Influence of Wall Thickness and Water Content on the Out-of-Plane Stability of Adobe Walls

This paper presents the results of a numerical study using the finite element method to assess the effect of wall thickness and a moist region along the wall base on the out-of-plane lateral stability of adobe walls. The models consisted of cantilever unreinforced adobe walls with a thickness of 25 cm, 30 cm, and 40 cm. Numerical analyses were performed for four lengths (L) of the moist region and for dry walls. The moist region was modelled with material characteristics corresponding to partially saturated and nearly saturated adobe. The results showed the detrimental effect of moisture along the lower portion of the wall on the out-of-plane lateral strength. The out-of-plane lateral strength reduction ratio dropped significantly as L increased. A high slenderness ratio was found to be critical for cantilever adobe walls when L extended beyond half of the wall length. The results provided insights concerning the role of wall thickness in the stability against overturning.


Introduction
Adobe structures are becoming an important part of the architectural identity of many cities in the southwestern region of the United States of America (USA). In addition, many landmarks with historic significance were built with adobe. Adobe bricks are made by forming mud into bricks and drying them under the sun. Traditionally, adobe walls are constructed by laying the brick units in alternating courses, filling the bed and head joints with mud mortar that has the same soil constituents as the bricks. The soils used to make adobe bricks typically contain fractions of clay, silt, and sand. Cut straw is often added to the adobe mixture to help obtain even drying and in turn reduce the shrinkage cracking of the bricks.
Adobe walls are usually thick, load bearing walls. They have high thermal mass that results in an excellent passive temperature control inside of the building [1]. However, adobe walls are susceptible to water damage [2][3][4][5]. Heavy rain, shallow ground water, and poor runoff drainage system may cause the increase of water content in adobe walls. Water can penetrate through cracks in the plaster and move up in the wall by capillary action. When the wall plaster has low permeability, such as cement-based stucco, water remains entrapped inside the wall [5]. Problems related to moisture and the associated material weathering, like basal erosion, coving, and bulging, have been observed in the walls of many adobe buildings in the Rio Grande region, southern New Mexico, USA.
In historic and older adobe buildings that generally include adobe footings, uncontrolled or excessive moisture in the adobe walls has been reported as the most prevalent cause of deterioration [3,5]. Capillary rise from ground water was the primary cause of wall damage in the Mission San Cayetano de Tumacácori (1691), located in Santa Cruz County, Arizona, USA. Water content values up to 16% and moisture as high as 90 cm above the ground were recorded in the walls of this historic church Infrastructures 2020, 5, 78 3 of 13 could also happen in the absence of seismic activity. Vertical cracks due to improper corner stitching of bricks or differential settlements can cause a wall to disconnect from the rest of the structure. Material deterioration or moisture damage along the wall base (i.e., area of the wall immediately above the footing) can create a weak longitudinal rotational axis that promotes out-of-plane instability [31]. Under this condition, relatively small loading, such as the lateral thrust from a roof frame, could lead to wall collapse. The presence of a moist region above the footing was found to decrease significantly the in-plane lateral strength of adobe walls [21,31].
The literature focuses mostly on the in-plane behaviour of adobe walls, especially when subjected to seismic loads. However, understanding the out-of-plane behaviour of adobe walls is as important [32]. This paper presents and discusses the results of a numerical study to assess the effect of the wall thickness and slenderness ratio and the presence of moisture in the lower region of adobe walls on the out-of-plane lateral stability. Three cantilever adobe walls with different slenderness ratios were modeled using finite element method. Pushover curves and maximum in-plane plastic strains from the numerical analysis were used to determine and compare the stability and out-of-plane lateral strength of the walls.

