1 6. NON-LINEAR PSEUDO-STATIC ANALYSIS OF ADOBE WALLS Blondet et al.  carried out a cyclic test on an adobe wall to reproduce its seismic response and damage pattern under in-plane loads. The displacement was applied at the wall top and at low increments to simulate a static analysis. In this work, the forcedisplacement curve obtained from the experimental test (see section 3.5) is used for calibrating preliminary numerical models of adobe walls. In the previous chapter different modelling approaches were described for representing the cracking and damage on concrete panels, which are also applied to masonry panels, including adobe walls. These numerical approaches are divided into discrete and continuum models. The numerical modelling of adobe structures is not simple since there is scarce material information, especially the fracture energy in tension and compression for modelling the inelastic behaviour of the adobe material. In this chapter three finite element models of adobe walls are built and, taking advantage of the experimental results shown in 3.5.3, the adobe material properties are calibrated within a discrete and continuum approach. For this, two finite element programmes are used: Midas FEA and Abaqus/Standard. In Midas FEA two finite element models are built, one following the simplified micromodelling (discrete model) and the second one following a smeared crack model (continuum model). The adobe wall built in Abaqus/Standard is modelled following a damaged plasticity model (continuum model). The finite element models are solved following an implicit solution, and are described as follows. 6.1 IMPLICIT SOLUTION METHOD FOR SOLVING QUASI-STATIC PROBLEMS In this analysis the load/displacement is applied slowly to the body so that the inertial forces can be neglected (acceleration and velocities are zero). It follows that the internal forces I (for a given displacement u ) must be equal to the external forces P at each time step t, or the residual force R u must be: R u =P I u (6.1)
2 114 Sabino Nicola Tarque Ruíz Amongst the different solution procedures used in the implicit finite element solvers, the Newton-Raphson solution procedure is the faster for solving non-linear problems under force control, though no convergence procedure is full proof. When solving quasi static problems, a set of non-linear equations are generally expressed as: T T dv ds G u B u N t (6.2) V S where G is a set of non-linear equations in u which are updated at each iteration and are function of the nodal displacements vector u, is the stresses vector, B is the matrix that relates the strain vector to the displacements, N is the matrix of element shape functions and t is the surface traction vector. V and S represent the volume and surface of the body, respectively. The right-hand side of Equation (6.2) represents the difference between the internal and the external forces. Equation (6.2) is solved for a displacement vector that equilibrates the internal and external forces, as explained in Harewood and McHugh . Besides, Equation (6.2) is solved by incremental methods where load/displacements are applied in time steps t, t t t u is solved from a known state u. In the following equations the subscript denotes iteration number and the superscript denotes an increment step. At the beginning, Equation (6.2) is solved to obtain the displacement correction u t t based on information of u as: u 1 t t G u t t i t t i 1 G ui u i The partial derivative on the right side is the so called Jacobian matrix expressed as the global stiffness matrix K : G u t t Ktan u i = u Thus: t t i tan 1 t t i 1 (6.3) (6.4) t t t t t t ui 1 K tan ui G u i (6.5) The previous equation involves the inversion of the global stiffness matrix (no singular matrix), which ends in a computationally expensive operation, but it ensures that a
3 Numerical modelling of the seismic behaviour of adobe buildings 115 relatively large time increment can be used while maintaining the accuracy of the solution [Harewood and McHugh 27]. t t The displacement correction u i 1 is added to the previous state, thus the improved displacement solution is given with Equation (6.6) and t u t i 1 is used as the current approximation to the solution for the subsequent iteration i 1 in Equation (6.2) until the equilibrium is reached. t t t t t t i 1 i i 1 u u u (6.6) Convergence is measured by ensuring that the difference between external and internal forces G u, displacement increment t t t t ui and displacement correction u i 1 are sufficiently small. The difference between several solution procedures is the way in which t t u i 1 is determined. As an example, Figure 6.1 shows the iteration process followed with a Newton-Rapshon procedure for reaching equilibrium in an implicit procedure. Figure 6.1. Iteration process for an implicit solution. 6.2 DISCRETE MODEL: MODELLING THE PUSHOVER RESPONSE OF AN ADOBE WALL The adobe wall I-1 presented in section 3.5 is now created in Midas FEA using linear elastic solid elements and zero thickness non-linear interface elements. The combined cracking-shearing-crushing model is used for representing the non-linear behaviour of the adobe masonry lumped at the mortar joints. A total of 17 courses were placed to model the adobe masonry. The original dimensions of the unit bricks (.3x.13x.1 and.22x.13x.1 m) were extended to take into account the 13~15 mm mortar thickness. The model includes the reinforced concrete beams (at the top: crown beam, and at the base: foundation), the adobe walls and the timber lintel. The base of the foundation is
4 116 Sabino Nicola Tarque Ruíz fully fixed; the top part of the crown beam is free. Since the test was displacement controlled, in the numerical model a monotonic top-displacement was applied at one vertical edge of the top concrete beam to reach a maximum displacement of 1 mm. The material parameters are changed in the finite element model for a parametric study. a) Complete model view b) View of the interface elements c) In red the applied loads Figure 6.2. Finite element model of the adobe wall subjected to horizontal displacement loads at the top. Discrete model, Midas FEA. The configuration of the numerical model in Midas FEA is shown in Figure All the solid elements are considered elastic and isotropic and the properties, shown in Table 6.1,
