionally, it aims to highlight the ductility requirements for structures.The primary goals of this study are (a) to provide readers with an understanding of the range of ductility methods available and (b) to assist in selecting suitable methods for elements and structures under varying conditions.

BEHAVIOUR OF CONCRETE MEMBER
Concrete members can be reinforced by various methods, such as using steel bars, prestressing tendons, and fibre-reinforced polymer (FRP).Each of these reinforcements has unique characteristics that significantly impact the behaviour of the members.
Under load, concrete members undergo displacement, rotation, or curvature.Figure 1 depicts the typical load-displacement responses of a member.Elastic behavior is shown in Figure 1(a), where displacement is proportional to the load throughout.This brittle response is prevalent in concrete members reinforced with fibre-reinforced polymer (FRP) (Abdelraham et al., 1995).Figures 1(b) and 1(c) show the yielding responses of a member with and without post-yield stiffness.When a member yields, its stiffness decreases dramatically, and a large deformation occurs.This post-yield defor-mation contributes significantly to the member's ductility, and can be observed in RC and PC members (Lim et al., 2021;Ghallab, 2014).Figure 1(d) demonstrates the cracking response of a member before yielding.Cracks appear when the concrete's tensile strain limit is exceeded, reducing the bearing area of the cracked section and subsequently deteriorating its section inertia (Fu et al., 2020).This slightly reduces the member's stiffness (Ling et al., 2019), which further decreases as reinforcements yield (Wang et al., 2021).

Conventional Ductility Ratios
The ductility ratio, which is used to measure a member's ductility, can be expressed as a function of curvature, rotation, displacement, or twist, as shown in Table 1.Historically, it was defined based on the behavior of the reinforcement, as de-scribed in studies by Zou (2003); Dancygier and Berkover (2016); Ashour (2000).However, this approach is applicable conditional on (a) the reinforcement yields before the member fails and (b) its yield point accurately determined.
The concept of yield deformation has been expanded for larger applications.Figure 2 shows that the yield deformation can be determined based on (a) the cracking response of the concrete (method Y1), (b) the stress-strain response of the reinforcement bar (methods Y2 and Y7), (c) the equivalent stiffness of the member (methods Y3, Y4, Y8, and Y9), (d) the compressive strain limit of the concrete (method Y5), and (e) the elastoplastic energy absorption principle (method Y6).
The yield point can then be determined by (a) constructing a tangential or linear regression line over the elastic part of the curve (method Y3), (b) drawing straight lines intercepting the critical points on the curve (methods Y1, Y4, Y8, and Y9), (c) identifying a point on the curve where the elastic stiffness has decreased by more than 5% (ASTM E2126, 2011), and (d) constructing a straight line that gives equal areas above and below the curve (method Y6).It is important to note that these yield points are hypothetical and that the reinforcement may not necessarily yield because they sometimes remain in the elastic state of the member.This may not be a member's true response, but it can be a conservative estimate.
Figure 3 illustrates several methods for obtaining the ultimate displacement, ∆u.These include (a) considering the compressive strength limit of the concrete (U 1 ), (b) determining the peak load of the member (U 2 ), (c) analyzing a certain percentage of reduced load after the peak (U 3 and U 4 ), and (d) identifying the failure of the reinforcement, such as the fracture of the transverse or longitudinal reinforcement or the buckling of the longitudinal compression reinforcement (U 5 ).
The ductility ratio is obtained by dividing the ultimate displacement, ∆u, by the yield displacement, ∆y.Table 2 outlines at least 12 combinations of ∆u and ∆y that researchers have used to determine the ductility of RC and PC members.Different methods may produce varying values for the same member.For that, the ductility ratio is regarded as an indication of ductile behavior.The absolute value is less significant than the relative comparison among members (Tann et al., 2004).The comparison should be made using the same method to ensure consistency.
A precedent was established in preliminary studies using the ductility methods listed in Table 2.However, this list may not be comprehensive, and other methods could be appropriate when properly justified.Tables 3 and 4 outline the circumstances in which the methods are applicable.

