THEORETICAL BACKGROUND OF MASONRY MODEL

THEORETICAL BACKGROUND OF MASONRY MODEL 

1. Introduction 

Masonry, though a traditional material which has been used for construction for ages, is a complex 

material. It is a complex composite material and its mechanical behavior, which is influenced by a large 

number of factors, is not generally well understood. In engineering practice, many engineers have adopted 

an elastic analysis for the structural behavior of masonry using rather arbitrary elastic parameters and 

strengths of masonry. Such analyses can give wrong and misleading results. The proper way to obtain 

elastic parameters of masonry is through a procedure of homogenization described in the next section. 

The effect of nonlinearity (i.e., tensile crack, compressive failure, and so on.) to the behavior of masonry 

model is very significant and must be accurately taken into account in analyzing the ultimate behavior of 

masonry structures. Having their own advantages and restrictions, many researches have been conducted, 

for instance, “Equivalent nonlinear stress-strain concept” of J. S. Lee & G. N. Pande1, Tomaževic’s “Story- 

Mechanism”2, the finite element analysis approach of Calderini & Lagomarsino3, and “Equivalent frame 

idealization” by Magenes et al.4. Thus, in practical application of crack effect to the masonry structure, one 

must be well aware of unique characteristics of each of the nonlinear models for masonry structure. The 

main concept of the nonlinear masonry model adopted in the masonry model of MIDAS is based along the 

line of theory of J.S. Lee & G. N. Pande and described in later. 

1 J. S. Lee, G. N. Pande, et al., Numerical Modeling of Brick Masonry Panels subject to Lateral Loadings, Computer & Structures, Vol. 

61, No. 4, 1996. 

2 Tomaževič M., Earthquake-resistant design of masonry buildings, Series on Innovation in Structures and Construction, Vol. 1, Imperial 

College Press, London, 1999. 

3 

Calderini, C., Lagomarsino, S., A micromechanical inelastic model for historical masonry, Journal of Earthquake Engineering (in print), 

2006. 

4 Magenes G., A method for pushover analysis in seismic assessment of masonry buildings, 12 th World Conference on Earthquake 

Engineering, Auckland, New Zealand, 2000. 

THEORETICAL BACKGROUND OF MASONRY MODEL page 1 / 20

2. Homogenization Techniques in Masonry Structures 

Masonry structures can be numerically analyzed if an accurate stress-strain relationship is employed for 

each constituent material and each constituent material is then separated individually. However, a threedimension-analysis 

of a masonry structure involving even a very simple geometry would require a large 

number of elements and the nonlinear analysis of the structure would certainly be intractable. To overcome 

this computational difficulty, the orthotropic material properties proposed by Pande et al.5,6 can be 

introduced to model the masonry structure in the sense of an equivalent homogenized material. The 

equivalent material properties introduced in Pande et al. are based on a strain energy concept. The details 

of the procedure to obtain equivalent elastic parameters based on the homogenization technique are given in 

the following. The basic assumptions made to derive the equivalent material properties through the strain 

energy considerations are: 

1. Brick and mortar are perfectly bonded 

2. Head or bed mortar joints are assumed to be continuous 

The second assumption is necessary in the homogenization procedure and it has been shown7 that the 

assumption of continuous head joints instead of staggered joints, as they appear in practice, does not have 

any significant effect on the stress states of the constituent materials. 

Let the orthotropic material properties of the masonry panel be denoted by 

E 

x 

, 

E 

y 

, 

E 

z 

, ν 

xy 

, ν 

xz 

, ν 

yz 

, 

G 

xy 

, 

by 

G 

yz 

, 

G 

xz 

, Fig. 1. The stress/strain relationship of the homogenized masonry material is represented 

σ = ⎡ ⎣ D⎤ 

⎦ ε 

(1) 

or 

5 G. N. Pande, B. Kralj, and J. Middleton. Analysis of the compressive strength of masonry given by the equation 

α 

( ′) ( ) 

β 

k 

= 

b m . The Structural Engineer, 71:7-12, 1994. 

f K f f 

6 G. N. Pande, J. X. Liang, and J. Middleton. Equivalent elastic moduli for brick masonry. Comp. & Geotech., 8:243-265, 1989. 

7 

R. Luciano and E. Sacco. A damage model for masonry structures. Eur. J. Mech., A/Solids, 17:285-303,1998. 


ε = ⎡ ⎣ C⎤ 

⎦ σ 

(2) 

where, 

σ = 

T 

{ σxxσ yyσzzτxyτ yzτxz} 

{ xx, yy, zz, xy, yz, 

xz} 

ε = ε ε ε γ γ γ 

T 

(3) 

are the vectors of stresses and strains in the Cartesian coordinate system. 

