Concrete walls weakened by openings as compression members: A review

The purpose of this paper is to review the advances that have been made in the design of monolithic and precast reinforced concrete walls, both with and without openings, subject to eccentrically applied axial loads. Using the results of previous experimental studies, a database was assembled to enable statistical assessment of the reliability of existing design models. Several design aspects are highlighted, including the size and position of openings, and the roles of boundary conditions and geometric characteristics. In addition, the performance of ﬁber-reinforced polymers in strengthening wall openings is discussed. Overall it is found that design codes provide more conservative results than alternative design models that have been proposed in recent studies. Research into the strengthening of walls with openings is still in its early stages, and further studies in this area are needed. The paper therefore concludes by highlighting some areas where new investigations could provide important insights into the structural behaviour of strengthened elements. (cid:2) 2015 The Authors. Published Elsevier Ltd.


Introduction
Sustainable social development requires a safe, functional and durable built environment. Many structures around the world are made of reinforced concrete (RC), most of which were built before 1970 [1]. Functional modifications of these structures are common because existing structures must often be adapted to comply with current living standards. Such modifications may include the addition of new windows or doors and paths for ventilation and heating systems, all of which require openings to be cut into structural walls.
These openings can be divided into three types, namely already existing openings, existing openings that have been enlarged and newly created openings. Creating or modifying openings in walls may change the stress distribution within the wall, adversely influencing its behaviour. It is generally believed that the effects of small openings can often be neglected, while the presence of a large opening usually significantly alters the structural system [2]. However, in the existing literature there is currently no clear delimitation between small and large openings.
Experimental investigations have shown that cutting an opening into an RC wall decreases its ultimate load capacity, requiring the wall to be upgraded [3,4]. Traditionally, two methods have been used to strengthen RC walls with openings, these being either to create a frame around the opening using RC/steel members [5] or to increase the cross-sectional thickness [6]. Both methods increase the weight of the strengthened elements and may cause significant inconvenience by limiting the use of the structure during repairs. A superior alternative that has been used successfully in diverse contexts [7][8][9][10] is to use fiber-reinforced polymers (FRP) as the externally bonded material. This technique requires that thin laminates or bars be bonded to the surface of the structure using an adhesive to form a composite material.
The following sections provide a review of contemporary wall design methods that have been included in various design codes [11][12][13][14]. Two different design methods can be identified in these documents: (1) a simplified design method and (2) a method based on column theory; the latter is arguably the more rational approach. Although the simplified method is straightforward to implement, its applicability becomes limited when lateral loads need to be considered because in such cases the resultant of all loads on the wall must be located within the middle third of its overall thickness. As a result, the total load eccentricity must not exceed one sixth of the wall's thickness. In this way the walls may be considered as reasonably concentrically loaded [15]. The column method represents a viable alternative that provides more accurate results.
The purpose of this paper is to review the considerable advances that have been made in the design of concrete walls, both with and without openings that are subjected to eccentric axial loads. Additionally, the performance of FRP-strengthened walls is discussed on the basis of earlier studies. Design codes and research studies from across the world were taken into consideration in the analysis. Several aspects are highlighted, including the size and position of the openings, and the roles of boundary conditions and the wall's geometric characteristics (i.e. slenderness k = H/t, aspect ratio d = H/L and thickness ratio g = L/t, where H, L and t represent the wall's height, length and thickness, respectively).
A statistical analysis of available models was performed on a database collected by the authors, and is presented in this paper. The outcome of this study provides an overview of the performance of current design models and identifies research gaps. Overall, design codes were found to provide more conservative results than recent design models proposed in other studies. Research into the strengthening of RC walls with openings is still at an early stage, and further studies are undoubtedly required in this area. The findings presented herein will be used to define a new experimental programme that aims to characterize the behaviour of axially loaded RC walls strengthened with FRP; the results of these investigations will be presented in a future publication.

Previous experimental work
The results of 253 experimental tests on RC walls reported in the literature were compiled in a database, which is presented in Appendices A1-A3.
In line with the aim of this study, the database contains information on walls that were loaded gravitationally with uniformly distributed forces applied eccentrically at a maximum of 1/6 of their thickness. Tests on walls loaded gravitationally with eccentricities greater than 1/6 of their thickness have also been reported in the literature [16,17]. However, these results are omitted from the database because the design of such walls is not compatible with current industry standards. Data for walls that failed before reaching their ultimate capacity due to incorrect laboratory manipulation were also omitted.

Database description
The database is organized into six different sections:  (d) Derived geometrical parameters of the tested wall: slenderness (k), aspect ratio (d) and thickness ratio (g). (e) The location(s) of opening(s) in the wall, given in Cartesian coordinates relative to the point at which the wall's centre of gravity would have been located if were completely solid with no openings. (f) Material properties of the tested wall: compressive strength of concrete (f c ), yield strength of steel reinforcement (f y ) and steel reinforcement ratio (q h -horizontal, q v -vertical).
(g) Ultimate axial capacity of the tested wall (N u ) as reported in the original reference.
It should be noted that some of these parameters are referred to by different names in the original references. However, as shown in Fig. 1, a unified naming system was adopted in this work for the sake of clarity.
Because both the experimental boundary conditions and the presence of openings influence the failure modes of stressed walls, the walls listed in the database were initially divided into four main categories: (1) one-way (OW) solid walls (41.1%); (2) twoway (TW) solid walls (26.1%); (3) OW walls with openings (19.1%) and (4) TW walls with openings (13.1%). Fig. 2 summarizes the ranges (frequency distributions between different types of walls) covered by some of the most important parameters recorded in the database. For example, Fig. 2e shows that 60% of OW solid walls included in the database had slenderness values of less than 20, 26% had a slenderness between 20 and 30, and only 14% had a slenderness higher than 30.