Deformation-Based Ductility Indices
The conventional ductility ratios outlined in Table 2 are applicable for members with clear plastic deformation, specifically Types 2, 3, and 4 in Figure 1.This plastic deformation typically results from the yielding of steel reinforcement (Wang and Belarbi, 2011).However, these ratios are not suitable for concrete members with FRP reinforcement that remain elastic throughout (Type 1 in Figure 1) (Zou, 2003;Grace et al., 1998;Wang and Belarbi, 2011;Abdelraham et al., 1995;El Zareef and El Madawy, 2018).In response, deformationbased ductility indices were introduced and are outlined in Table 5.These indices use more complex equations than the conventional ductility ratios and are expressed in deflection, rotation, and curvature.
The ductility indices compare a member's ultimate state against its serviceability.The ultimate deformation corresponds to the peak load, as shown in Figure 4.The serviceability deformation can be any of the following illustrated in Table 5.
1. for a beam that fails by concrete crushing, the concrete strain at the top compression fibre is about 0.001 (D13), 2. the propagation of the first crack (D14 and D15), 3. 2/3 of the peak load (D16).
Ductility indices D 13 , D 14 , and D 17 (Table 5) comprise load factors like Mu M 0.001 , Mu Mcr , and Mmax My .Despite the deformation, the load is considered a part of ductility.However, this differs from the conventional ductility ratios, which only include deformation.
It is worth noting the difference between ductility and deformability, as explained by Oudah and El- a) θ τ,u is the ultimate twist b) θ τ,y is the yielding twist

Method Equation Description
D13 1. µ is the ultimate moment 2. φ u is the curvature at the ultimate state.
3. φ 0.001 is the curvature when the concrete strain at the outermost compression fiber is 0.001 (Figures 4(a) and 4(b)).4. M 0.001 is the moment the concrete strain at the outermost compression fiber is 0.001.
∆ cr is deflection at first cracking (Figure 5). 3. M u is the ultimate moment.4. M cr is the cracking moment.
Ghallab (2014); Tann et al. ( 2004) 1. φ u is curvature when (a) the postpeak remaining moment capacity of the column reduces to 80% of the maximum moment capacity M max , or (b) the longitudinal reinforcement steel strain reaches the ultimate strain su , or (c) the strain in concrete reaches the maximum confined strain ccu .2.
3. φ yc is the curvature when the concrete strain reaches the strain at peak stress in unconfined concrete 4. M yc is the moment corresponding to φ yc . 5. φ ys is the curvature at the onset of the first yielding of the longitudinal bars.6. M ys is the moment corresponding to φ ys .(2004).Ductility refers to the plastic work that a member can undergo before it fails, and it requires yielding and plastic behavior.In contrast, deformability refers to the amount of deformation that a member can experience before failure, regardless of whether it exhibits yielding and plastic behavior.Several methods, including D1, D13, D14, D15, and D17, are considered deformability indices.They compute either the member's first crack load or the concrete's elastic strain (i.e., 0.001) without necessarily requiring plastic failure.

Paultre and Légeron (2008)
According to Tann et al. (2004), high deformability is a necessary but insufficient prerequisite for ductile behaviour.Although a ductile member typically exhibits high deformability, a member with high deformability can still experience brittle failure (Tann et al., 2004).Nevertheless, a highly deformable member can provide early warnings before failure by showing significant elastic deformations.For adequate ductility, an elastic member should possess a higher reserve of strength than a ductile member (ACI Committee 440, 2015), as shown in Figure 5.

Energy-Based Ductility Indices
The energy-based ductility indices adopt the concept of energy in ductility.They apply to structures subjected to earthquake loads.These ductility indices can also be used for members subjected to static loads (Antonius and Imran, 2012; Hason et al., 2021).The ductility indices typically deal with a member's total energy at ultimate and its energy at service, as shown in (Table 6).The area beneath the load-displacement curve represents the total energy possessed by a member (Figures 6,7,and 8).The energy at service can be one of the following: (a) the area under the elastic region of the load-displacement curve (method D18), (b) the area under the load-displacement curve up to 0.75 times the ultimate load (method D19), (c) the area under the load-displacement curve at the yielding of tension reinforcement (method D20), and (d) the area under the load-displacement curve at service (method D21).
The method D17 was originally introduced by Naaman and Jeong (1995) as a means of analyzing RC members subjected to cyclic loading.In this method, the inelastic energy and elastic energy are separated by a line, as shown in Figure 6.Its slope, S, is computed using equation E1 in Table 7, which covers only the member's load response.Grace et al. (1998) further included the effects of reinforcement, failure mode, and stirrup in the function as given in equation E2 (Table 7).