5 Numerical modelling of the seismic behaviour of adobe buildings 117 are taken from the published literature, where E is the elasticity modulus, is the Poisson s ratio, and m is the weight density. Figure 6.2b shows the interface elements, placed around each adobe brick for the in-plane wall, and just at the top and bottom part of the adobe courses in the transverse walls. This assumption intends to simulate horizontal failure planes in the transverse walls. The interface layers that join the adobe blocks with the top reinforced concrete beam are numerically stiffer than the mud mortar joints; this was considered to avoid sliding between the concrete beam and the adobe layer. Figure 6.2c shows the point where the displacement load is applied. Table 6.1. Elastic material properties of the adobe blocks, concrete and timber materials. Adobe blocks Concrete Timber E (MPa) m (N/mm 3 ) E (MPa) m (N/mm 3 ) E (MPa) m (N/mm 3 ) e e e-6 The parameters calibration is done by comparing the numerical pushover curve with the experimental envelope of the cyclic Force-Displacement curve (Figure 3.15). First, the elastic behaviour is calibrated and then the inelastic behaviour is discussed Calibration of material properties As previously mentioned, a complete database of material properties of the adobe typically used in Peru is not available. The scarce available data refers to compression strength and elastic properties only (e.g. elasticity modulus). The lack of available data for defining the inelastic properties of adobe, such as fracture energy in compression and tension, introduces large uncertainties in the analyses. A correct parametric study should vary all parameters (material properties) at the same time to look for a non-linear optimization. However, since in this work it is not possible due to the lack of information about the material properties, here each parameter is varied one by one, and the best value is determined by comparison between the numerical with the experimental pushover curve. The values of the elasticity modulus for the adobe masonry (bricks plus mortar joints) vary from 17 to 26 MPa, as discussed in section 3.2. The first parameters to define a discrete model are the penalty stiffness k n and k s, which refers to normal and shear stiffness at the interface, respectively [CUR 1997; Lourenço 1996]. These penalty values
6 118 Sabino Nicola Tarque Ruíz are related to the elasticity modulus of the adobe bricks and the mortar joints, as seen in Equation (5.12) and repeated here for convenience: k n Eunit Emortar h E E mortar unit mortar (6.7) Gunit Gmortar ks h G G mortar unit mortar The elasticity modulus for the adobe bricks is assumed as 23 MPa. The elasticity modulus for the mud mortar is assumed to be lower than that for bricks, so values of 79.25, 113.4, and MPa are considered. The mortar thickness, h mortar, is around 15 mm. The shear modulus is taken as.4e for the bricks and mortar. With the previous values, the normal and shear penalty stiffness are those reported in Table 6.2. (6.8) Table 6.2. Variation of elasticity of module of the mortar joints for evaluating the penalty stiffness kn and k s. Mortar thickness (mm) E unit (MPa) E mortar (MPa) k n (N/mm 3 ) K s (N/mm 3 ) In order to make the first round of analysis the other material parameters are defined as shown in Table 6.3, where c is the cohesion, o is the friction angle, is the dilatancy I angle, r is the residual friction angle, f t is the tensile strength, G f is the fracture energy for Mode I (related to tension softening), a and b are factors to evaluate the fracture energy for Mode II (expressed as G II = a. +b and related to the shear behaviour), f c is the compression strength C s is the shear tension contribution factor (which is 9 according to Lourenço , G c f is the compressive fracture energy, and k p is the peak equivalent plastic relative displacement. The parameters which are marked with * are considered for the parametric study and those shown in Table 6.3 are the ones which gave good numerical experimental agreementin terms of global response of the masonry wall (as discussed later). The inelastic parameter values for the adobe masonry were taken lower than those proposed by Lourenço  for clay masonry based on the clear difference in the material strengths.
7 Numerical modelling of the seismic behaviour of adobe buildings 119 k n (N/mm 3 ) Table 6.3. Preliminary material properties for the interface model (mortar joints). k t (N/mm 3 ) Structural c (N/mm 2 ) o (deg) (deg) r (deg) f t (N/mm 2 )* Mode I I G (N/mm)* f Table 6.2 Table Mode II Compression cap a (mm)* b (N/mm)* f c (N/mm 2 ) * C s c G f (N/mm) * k p (mm) * Figure 6.3 shows the effect of the variation of the penalty stiffness k n and k s, on the elastic part of the pushover curve. As expected, it is seen that greater stiffness values lead to a stiffer elastic response. It is shown that k n 8.9 N / mm 3 and ks 3.23 N / mm 3, which implies E unit 23MPa and Emortar 8MPa, match well the elastic part and the yielding initiation, so these values were fixed to analyze other material parameters. 1 8 Force (kn) 6 4 kn= 8.9 N/mm³ 2 kn= N/mm³ kn= N/mm³ Kn= 253 N/mm³ Figure 6.3. Comparison of the pushover curves in models with different penalty stiffness values. Discrete model, Midas FEA. Variation of the tensile strength f t of the mortar was applied to study its influence. The fracture energy G I f was maintained constant and equal to.8 N/mm, which means that the area under each curve is the same. So, due to this the crack displacement values related to low tensile strength are greater than those related to high tensile strength (e.g. see curve for f t.25 MPa in Figure 6.4). The crack displacement is equal to the tensile strain times the thickness of the mortar joint, which is around 15 mm.
8 12 Sabino Nicola Tarque Ruíz.25 Tensile strength (MPa) ft=.2 MPa ft=.1 MPa ft=.5 MPa ft=.25 MPa Crack displacement (mm) Figure 6.4. Variation of tensile strength I f t at the interface, G f =.8 N/mm, h = 15 mm. Figure 6.5 shows the pushover curves obtained varying the tensile strength. It is seen that the global strength of the masonry depends on the crack displacement values in the tensile softening. Greater crack displacement values produce an increment on the lateral resistant at the post-yield peak. When the tensile softening part of the constitutive law descends abruptly (e.g. f t.2 MPa curve in Figure 6.4), the pushover curve stops after the yielding initiation because can not develop greater crack displacements. All the curves, except the one related to f t.2 MPa, have similar global behaviour until 6.2 mm top horizontal displacement. The best match in terms of crack pattern and strength was obtained with f t.1mpa Force (kn) ft=.25 MPa ft=.5 MPa ft=.1 MPa ft=.2 MPa Figure 6.5. Influence of the tensile strength t f on the pushover response. Discrete model, Midas FEA.