Boundary conditions
Walls restrained along top and bottom edges are referred to as OW action panels. Walls that are restrained in this fashion tend to develop a single out-of-plane curvature in parallel to the load direction, and are usually encountered in tilt-up concrete structures. Panels restrained along three or four sides are referred to as TW action panels. Walls restrained in this way generally deform along both the horizontal and vertical directions and are usually encountered in monolithic concrete structures. In all experimental tests found in the literature, restraining elements that were applied along the top and bottom edges were designed as hinged connections that prevented translation while allowing free rotation. The restraining elements applied along the lateral sides were also fixed to prevent translation without restricting rotation.
Restraints can reduce the wall's deformation and increase its ultimate strength. The use of lateral restraints increased ultimate strength by up to 29% for walls with d 6 1; increases of up to 68% were achieved for walls of d > 1 [19]. The data gathered in [20] suggest that even greater increases of up to 300% are possible when d = 1.
Boundary conditions have a dominant influence on cracking patterns and failure modes. Tests on OW walls usually reveal the development of a horizontal main crack along the middle of the wall. According to Swartz et al. [21], TW walls behave similarly to transversely loaded slabs with simple supports. Typical crack patterns for walls both with and without openings are shown in Fig. 3.

Slenderness and aspect ratio
In general, slender walls will have a lower ultimate strength [17,19,20,[22][23][24]. Saheb and Desayi [22] and Saheb and Desayi [19] proved that increasing the slenderness ratio from 9 to 27 reduces strength by 35% for OW walls and 37% for TW walls. A separate study showed that the reduction in strength with increasing slenderness was more pronounced in walls made out of highstrength concrete than in those made of normal strength concrete [20]. El-Metwally et al. [25] subsequently showed that the failure mode is sensitive to both slenderness and end eccentricity.
In general, walls with a low slenderness may fail by crushing on the compressed face and bending on the tension face, while those with high slenderness may additionally fail through buckling. In either case, brittle types of failure have been observed in all experimental studies performed to date [15,16,19,20,22,24,[26][27][28][29].
For OW walls the ultimate strength tends to decrease with an increase in aspect ratio, while for TW walls the opposite trend is found. For an increase in aspect ratio from 0.67 to 2, Saheb and Desayi reported a 16.6% decrease in the ultimate strength of OW walls, [22], and a 26% increase for TW walls [19].

Reinforcement ratio
When RC walls are subject to axial loads, reinforcement is mainly required to offset creep and shrinkage effects in the Typical crack pattern and deflection shape of axially loaded RC walls.
concrete, and additionally due to accidental eccentricities in the applied loads [11][12][13][14]. When walls act as compression members it is generally believed that the contribution of the steel reinforcement should be neglected. Indeed, one of the first experimental studies conducted in Sweden [30] found that RC walls with the minimum level of steel reinforcement exhibited lower than expected ultimate strengths due to difficulties in compacting the concrete. However, no such effect was observed in subsequent studies on this phenomenon [2,15]. Pillai and Parthasarathy [15] found that varying the steel reinforcement ratio had a negligible influence on the ultimate strength when the reinforcement is placed centrally within one layer. When the reinforcement is placed in two layers, however, a significant increase in ultimate strength can be achieved [2,31]. With the reinforcement being placed in two layers it was found that an increase in vertical reinforcement ratio from 0.175% to 0.85% caused an increase in ultimate strength of 54-55% for panels with a slenderness of 12, and about 43-45% for slenderness equal to 24 [19,22]. Increasing the horizontal steel amount, on the other hand, has no influence on the ultimate strength of the walls [19,22]. These observations are valid for both OW and TW walls.

Openings
The presence of openings in a wall considerably reduces its ultimate load capacity relative to the equivalent solid wall. Saheb and Desayi [18] showed that although at 75% of the ultimate load cracking loads are higher for TW than OW walls, at ultimate load the presence of the openings negates the advantage of having restraints on all sides. On the other hand, Doh and Fragomeni [27] and Fragomeni et al. [28] observed that taking the side restraints into consideration could achieve significant gains in the ultimate capacity. It is believed that the differences between the above studies, even though they are studying the same parameter (the effects of restraints on walls with openings), can be explained in terms of the different layouts and opening sizes that the studies investigated. Furthermore, it is not clear whether the lateral restraints were able to function correctly in providing the desired restraining effect.
The magnitude of the ultimate load is governed by the premature failure of the column or beam strips that enclose the opening, however, how large the opening must be for the side restraints to play an important role in the ultimate capacity is currently unknown.