Model Geometry and Boundary Conditions
A finite element model was created using the FE analysis software Abaqus ® [33] to simulate a freestanding cantilever wall subjected to lateral loading ( Figure 1). The wall dimensions are 700 cm long and 300 cm high, which are within the range of the typical dimensions of exterior walls of one-story adobe dwellings in the southwestern USA. Three values of wall thickness (t) are considered, 25 cm, 30 cm, and 40 cm, corresponding to slenderness ratios (S l ) of 12, 10, and 7.5, respectively. The wall is bounded at the top and bottom by two concrete beams with no interaction. The base beam serves as the wall footing with fixed boundary conditions. The beam at the top serves as a bond beam to ensure a uniform distribution of the applied loads. The top and side boundaries of the wall are unrestricted. The concrete beams and the adobe wall are represented by linear brick solid elements (see Figure 1a). The size of each solid element is 100 × 100 × 100 mm.
Infrastructures 2020, 5, x FOR PEER REVIEW  3 of 13 improper corner stitching of bricks or differential settlements can cause a wall to disconnect from the rest of the structure. Material deterioration or moisture damage along the wall base (i.e., area of the wall immediately above the footing) can create a weak longitudinal rotational axis that promotes outof-plane instability [31]. Under this condition, relatively small loading, such as the lateral thrust from a roof frame, could lead to wall collapse. The presence of a moist region above the footing was found to decrease significantly the in-plane lateral strength of adobe walls [21,31]. The literature focuses mostly on the in-plane behaviour of adobe walls, especially when subjected to seismic loads. However, understanding the out-of-plane behaviour of adobe walls is as important [32]. This paper presents and discusses the results of a numerical study to assess the effect of the wall thickness and slenderness ratio and the presence of moisture in the lower region of adobe walls on the out-of-plane lateral stability. Three cantilever adobe walls with different slenderness ratios were modeled using finite element method. Pushover curves and maximum in-plane plastic strains from the numerical analysis were used to determine and compare the stability and out-ofplane lateral strength of the walls.

Model Geometry and Boundary Conditions
A finite element model was created using the FE analysis software Abaqus ® [33] to simulate a freestanding cantilever wall subjected to lateral loading ( Figure 1). The wall dimensions are 700 cm long and 300 cm high, which are within the range of the typical dimensions of exterior walls of onestory adobe dwellings in the southwestern USA. Three values of wall thickness (t) are considered, 25 cm, 30 cm, and 40 cm, corresponding to slenderness ratios (Sl) of 12, 10, and 7.5, respectively. The wall is bounded at the top and bottom by two concrete beams with no interaction. The base beam serves as the wall footing with fixed boundary conditions. The beam at the top serves as a bond beam to ensure a uniform distribution of the applied loads. The top and side boundaries of the wall are unrestricted. The concrete beams and the adobe wall are represented by linear brick solid elements (see Figure 1a). The size of each solid element is 100 × 100 × 100 mm. The FE model includes an area with higher water content located in the lower portion of the adobe wall, immediately above the footing (See Figure 1b), herein referred to as the moist region. It  The FE model includes an area with higher water content located in the lower portion of the adobe wall, immediately above the footing (See Figure 1b), herein referred to as the moist region. It is assumed that the moist region starts from one of the wall edges and extends towards the opposite side along the wall base for a specified length (L). The height of the moist region is set to be 30 cm, based on common field observations of moisture height in adobe walls. The thickness of the moist region is assumed to be constant across the wall thickness (t). Despite the fact that moisture may vary across the wall thickness depending on the water source, the assumption of constant water content and material properties across the wall thickness represents the worst-case scenario and offers a baseline against which the detrimental effect of moisture in adobe walls can be compared. For each case of t, four L values are considered, namely 100 cm, 300 cm, 500 cm, and 700 cm, in addition to the dry wall case (L = 0 cm).

