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									            Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012      ISSN: 2251 - 0478
            © Wilolud Journals, 2012                              
                                       Printed in Nigeria


                                               T.C. Nwofor
Department of Civil and Environmental Engineering, University of Port Harcourt, P.M.B. 5323, Port Harcourt,
                               Nigeria. e-mail:

    This paper is a presentation of the finite element modeling of stress in a brick-mortar couplet. The
    brick-mortar couplet here serves as a meaningful substitute for a brick wall structure. In this work the
    basic steps in derivation of the element stiffness matrix for a plane stress problem using the
    displacement approach is presented. The main objective in the work is to determine the nodal
    displacements, which in turn would be used to determine the stress in each element, from which the
    general stress pattern for the overall structural model can be evaluated. The process involves
    voluminous numerical work and a computer code is used to do the analysis. The result in this work
    shows that the stresses obtained are directly affected by the value of the modulus of elasticity for mortar
    (Em) assuming that the modulus of elasticity for the brick region (EB) remains constant. Hence the
    usual failure theory formulated by previous investigators by considering a homogeneous structure is
    limited as it is well noticed that the variation of modulus of elasticity for the mortar region is a
    significant factor that should not be ignored.

    KEYWORDS: Brick, mortar, masonry, compression, stress, and strain.

Brickwork construction has come to be the most commonly used in the construction industry. They are mostly
used as infill in framed structures or load bearing elements (compression elements) in unframed structures. The
importance of bricks in construction lies in the fact of its numerous advantages, such as durability, appearance,
economy, fire protection, heat insulator and workability. In these modern times where we have tall buildings,
much load comes on the bricks in form of compression and that is when we consider that the bricks act as load
bearing elements (unframed structure) and as a result there is higher stress on the brick wall. Even in framed
structure, the brick which serves as infill is also subjected to little stiffening effects which have long been
neglected in the design of framed constructions.

Brick walls subjected to axial compression and in plane shear has been investigated (Hendry, 1985 and
Borchelt, 1970). The state of the art review of seismic shear strength of masonry has also been published
(Mayes and Clough, 1975). The first category of researchers (Borchelt, 1970; Mayes and Clough, 1975; Blume
and Proulx, 1968) obtained theoretical and empirical expressions for shear strength in terms of principal tensile
stress on the assumption that diagonal tensile failure was initiated when the principal tensile stress reached a
critical value. The second category (Sinha and Hendry, 1967; Pieper and Trantsch, 1970) showed that shear
strength is dependent on bond and friction between the mortar and bricks.

Another set of researchers in their paper which is also a proceeding of the second international conference on
structural Engineering Analysis and Modeling worked on the plane of weakness theory for masonry Brick
Elements (Graham and Andam, 1990). The paper is an experimental verification of the fundamental behaviour
of brick elements subjected to in-plane uniaxial compressive stress forces. In their work they saw a brick
masonry assemblage as a two – phase composite material. The phases do not only have different strengths but
also different deformational characteristics. The mortar phase which acts as a binding medium for the
assemblage, introduces two distinct planes of weakness, orthogonal to each other. Hence, the behaviour of brick
masonry panels in response to uniaxial compressive stress applications was investigated in relation to the plane
of weakness theory.

Another work on single plane of weakness theory was also carried out (Orr, 1981). The biaxial compressive
strength of brick masonry was studied (Page, 1981; Dhanasekar et al., 1985). Studies were also done on
sandcrete block wall subjected to uniaxial stress (Perera, 1993).

             T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012

The condition of masonry in framed structure where the bricks serves as infill has also received attention
(Madan et al., 1997; Ephraim, 1990). In these studies several failure mechanisms were identified. Also the
failure of masonry due to stress released under compression load have been studied (Stroven, 2002) while the
failure of brick wall due to acid rain from radioactive emission has been investigated (Adedeji, 2002). Other
comprehensive studies on the resistance of masonry wall to compressive and transverse loads were conducted
(Chinwah, 1973, 1985; Abrams, 1996).