9 Numerical modelling of the seismic behaviour of adobe buildings Tensile strength (MPa) Gf=.15 N/mm Gf=.8 N/mm Gf=.5 N/mm Crack displacement (mm) Figure 6.6. Variation of fracture energy I G f at the interface, constant f t =.1 MPa, h = 15 mm. I A variation of the tensile fracture energy G f with constant tensile strength f t was applied to study its influence on the structural response (Figure 6.6). An increment in the lateral strength of the masonry is observed when the tensile softening part does not descent abruptly (Figure 6.7), which partially concludes that greater fracture energy gives greater lateral strength Force (kn) Gf=.5 N/mm Gf=.8 N/mm Gf=.15 N/mm Figure 6.7. Influence of the fracture energy in tension model, Midas FEA. G I f on the pushover response. Discrete A parametric study was carried out for evaluating the compression strength of adobe masonry. Since there is not information about the inelastic material properties, a hardening/softening curve was assumed taking into consideration the experimental results obtained by Lourenço  for clay masonry. It was decided to keep the peak compressive plastic strain around.8 mm/mm (related to k p =.9 mm) and the
10 122 Sabino Nicola Tarque Ruíz compression fracture energy of.2 N/mm, which can be considered a lower bound value of a real compression curve (Figure 6.8)..5 Compression strength (MPa) fc=.4 MPa fc=.25 MPa fc=.15 MPa Strain (mm/mm) C Figure 6.8. Variation of compression strength f c at the interface, constant G f =.2 N/mm, h = 115 mm. The characteristic length for compression, h, is taken as twice the half height of the adobe brick plus the mortar thickness, that is 115 mm. The relative peak compressive displacement, k p, is taken as the peak compressive plastic strain times the characteristic length. Figure 6.9 shows that the failure process does not depend at all on the compression behaviour Force (kn) fc=.15 MPa fc=.25 MPa fc=.4 MPa Figure 6.9. Influence of compression strength FEA. f c on the pushover response. Discrete model, Midas A variation of the compressive fracture energy G was applied to analyze its effect on the global response of the masonry wall. The plastic strain related to f c.25 MPa was kept f c
11 Numerical modelling of the seismic behaviour of adobe buildings 123 constant and close to.8 mm/mm (Figure 6.1). The results showed no variation on the global response of the adobe wall..3 Compression strength (MPa) Gf=.4 N/mm Gf=.2 N/mm Gf=.1 N/mm Figure 6.1. Variation of fracture energy Strain (mm/mm) C G f at the interface, constant f c =.25 MPa, h = 115 mm. Another studied considered a variation of the relative peak compressive displacement k p while maintaining a constant compressive strength (f c.25 MPa ) and a constant f compressive fracture energy ( Gc.2 N / mm), Figure As in the previous cases, no variation in the global response was observed. The pushover curves were more or less the same as the one shown in Figure Compression strength (MPa).2.1 kp=.9 mm kp=.15 mm kp=.2 mm Figure Variation of N/mm, h = 115 mm. Strain (mm/mm) k p for the compression curve, constant f c =.25 MPa and G C f =.2 It should be said that Midas FEA does not include the compression cap model when the analysis refers to a three-dimensional interface model. For this reason no variation on the pushover curves was observed varying the compression strength. Finally, a variation of
12 124 Sabino Nicola Tarque Ruíz the Mode II fracture energy (shear) was applied to study the shear behaviour of the mortar joints. The dilatation angle was assumed zero. In this case it seems that fracture energy of.1 N/mm can be considered for adobe masonry (Figure 6.12) Force (kn) Gf=.2 N/mm Gf=.5 N/mm Gf=.1 N/mm II Figure Influence of the shear fracture energy G f at the interface on the pushover response. Discrete model, Midas FEA Results of the pushover analysis considering a discrete model Figure 6.13 shows the sequence of damage obtained with the selected parameters specified in Table 6.1 and Table 6.3. The crack pattern follows the experimental results: the cracks go from the top left (where the load is applied) to the right bottom of the wall (Figure 6.14). Also, the horizontal cracks at the transversal walls are in agreement with the experimental results. Since the load applied is monotonic, the FE model cannot capture the X-shape failure observed in the experiments. The presence of the two transversal walls prevents the rocking behaviour, representing correctly the tested wall. The maximum displacement reached at the top of the wall is around 6.2 mm, after which the program stopped due to convergence problems. The comparison between the numerical and experimental pushover response is shown in Figure 6.15.
13 Numerical modelling of the seismic behaviour of adobe buildings 125 a) Top displacement= 1 mm b) Top displacement= 2 mm c) Top displacement= 4 mm d) Top displacement= 6.26 mm Figure Damage pattern of the adobe wall subjected to a horizontal top displacement. e) Top displacement= 6.26 mm, isometric view Figure Continuation. Damage pattern of the adobe wall subjected to a horizontal top displacement. Discrete model, Midas FEA.