Numerical Approach
Traditionally, the same soil used to form the adobe bricks has been used for the mortar. Hence, the adobe wall is modeled herein as a homogeneous, isotropic continuum. The concrete damage plasticity (CDP) model is used in this study to account for the nonlinear behaviour of the adobe material. The CDP model is a plasticity-based damage material model in which failure is assumed to happen through tensile cracking or compressive crushing [34,35]. The uniaxial tension and uniaxial compression behaviours of the adobe material are defined by the curves illustrated in Figure 2. In uniaxial compression, the input material parameters for the plastic range of the CDP model were obtained from the results of unconfined compression tests performed on adobe soil specimens with water content (w) values ranging from 1% (air-dry adobe) to 15% (nearly saturated condition) [31]. The initial compressive failure stress (σ co ) was assumed to occur at 50% of the ultimate compression strength (σ cu ) [17,21]. The tensile failure stress (σ to ) values were calculated as 0.1 σ co [17,21,36].
The compressive equivalent plastic strain (ε pl c ) was calculated as the total strain minus the elastic strain (ε el c ). The nonlinear stress-strain curve for each water content was input into the FE analysis as a set of stress-strain values that describe the shape of the curve. To complete the description of the yield surface of the CDP model, the values of the dilation angle (ψ), the flow potential eccentricity (∈), the relation between the initial equibiaxial compressive yield stress to the initial uniaxial compressive yield stress (f bo /f co ), and a parameter that defines the shape of the yield surface in the deviatoric plane (K) are 1 degree, 0.1, 1.16, and 2/3, respectively [17,21].   [21,31].
In the numerical model, the moist region is defined as a set of nodes in an area with dimensions L, H, and t (length, height, and thickness, respectively). This node set (moist region) is assumed to have higher gravimetric water content (w), and consequently weaker material properties, than the rest of the dry adobe in the wall. Table 1 contains the values of the initial compressive failure stress (σco) and the tensile failure stress (σto) of adobe relative to w of the adobe soil. Two scenarios for the soil conditions in the moist region are considered; one scenario with w = 7-9% (referred to as partially saturated condition or Scenario wps) and another scenario with w = 13-15% (referred to as nearly saturated condition or Scenario wns). The rest of the wall is assigned nearly dry conditions with w = 1-3%. Even though the strength properties may vary among adobe bricks (depending on characteristics of the soil constituents, proportions, regional brick making techniques, etc.), the values of dry adobe used in the study are reasonable and within the expected range for the adobe of the In uniaxial tension, the tensile failure is assumed to have no strain softening, i.e., brittle failure. Strain softening in tension could not be considered due to the lack of literature and data about the post cracking behaviour of adobe at various values of water content. To address this limitation, a small out-of-plane lateral displacement is applied to the wall in small increments to allow for the comparison of pushover curves. The elastic parameters of the adobe material were adopted from the literature [17]. The elastic modulus (E), Poisson's ratio (υ), and density (ρ) of the adobe material are 200 MPa, 0.2, and 1700 kg/m 3 , respectively. The top and bottom beams are assigned the properties of concrete: E = 22,000 MPa, υ = 0.2, and ρ = 2500 kg/m 3 . The input material parameters for the elastic range are assumed to be constant regardless of any change in water content (w). From preliminary analyses, it was found that changing E as a function of water content does not affect significantly the results or the lateral behaviour of the wall as long as reasonable E values for the dry adobe are used. A similar observation was made for υ. Regarding the moist adobe, the elastic range is very narrow and the selection of the E value in the analyses does not affect the results at relatively large deformations.
In the numerical model, the moist region is defined as a set of nodes in an area with dimensions L, H, and t (length, height, and thickness, respectively). This node set (moist region) is assumed to have higher gravimetric water content (w), and consequently weaker material properties, than the rest of the dry adobe in the wall. Table 1 contains the values of the initial compressive failure stress (σ co ) and the tensile failure stress (σ to ) of adobe relative to w of the adobe soil. Two scenarios for the soil conditions in the moist region are considered; one scenario with w = 7-9% (referred to as partially saturated condition or Scenario w ps ) and another scenario with w = 13-15% (referred to as nearly saturated condition or Scenario w ns ). The rest of the wall is assigned nearly dry conditions with w = 1-3%. Even though the strength properties may vary among adobe bricks (depending on characteristics of the soil constituents, proportions, regional brick making techniques, etc.), the values of dry adobe used in the study are reasonable and within the expected range for the adobe of the Southwestern United States and meet the New Mexico Earthen Building Materials Code [7] requirements. Lower compressive and tensile strengths of the moist adobe are associated with changes in the consistency of the clay fraction in the adobe bricks and mortar as a function of the water content. As the water content of the adobe increases, the matric suction of the soil drops considerably leading to substantial loss of compressive, tensile and shear strengths [9]. For the adobe soil used in this study, the matric suction decreases with increasing water content from 30 MPa at w = 2.8% to 0.5 MPa at w = 8.0%, and to 0.03 MPa at w = 15.0% [9].