Brick-mortar elements which are under various stress conditions are primarily vertical load bearing elements in
which the resistance to compressive stress is the primary factor in design. Most of the design values at the
present have been obtained by empirical test on walls and smaller specimen, without considering the influence
of the varying values of the mechanical properties of mortar region. Hence in this present work, it is necessary
to consider the effect of the varying mechanical properties of the mortar region on the stress state of a brick-
mortar couplet which in this case serves as meaningful substitute for a brickwork structure by using the finite
element method of analysis.

This investigation shall be carried out using the constant strain three node triangular elements. Hence the brick
and mortar couplet continuum shall be divided into constant strain triangular elements which shall include
elements in the mortar joints. By the use of this method the distribution of stress in the brick-mortar continuum
can be obtained easily, and with the result obtained, areas of critical stress states will be obvious.

The basic principle of this method is that the continuum is divided into a finite number of elements
interconnected at node points situated on their boundaries. The structure thus idealized can be analysed by any
of the standard method of structural analysis. The development and application of the finite element method are
well published.

For the purpose of this brick-mortar analysis, the formulation used will be the displacement approach. In using
this method the nodal displacements are the basic unknown, while the stresses and strains are assumed constant
for each element.

Finite Element Method
The basic concept of the finite element method of analysis is that the structure can be considered to be an
assemblage of individual structural elements. Hence the idea of the finite element method is the use of two or
three dimensional elements for the idealization of a continuum, where accuracy of the solution increases with
the number of elements taken.

The basic approaches to obtain the triangular element stiffness matrix for a plane stress problem are well
documented. I will only present some essential features in this paper.
Figure 1, represents the coordinates and node numbering for a plane elasticity triangular element

                                                       Fy3, v3

                                                                  Fx3, u3

                                                  v1                        Fy2,
                                          Fy1                      22
                                                   1                               Fx2, u2
                                        x1              Fy1, u1

         Figure 1: Nodal forces and displacements displayed in the Cartesian co-ordinate system
                T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012

A typical node will have two components of displacements
                      u 
         {δ }
            i       =  i
                       vi 
and would produce corresponding force vector
                      f 
         {F }
            i       =  xi 
                       f yi 
Therefore, the complete displacement and force vector for a particular triangular element can be represented as
                  u1 
                  v 
                   1
         {δ e } = u2 
                  v                                                                                           (1)
                   2
                  u3 
                   
                   v3 
                   Fx1 
                  F 
                   y1 
                  F 
         {F e } = Fx 2                                                                                        (2)
                   y2 
                   Fx 3 
                   
                   Fy 3 
Hence the element stiffness matrix [Ke] would be a 6 x 6 matrix for this plane elasticity triangle.
         {F }   e
                    = [K e ] { e }
                              δ                                                                                 (3)

Note that equation (1) and (2) shows contributions from the three nodes shown in figure 1. A suitable
displacement function is chosen to define the displacement at any point in the element. This is simply
represented by two linear polynomials functions containing six unknown coefficients              (α , α
                                                                                                      1   2   Lα 6 )
representing the six degrees of freedom in the case of a plane triangular element.
         u = α1 + α 2 x + α 3 y 
                                                                                                               (4)
         v = α 4 + α 5 x + α 6 y
For plane elasticity problems the [D] matrix which represent the contribution of modulus of elasticity E and
poisons ratio v can be expressed as
                d11             d12     0
         [D] = d 21
                                 d 22    0
                                 0     d 33 

               T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012

where for plane stress problem
                    d11 = d22 =     E (1 − v 2 )
                    d12 = d21 =     vE (1 − v 2 )
                    d33 = d22 =     E 2(1 + v )
and for plane strain problem
                    d11 = d22 =     E (1 − v ) (1 + v )(1 − 2v )
                    d12 = d21 =     vE (1 + v )(1 − 2v )
                    d33 = d22 =     E 2(1 + v )