Design for ultimate strength
To the authors' knowledge, the design of axially loaded RC walls is generally based on column theory. This approach involves an analytical derivation that considers stress-strain compatibilities and the equilibrium of forces over the wall's cross section, as shown in Fig. 4. Two conditions are required for this method to be applicable: (1) the steel reinforcement ratio has to be higher than 1% and (2) the total amount of reinforcement has to be placed in two layers [17]. If treated as columns, walls can be regarded as compression members that carry mainly vertical loads. However, pure axial loads rarely occur in practice; a small eccentricity usually exists. In such cases the walls can still be regarded as compression members because compression forces control their failure. Fig. 4 shows a cross-section of an axially loaded wall with an eccentricity, e, from its centreline. The distribution of the strains along its thickness is also shown, together with the corresponding rectangular stress distribution proposed by Mattock et al. [32]. The width of the stress block is taken to be 0.85f c acting on the uncracked depth, x.
The equilibrium between internal and external forces is described by Eq. (1), together with Eq. (2), which describes the equilibrium between internal and external moments. These expressions can be used to compute the interaction between the ultimate axial load, N u , which is given by Eq. (3), and the capable moment, M u , which is given by Eq. (4), at any eccentricity e. thus, and, Eq. (3) is valid for walls whose slenderness does not significantly affect their ultimate capacity. These walls are generally described in the literature as stocky or short walls with a slenderness of less than 15. Macgregor et al. [33] indicated that 98% of the columns in braced frames have a slenderness of less than 12.5, while 98% of the columns in unbraced frames have a slenderness of less than 18. With the increased use of high-strength materials and advanced methods for dimensioning, however, slender elements are becoming more common in current building practices [28].
For slender elements, the predicted ultimate capacity has to be reduced through a second-order analysis that takes into consideration the material nonlinearity, cracking stages and member curvature. A second-order analysis that takes into account variable wall stiffness, as well as the effects of member curvature and lateral drift, is proposed in all international design codes [11][12][13][14]. As an alternative to the refined second-order analysis, design may be based on axial forces and moments obtained from the moment magnifier approach. Through this method, the total design moment according to EN 1992-1-1 [14] may be expressed as, Fig. 4. Forces acting on the cross-section of a wall at equilibrium.
where M 0Ed is the first order moment, N Ed is the design value of the axial load, N B is the critical buckling load based on nominal stiffness and b = p 2 /c 0 is a factor accounting for the curvature distribution along the member, assuming that the second order moments have a sinusoidal distribution. The c 0 factor depends on the distribution of the first order moments and, according to EN 1992-1-1 [14], can be approximated as c 0 = 8 for a constant distribution, c 0 = 9.6 for a parabolic distribution, c 0 = 12 for a symmetric triangular distribution. Parme [34] has suggested simplifying Eqs. (5) and (6), this form of expressing the second order effects being currently adopted in the European norm EN 1992-1-1 [14].
Robinson et al. [17] concluded that the equivalent column procedure should not be used to design RC walls with steel reinforcement ratios lower than 1%, or those with central reinforcements regardless of the reinforcement ratio (as observed in [15]). This is because in these cases the axial capacity depends mainly on the un-cracked wall section stiffness and the tensile strength of the concrete in flexure [17].
While the above method explicitly accounts for parameters such as eccentricity, slenderness and creep, it tends to not be used in practice because of its generalized form and complexity. Instead, numerous models have been derived empirically that are simpler but less accurate. Some of these models have been implemented in design codes, and the details of these models are presented in the following section.

Design models in codes
Currently the practical design of RC walls, described in standards such as ACI318 [11], AS3600 [12] or CAN/CSA-A23.3 [13] is based on empirical models whereas EN 1992-1-1 [14] is based on calibration against the results of non-linear analysis. The design equation of ACI318 [11] was developed over the time with contributions from several studies [15,[35][36][37][38]. Its current form was first proposed by Kripanarayanan [35], and adopted by ACI Committee 318 [39]. Despite the subsequent completion of numerous studies, no modifications to this formula have been implemented. The design equation found in CAN/CSA-A23.3 [13] is similar to that of [11], the only differences being in the design factors. Doh [40] suggested that the simplified design method found in AS3600 [12] is based on the complementary moment method recommended in the British Concrete standard [41].
According to Hegger et al. [42] the EN 1992-1-1 [14] approach was adapted from the work of Haller [43], a method that was originally developed for masonry elements.
The empirical method is based on the following assumptions: (1) the steel reinforcement will not bring any contribution to the load capacity; (2) the tensile strength of concrete is disregarded; (3) the wall is loaded with an eccentricity applied only at the top. The most important differences between the design models discussed above will now be highlighted.
Differences exist between design codes regarding how they deal with the following parameters: variation of the compressive forces within the stress block, eccentricities, slenderness and creep. The first important difference in development occurs in the assumptions made on the distribution of stresses within the compressed concrete block. ACI318 [11], AS3600 [12], CAN/CSA-A23.3 [13] define a linear stress distribution, as shown in Fig. 5a, whereas EN 1992-1-1 [14] assume a rectangular stress distribution ( Fig. 5b). The ultimate capacity is then defined as the resultant force of the stress distribution, where r c is the allowable compressive stress and x is the un-cracked depth of the concrete section.
Furthermore, the initial eccentricity caused by the applied loads, e i , is further increased by an additional one, e a , due to the lateral deflection of the wall. This factor accounts for the effect of slenderness, known also as second order effects (or P-D effects). The procedure described in [11][12][13] to find the maximum deflection at the critical wall section uses a sinusoidal curvature (Fig. 5a), using deflections obtained from the bending-moment theory [44]. Conversely a triangular curvature is assumed in [14], a consequence of a concentrated horizontal force at the critical point of the wall (Fig. 5b). This approach results in a linear, rather than parabolic, deformation, which acts to reduce the predicted ultimate capacity of slender walls [45]. For the sake of brevity the derivations of these models are not presented in this paper, and can be found in [40].
Most of the experimental studies involved short-term tests and so their results are not very relevant to real walls, which are always subject to a relatively high sustained load. Macgregor et al. [46] showed that sustained loads tend to weaken the performance of slender columns by increasing their deflections. The creep due to sustained loads may also decrease the column's ultimate capacity. Consequently, the effects of creep should always be considered for safety reasons. As shown by Doh [40], the AS3600 [12] standard accounts for the effects of creep by increasing the first-order eccentricity by 20%. Similarly, the EN 1992-1-1 [14] standard states that the normal effects of creep are included in its underlying model. However, Westerberg [47] has demonstrated that the effects of creep are not properly described in the EC model because it produces results that are inconsistent with those obtained using a general method that explicitly accounts for creep effects.
For OW walls, restrained against rotation provided at both ends, k, takes different values for different codes, i.e. k = 0.75 [12], k = 0.8 [11,13], k = 0.85 [14]. Unless no restraint against rotation is provided at one or both ends, the slenderness factor k equals 1.
Both Australian [12] and European [14] design codes include the effect of the side restraints, applied to TW walls, through the effective height factor k (Eqs. (11) and (12)). This factor is dependent on the aspect ratio of the wall and is given by Eq. (11) for walls restrained on three sides and Eq. (12) for walls restrained on all four sides.