Numerical Analysis
The FE analysis included four loading steps as described in detail by Al Aqtash et al. [21]. The wall self-weight was applied in the first step. In the second step, 12-kN/m (800-lb/ft) uniformly distributed (vertical) load was applied along the bond beam, representing a typical downward thrust from the roof of a one-story adobe house ( Figure 1). The initial material properties of the wall were those of the adobe in nearly dry condition. In the third step, the material properties of the node set in the moist region were changed to those of Scenario w ps or w ns , whereas the rest of the wall maintained the material properties of dry adobe. In the last step, 70-mm out-of-plane lateral displacement was progressively imposed on the bond beam ( Figure 1).
Graphs of out-of-plane lateral force vs. lateral displacement (i.e., pushover curves) were prepared for all the cases. For a given wall thickness, the pushover curves of walls with a moist region along the base were compared to the pushover curve of the dry wall. The out-of-plane lateral strength of the adobe walls was defined as the maximum lateral force corresponding to a 70 mm displacement (applied along the top of the wall). When the FE analysis did not converge before reaching 70 mm, the last value of the applied lateral force was used. For the purposes of this paper, the wall behaviour up to the peak lateral force was considered. It was assumed that plastic strains appeared at the locations of tensile failure, i.e., wall cracks. Thus, the initiation and propagation of tensile cracks on the main wall face was observed by mapping contours of the maximum in-plane plastic strains [19,21].
The stability of the walls was investigated by relating the top wall displacement (d) when the maximum in-plane plastic strains appeared to the slenderness ratio. The effect of wall thickness and length and water content of the moist region on the wall ability to carry out-of-plane lateral loads was determined using the strength reduction ratio (R), defined as the ratio of the lateral strength of a wall with a moist region to the lateral strength of the dry wall with the same thickness. Additional strength assessment was conducted by comparing the out-of-plane lateral strength to the typical roof thrust on a one-story adobe building. The horizontal out-of-plane uniformly distributed load (i.e., roof thrust from a light wood frame) was estimated at approximately 0.75 kN/m (50 lb/ft) [31]. For the 7-m long wall model used in this research, the out-of-plane lateral force induced by the roof thrust was considered as 5 kN.

Dry Walls
The pushover curves of the dry walls for the three values of wall thickness (t) are presented in Figure 3. These curves have an initial linear portion and become nonlinear with further lateral displacement (d). The pushover curves for t = 25 cm, 30 cm, and 40 cm are linear up to d corresponding to 45%, 35%, and 23% of the prescribed maximum displacement, respectively. The maximum lateral force, a representation of the lateral strength, is 3.7 kN, 6.7 kN, and 14.3 kN when the wall thickness is t = 25 cm, 30 cm, and 40 cm, respectively.
Infrastructures 2020, 5, x FOR PEER REVIEW 6 of 13 (applied along the top of the wall). When the FE analysis did not converge before reaching 70 mm, the last value of the applied lateral force was used. For the purposes of this paper, the wall behaviour up to the peak lateral force was considered. It was assumed that plastic strains appeared at the locations of tensile failure, i.e., wall cracks. Thus, the initiation and propagation of tensile cracks on the main wall face was observed by mapping contours of the maximum in-plane plastic strains [19,21]. The stability of the walls was investigated by relating the top wall displacement (d) when the maximum in-plane plastic strains appeared to the slenderness ratio. The effect of wall thickness and length and water content of the moist region on the wall ability to carry out-of-plane lateral loads was determined using the strength reduction ratio (R), defined as the ratio of the lateral strength of a wall with a moist region to the lateral strength of the dry wall with the same thickness. Additional strength assessment was conducted by comparing the out-of-plane lateral strength to the typical roof thrust on a one-story adobe building. The horizontal out-of-plane uniformly distributed load (i.e., roof thrust from a light wood frame) was estimated at approximately 0.75 kN/m (50 lb/ft) [31]. For the 7-m long wall model used in this research, the out-of-plane lateral force induced by the roof thrust was considered as 5 kN.