14 126 Sabino Nicola Tarque Ruíz Figure damage pattern for wall I-1 due to cyclic displacements applied at the top. Just the adobe wall is shown here, the concrete beam and foundation are hidden, [Blondet et al. 25]. All the models were run in Midas FEA with arc-length method with initial stiffness. The number of load steps was specified as 1, the initial load factor was.1, and the maximum number of iteration per load step was 3. The convergence criteria were given by an energy norm and displacement norm of Force (kn) Numerical Figure Load-displacement diagrams, experimental and numerical. 6.3 TOTAL-STRAIN MODEL: MODELLING THE PUSHOVER RESPONSE In this part the adobe wall tested by Blondet et al.  is modelled using a continuum approach. A plane stress finite element model is created in Midas FEA using 4-node rectangular shell elements (Figure 6.16a) and considering drilling DOFs and transverse shear deformation. The size of the mesh is usually kept at 1 x 1 mm, which is related to a characteristic length dimension h= mm, obtained from the square root of the area of the shell element [Bažant and Oh 1983]. The thickness of the shell is 3 mm. The adobe masonry includes the adobe bricks and the mud mortar joints; in this case, a
15 Numerical modelling of the seismic behaviour of adobe buildings 127 homogeneous material is assumed and the cracks are smeared into the continuum. The top and bottom reinforced concrete beams and the timber lintel are considered elastic. The foundation is fully fixed at the base. The crown beam is at the top. The numerical model is subjected to a unidirectional displacement imposed at the two ends of the top crown beam (Figure 6.16b). The elastic material properties for the concrete beam and the timber lintel are given in Table 6.1, while those for the adobe masonry are specified in Table 6.4. The material properties marked with * are calibrated based on the experimental pushover curve as discussed later. Equation (5.29) and Equation (5.37) are considered for computing the inelastic part of the tension and compression constitutive law, respectively. i is taken greater than f c /3 to maintain a parabolic shape of the compression curve. E (N/mm 2 )* Table 6.4. Material properties for the adobe masonry within total-strain model. Elastic Tension Compression m (N/mm 3 ) h (mm) f t (N/mm 2 )* I G f (N/mm)* f c (N/mm 2 )* c G f (N/mm)* p (mm/mm)* 2.2 2e a) Complete view of the model b) Position of horizontal applied loads Figure Finite element model of the adobe wall subjected to horizontal displacement loads at the top. Total-strain model, Midas FEA.
16 128 Sabino Nicola Tarque Ruíz Calibration of material properties The first parameter that was calibrated is the elasticity modulus E of the adobe masonry. According to section and 3.7, the E value can be considered between 2 and 22 MPa. Besides, [Blondet and Vargas 1978] suggests to use E= 17 MPa; however, this value seems to be too conservative. In this work E= 2 MPa has been considered for all the numerical analyses since it yields a good agreement between the numerical and experimental curves. Figure 6.17 shows the variation of the in-plane response of the adobe wall due to the variation of elasticity modulus. The other, elastic and inelastic, material properties used were the ones specified in Table Force (kn) E= 25 MPa E= 22 MPa E= 2 MPa E= 15 MPa Figure Comparison of the pushover curves in models with different E. Total-strain model. The next parameter that was calibrated was the tensile strength f t of the masonry, which can be roughly assumed around 1% of the compression strength f c. The tensile fracture energy G is maintained in all cases as.1 N/mm (Figure 6.18). I f.7.6 Tensile strength (MPa) ft=.2 MPa ft=.4 MPa ft=.6 MPa Crack displacement (mm) I Figure Variation of tensile strength for total-strain model, constant G f =.1 N/mm, h = mm.
17 Numerical modelling of the seismic behaviour of adobe buildings 129 Since the area under the tensile softening curve is fixed for all values of the tensile strength, the crack displacement values are greater for lower tensile strengths (Figure 6.18). Figure 6.19 shows that the most accurate pushover curve is obtained when f t.4 MPa. Lower values of f t reduce the in-plane strength of the masonry. However, they also give more stable pushover curves due to the large values of the crack displacement (see Figure 6.18). On the other hand, larger values of f t increase the seismic in-plane capacity, but the pushover curve stops due to convergence problems Force (kn) ft=.2 MPa ft=.4 MPa ft=.6 MPa Figure Influence of the tensile strength FEA. ft on the pushover response. Total-strain model, Midas Tensile strength (MPa) Gf=.5 N/mm Gf=.1 N/mm Gf=.15 N/mm Crack displacement (mm) Figure 6.2. Variation of fracture energy G I f for total-strain model, constant f t =.4 MPa, h = mm.
18 13 Sabino Nicola Tarque Ruíz I A variation of the tensile fracture energy G f was considered.5,.1 and.15 N/mm (Figure 6.2). The tensile strain values are equal to the crack displacements divided by the characteristic element length h (141.4 mm). It is seen that greater the fracture energy, the larger the crack displacement values. The best fit of the experimental pushover curve is obtained with G.1MPa (Figure 6.21), in terms of both wall strength and crack pattern. This preliminary study concludes that even though adobe is a very brittle material, it still retains some tension fracture energy, which controls the crack formation process. I f Force (kn) Gf=.5 N/mm Gf=.1 N/mm Gf=.15 N/mm I Figure Influence of the fracture energy in tension G f on the pushover response. Total-strain model, Midas FEA. The smeared crack model takes into account the effect of shear through a reduction factor that multiplies the shear stiffness. This is only possible when a fixed crack model is used, as is the case for this work (see section 4.2.2). A variation of is analyzed to see how much this can influence the global numerical response. As shown in Figure 6.22, the best numerical result was obtained considering.5. The compression strength f c of the adobe masonry was varied from.3 to.8 MPa. The hardening/softening curve is similar to the one used by Lourenço  for clay masonry C but proportionally scaled for adobe masonry. For this reason the ratio Gf / fc is kept at about.344 mm. The peak plastic compression strain p is kept at.2 mm/mm, and the plastic strain at 5% of the compression strength m is kept at.5 mm/mm in all cases. The different compression curves are shown in Figure According to the experimental data, it is seen that a reasonable f c value can be greater than.5 MPa; however, this value is calibrated for within a smeared crack approach.
