Analysis of Brick Masonry Wall using Applied Element Method

The Applied Element Method (AEM) is a versatile tool for structural analysis. Analysis is done by discretising the structure as in the case of Finite Element Method (FEM). In AEM, elements are connected by a set of normal and shear springs instead of nodes. AEM is extensively used for the analysis of brittle materials. Brick masonry wall can be effectively analyzed in the frame of AEM. The composite nature of masonry wall can be easily modelled using springs. The brick springs and mortar springs are assumed to be connected in series. The brick masonry wall is analyzed and failure load is determined for different loading cases. The results were used to find the best aspect ratio of brick to strengthen brick masonry wall.


Introduction
Brick masonry walls are commonly used in most of the buildings in India. Bricks of various materials and aspect ratio are available in the market. It will be of immense use if the best aspect ratio which provides maximum strength to the structure is known. Applied Element Method enables the analysis of brick masonry wall by appropriately considering its composite nature. The background to AEM is discussed by Kimiro Meguro in [1] and [2].

Methodology.
The brick masonry wall is discretized along both length and height direction as shown in Fig. 1. In this paper a half brick is taken as an element. Springs are provided at every 5 mm distance. Springs those accommodate mortar joints are treated as 'joint springs'. Stiffness of joint spring depends upon the properties of mortar and brick. Hence, the simple stiffness formula has to be modified. For the joint springs, equivalent normal and shear stiffness is calculated by assuming that these springs are arranged in series [3] as shown in Fig. 2  The equivalent stiffness of normal and shear springs, Kneq and Kseq, are defined as follows:- where, Kneq is the equivalent stiffness of normal spring, Kseq is the equivalent stiffness of shear spring, Knb is the stiffness of normal spring (brick), Ksb is the stiffness of shear spring (brick), Knm is the stiffness of normal spring (mortar), Ksm is the stiffness of shear spring (mortar). Since a half brick is treated as an element, two types of element connection occur in the length direction and three types of element connection occur in the height direction. These connections are indicated in Fig. 1.

Element connections along length.
Two types of element connection occur along length. One is brick-to-brick connection and the other is brick-mortar-brick connection. They are shown in Stiffness of mortar springs:- In element connection 2, top and bottom springs represent mortar where as intermediate springs represent brick-mortar-brick joint. The stiffness of mortar springs and brick-mortar-brick springs are given by Eq. 4 and Eq. 5. Stiffness of mortar springs:-

Kn=
Em×5×T Stiffness of brick-mortar-brick springs:- For the left end element, leftmost spring has the property of mortar whereas the rightmost spring has the property of mortar in the case of right end element. The leftmost and rightmost springs are treated as brick-mortar joint in the case of middle element. All the interior springs are brick-mortarbrick joint for all the connections. The stiffness of mortar springs, brick-mortar springs and brickmortar-brick springs are given by Eq. 6, Eq. 7 and Eq. 8.

Stiffness of mortar springs
Stiffness of brick-mortar springs:- Stiffness of brick-mortar-brick springs:- Since five types of element connections occur, five different types of local stiffness matrices will be obtained. The global stiffness matrix can be determined by appropriately assembling these five local stiffness matrices.

Validation.
Brick Masonry wall with opening [4] is considered for validation. The wall is made of 12 bricks along length direction and 17 bricks along depth direction as shown in Fig. 5 Fig. 6 shows that the analysis by AEM can give the load-displacement curve with a maximum error of 20%.

Determination of best Aspect Ratio.
To find the best aspect ratio which gives maximum strength to masonry, two cases of support condition and loading were considered for analysis. Aspect ratio represents the length to depth ratio of brick.10 number of bricks were adopted along length and depth directions. Size of the brick used is 190 × 90 × 90 mm. Case 1. In case 1, the lower part of the brick masonry is regarded to be fixed. Therefore, all the elements in the lower row are constrained in translation and rotation. Lateral load is applied at the top left end which is uniformly distributed along the top layer. The support conditions and loading of this case is shown in  The graph showing the relation between aspect ratio of brick and first cracking load of case 1 is shown in Fig. 8. The graph showing the relation between aspect ratio of brick and first cracking load of case 2 is shown in Fig. 10.  Fig. 10 shows that the strength of brick masonry wall will be a minimum for the aspect ratio 1.9 for this case of support and loading conditions. The strength increases as the aspect ratio increases or decreases from this value. The aspect ratio corresponding to the minimum strength of brick masonry varies with the number of bricks arranged in horizontal as well as vertical directions.

Conclusions
In this paper Applied Element Method was used for the analysis of Brick Masonry wall. From the study the following conclusions were arrived:- x AEM could predict the load-deflection curve with less than 20% error.
x For the support and loading conditions shown in Fig. 7, the brick masonry wall can be strengthened by increasing the aspect ratio and thickness of brick. x If the brick masonry wall is supported and loaded as shown in Fig. 9, the wall will give minimum strength at an aspect ratio of 1.9.