Keywords

1 Introduction

Analytically, the masonry structure behaviour is quite complex because of the heterogeneity in construction. Several numerical modelling techniques were developed to study the crack patterns in masonry walls. There are several limitations in numerical methods to approximate the behaviour of the infilled wall structure. Several numerical procedures were adapted to reduce approximation levels. In this context, the most popular method, i.e., the Finite Element Method (FEM), implemented for the masonry structures and constitutive laws for nonlinear behaviour in account with the material homogenisation process. Continuum idealisation, or macro-modelling, and system discontinuity, or micro-modelling, are two numerical approaches that are mainly used. In Roca et al. [1], the overview of these methods has been presented. Although, there are several extensions and hybrid methods developed for FEM, it is very difficult to achieve the element separation, rotation. The formulation for the distinct nature of masonry was started in the early seventies [2]. In the Discrete Element Method (DEM), the elements that can be modelled as rigid, partial-rigid, or deformable were considered. Unlike FEM, DEM elements remeshing is not required at the stress concentration region [3]. Cundall and Hart [4], the classical DEM have two main hypotheses, i.e. Element finite displacement and rotations, element contact or collision. In Lourenço [5], recent improvements of various methods such as MDEM, EDEM, DEM compared with FEM were depicted. The collapse behaviour of structures can be achieved in DEM. Because of this feature, many DEM related applications were developed. However, attaining real collapse behaviour of structures is difficult because inefficiency of damping factors. The examples of successful applications of DEM for earthquake analysis of masonry structures were well presented [6, 7] in their research.

Recently the Non-Smooth Contact Dynamics method adapted for the masonry structures under in-plane dynamic loading to study the behaviour [8, 9].

The rigid body spring model (RBSM) [10] partially overcome these issues in DEM. In this method, a masonry elements assembly is assumed as rigid elements connected with normal and shear springs. The structural response can be achieved by the deformation of springs. Although RBSM is well handed for static problems, dynamic analysis can also be applied; under several assumptions, adequate results can be obtained.

Meguro and Tagel-Din [11] came with a similar approach named as Applied Element Method (AEM), in which rigid elements connected by material springs. The stiffness of the springs in both normal and shear directions need to be calculated for the material behaviour. In the AEM [12], analysis can be carried out until the complete collapse of a structure. In RBSM, it is not possible to follow the complete collapse behaviour of the structure.

In the current research, Single-storey and Multi-storey RC bare frames with masonry infill walls were considered to study the behaviour of structures using the AEM.

2 Discretisation of Masonry Elements in Applied Element Method

According to Malomo [13], assembly by zero thickness springs of masonry elements as rigid units. In between the masonry elements, unit interface mortar properties are lumped. The simplified micro-modelling strategy of masonry with mortar shown in Fig. 1a.

Fig. 1
figure 1

Masonry assembly in AEM: a AEM discretisation of masonry elements, b σ-ε relationship of normal springs, and c σ-ε relationship of shear springs in cyclic loading, d bending and twisting of elements [13]

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The arrangement of interface springs and unit springs in series, stiffness kni and ksi shown in Eq. (1). knu and ksu, are unit deformation stiffnesses of springs connected between the masonry elements as shown in Eq. (2),

$$ k_{{nu}} = \left( {\frac{{l_{i} - t_{{mo}} }}{{E_{u} dt_{u} }} + \frac{{t_{{mo}} }}{{E_{{mo}} dt_{u} }}} \right)^{{ - 1}} ,\quad k_{{su}} = \left( {\frac{{l_{i} - t_{{mo}} }}{{G_{u} dt_{u} }} + \frac{{t_{{mo}} }}{{G_{{mo}} dt_{u} }}} \right)^{{ - 1}} $$
(1)
$$ k_{{ni}} = \frac{{E_{u} - dt_{u} }}{{l_{u} }},\quad k_{{si}} = \frac{{G_{u} - dt_{u} }}{{l_{u} }} $$
(2)

where li is the distance between the centroidal nodes of interface elements, lu is the distance between the centroidal nodes of masonry elements, tmo is the mortor bond thickness, tu is the unit thickness of mortor, d is spacing of transverse springs, and Eu, Emo are Young’s modulus of unit mortar and actual mortar respectively, Gu and Gmo are Rigidity modulus of unit mortar and actual mortar, respectively.

3 Experimental Data

Ahmed Sayed [14] was conducted the experimental work on four half scaled Single-storey Reinforced Concrete frame specimens such as bare frame, hollow red brick infill wall of thickness 120 mm, hollow red brick infill wall of thickness of 60 mm and cement bricks infill wall of thickness 120 mm. From his study, it was concluded that the lateral load resistance of the frame with second specimen i.e., holes red bricks masonry of thickness 120 mm was 184% greater than the first specimen i.e., bare frame, whereas for the third and fourth specimen are by 61% and 99% respectively. The initial stiffness of the second specimen is much greater than the remaining specimens.