19 Numerical modelling of the seismic behaviour of adobe buildings Force (kn) bheta=.2 N/mm bheta=.5 N/mm bheta=.1 N/mm Figure Influence of the shear retention factor on the pushover response. Total-strain model, Midas FEA. The total peak compression strain values shown in Figure 6.23, which are the sum of the elastic plus the peak plastic strain, are.345,.41,.5275 and.56 mm/mm for f =.3,.45,.7 and.8 MPa, respectively. c.9 Compression strength (MPa) fc=.3 MPa fc=.45 MPa fc=.7 MPa fc=.8 MPa Strain (mm/mm) Figure Variation of compression strength f c for total-strain model. Relation mm in all cases, h = mm. G C f / f =.344 c Figure 6.24 shows the numerical pushover curves obtained with different compression strength values. It can be observed that the best results are obtained with f c =.3 MPa. The pushover curves obtained with other compression strengths are superimposed and give higher strength, but stop earlier due to convergence problems due to the
20 132 Sabino Nicola Tarque Ruíz concentration of compression stress at the right top window corner. If convergence is reached, so the pushover curve will down and continues closes to the experimental curve Force (kn) fc=.45 MPa fc=.3 MPa fc=.7 MPa fc=.8 MPa Figure Influence of the compression strength Midas FEA. fc on the pushover response. Total-strain model, Results of the pushover analysis considering a total-strain model Figure 6.25 shows the sequence of damage obtained with the parameters specified in Table 6.4. Only the adobe walls are shown here; the ring concrete beams and the lintel are hidden. The maximum top displacement reached was around 9.34 mm. Similar to the experimental response (Figure 6.13), the numerical results show a diagonal crack forming from the corners of the opening. Horizontal cracks are also detected in the perpendicular walls. a) Top displacement= 1 mm b) Top displacement= 4 mm Figure Damage evolution of the wall subjected to a horizontal top load (4 displacement levels). The top and bottom concrete beam and the lintel are hidden. Total-strain model, Midas FEA.
21 Numerical modelling of the seismic behaviour of adobe buildings 133 c) Top displacement= 6 mm d) Top displacement= 9.34 mm Figure Continuation. Damage evolution of the wall subjected to a horizontal top load (4 displacement levels). The top and bottom concrete beam and the lintel are hidden. Total-strain model, Midas FEA. Figure 6.26 shows the evolution of the maximum principal stresses on the in-plane adobe wall (the concrete beams, lintel and perpendicular walls are not shown). In the legend the maximum tensile stress value is limited to.5 MPa. A good agreement is seen between the maximum tensile stresses and the experimental damage pattern (Figure 6.14). Large tensile stresses correspond to large crack openings. The principal stresses reach their maximum values at the opening corners and then travel to the wall corners. The white zones inside the adobe wall show the parts where the tensile strength has been exceeded; this is possible when dealing with a fixed crack [de Borst and Nauta 1985; Feenstra and Rots 21; Noghabai 1999]. The good global agreement between the numerical results and the experimental tests leads to the conclusion that the total-strain model can be successfully applied to the analysis of adobe masonry, and the assumption of a homogeneous material is reasonable. Figure 6.27 shows the deformation pattern at the last computation step. The models were run in Midas FEA with arc-length method with initial stiffness. The number of load steps was specified as 1, the initial load factor was.1, and the maximum number of iterations per load step was 8. The convergence criteria were given by an energy norm equal to.1 and a displacement norm equal to.5.
22 134 Sabino Nicola Tarque Ruíz a) Top displacement= 1 mm b) Top displacement= 4 mm c) Top displacement= 6 mm d) Top displacement= 9.34 mm Figure Evolution of maximum principal stresses in the adobe wall subjected to a horizontal top displacement. Total-strain model, Midas FEA. Figure Deformation of the adobe wall due to a maximum horizontal top displacement of 9.34 mm. Total-strain model, Midas FEA.
23 Numerical modelling of the seismic behaviour of adobe buildings CONCRETE DAMAGED PLASTICITY: MODELLING THE PUSHOVER RESPONSE. A finite element model was created in Abaqus/Standard using 4-node rectangular shell elements without integration reduction (Figure 6.28a). For element controls, a finite membrane strain and a default drilling hourglass scaling factors were selected. The numerical model is similar to the one created in Section 6.3 with Midas FEA, so the shell elements are 1 x 1 mm with 3 mm thick, and the characteristic length is equal to the diagonal of the shell element. The reinforced concrete beams (top and bottom) and the wooden lintel are modelled using linear material properties. The adobe masonry is represented by the concrete damaged plasticity model, which takes into account the tension and compression constitutive laws for adobe. The displacement history is applied at one edge of the top concrete beam as seen in Figure 6.28b. The base of the foundation is fully fixed, while the top part of the crown concrete beam is free of movement. a) Complete view of the model b) Position of horizontal applied loads at the top beam Figure Finite element model of the adobe wall subjected to horizontal displacement loads at the top. Concrete Damaged Plasticity model, Abaqus/Standard. The material parameters used in Abaqus/Standard are essentially the ones used for Midas FEA, though the compression strength is increased to.45 MPa as shown in Table 6.5. Table 6.5. Material properties for the adobe masonry within concrete damaged plasticity model. E (N/mm 2 ) Elastic Tension Compression m (N/mm 3 ) h (mm) f t (N/mm 2 ) I G f (N/mm) f c (N/mm 2 ) c G f (N/mm) p (mm/mm) 2.2 2e
24 136 Sabino Nicola Tarque Ruíz The following default additional parameters are required for the concrete damaged plasticity model: dilatation angle= 1, eccentricity=.1, ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress= 1.16, k parameter related to yield surface= 2/3, and null viscosity parameter. Again, a parametric study for selection of the tensile and compression strength is carried out. The tensile strength is between.2 to 6 MPa and the compression strength is varied between.3 to.8 MPa (see Figure 6.18 and Figure 6.23). As in the previous analysis, it is seen from the numerical pushover curves that the tensile strength is the parameter that controls the global behaviour of the adobe masonry; low values of tensile strength allows to a fast inelastic excursion and will end with convergence problems, high values of tensile strength make more brittle the adobe masonry but increase its lateral strength. According to Figure 6.29 a value of f t equal to.4 MPa should be selected Force (kn) ft=.2 MPa ft=.4 MPa ft=.6 MPa Figure Influence of tensile strength model, Abaqus/Standard. f t on the pushover response. Concrete damaged plasticity Figure 6.3 shows the numerical pushover curves analyzed with different compression strength values. It is seen that the compression strength increment influences on the maximum lateral strength of the adobe wall, but it maintains similar post peak behaviour and failure pattern. The main difference in lateral strength is seen from 2 to 4 mm of top displacement and it is due to the biaxial interaction between tensile and compressive strength. Less difference it is seen for the pushover curves computed with f c =.45 MPa to.8 MPa. A lower bound of f c =.3 MPa can be considered without loss of accuracy on the global response for the plasticity damage model implemented in Abaqus. In this case, there was a not convergence problem as seen in Midas FEA.