For a plane elasticity problem, the state of stress      {σ (x, y )} at any point within the element related to the
strain   {ε ( x, y )} can be represented by three stress-strain components as follows:
                             σ x                      1       v       0  εx 
                                      E                                    
           {σ (x, y )}     = σ y  =                   v       1       0  εy                             (5)
                             τ      1−V 2                            1− v  
                              xy                      0       0             σ xy
                                                                        2   
which can be represented simply as
           {σ (x, y )}    = [D ] {ε ( x, y )}                                                                 (6)
           {ε ( x, y )}   = [B ] { e }
                                  δ                                                                           (7)
Hence the relationship between element stresses and the nodal displacements is obtained as
           {σ (x, y )}    = [D ][B ]{ e }
                                     δ                                                                        (8)
where matrix [B] contains constant linear dimensional values.
The statically equivalent nodal forces {Fe} related to the nodal displacements       {δ } and hence the element

stiffness matrix [Ke] can be obtained as
           {F } = [∫ [B] [D][B]d (vol )] {δ }
               e                T                        e
Carrying out the integration and noting that ∫d(vol) can be represented by the area of the triangle ∆ multiplied
by constant thickness t
           {F } = [[B] [D][B]∆t ] {δ }
               e            T                   e
Hence the triangular element stiffness matrix [K ] is represented by
           {K } = [[B] [D][B]∆t ]
                e           T
Through rigorous calculations the nodal displacements for a whole continuum is obtained from equation 9, from
which the element stresses and strains are deduced from equation 8 and 7, hence a suitable computer programme
code will aid this analysis.

Analytical Model
The analytical brick-mortar model consists of the full size brick and mortar couplet. The couplet consists of
brick panels of average dimensions 215mm x 103mm x 65mm, fabricated from high quality perforated bricks of
standard dimensions. A typical representation of a brick-mortar couplet under uniform loading is shown in
Figure 2.

Two types of analytical models are shown in Figures 3 and 4. While Model 1 is divided into 100 triangular
elements, model 2 is a more typical representation of a practical couplet with 216 elements. It is also interesting
             T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012

to note that the elements within the mortar joint are made smaller in two rows. The idea in the smaller mesh
size seen in model 2 is to as much as possible keep the aspect ratio as near unity as possible, i.e. to avoid long
thin triangles since their use reduces accuracy.

The geometry and the loading condition of the structure made in this analysis in order to formulate a finite
element model for the analysis of the stress-strain in the brick-mortar couplet is shown in the figures below.
Other features of the analytical model are as follows.
         (i)       The thickness of the mortar joint is 19mm
         (ii)      A compressive force consisting of a unit of uniformly distributed load (udl) acts on it.
         (iii)     The support reaction of the couplet will consist of a pin support of one end and a roller
                   support on the other end and this is to ensure a determinate structure.

The elastic properties of brick and mortar component are given in Table 1.

                     Table 1: Average values for Modulus of Elasticity and Poisson’s Ratio
                                                   E(N/mm) x 103                           ν
          Brick                                          8.83                            0.060
          Mortar                                   See comment*                          0.170

    *Comment:- The value of modulus of elasticity E for mortar varies with the actual water/cement ratio or
    cube crushing strength fcu while the Poisson’s ratio ν for mortar can be taken as a average of 0.170. The
    value for Em from literature varies from    1.24 x 103N/mm2 to about 37.23 x 103N/mm2 (Page, 1981).

               Figure 2: Typical representation of a Brick-mortar couplet under uniform loading.

T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012

  Figure 3: Analytical model of the Brick-mortar couplet for finite element analysis

  Figure 4: Analytical model of the Brick-Mortar couplet for finite element analysis

             T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012

Using the mechanical property from Table 1 and a computer programme code, a finite element analysis of the
two models are carried out, varying the modulus of elasticity of the mortar joint. From the results, only the
maximum vertical compressive stress and strain in the models have been tabulated and shown in Table 2 and 3.