For this reason, the current research has chosen hollow red brick infill walls of 120 mm thick to study the behaviour of two-storey two-bay RC frame with and without infill walls subjected to in-plane cyclic loading using the Applied Element Method.

4 Modelling of RC Frames

Five half scaled Reinforced Concrete models, namely SBF1, SIF1, MBF, MSF, MIF shown in Fig. 2. were considered for the nonlinear behaviour under cyclic load analysis using Applied Element Method. The first two RC models, i.e., SBF1 & SIF1 details, were taken from Ahmed Sayed’s experimental work [14], namely SBF2 & SIF2, to validate numerical models. The Single-storey and Multi-storey RC bare frames were considered for infill wall models also.

Fig. 2
figure 2

Elevation of models: a single-storey bare frame—SBF1 & SBF2, b single-storey infill frame—SIF1 & SIF2; c multi-storey bare frame—MBF, d multi-storey soft-storey frame—MSF, e multi-storey infill frame—MIF

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5 Sectional & Reinforcement Details

The beams of cross-section 120 × 200 mm, columns of cross-section 200 × 120 mm and base of cross-section 300 × 500 mm are considered. Hollow Red Bricks of size 215 × 102.5 × 65 mm is used for masonry infill wall. High yield strength deformation bars as main reinforcement and Mild steel bars as shear reinforcement are considered. The sectional and reinforcement details of all models are shown in Table 1.

Table 1 Sectional and reinforcement details of all models
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6 Material Properties

High Strength Concrete (HSC) is considered for beams and columns, and Normal Strength Concrete for Base is considered for all the models. Hollow Red Bricks is considered for Masonry infill wall (Table 2).

Table 2 Material properties of SBF1, SIF1, MBF, MSF, MIF [14]
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7 Loading Configuration

All the models were subjected to cyclic displacement controlled loading condition, as shown in Fig. 3. The displacement history was the same for all tested models; Incremental Displacement considered in five stages viz., Stage 1: ±0.5 mm until 3 mm, Stage 2: ±1.0 mm until 10 mm, Stage 3: ±2.0 mm until 20 mm, Stage 4: ±4.0 mm until 40 mm, and Stage 5: ±10.0 mm up to 100 mm.

Fig. 3
figure 3

Displacement history considered for all the models

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8 Results and Discussions

8.1 Mode of Failure

For frame SBF1, hair cracks observed in the columns at beam-column juncture at the end of cycle 2 mm, but for SBF2, it was observed at cycle −2 mm. The cracks were closed and opened for the increased in-plane load for each cycle. Upon application of repeated cyclic load, the flexural and shear compression cracks observed at one-fourth of the column base and beam-column connection, respectively. At higher amplitudes of repeated cyclic load, diagonal cracks initiated and large crack openings can be seen at the beam-column junction after cycle +50 mm. The point of failure observed at +61.28 mm.

For frame SIF1, minor cracks observed in the masonry wall diagonally at the end of cycle −2 mm, but for SBF2, it was observed at cycle −1.5 mm. The cracks were closed and opened for the increased in-plane load for each cycle. Upon application of repeated cyclic load, diagonal crack further extended, and the brick element starts separating from each other at the centre at low-level loading. Lateral strength improved when compared with SBF1 but failed at cycle +12 mm. The Ultimate lateral load and Relative Displacement of SBF1, SBF2, SIF1, SIF2 shown in Table 3.

Table 3 Numerical and experiment comparision of ultimate lateral load and relative displacement of single-storey bare frame and infill frame specimens
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The mode of failure of SIF1 and SIF2 almost matching each other to the maximum extent.

From Table 3 and Fig. 4, it was observed that the AEM results were well accepted with experimental results. Further, the displacement controlled load condition is applied to MBF, MSF, MIF to study the behaviour.

Fig. 4
figure 4

Mode of failure of: a experimental infill frame (SIF2) [14] and b numerical infill frame (SIF1)

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From Table 4, the Ultimate lateral load of MSF is more compared with MIF has been observed. It is because of the weak column and strong infill wall.

Table 4 Relative displacement and ultimate in-plane load of all models
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From Table 4,

  • For frame SIF1, the ultimate in-plane load resistance is 164% more, compared with the frame SBF1

  • For frame MSF, the ultimate in-plane load resistance is 411% more, compared with the frame MBF

  • For frame MIF, the ultimate in-plane load resistance is 396% more, compared with the frame MBF

The shear failure observed at the base of either side column for MSF, whereas for MIF, the flexural failure observed at the beam-column juncture of either side columns of the first floor and shear failure observed at the base of the middle column.