25 Numerical modelling of the seismic behaviour of adobe buildings Force (kn) fc=.3 MPa fc=.45 MPa fc=.7 MPa fc=.8 MPa Figure 6.3. Influence of compression strength f c on the pushover response. Concrete damaged plasticity model, Abaqus/Standard. For the cyclic analysis done in Section 6.5 a compression strength of.3 MPa is assumed for the adobe masonry, while for the dynamic analysis performed in Section 7 the compression strength was increased to.45 MPa. In both cases the relation G C f / fc is maintained as.344 mm Results of the pushover response considering the concrete damaged plasticity model The displacement pattern at the last stage is shown in Figure The global behaviour of the numerical analysis on the adobe wall represents well the experimental test in terms of crack pattern and lateral capacity (see Figure 6.14). It is preliminarily concluded that the calibrated material properties can be used for further numerical analyses. Figure Deformation of the adobe wall due to a maximum horizontal top displacement of 1 mm. Concrete damaged plasticity model, Abaqus/Standard.
26 138 Sabino Nicola Tarque Ruíz Furthermore, the damage pattern is analyzed based on the formation of plastic strain in tension at different levels of top displacement (Figure 6.32). It is observed that the formation of cracks starts at the opening corners and at the contact zone of the lintel with the adobe masonry. Horizontal cracks are also observed at the perpendicular walls, similarly to the ones observed in the experimental test. a) Top displacement= 1 mm b) Top displacement= 4 mm c) Top displacement= 6 mm d) Top displacement= 1 mm Figure Evolution of maximum in-plane plastic strain in the adobe wall subjected to a horizontal top displacement. Concrete damaged plasticity model, Abaqus/Standard. Figure 6.33 shows the evolution of the maximum principal stresses in the in-plane adobe wall, without the two concrete beams, the timber lintel and the perpendicular walls. It is observed that the maximum tensile zones (shown in red) are reached first at the opening corners and evolve diagonally to the wall corners. After any integration point reaches f t, so the tensile stress value descends but increasing the crack displacement (softening part of the tensile constitutive law). The models were run in Abaqus/Standard specifying a direct method -for equation solver- with full Newton solution technique. The total displacement load is applied in 1s, having a minimum increment size of.1 s with a maximum of.5 s. The maximum number of increments is 2. Non linear geometric effects are considered for the
27 Numerical modelling of the seismic behaviour of adobe buildings 139 analysis of equilibrium even though they are not expected to affect the results of such a stiff wall. Two control parameters are also specified, the automatic stabilization for a dissipated energy fraction of.1, and the adaptive stabilization with maximum ratio of stabilization to strain energy of.1. a) Top displacement= 1 mm b) Top displacement= 4 mm c) Top displacement= 6 mm d) Top displacement= 1 mm Figure Evolution of maximum principal stresses in the adobe wall subjected to a horizontal top displacement. Concrete damaged plasticity model, Abaqus/Standard. 6.5 CONCRETE DAMAGED PLASTICITY: MODELLING THE CYCLIC BEHAVIOUR The finite element model used for the pushover analysis in Abaqus/Standard was then used for calibration of parameters for cyclic behaviour. The idea is to calibrate the damage factors d t and d c (which control the closing of cracking and reduces the elastic stiffness during unloading), and the stiffness recovery w t and w c, for tension and compression respectively, to reproduce the behaviour of adobe masonry under reversal loads. The tensile and compression strength are kept as.4 and.3 MPa, respectively. According to Abaqus 6.9 SIMULIA , it was seen from experimental tests on concrete that the compressive stiffness can be recovered upon crack closure as the load changes from tension to compression. On the other hand, the tensile stiffness is not
28 14 Sabino Nicola Tarque Ruíz recovered as the load changes from compression to tension once crushing micro-cracks have developed. This assumption is kept for the adobe masonry. The sequence of applied displacements consists two cycles for top displacement of 1 mm and one cycle for top displacements of 2, 5 and 1 mm, as shown in Figure In the experimental test, each displacement limit was repeated twice Time step (s) Figure History of static horizontal displacement load applied to the numerical model Calibration of stiffness recovery and damage factors Since no data tests exist for adobe masonry under reversal loads, stiffness recovery and damage factors were assumed. The idea was to numerically represent the closing of cracking during the change of tension to compression by selecting appropiate damage factors. The numerical force-displacement curves are compared with the results of the cyclic experimental test. By default, Abaqus assigns w c = 1 and w t =, which indicate full stiffness recovery when the integration point is under compression stress, and no stiffness recovery when it is subjected again to tensile stress (see Figure 5.18). Much attention was given to the tensile behaviour rather than to the compression one because it seems that the tension controls the global response of the in-plane adobe walls. As Table 6.6 to Table 6.16 show it is not possible to reach zero crack displacement when the load changes from tension to compression -also with large values of the tensile damage factors- so a residual crack deformation remains when the load is revearsed. Table 6.6. Proposed compression damage factor: Dc-1. Damage factor dc Plastic strain (mm/mm)