Other models proposed by researchers
Numerous studies have attempted to further improve the design models. Their proposed models incorporate the effects of the slenderness, aspect and thickness ratios, boundary conditions and steel reinforcement. The early studies that modelled RC walls as compression members were performed by [2,15,21,35,37,38], and subsequently reviewed by [48]. In this section only the most recently developed models are presented, although the results obtained from the earlier studies are included in the database and used for the performance assessment of the current models.
In the next section, the models proposed by recent studies will be given in chronological order. All models are abbreviated as OWM -one-way model for solid walls, TWM -two-way model for solid walls and OM -model for walls with openings.
3.2.1. Design equations for solid walls 3.2.1.1. Saheb and Desayi model (OWM1) [22]. To the best of the author's knowledge the first systematic study of solid concrete walls tested under both OW and TW actions was reported by Saheb and Desayi [22]. The influence of the aspect, thickness and slenderness ratios, as well as the vertical and horizontal steel reinforcement ratios, on the ultimate load was studied. Based on their own experimental results and those reported in [15,37,38], an empirical equation was proposed (Eq. (13)), valid for OW walls.
In the assessment chapter (Section 4) this model is abbreviated as OWM1.
where A sv is the area of vertical steel reinforcement. When compared to Eq. (7) this model additionally takes into account both the effect of the steel reinforcement and that of the aspect ratio. However, for walls with an aspect ratio higher than 2, the effect of the aspect ratio is not accounted for (i.e. the term (1.20d/10) = 1). This model has been validated for axially loaded walls with an eccentricity of t/6 and a slenderness of up to 27. Another important assumption was that the minimum amount of steel reinforcement placed in two layers yields at ultimate load. Therefore, this model may not be suitable for walls that are centrally reinforced, or when the eccentricity is less than t/6. [19]. In the same way as the authors did for OW panels, the influence of the aspect, thickness and slenderness ratios, as well as the vertical and horizontal steel reinforcement ratios, on the ultimate load was studied for TW action panels. It was found that the ultimate load increased as the percentage of vertical steel increased, this was due to the reinforcement being placed in two layers. From their results it can also be concluded that the steel ratio has a more pronounced effect on the ultimate capacity when the panels have a high slenderness.