Dry Walls
The pushover curves of the dry walls for the three values of wall thickness (t) are presented in Figure 3. These curves have an initial linear portion and become nonlinear with further lateral displacement (d). The pushover curves for t = 25 cm, 30 cm, and 40 cm are linear up to d corresponding to 45%, 35%, and 23% of the prescribed maximum displacement, respectively. The maximum lateral force, a representation of the lateral strength, is 3.7 kN, 6.7 kN, and 14.3 kN when the wall thickness is t = 25 cm, 30 cm, and 40 cm, respectively. The maximum in-plane plastic strain distribution in the dry walls is presented in Figure 4. The plastic strain contours correspond to the largest applied lateral displacement in each case and are shown for the inside wall face (same side on which the out-of-plane force is applied) and for a wall cross section at the wall edge. The first appearance of in-plane plastic strains is along the wall base (i.e., immediately about the footing). The lateral displacement at which the pushover curve changes from linear to nonlinear in Figure 3 corresponds to the initial appearance of plastic strains. With additional lateral displacement, the plastic strain contours extend upward on the wall, but no more than about one third of the wall height. The moment about the fixed wall base generated by the applied displacement causes the interior face of the wall (on which the displacement is applied) to be The maximum in-plane plastic strain distribution in the dry walls is presented in Figure 4. The plastic strain contours correspond to the largest applied lateral displacement in each case and are shown for the inside wall face (same side on which the out-of-plane force is applied) and for a wall cross section at the wall edge. The first appearance of in-plane plastic strains is along the wall base (i.e., immediately about the footing). The lateral displacement at which the pushover curve changes from linear to nonlinear in Figure 3 corresponds to the initial appearance of plastic strains. With additional lateral displacement, the plastic strain contours extend upward on the wall, but no more than about one third of the wall height. The moment about the fixed wall base generated by the applied displacement causes the interior face of the wall (on which the displacement is applied) to be in tension and the outer face to be in compression. The plastic strains appear in the interior wall face, which is in tension.

Walls with a Moist Region along the Base
The pushover curves for Scenarios w ps and w ns are shown in Figures 5 and 6, respectively, to illustrate trends and general observations. These figures compare the out-of-plane pushover curves in terms of L for the three wall thicknesses. The curves follow a similar trend as the dry curves including an initial linear portion followed by a nonlinear part. For a given t, the displacement d at which the curves turn nonlinear is smaller when the wall has a moist region compared to the dry wall. The values of the out-of-plane lateral strength obtained from the pushover curves for all cases analyzed are provided in Table 2.
Infrastructures 2020, 5, x FOR PEER REVIEW 7 of 13 in tension and the outer face to be in compression. The plastic strains appear in the interior wall face, which is in tension.

Walls with a Moist Region along the Base
The pushover curves for Scenarios wps and wns are shown in Figures 5 and 6, respectively, to illustrate trends and general observations. These figures compare the out-of-plane pushover curves in terms of L for the three wall thicknesses. The curves follow a similar trend as the dry curves including an initial linear portion followed by a nonlinear part. For a given t, the displacement d at which the curves turn nonlinear is smaller when the wall has a moist region compared to the dry wall. The values of the out-of-plane lateral strength obtained from the pushover curves for all cases analyzed are provided in Table 2.       Figure 7 presents the maximum in-plane plastic strain distribution for the three wall thicknesses and Scenario wps at the end of loading for different L values. Location of the moist region is shown with dashed lines. This figure shows the interior face of the wall on which the out-of-plane displacement is applied (i.e., face in tension). The in-plane plastic strains in Scenario wps do not appear on the wall during the first three loading steps of the FE analysis. As the lateral displacement is progressively applied, in-plane plastic strains develop initially within the moist region regardless of the wall thickness. As loading continues, the zone of in-plane plastic strains spreads throughout the moist region and extends laterally to the dry adobe along the wall base, starting from the inner end of the moist region towards the wall edge. The plastic strain zone does not propagate upward to the    Figure 7 presents the maximum in-plane plastic strain distribution for the three wall thicknesses and Scenario wps at the end of loading for different L values. Location of the moist region is shown with dashed lines. This figure shows the interior face of the wall on which the out-of-plane displacement is applied (i.e., face in tension). The in-plane plastic strains in Scenario wps do not appear on the wall during the first three loading steps of the FE analysis. As the lateral displacement is progressively applied, in-plane plastic strains develop initially within the moist region regardless of the wall thickness. As loading continues, the zone of in-plane plastic strains spreads throughout the moist region and extends laterally to the dry adobe along the wall base, starting from the inner end of the moist region towards the wall edge. The plastic strain zone does not propagate upward to the   Figure 7 presents the maximum in-plane plastic strain distribution for the three wall thicknesses and Scenario w ps at the end of loading for different L values. Location of the moist region is shown with dashed lines. This figure shows the interior face of the wall on which the out-of-plane displacement is applied (i.e., face in tension). The in-plane plastic strains in Scenario w ps do not appear on the wall during the first three loading steps of the FE analysis. As the lateral displacement is progressively applied, in-plane plastic strains develop initially within the moist region regardless of the wall thickness. As loading continues, the zone of in-plane plastic strains spreads throughout the moist region and extends laterally to the dry adobe along the wall base, starting from the inner end of the moist region towards the wall edge. The plastic strain zone does not propagate upward to the dry adobe above the moist region. In some cases of Scenario w ns , the maximum in-plane plastic strains appear in the moist region due to the wall self-weight and the vertical component of the roof thrust, before any out-of-plane lateral displacement is applied, particularly for L = 300 cm or greater, when t = 25 cm and 30 cm. This observation indicates that the adobe material with high water content (near saturation) is very weak and cannot contribute considerably to carrying lateral loading in the subsequent step of the FE analysis.
Infrastructures 2020, 5, x FOR PEER REVIEW 9 of 13 dry adobe above the moist region. In some cases of Scenario wns, the maximum in-plane plastic strains appear in the moist region due to the wall self-weight and the vertical component of the roof thrust, before any out-of-plane lateral displacement is applied, particularly for L = 300 cm or greater, when t = 25 cm and 30 cm. This observation indicates that the adobe material with high water content (near saturation) is very weak and cannot contribute considerably to carrying lateral loading in the subsequent step of the FE analysis.