8.2 Relationship Between In-Plane Load and Displacement

Plots of the hysteresis envelope of lateral load versus lateral displacement till the collapse of all the specimens shown in Fig. 5. The plots show the behaviour of all the specimens from the elastic stage to the collapse stage.

Fig. 5
figure 5

Hysteresis envelope of all the specimens till the collapse

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8.3 Ductility

According to Comite Euro-International Du Beton, 1996 [14], the Displacement ductility factor (Rµ) is the ratio between displacement at failure point (Δf) to displacement at the yield point of the specimen (Δy) shown in Eq. (3) and the Displacement Ductility is the ratio of maximum displacement in each cycle interval (Δi) to the displacement at the yield point (Δy) shown in Eq. (4) where these displacements are obtained from the plot of lateral force and lateral displacement of the specimen. The yield displacement is 80% of the ultimate load on the rising curve of lateral load versus lateral displacement plot of the specimen. The failure displacement is 80% of the ultimate load on the falling curve of load versus the displacement plot of the specimen.

$$ {\text{Displacement}}\,{\text{ductility}}\,{\text{factor}},R\mu = \frac{{\Delta _{f} }}{{\Delta _{y} }} $$
(3)
$$ {\text{Displacement}}\,{\text{Ductility}} = \frac{{\Delta _{i} }}{{\Delta _{y} }} $$
(4)

where Δf is the displacement at failure point, Δy is the displacement at the yield point, Δi is the maximum displacement in each cycle interval.

The Accumulated Displacement Ductility is the accumulation of the lateral displacement of first cycle until the failure load cycle and expressed by Eq. (5).

$$ {\text{Accumulated}}\,{\text{Displacement}}\,{\text{Ductility}} = \sum {\left( {\frac{{\Delta _{i} }}{{\Delta _{y} }}} \right)} $$
(5)
Table 5 Accumulated displacement ductility and displacement ductility factor Rµ
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From Table 5,

  • For frame SIF1, the accumulated displacement ductility is 67% lesser than that of frame SBF1

  • For frame MSF, the accumulated displacement ductility is 13% lesser than that of frame MBF and also 1.15% lesser than of frame MIF

  • For frame MIF, the accumulated displacement ductility is 12% lesser than that of frame MBF but 1.16% greater than of frame MSF

8.4 Stiffness

According to Comite Euro-International Du Beton, 1996 [14], the stiffness of structure can be computed for the each interval of cyclic loading as the ratio of peak load at a particular cycle to the peak displacement of the same cycle.

$$ {\text{Cracked}}\,{\text{Stiffness,}}\,k_{i} = \frac{{P_{i} }}{{\Delta _{i} }} $$
(6)

where, Pi is the peak load at cycle i, Δi is the peak displacement at cycle i (Fig. 6).

Fig. 6
figure 6

Stiffness degradation versus the number of cycles

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The initial stiffness is the ratio of change in lateral load to change in lateral displacement for the tangent drawn at the 5th cycle. Table 6 shows the initial stiffness for all models.

Table 6 Initial stiffness (ton/mm) of all models
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From Table 6,

  • For frame SIF1, the initial stiffness is 396% greater than that of frame SBF1

  • For frame MSF, the initial stiffness is 341% greater than that of frame MBF

  • For frame MIF, the initial stiffness is 352% greater than that of frame MBF

9 Conclusions

From this study, it can be concluded that;

  1. 1.

    Applied Element Method results are well accepted with experimental results.

  2. 2.

    The Displacement Ductility Factor of bare frames is greater than the infill frames in both Single-storey and Multi-storey frames i.e., SBF1 = 2.71, SIF1 = 1.17, MBF = 2.05, MSF = 1.90, and MIF = 1.93.

  3. 3.

    The accumulated displacement ductility for SIF1 with an infill wall of thickness 120 mm was less than the bare frame SBF1 by about 67%. The accumulated displacement ductility of frames MSF and MIF with infill walls is less than bare frame MBF by about 13% and 12% respectively.

  4. 4.

    The strength of frame SIF1 with an infill wall was 164% stronger than the bare frame SBF1. The strength of frames MSF and MIF with infill wall of 411% and 396% respectively were stronger than the bare frame MBF.

  5. 5.

    The ultimate in-plane load resistance for frame MSF was greater than the frame MIF by about 3.79% because of the weak column, strong infill wall behaviour.

  6. 6.

    The shear failure observed at the base of either side columns for MSF, whereas for MIF, the flexural failure observed at the beam-column juncture of either side columns of the first floor and shear failure observed at the base of the middle column.

  7. 7.

    Stiffness degradation of masonry infilled frames is very rapid compared to bare frames. Whereas the initial stiffness of SIF1 is 396% greater than that of frame SBF1 and the initial stiffness of MSF and MIF is greater than MBF by about 341% and 352%, respectively.