Saheb and Desayi model
Before the Saheb and Desayi study there were no equations for predicting the ultimate strength of TW wall panels, because of this the authors proposed both an empirical and a semi-empirical model. The first one (TWM1) is an empirical formulation that was validated using their own experimental data and that published by Swartz et al. [21]. Shown in Eq. (14), it is limited to those panels whose aspect ratio is between 0.5 and 2, and the maximum limit of the thickness ratio is 60.
The second proposal (TWM1 ⁄ ) is a semi-empirical method (Eq. (15)), developed from a modification of the buckling strength theory of thin rectangular metal sheets, proposed by Timoshenko and Gere [49]. The original formulation of Timoshenko and Gere [49] was modified by substituting the yield strength of the metal plate with the compressive strength of the concrete wall.
A sv , A sh are the areas of vertical and horizontal steel reinforcement, respectively. Unlike the model for OW panels, the effect of the steel reinforcement could not be directly accounted for because of limited available data; however, it was included indirectly through the term R.
3.2.1.3. Aghayere and MacGregor model [50]. A procedure for obtaining the maximum eccentricity, e y , for a given set of loads, N x and N y , was proposed by Aghayere and MacGregor [50]. By obtaining the M-N-u relationship for sections of unit length at the centre of the plate, one can determine the internal resisting moments per unit length M xi and M yi , corresponding to the maximum curvatures u xo and u yo , respectively. Different eccentricities can be obtained for various load levels, and through interpolation the maximum in-plane load for a given eccentricity can be obtained [50].
where e x and e y give the eccentricity of the in-plane load in the x and y directions, respectively, N x and N y are compressive forces per unit length in the x and y directions, respectively, M xi and M yi are the internal resisting moments per unit length in the x and y directions, respectively, and u xo and u yo give the maximum curvature in the x and y directions, respectively. The model proposed by Aghayere and MacGregor [50] takes into account material nonlinearities including tension-stiffening effects. Owing to its complexity and the limited information reported in previous experimental tests, however, this model was not included in the assessment.
3.2.1.4. Fragomeni and Mendis model (OWM2) [51]. The experimental programme undertaken by Fragomeni [52] focused on investigating the axial load capacity of normal and high-strength concrete OW walls. It was found that the ultimate load capacity is not influenced by the minimum amount of steel reinforcement when this reinforcement is placed centrally in one layer. It was also found that the ultimate load capacity did increase for aspect ratios higher than 2, in contradiction to the results reported in [22]. Significant differences exist, however, between these two studies, for instance the concrete compressive strength and steel ratio are both higher for the specimens tested in [52].
The proposed model [51], that suggests modifications to the Australian code, accounts for the high strength concrete contribution through Eqs. (17a) and (17b).  [20]. Following the suggestions of Fragomeni [52], who took high concrete strength values into account to increase the wall strength, Doh [40] attempted to refine the existing equation through an extensive experimental study on OW concrete walls. The design equation that this research produced, shown in Eq. (18), applies to walls with larger slenderness ratios and a variety of concrete strengths.
where the effective length factor k is k ¼ 1 for k < 27 and k ¼ 18=k 0:88 for k P 27. This model omits the centrally placed reinforcement and the aspect ratio effects.
3.2.1.6. Doh and Fragomeni model (TWM2) [20]. In addition to the above tests performed on OW wall panels, Doh [40] tested walls in TW action in order to extend the applicability of their design equation. In this way they were able to extend Eq. (18) to include the effects of side restraints, through the effective length factor k.  [42]. Hegger et al. [42] have proposed a new model valid for OW walls, based on the methodology presented in [43]. Their model is similar to [14] and, by taking into account the concrete tensile strength and material nonlinearity, predicts an increase in ultimate capacity. This increase is more pronounced when considering specimens of high slenderness and eccentricity. Chen and Atsuta [53] suggested that the tensile strength of normal concrete has a significant effect on the wall strength, and should therefore be taken into account when computing ultimate strengths.
In the study Hegger et al. [42] proposed two functions for the purposes described above, one to describe the nonlinear behaviour of concrete material, Eq. (21), and the other to describe the linearelastic behaviour, Eq. (22). Eq. (21) is in accordance with the paper of Kirtschig and Anstötz [54], and is valid only for normal strength concrete. Eq. (22) was first proposed by Glock [55], who showed that the formulation is valid only for high slenderness and eccentricity values, i.e. e P 0.2t.
with e c2 ¼ 2f cd =E c0d , A = 1.25 and B = 1.70. Here e c2 is the strain in the concrete at the peak stress f cd and E c0d is the design value of the modulus of elasticity of concrete.
The maximum value between U non lin and U lin-el has to be used in connection with Eq. (23) with a minimum eccentricity of e = 0.2t, suggesting that the formula would be suitable for higher eccentricities as well.
This model requires specific material characteristics, namely the tensile strength and modulus of elasticity of concrete, and as such information is limited in the experimental test reports it is difficult to test the model precisely. Instead the required characteristics where estimated using the equations proposed in fib Model Code 2010 [56].  [23]. In two recent studies, Ganesan et al. [23,24] tested wall panels under OW action to study the axial strength of steel fiber reinforced concrete and geopolymer concrete. The authors reported that if the slenderness is kept constant, the ultimate strength of the concrete panels decreases as the aspect ratio increases. Their proposed model is similar to the one developed by Saheb and Desayi [22], including both the effect of the steel reinforcement and that of the aspect ratio. The specimens used to derive the model, however, had aspect ratios lower than 2, meaning that for higher values the model may not be valid.
Furthermore, due to the differences between the material characteristics of the concrete, the authors suggested new modifications to Eq. (24). Eq. (25) is suitable for reinforced geopolymer concrete walls under OW action.  [17]. From experimental results obtained in a series of tests performed on slender OW wall panels Robinson et al. [17] found that current design methodologies are considerably conservative. The authors devised a new model using the semi-empirical semi-probabilistic DAT (Design Assisted by Testing) methodology [57], based on the ''lumped plasticity'' concept. This concept allows the entire inelasticity of the element to be concentrated at the critical section, using a ''non-linear'' fibre hinge [17]. The model (Eq. (26)) has been validated using their experimental data, and was calibrated using statistical techniques.