                         Table 2: Finite element results for models idealized with 100 elements
             Modulus of Elasticity for       Maximum Compressive Stress          Maximum Compressive
             Mortar (Em) (N/mm2) x 103                   (N/mm2)                     Strain (mm/mm)
                       1.24                                 6.10                         0.00290
                       3.72                                 6.90                         0.00320
                       13.10                                7.85                         0.00370
                       24.82                                9.01                         0.00400
                       31.03                                9.85                         0.00405
                       37.23                               10.35                         0.00415

                         Table 3: Finite element results for models idealized with 216 elements
             Modulus of Elasticity for       Maximum Compressive Stress          Maximum Compressive
             Mortar (Em) (N/mm2) x 103                   (N/mm2)                     Strain (mm/mm)
                       1.24                                5.75                          0.00285
                       3.72                                6.45                          0.00315
                       13.10                               7.23                          0.00330
                       24.82                               8.75                          0.00340
                       31.03                               9.45                          0.00402
                       37.23                               9.98                          0.00413

The comparison of stress-modulus of elasticity for mortar (Em) curves for both models, and experimental
study are shown in Figure 5.

It should be pointed out that the numerical results presented above were obtained after collection of the
mechanical property data for brick and mortar. The first FEM calculation with the mortar, masonry, and
brick's mechanical properties taken from the experimental study results led to big discrepancies between the
results from FEM, and testing of the brick mortar couplet with different elasticity property for mortar joint.
This confirms that the properties of mortar inside the joints of the couplet were by far different from those
exhibited by the mortar samples, which were tested separately. Therefore, in the next FEM, the values of the
mechanical properties were increased significantly and according and the results obtained were much closer
to the experimental results.

The stress strain curve obtained from both analyses shows that the maximum compressive strength of
brickwork was slightly higher for the case of higher value of elastic property of the mortar region. The
mortar joint having lower strength than the brick strength, will cause a reduction in the compressive
strength of masonry. The actual compressive strength of the masonry determined by experimental method
was much higher than the strength obtained by numerical method. However, in reality, the brickwork was
constructed from two layered materials, namely brick and mortar, therefore, the idealization of typical brick
work material for the analysis should be adopted. Models with 216 elements show about 4% higher strength
than the model with 100 elements which shows a typical convergence of the finite element result.

From the foregoing in order to get maximum compressive strength in brickworks, it may be recommended
that the average value of elasticity modulus greater than 13.10 x 103(N/mm2) for the mortar region should
be use in brick work construction. The finite element result compared favorably with the experimental result
(Page, 1981; Bakhteri and Sambasivam, 2003) as they both show increase strength for corresponding
increase in the elastic property of the mortar region.

However, comparison of experimental and numerical results indicates that the compressive strength of
masonry obtained by experimental method was 40% higher than those obtained by finite element method.
Therefore, to get the actual compressive design strength of brick masonry, the finite element analysis results
should be enhanced by a factor of 1:4.
                                              T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012

                                         16                      100 elements             216 elements         Experimental
    Maximum compressive stress (N/mm )








                                              0         5          10        15           20        25         30         35        40
                                                                Modulus of elasticity for Mortar (Em ) (N/mm )

                                         Figure 5: Graph of modulus of elasticity for mortar versus maximum compressive stress for brick
                                                          mortar couplet

Numerical and experimental analyses results showed that, by increasing the Em value, the strength of the
masonry will increase it was proposed that in idealization of numerical analysis, the ideal brickwork should be
assumed, since brick masonry is in reality, a layered material. For the design purposes, the compressive strength
of brick masonry obtained from finite element analysis should be factored by a value of 1.4, in order to get the
actual strength of the brickwork.

The comparison of the results from present study with that of the experimental investigation carried out earlier
showed that there is a significant difference in some cases. Since the experimental result in each set is the
average value obtained from the testing of four to five couplets, therefore, by testing large number of couplets in
each set, having a unique Em may give a better average value.