29 Numerical modelling of the seismic behaviour of adobe buildings 141 Table 6.7. Proposed tensile damage factor: Dt-1. Damage factor d t Plastic disp. (mm) Tensile strength (MPa) Crack displacement (mm) Tensile curve Degradated stiffness for unloading Table 6.8. Proposed tensile damage factor: Dt-2. Damage factor d t Plastic disp. (mm) Tensile strength (MPa) Crack displacement (mm) Tensile curve Degradated stiffness for unloading Table 6.9. Proposed tensile damage factor: Dt-3. Damage factor d t Plastic disp. (mm) Tensile strength (MPa) Crack displacement (mm) Tensile curve Degradated stiffness for unloading Table 6.1. Proposed tensile damage factor: Dt-4. Damage factor d t Plastic disp. (mm) Tensile strength (MPa) Crack displacement (mm) Tensile curve Degradated stiffness for unloading
31 Numerical modelling of the seismic behaviour of adobe buildings 143 Damage factor d t Table Proposed tensile damage factor: Dt-9. Plastic disp. (mm) Tensile strength (MPa) Crack displacement (mm) Table Proposed tensile damage factor: Dt-1. Tensile curve Degradated stiffness for unloading Damage factor d t Plastic disp. (mm) Tensile strength (MPa) Crack displacement (mm) Tensile curve Degradated stiffness for unloading Figure 6.35 shows the results of the 21 models run in Abaqus under cyclic loading. Models 1 to 4 show the effect of stiffness recovery in tension and compression without taking into account damage factors. The unloading branch beyond 5 mm and the loading branch for 1 mm do not match well the experimental curve. The numerical branches seem to dissipate more energy that the experimental one. Models 5 to 7 show the influence of damage factors with variation of the stiffness recovery; these models show some improvement matching the experimental results with respect to the previous models, especially the loading branch for 1 mm of displacement. However, due to convergence problems, none of the models reached the last displacement cycle. It is preliminarily concluded that the inclusion of damage factors allows a better approximation of the actual test results. The best result was obtained with w c.5 (Model 6). Models 8 to 12 analyze the influence of the tensile damage factor. In these cases wc is kept at 1. It is seen that the compression stiffness recovery is needed to match the experimental results, especially for the loading branch at 1 mm. Models 13 to 17 analyze the variation of the compression recovery stiffness from.7 to.9 and the tensile damage factors. It is seen that lower values of w c should be used for a better match with the experimental result in combination with the tensile damage parameter from Table 6.7 or Table 6.8. The unloading branch after the 5 mm displacement still shows large residual deformations for a lateral load equal to kn, which is not in agreement with the experimental observations. It is understood that this phenomena depends basically on the tensile damage factors applied to the masonry; however, special attention should be paid to the selection of d t values in order to avoid convergence
32 144 Sabino Nicola Tarque Ruíz problems. Models 18 to 21 maintain the tensile damage factor specified in Table 6.7, which are close to the ones specified in Table 6.9, but with a variation of the compression stiffness recovery from.5 to.8. The best match is obtained with Model 2, which considered wc.6, despite the large residual deformations Numerical model Numerical model 2 a) w c = 1, w t =, no damage factor b) w c = 1, 5 w t =.9, no damage factor Numerical model Numerical model c) w c =.75, w t =, no damage factor d) w c =.5, w t =, no damage factor Numerical model Numerical model e) w c = 1, w t =, d c =Table 6.6, d t =Table 6.7 f) w c =.5, w t =.5, d c =Table 6.6, d t =Table 6.8
33 Numerical modelling of the seismic behaviour of adobe buildings Numerical model 7 g) w c =.5, -4 w t =.5, d c = Table 6.6, d t = Table Numerical model Numerical model 9 h) w c = 1, w t =, d t = Table 6.7 i) w c = 1, w t =, d t = Table Numerical model Numerical model j) w c = 1, w t =, d t = Table 6.11 k) w c = 1, w t =, d t = Table 6.12 Figure Comparison of the experimental and numerical cyclic behaviour of the adobe wall taking into account variability in the recovery stiffness and damage factors, in tension and compression. Horizontal load applied on the left part of the top concrete beam.
34 146 Sabino Nicola Tarque Ruíz Numerical model 12 l) w c = 1, -5 w t =, d t = Table m) w c =.9, Numerical model 13 w t =, d t = Table 6.7 n) w c =.9, Numerical model 14 w t =, d t = Table Numerical model Numerical model o) w c =.9, w t =, d t = Table 6.14 p) w c =.8, w t =, d t = Table 6.4 Figure Continuation. Comparison of the experimental and numerical cyclic behaviour of the adobe wall taking into account variability in the recovery stiffness and damage factors, in tension and compression. Horizontal load applied on the left part of the top concrete beam.
35 Numerical modelling of the seismic behaviour of adobe buildings Numerical model 17 q) w c =.7, -5 w t =, d t = Table Numerical model Numerical model r) w c =.8, w t =, d t = Table 6.7 s) w c =.7, w t =, d t = Table Numerical model Numerical model 21 t) w c =.6, w t =, d t = Table 6.7 u) w c =.5, w t =, d t = Table 6.7 Figure Continuation. Comparison of the experimental and numerical cyclic behaviour of the adobe wall taking into account variability in the recovery stiffness and damage factors, in tension and compression. Horizontal load applied on the left part of the top concrete beam.