Design equations for walls with openings
The design codes that have been reviewed above [11][12][13][14] do not provide design equations to evaluate the axial strength of a concrete wall that contains openings. There is very little information in the research literature, therefore, probably due to the complex failure mechanisms of such elements. Some guidelines are provided, such as in AS3600 [12] and EN 1992-1-1 [14]. These state that if the walls are restrained on all sides, and enclose an opening with an area less than 1/10 of the total, the effects of this opening on the axial strength can be neglected. The height of the opening should also be less than 1/3 of the wall height. If these conditions are not accomplished, the portion between restraining member and opening has to be treated as being supported on three sides, and the area between the openings (if more than one) has to be treated as being supported on two sides. This approach is only valid if the openings are included at the early stages of the design, as special reinforcement bars have to be placed around openings to avoid premature failure. No recommendations are given, therefore, if the openings are created in an existing wall.
3.2.2.1. Saheb and Desayi model (OM1) [18]. The effect of one or two openings, placed either symmetrically or asymmetrically, and combinations of door or window openings, have been studied by Saheb and Desayi [18]. To extend the usefulness of their empirical method to account for the presence, size and location of an opening, the authors proposed a new equation that is given below.
where N u is the ultimate load of an identical panel without openings under OW (Eq. (13)) or TW action (Eq. (14)). The constants k 1 and k 2 were obtained using curve-fitting techniques. Under OW action this procedure yields k 1 = 1.25 and k 2 = 1.22, while under TW action k 1 = 1.02 and k 2 = 1.00. The effect of the size and location of the opening in the wall is taken into account through a dimensionless parameter, a x , defined as, where A 0x and A x represent the horizontal wall cross-sectional area of the opening (i.e. A 0x = L 0 t) and of the solid wall (i.e. A x = Lt), respectively. All parameters involved in Eq. (28) can be easily determined from Fig. 1, however, for simplicity the term a is calculated according to Eq. (29). a ¼ Again, the constants k 1 and k 2 were obtained using curve-fitting techniques, this time through a larger number of tests. For OW panels this yielded k 1 = 1.175 and k 2 = 1.188, while for TW panels k 1 = 1.004 and k 2 = 0.933.
While the differences between these constants are not large, the main contributor to the ultimate load comes from the load capacity of the solid wall, which is calculated in a different way. Both models take into account the size and position of an opening through the parameter a x , allowing a reduction in the ultimate capacity.
Fragomeni et al. [28] found that this model gives results in good agreement with the test results from another experimental study [58]. [59]. Guan et al. [59] found that increasing both the length and the height of an opening has the most significant effect on the capacity, and proposed a new model to account for this effect. Having established a benchmark model, the authors performed a parametric study by varying the parameters that the capacity was most sensitive to. Their analysis proceeded through a nonlinear finite element method. In the model a three-dimensional stress state was used with elastic brittle fracture behaviour for concrete in tension, and a strain hardening plasticity approach was assumed for concrete in compression. Their model is nearly identical to that proposed by Doh and Fragomeni (OM2), the only difference being that a x was changed to a xy to account for the opening height.

Guan et al. model (OM3)
in which A 0y represents the vertical cross-sectional area of the opening (i.e. A 0y = H 0 t), A y represents the vertical cross-sectional area of the solid wall (i.e. A y = Ht) and d represents the distance between centres of gravity (G 1 and G 3 ) of the wall with and without the opening, in the vertical direction (Fig. 1). In Eq. (30), c represents ''the weighting ratio indicating the percentage of a y in relation to a x ''. Using regression analysis, a new set of constants was determined; c = 0.21, k 1 = 1.361 and k 2 = 1.952 for OW walls and c = 0.40, k 1 = 1.358 and k 2 = 1.795 for TW walls. It should be noted that this model was derived from walls with a fixed slenderness ratio (i.e. k = 30) and an aspect ratio of unity. [4]. In a more recent study, Mohammed et al. [4] tested OW walls with cut-out openings. The size of the openings was varied from 5% to 30% of the solid wall. It was found that the presence of a cut-out opening in a solid OW wall led to the formation of disturbance zones. Discontinuities in these disturbance zones cause high stresses in the concrete, and cracks will form at the corners of the opening if improperly reinforced.

Mohammed et al. model
For this case, Mohammed et al. [4] suggested a new set of constants to be used in Eq. (27). The authors tested one-way panels only, obtaining k 1 = 1.281 and k 2 = 0.737. It should be noted that Eq. (27), proposed by Saheb and Desayi [19], considers steel reinforcement placed in two layers that yields at ultimate, whereas the experimental programme presented in [4] consisted only of centrally reinforced panels.
Since the model was calibrated on walls with cut-out openings (i.e. no diagonal bars around corners) it cannot be assessed through the current database. However, the results of the experiments by Mohammed et al. [4] were incorporated into the current database and used in the assessment of other models (i.e. OM1, OM2, and OM3).

Assessment of existing design models
The empirical design models reviewed above were derived using a limited number of either experimental tests or numerical simulations. Some models were developed solely from tests performed by the researchers themselves, while others additionally used tests from other sources, therefore the predictions of the latter may give more reasonable outcomes by covering a broader spectrum of designs. The studies focussed on either the variation of geometric characteristics (i.e. slenderness, aspect ratio, size and position of the opening) or the variation of material properties (i.e. concrete strength, influence of steel reinforcement). If one has to design a compression member under conditions that were not specifically covered by any of the available design models, then it remains unclear how accurate the models will be. In order to quantify this a statistical analysis was performed on each model in turn, using all of the experimental results available (these are included in Appendices A1-A3), unless the model explicitly specifies its limiting parameters.
The accuracy of the models was evaluated using the following statistical indicators; the average (Avg), the standard deviation (St Dev) which measures the amount of variation from the average, the coefficient of variation (CoV) which shows the extent of variation and the coefficient of determination (R 2 ) that indicates how well the data fit a model within a 95% confidence interval.
The analysis was conducted separately for solid OW action, solid TW action and for walls with openings. For all models, the material strength reduction factor, u, was set to 1.0. Fig. 6 shows the normalized strength versus slenderness, as predicted by the investigated design codes for a typical wall that is assumed to be loaded axially with an eccentricity of t/6 and has a strength reduction factor u = 1. ACI318 [11] model provides higher loads for slenderness values above 10 when compared to EN 1992-1-1 [14].   The AS3600 [12] and CAN/CSA-A23.3 [13] models predict the lowest load values for slenderness values lower than 15, above this value the load value predictions increase above those of EN 1992-1-1 [14], while remaining lower than ACI318 [11].