Discussion
Adobe masonry walls rely on their own weight to resist lateral loads. The thicker the wall, the larger the load required to cause overturning about its base. In Figure 3, the out-of-plane lateral strength of the dry wall with t = 40 cm is more than twice the lateral strength of the dry wall with t = 30 cm and almost four times the lateral strength of the dry wall with t = 25 cm. Because the wall selfweight is the major contributor to the resistance against the overturning moments, a relatively small increase in the wall thickness increases the wall weight significantly. The point at which the top displacement of the wall starts to increase nonlinearly occurs at a smaller d for the thicker walls. At this transition point, the initial signs of cracking start to appear at the location of tensile stresses on the wall face. This observation provides an indication of the effect of the slenderness ratio on the wall stability. The thinnest wall (t = 25 cm), which has the highest slenderness ratio (Sl = 12), reaches the maximum lateral strength (peak strength) shortly after the initial signs of cracking. Beyond that point, the top displacement of the wall continues to increase without any increase of the lateral force ( Figure  3). In practice, a slender wall with a moist region at the base might undergo sustained plastic displacement with no further change in the loading, so shoring the wall prior to any work in the site is necessary. On the other hand, 70% of the wall maximum strength (peak strength) of the thickest wall (t = 40 cm), which has the lowest slenderness ratio (Sl = 7.5), is reached before the first signs of cracking. With additional displacement, the lateral capacity of the wall continues to increase nonlinearly until the maximum lateral strength is reached (Figure 3). This explains why many

Discussion
Adobe masonry walls rely on their own weight to resist lateral loads. The thicker the wall, the larger the load required to cause overturning about its base. In Figure 3, the out-of-plane lateral strength of the dry wall with t = 40 cm is more than twice the lateral strength of the dry wall with t = 30 cm and almost four times the lateral strength of the dry wall with t = 25 cm. Because the wall self-weight is the major contributor to the resistance against the overturning moments, a relatively small increase in the wall thickness increases the wall weight significantly. The point at which the top displacement of the wall starts to increase nonlinearly occurs at a smaller d for the thicker walls. At this transition point, the initial signs of cracking start to appear at the location of tensile stresses on the wall face. This observation provides an indication of the effect of the slenderness ratio on the wall stability. The thinnest wall (t = 25 cm), which has the highest slenderness ratio (S l = 12), reaches the maximum lateral strength (peak strength) shortly after the initial signs of cracking. Beyond that point, the top displacement of the wall continues to increase without any increase of the lateral force ( Figure 3). In practice, a slender wall with a moist region at the base might undergo sustained plastic displacement with no further change in the loading, so shoring the wall prior to any work in the site is necessary. On the other hand, 70% of the wall maximum strength (peak strength) of the thickest wall (t = 40 cm), which has the lowest slenderness ratio (S l = 7.5), is reached before the first signs of cracking. With additional displacement, the lateral capacity of the wall continues to increase nonlinearly until the maximum lateral strength is reached (Figure 3). This explains why many historical adobe buildings with considerably thick walls are still standing despite having a partially damaged base.
When considering Scenario w ps (Figure 5), the difference between the dry lateral strength and the lateral strength of the walls with a moist region is greater for t = 40 cm. For example, when t = 40 cm, there is a 6.6-kN drop in the out-of-plane lateral strength between the dry wall and the wall with L = 700 cm. For t = 30 cm and t = 25 cm, the drop in out-of-plane lateral strength is 3.5 kN and 1.9 kN, respectively. The pushover curves for all the cases with t = 40 cm show a greater out-of-plane lateral strength than for the cases with t = 30 cm and 25 cm even when dry. This observation is consistent with the practice of constructing thick adobe walls even though the local building code [7] may allow the construction of a wall with t = 25 cm. The results on this work are for free-standing walls separated from their lateral support. For any given case, the difference between the dry lateral strength and the lateral strength of the wall with a moist region in Scenario w ns (see Figure 6) is greater than in Scenario w ps ( Figure 5). These results confirm that the wall thickness is a key factor when designing adobe buildings or assessing their structural stability. Adobe walls are more likely to perform better structurally over time if they have enough mass to support out-of-plane loading despite having moisture damage along their base and compromised support conditions.
The pushover curves of the adobe walls with t = 40 cm for Scenarios w ns and w ps are compared in Figure 8. For a given L value, the curves for Scenario w ps are lower than those for Scenario w ns . When L = 100 cm, the curves for these two cases are close to the pushover curve of the dry wall, indicating that a short moist region has a limited effect on the out-of-plane lateral strength of the wall. As the length of the moist region extends, the difference between the curve of the dry wall and the curve of the wall with a moist region becomes larger (i.e., the wall becomes weaker). Similar observations are noted for t = 30 cm and 25 cm.