Abrams, D.P. (1996). “Effects of Scale and Loading Rate with Test of Concrete and Masonry structures,”
Earthquake spectra, Journal of Earthquake Engineering Research Institute, vol.12, no.1, pp.13-28.

Adedeji, A.A. (2002). “Evaluation of Brick-wall mortar joint in Acid Rain from Radioactive Emission,”
Proceedings of the fourth International Conference on Structural Engineering, Analysis and Modelling (SEAM
4), University of Science and Technology, Kumasi Ghana, February 26-28, vol.1, pp.67-74.

Bakhteri, J. and Sambasivam, S. (2003). “Mechanical Behaviour of Structural Brick Masonry: An Experimental
Evaluation,” Proceeding of 5th Asia-Pacitic Structural Engineering and Construction Conference Malaysis. 305-

Blume, J. A. and Proulx, J. (1968). “Shear in grouted brick masonry wall elements,” Western state clay product
Association, San Franscico, California

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Borchelt, J.G. (1970). “Analysis of brickwalls subjected to axial compression and in-plane shear,” Proceedings
of second international brick masonry conference

Chinwah, J.G. (1973). “Shear resistance of brick walls,” Ph.D thesis, University of London.

Chinwah, .J.G. (1985). “Shear stress path failure criterion for brickwork,” International Journal of masonry
construction, No.4, PP, 22-28.

Dhanasekar, M., Page, A.W. and Kleeman, P.W. (1985) “The failure of brick masonry under biaxial stresses,”
Proc. Inst. Civil Engineers, Part 2, 79. June 195-313.

Ephraim, M.E., Chinwah J.G., Orlu I.D. (1990) “Mechanisms Approach to Composite Frame and infill,”
Proceedings of the Second International Conference on Structural Engineering Analysis and Modelling (SEAM
2), University of Science and Technology, Kumasi, Ghana, July 17-19, ol.1, pp.13-26.

Graham, E.K. Andam, K.A. (1990). “Plane of Weakness theory for masonry Brick,” Proceeding of the second
International Conference on structural Engineering and Modelling, pp.27-36.

Hendry, A W. (1985). “Recent research on load bearing brickwork,” The British Ceramic Research
Association, Special Publ. No.38.

Madan, A. Reinhoru, A.M. Mander, J. and Valles, R. (1997). “Modelling of masonry infill panels for structural
analysis,” ASCE Journal of Structural Engineering, vol.123, no.10, pp.1295-1302

Mayes, R .C and Clough, R.W. (1975). “State-of-the art in seismic shear strength of masonry -an evaluation and
Review,” EERC Report No.75-21

Orr, D.M.F. (1981). “Single plane of weakness theory applied to masonry,” International Journal of Masonry
Construction, Vol. 2, No.1, 1981.

Page, A.W. (1981). “The biaxial compressive strength of brick masonry,” Proc. lnst. Civil Engineers, Part 2, 71,
Sept., 893-906.

Perera, A. (1993). “Low cost House Using Cement Stabilized Blocks and Timber Poles,” Department of Civil
Engineering, University of Moratuwa, Sri Lanka.

Pieper, K and Trantsch,W. (1970). “Shear tests on wall,” Proceedings of the seconding international brick
masonry conference.

Sinha, B. P. and Hendry, A. W. (1967). “Racking tests on storey height shear wall structures with openings
subjected to pre compression,” Proc. 3rd International Brick Masonry Conference, Essen, West Germany, 1967,
pp. 141-145.

Stroven P. (2002) “Damage in Compressed masonry due to stress release,” Proceedings of the fourth
International Conference on Structural Engineering, Analysis and Modelling (SEAM 4), University of Science
and Technology, Kumasi Ghana, February 26-28, vol.1, pp.162-171.

Received for Publication: 14/02/2012
Accepted for Publication: 20/04/2012


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