Assessment of predicted values for OW solid walls
The limits of the slenderness values given in the codes are also plotted in Fig. 6. Beyond these limits, presumably imposed by the data available at the time of development, the models are not accurately calibrated and can yield negative values for the normalized strength. Recent studies have shown that the slenderness limit can be increased with confidence [15,17,23,40], however, suggesting that there is a need to update the current design codes.
How these models perform when assessed using experimental tests from the database is shown in Fig. 7a and b. While code models [11][12][13][14] present a natural degree of conservationism, due to statistical calibration, the trend is opposite for the models presented in the literature [17,20,22,23,51] (see Fig. 7b). A statistical summary for these models is presented in Table 1. Overall, the most conservative model is that proposed by CAN/CSA-A23.3 [13], with an average ratio between theoretically and experimentally determined capacity of 0.57 and a standard deviation of 0.20. The least conservative model is OWM6 [17], with an average ratio between theoretically and experimentally determined capacities of 1.24 and a standard deviation of 0.68. However, most of the extreme nonconservative results for the later model come from walls made of high-strength concrete. Since this aspect was not discussed in [17], the authors assumed that using the OWM6 model for normal strength concrete would provide better results. The new results obtained excluding high-strength concrete values are abbreviated as OWM6 ⁄ and are listed in Table 1. The model proposed by Hegger et al. [42] (OWM4) is the most statistically accurate, with an average ratio between the theoretically and experimentally determined capacities of 0.89 and a standard deviation of 0.17.

Assessment of predicted values for TW solid walls
In the case of TW walls, EC2 and AS3600 are the only major codes that provide a methodology to account for a higher capacity due to restraints on all sides. It remains unclear whether the limitations placed on the slenderness values in these models (k = 25 [14] and k = 30 [12]) apply only to OW walls or to both OW and TW walls. By plotting both models with aspect ratios usually encountered in practice, one can observe that such limitations would be highly restrictive. The way that these codes account for lateral restraints is by using a subunitary coefficient (k), based on the end-restraint of the wall and its aspect ratio (Eqs. (11) and (12)). Significant increases in strength can be achieved by restraining the walls on all their sides, as can be observed in Fig. 8.
Fewer testes were carried out on walls restrained on all sides; correspondingly less models are also available. The performances of these models are shown in Fig. 9a for design codes and Fig. 9b for models found in the literature.
The outliers in Fig. 9a (EC2 and AS3600) and Fig. 9b (TWM1 and TWM1 ⁄ ), enclosed by the ellipsoids, originated from walls made of high-strength concrete. In addition, the tests were performed in a horizontal position with the eccentricity acting in favour of the strength, due to effect of gravity, and are consequently extremely non-conservative. A statistical summary for these models is presented in Table 2. The most conservative model is that proposed by AS3600 [12], with an average ratio between the theoretically and experimentally determined capacities of 0.71 and a standard deviation of 0.40. The least conservative model is TWM1 ⁄ [19], with an average ratio between the theoretically and experimentally determined capacities of 1.44 and a standard deviation of 0.87. The most accurate model in terms of average ratio is TWM2 [20], however, a relatively high standard deviation of 0.30 weakens its precision. Fig. 8. Comparison of different design models in the investigated codes [12,14] for TW solid walls. Fig. 9. Assessment of the current design models of two-way solid walls: (a) design codes; (b) design equation from different studies.

Assessment of predicted values for walls with openings
The first model to include the effect of the openings, OM1 [18], was derived using six OW and six TW specimens, while model OM2 [27] was derived using ten OW and ten TW specimens. The model OM3 [59] was calibrated on thirty-six OW and thirty-seven TW specimens. The number of tests used to calibrate these models, therefore, is rather limited. This means that their predictive value may not extend to the design of openings in walls with different material and geometric characteristics.

OW walls with openings
The OM1 model provides the most conservative results, with the smallest value of the average ratio between the theoretically and experimentally determined capacities of 0.77 and a standard deviation of 0.16, while the best model in terms of average is OM2, i.e. 0.95 with a standard deviation of 0.19. The performances of these models are shown in Fig. 10 and the statistical summary is presented in Table 3.

TW walls with openings
Owing to its limited number of tests, OM1 model shows a large scatter from the bisector for those walls restrained on all their sides. A significantly more accurate model is OM2, with an average of 0.90 and a standard deviation of 0.13, proposed by Doh and Fragomeni [27].