Conclusions
This paper studied the strength and stability of adobe walls under static, out-of-plane lateral  Table 3 shows the strength reduction ratio (R) for the cases considered. An R value lower than but close to 1.0 means that the out-of-plane lateral strength of the wall is not considerably affected by the presence of a moist region along the base compared to the strength of a dry wall. This ratio is 0.90 or higher for all the cases with L = 100 cm. This is due to stress arching and load redistribution over the relatively short, weak (moist) region. The R values for all the cases in Scenario w ps are approximately 0.50 or greater. However, the R values for the cases with L = 300 cm or greater in Scenario w ns are lower than 0.50 regardless of the wall thickness. The strength reduction ratio R is practically zero for the cases in Scenario w ns when L = 700 cm for any t, which means that a free-standing adobe wall with a nearly saturated base is highly unstable and will collapse under the slightest lateral force. By comparing the strength values in Table 2 to the lateral roof thrust of 5 kN considered in this study, it is possible to identify the cases at which the wall will be able to support or resist that load. A cantilever adobe wall with t = 25 cm cannot support a 5 kN lateral load even if the wall is dry (i.e., case when L = 0). A cantilever adobe wall with t = 30 cm can support this roof thrust when L = 100 cm regardless of the water content of the moist region. On the other hand, a cantilever adobe wall with t = 40 cm can support the 5 kN roof thrust, even if the moist region extends the full length of the wall (L = 700 cm) and the adobe material is partially saturated (Scenario w ps ) or if L is no longer than 500 cm and the adobe material is nearly saturated (Scenario w ns ). In the construction of contemporary adobe structures, which are likely to be designed with relatively high slenderness ratio, attention should be placed in securing and maintaining the connection between perpendicular walls as they may not be able to carry the horizontal roof thrust if they become structurally disconnected from adjacent walls and a region along the base has a relatively high water content.

Conclusions
This paper studied the strength and stability of adobe walls under static, out-of-plane lateral load from the roof thrust of a light wood frame when the walls have a moist region along the base. The results showed that greater wall thickness provides sufficient out-of-plane stability despite having a moist region along the base. A moist region shorter than approximately one fourth of the wall length is not critical for the wall stability and has a relatively small effect on the out-of-plane lateral strength. When the moist region is longer, the out-of-plane lateral strength drops to 80% or lower compared to a dry wall with the same thickness. Because the walls of contemporary adobe buildings are likely to be constructed with a large slenderness ratio, caution is advised when a wall starts to lose support and detach from the rest of the structure.
While the strength properties may vary among adobe bricks depending on the characteristics of the soil constituents, proportions, or other factors related to fabrication, the values used in the study are reasonable and within the expected range for adobe in the Southwestern United States. The results help to stress the importance of maintaining proper surface drainage conditions around adobe structures and preventing the accumulation of moisture within the walls. Future studies will consider the variability of moisture across the wall thickness and boundary conditions of the walls to represent lateral support.