36 148 Sabino Nicola Tarque Ruíz A new cyclic analysis was performed in Abaqus considering a variation of the zones where the load displacements are applied. In the previous cases the load was applied at one vertical edge of the top concrete beam (Figure 6.28b), which can be good for a monotonic test but probably not good representative for a cyclic one. The same load pattern was later applied at both vertical edges of the top concrete beam and part of the adobe masonry, as shown in Figure 6.36, in order to simulate better the experimental test (see Figure 3.9). The results of the parametric study are shown in Figure 6.37 and demonstrate some improvements for the numerical results. Figure 6.36 Finite element model of the adobe wall considering both ends of the top concrete beam for application of the cyclic horizontal displacement. Concrete Damaged Plasticity model, Abaqus/Standard. Models 22 to 25 evaluate the variation of the tensile damage factors. Again, the needs to reduce the compression stiffness when the stress goes from tension to compression are observed to match the experimental curve, especially for the loading branch at 1 mm displacement. The new tensile factors specified in Table 6.15 and Table 6.16 do not show improvement in the reduction of residual deformations for unloading. Models 26 to 28 consider the tensile damage factors given in Table 6.7 and consider compression stiffness factors from.5 to.8. The best results are obtained with models 27 and 28, concluding that the compression stiffness factors w c should be specified between.5 and.6. The tensile damage factor can not be significantly different from those given in Table 6.7 or Table 6.9; otherwise, convergence problems may stop the analysis before the last stage, always in Abaqus.
37 Numerical modelling of the seismic behaviour of adobe buildings Numerical model Numerical model 23 a) w c = 1, w t =, d t = Table 6.7 b) w c = 1, w t =, d t = Table Numerical model Numerical model c) w c = 1, w t =, d t = Table 6.15 d) w c = 1, w t =, d t = Table Numerical model Numerical model 27 e) w c =.8, w t =, d t = Table 6.7 f) w c =.6, w t =, d t = Table 6.7 Figure Comparison of the experimental and numerical cyclic behaviour of the adobe wall taking into account variability in the recovery stiffness and damage factors, in tension and compression. Horizontal load applied at both ends of the top concrete beam.
38 15 Sabino Nicola Tarque Ruíz Numerical model 28 g) w c =.5, -5 w t =, d t = Table 6.7 Figure Continuation. Comparison of the experimental and numerical cyclic behaviour of the adobe wall taking into account variability in the recovery stiffness and damage factors. Horizontal load applied at both ends of the top concrete beam. The Model 28 (Figure 6.37g) is used for showing the cracking process. From the analysis of the plastic strain it is seen that after the first 2 cycles of 1 mm some regions of the adobe masonry already exceed the maximum elastic strain. This effect is seen at the opening corners, where a concentration of tensile stresses is expected to occur (Figure 6.38a). Figure 6.38 shows the formation process of tensile plastic strains at different values of the top displacement load in Model 28, where the most important aspect is the formation of the X-diagonal cracks, typical of the in-plane behaviour of masonry. The numerical results match the failure pattern seen in the experimental test (Figure 6.14). Horizontal cracks at the perpendicular walls are also formed due to bending. a) Plastic strain values at the end of the 2 cycles of 1 mm Figure Formation process of the tensile plastic strain on the adobe wall under cyclic loads. A non unique legend in placed each to each figure to visualize better the plastic strain. Concrete damaged plasticity model, Abaqus/Standard.
39 Numerical modelling of the seismic behaviour of adobe buildings 151 c) Plastic strain values at the end of the cycle of 5 mm d) Plastic strain values at the end of the cycle of 1 mm Figure Continuation. Formation process of the tensile plastic strain on the adobe wall under cyclic loads. A non unique legend in placed each to each figure to visualize better the plastic strain. Concrete damaged plasticity model, Abaqus/Standard. Another way for interpretation of the tensile damage occurred in the adobe masonry is to show the tensile damage factor (Figure 6.39). Figure Tensile damage factor for Model 28 at the end of the history of cyclic horizontal displacement load.
40 152 Sabino Nicola Tarque Ruíz The tensile damage factor is a non-decreasing quantity associated with the tensile failure of the material. In Figure 6.39, the zones which are not in blue ( d t = ) indicate the zones which already are in the softening part of the tensile constitutive law and can be interpreted as damage zones. 6.6 VIBRATION MODES An eigenvalue analysis is done to compute the vibration modes of the model, especially in the direction of the applied load (X-X direction). The reinforced concrete beam placed as foundation of the wall is removed, so the total weight of the model is 1.63 kn. The base of the wall is fully fixed. The analysis is performed with Abaqus/Standard through the linear perturbation option and considering Lanczos method for extraction of the frequency values. A 5% of the elasticity of modulus has been used according to Tarque  to take into account early cracking into the material. Figure 6.4 shows the effective mass related to the first 11 vibration modes, represented here by the frequency values. In theory the sum of all the effective masses should be equal to the total mass of the model. It is seen that 11 modes of vibration are required to reach the 9% of the total mass (Table 6.17), being the fundamental one the first mode. 8 7 Effective mass (%) Frequency (Hz) Figure 6.4. Contribution of the modes of vibration in the X-X direction until reaches the 9% of the total mass of the model. The deflected shapes given for each mode of vibration are shown in Figure The first vibration mode, which involves 74.24% of the total mass, is a translational mode; while the others are basically out-of-plane deformations of the flange walls. This analysis considers the use of the elastic material properties. However, the adobe material is brittle and goes into the inelastic range very early; therefore the frequencies are expected to shorten.
41 Numerical modelling of the seismic behaviour of adobe buildings 153 a) Mode 1. T 1 =.1157s b) Mode 2. T 2 =.533s c) Mode 3. T 3 =.391s d) Mode 4. T 4 =.344s e) Mode 5. T 5 =.275s f) Mode 6. T 6 =.242s Figure Modes of vibration in the X-X direction.
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