FRP -based strengthening
The successful application of FRP to strengthen solid concrete walls has been achieved in several studies [60][61][62]. All of them performed a rehabilitation of structural walls using externally bonded FRPs to increase the flexural and/or shear strength, stiffness and energy dissipation. The creation of large openings in walls removes a significant quantity of concrete and steel reinforcement, necessarily reducing the load capacity of the wall. FRPs are able to strengthen such walls by redistributing the stresses, allowing the wall to recover almost its full capacity before the opening was created, if not more [3,4,63,64].
As the size of the opening increases, the global behaviour of the wall will change to that of a frame, and consequently new failure modes may arise. This has an influence on the optimal strengthening configuration. The research conducted so far on strengthening structural members with openings, such as slabs, walls or beams, using FRPs is promising [3,4,63,[65][66][67]. The alignment of the fibres was based on observations of the failure modes of the un-strengthened elements. Usually the FRP material is placed around openings in a vertical, horizontal or inclined alignment, or a combination of these. In some cases the side strips were fully or partially wrapped to provide confinement. In general, the amount of FRPs were chosen intuitively, or by converting the amount of steel reinforcement  [63] have demonstrated that the optimization of the direction, width, and number of layers of the FRP strips by using a strut-and-tie model can provide rigorous results.
Mohammed et al. [4] tested 1/3-scale one-way RC walls with cut out openings, these openings having areas varying from 5% to 30% of the total wall area. The specimens were tested with a uniformly distributed axial load applied with an eccentricity of t/6. The introduction of small openings (5% area) reduced the axial capacity by 9%, while large openings (30% area) reduced the capacity by nearly 33%. While keeping the same geometric characteristics and applying two different CFRP patterns (see Fig. 11), the capacity was increased as the principal stresses on the opening corners were reduced. When applied to small openings the first pattern, in which the CFRP was applied around the corners, increased the axial strength by 49.9%. The second pattern, with CFRP placed at the corners, performed better on small openings, causing an increase in axial strength of 75.4%. When applied to large openings, however, the efficiency of these reinforcements was significantly reduced, with 11.3% and 15.1% increases for the first and second patterns, respectively. This confirms the afore-mentioned claim that different sized openings lead to different failure modes, and consequently require different strengthening patterns. A configuration that may yield better results for large openings would be to fully wrap the side chords, as their thickness ratio was slightly above 2. EN 1992-1-1 [14] emphasizes that elements with a thickness ratio below 4 should be considered as columns rather than walls.
The research conducted so far on the rehabilitation of walls using FRPs was promising, however, the repaired walls were loaded principally in the horizontal direction to simulate the effects of earthquakes. The proposed strengthening schemes, therefore, may not be suitable for the repair of gravitationally loaded walls, and more research is required with the loads applied vertically.
Just one study was found in the literature that focused on using FRPs to strengthen axially loaded RC walls with cut-out openings [4]. In order to better understand the structural behaviour of such a configuration, therefore, more studies are required. To this end a research programme at the Luleå University of Technology is currently underway. This study will test a number of concrete walls with different parameters, such as size opening and strengthening configurations, under TW action. The results are expected to be published upon completion of the study.

Conclusions and future directions
Through the statistical analysis of existing experimental studies this study indicated areas where further testing is required in order to enhance the reliability of current design models. It was found that most experimental studies have focussed on testing RC walls under OW action, with a fixed eccentricity of t/6. Fewer tests exist on walls under TW action, walls with openings or different eccentricities, and more tests are required in these experimental regimes to facilitate the development of appropriate design models. The current database is useful because it highlights areas where the current literature is lacking, and where systematic studies could provide important insights into the behaviour of wall types that are poorly understood (e.g. walls with eccentricities above t/6 or OW solid walls with high slenderness ratios) or the effects of parameters that are not well covered by existing design provisions (e.g. the presence of an opening or the influence of steel reinforcement).
The design of the experimental programme has a significant role in determining the accuracy of the regression-based models derived. Although the design is carried out assuming a perfect hinge, laboratory evidence shows that neither a perfect hinge nor a full rotation restraint could be achieved in the laboratory environment, much less in practice. All design models empirically derived from such tests, therefore, will necessarily contain a certain level of inaccuracy.
Since the simplified methods assume that the walls are unreinforced elements, the contribution of any steel reinforcement is disregarded. This occurs regardless of the location of the steel mesh layer, or if the reinforcement is placed in one or two layers. For centrally reinforced walls this seems to be valid, although in some cases it may bring some ductility at higher loads. For double-reinforced walls, however, the enhanced capacity should be accounted for, even when the steel ratio is at a minimum level.
The design models found in established design codes provide the most conservative results, while those proposed in other studies showed a certain level of non-conservatism. However, all design models were plotted using u = 1, while a carefully chosen safety factor should be used in practice.
FRPs have been recognised as a viable alternative for the strengthening of concrete structures. The potential applications of FRPs in strengthening walls that have been weakened by new openings need to be further studied. There are only a few research studies in the literature on the FRP strengthening of walls with openings, and almost all the experimental tests involved wall openings that were initially planned. The case of RC walls with cut-out openings is still largely unexplored, with just one research study focussing on this problem [4]. Currently there are no design philosophies or reliable theoretical guidelines for calculating the capacity of strengthened walls in the literature. Safe and clear design procedures for strengthening walls with openings are needed. In the bullet points listed below the main gaps in the research literature, that require further study, are presented.
1. Openings can be of different sizes and may have different positions with respect to a reference point of the RC wall. Therefore, it is natural to ask: How do these parameters influence the FRP contribution to the overall capacity of the wall? 2. What are the efficiencies of different FRP strengthening configurations and systems (sheets, plates or bars) when strengthening RC walls with openings? 3. How does the failure mechanism of an RC wall with an opening change after strengthening with FRP?
4. When designing RC walls with openings, engineers tend to adopt a simplified method by dividing the wall openings into isolated columns connected by beams. While this method provides acceptable results it is overly conservative, and it would be beneficial to know how to delineate small and large openings in walls, and where the transition from RC walls to RC frames should occur in the design of structural elements.