DUCTILITY REQUIREMENTS
Ductility plays a crucial role in the integrity of a structure.Compared to non-ductile structures, ductile structures possess the ability to withstand unexpected or unforeseen forces more effectively µ en = 0.5 Etot E ela + 1 (5) a) E tot is the total energy, which is the area under the load-displacement curve up to the failure load (Figure 6) • ρ is the factor related to the failure mode effect, • S2 is the slope of the second line • γ is the factor related to the type of reinforcement effect • P c is the cracking load • Ef is the FRP modulus of elasticity, • P y is the yielding load.
• Es the steel modulus of elasticity • fy the steel yield stress • fds the design strength of FRP • Pu the ultimate load, • S1 is the slope of the first line • S2 is the slope of the second line • S3 is the slope of the third line  (Spadea et,al., 1997) Figure 8 Energy under a load-deflection curve (Ghallab, 2014).A ductile structure can (a) deform significantly prior to failure (El Zareef and El Madawy, 2018), (b) redistribute moments during excessive loads (Ashour, 2000), and (c) dissipate substantial amounts of energy prior to collapse (Park, 1988).
A member should satisfy ductility requirements for it to be used reliably in structural engineering applications (Naaman, 2003).The degree of ductility required can vary depending on the type of structure.NZS 3101.1 ( 2006) outlines the structural ductility factors, as per method D3, for different types of structures, as shown in Table 8.Structures with high ductility can better withstand seismic activities, whereas brittle structures require primary seismic-resisting members to resist earthquake forces.
Applying the structural ductility factors specified in NZS 3101.1 (2006) to other ductility indices may not be straightforward, as different calculation methods can yield significantly varying values, as shown in Table 9.Some methods, such as those that consider the first crack (e.g., method D1) and load factors (e.g., methods D13 and D14), can result in overestimated ductility indices, with values exceeding 8 and, in some cases, as high as 11 (Grace et al., 1998).Table 10 presents other ductility requirements found in previous studies utilized to evaluate the ductility of structures.
It is important to note that the ductility requirements for all methods may not be adequately specified in Table 10.As such, the appropriate range for ductile behavior may not be fully determined for these methods.In such cases, it is recommended to compare the ductility indices of a member to those of control specimens.
To achieve sufficient ductility, it is recommended that reinforced concrete (RC) members be underreinforced to ensure the reinforcement yields before the concrete crushes (Wu, 2006).This can be achieved by having a reinforcement ratio of less than 1.5% (Ashour, 2000).The tensile strength of the reinforcement should be at least 1.1 times or 1.25 times its specified yield strength (ACI-318, 2019;BS-EN 1992-1-1, 2004).Additionally, the ultimate elongation of the reinforcement should be at least 8% (Macchi et al., 1996).

CONCLUSION
This study presents a comprehensive overview of 21 different ductility indices that have been identified in the literature.This study summarises the ductility requirements for brittle and ductile structures.Only six methods have the ductility requirements explicitly stated and the requirements for one method cannot simply be used for another.This is to avoid misjudging a structure's ductile behaviour, which can be catastrophic.
The focus of this study is on the ductility indices that are computed from load-deformation curves.
This excludes those calculated using analytical equations or numerical methods.Future studies may consider examining this aspect in greater detail.

DISCLAIMER
The authors declare no conflict of interest.

Figure 1
Figure 1 Typical load-displacement curves of structural elements

Figure 3
Figure 3 Alternative definitions for ultimate displacement

Figure 5 Figure 6
Figure 5 Ductility of the members with elastic and elastoplastic reinforcements ela is the elastic energy computed as the area of the triangle formed at failure load by unloading the beam (Figure6) D19 µ e = Etot E 0.75pu (6) a) E tot is the total area under the loaddisplacement curve at ultimate failure (Figure7)Spadea et al. (1997);Alsayed and Alhozaimy (1999) b) E 0.75pu is the area under the loaddisplacement curve up to 0.75 times the ultimate load (Figure7) D20 µ e = Etot Ey (7) a) E tot is the total area under the loaddisplacement curve at ultimate failure (Figure8)Spadeaet al. (2001); Thomsen et al. (2004)b) E y is the area under the load-displacement curve at the yielding of tension steel (Figure8) D21 µ e = Etot Ey,s (8) a) E tot is the area under the load-displacement curve at ultimate (Figure8)Ghallab (2014); ACI Committee 440 (2015) b) E y,s is the area under the load-displacement curve at service (Figure8) Table7.Equations for the slope of the unloading branch to compute method D17 is the slope of the unloading branch• α is the factor related to the stirrup type effect, • S1 is the slope of the first line

Table 2 .
Methods to compute conventional ductility ratios

Table 3 .
Determining the yield displacement of a member under various circumstances

Table 4 .
Determining the ultimate displacement of a member under various circumstances