⎡ 1 ν 

xy ν 

⎤ 

xz 

⎢ − − 0 0 0 ⎥ 

⎢ 

Ex Ex Ex 

⎥ 

⎢ ν 

yx 1 ν 

⎥ 

yz 

⎢− 

− 0 0 0 ⎥ 

⎢ Ey Ey Ey 

⎥ 

⎢ 

ν 

⎥ 

⎢ ν 

zx zy 1 

− − 

0 0 0 ⎥ 

⎢ Ez Ez E 

⎥ 

z 

⎡ 

⎣C⎤ ⎦ = ⎢ ⎥ 

⎢ 

1 

0 0 0 0 0 ⎥ 

⎢ 

G 

⎥ 

xy 

⎢ 

⎥ 

⎢ 

1 

0 0 0 0 0 

⎥ 

⎢ 

G ⎥ 

yz 

⎢ 

⎥ 

⎢ 

1 ⎥ 

⎢ 0 0 0 0 0 ⎥ 

⎣ Gxz 

⎦ (4) 

The details of the derivation of orthotropic elastic material properties of masonry in terms of the properties 

of the constituents are given in Appendix I. In the mathematical theory of homogenization, there has been 

an issue relating to the sequence of homogenization, if there are more than two constituents. For example, 

if we homogenize bricks and mortar in head joints first and then homogenize the resulting material with bed 

joints at the second stage, then result may not be the same if we had followed a different sequence. 

However, it has been shown in the case of masonry, the sequence of homogenization does not have any 

significant influence. Here we present in Appendix I equations for equivalent properties if bricks and bed 

joints are homogenized first. It is noted that, in Pande et al., the equivalent material properties were derived 

with the brick and the head mortar joint being homogenized first. The equivalent orthotropic material 

properties derived from the homogenization procedure are used to construct the stiffness matrix in the finite 

element analysis procedure and, from this equivalent stress/strains are then calculated. The stresses/strains 

in the constituent materials can be evaluated through structural relationships, i.e., 


σ = 

σ 

σ 

b 

[ S ] 

b 

⎡S 

bj 

= ⎣ bj ⎦ 

⎡S 

σ 

⎤σ 

⎤σ 

hj 

= ⎣ hj ⎦ (5) 

where subscripts b, bj and hj represent brick, bed joint ad head joint, respectively. 

The structural relationships for strains can similarly be established. The structural matrices S are listed in 

Appendix II. From the results listed in Pande et al., it can shown that the orthotropic material properties are 

functions of 

Fig.1 Coordinate System used in Masonry Panel 

1. Dimensions of the brick, length, height and width 

2. Young’s modulus and Poisson’s ratio of the brick material 

3. Young’s modulus and Poisson’s ratio of the mortar in the head and bed joints 

4. Thickness of the head and bed mortar joints 

It must be noted that the geometry of masonry has to be modeled in the reference of the above figure. This 

is because the homogenization is performed based on the coordinate system. 


3. Criteria for Failure for Constituents 

Failure of masonry can be based on the micromechanical behaviour. At every loading step, once the 

equivalent stresses/strains in the masonry structure are calculated, stresses/strains of the constituent 

materials can be derived based on the structural relationship in eq. (5). The maximum principal stress is 

calculated in each constituent level (i.e., Brick, Bed joint, and Head joint) and is compared to the tensile 

strength defined by user. If the maximum principal stress exceeds the tensile strength at current step, the 

stiffness contribution of the constituent to the whole element is forced to be negligible. For the nonlinear 

stress-strain relation of constituents, even the elastio-perfectly plastic relation could be simulated. This can 

be numerically implemented by substituting the stiffness of the constituent with very small value as 

Ei 

zero 

(where the subscript ‘i’ could be brick, bed joint, or head joint). If users set the ‘Stiffness Reduction 

Factor’ as very small value, the masonry model will behave with nonlinearity. In the same reason, if the 

‘Stiffness Reduction Factor’ is set to be unit value, the masonry model will behave elastically (refer to the 

figure 2 below). 

Fig.2 Stress-Strain of a constituent of masonry model 

In this way, the local failure mode can be evaluated. For better understanding this kind of equivalent 

nonlinear stress-strain relationship theory, see Lee et al.(1996). Once cracking occurs in any constituent 

material, the effect is smeared onto the neighboring equivalent orthotropic material through another 

homogenization. 

Although there are a number of criteria for the masonry model such as Mohr-Coulomb and so on, the 

masonry model in MIDAS currently determines the tensile failure referring only to the user-input tensile 

strength. More advanced failure criteria are developed in the near future based on the abundant research. 

After the tensile cracks occur, the crack positions can be traced by post processor of solid stresses. 


4. Analysis methods of masonry structures 

For the performance assessment of masonry structure, it is generally suggested that the structure needs 

to be analyzed in both out-of-plane damage and in-plane damage concepts. 

Firstly, referring to the figure 3, the out-of-plane damage which is also called as “first-mode collapse” or 

“local damage” involves any kinds of local failure such as tensile failure and partial overturn of masonry wall. 

Fig.3 Example of out-of-plane damage mechanism 

For the precise analysis of out-of-plane damage of masonry structure, part of the structure is modeled 

with detailed finite elements such as material nonlinear models and interface elements to simulate discrete 

mortar cracking, interface interaction, shear failure, and etc. This type analysis is numerically expensive and 

difficult to simulate real structural response and is not the case in the masonry model of current MIDAS 

program. 

Secondly, in the reference of figure 4, the in-plane damage which is also called as “second-mode 

collapse” means the structural response to the external loading as a whole. MIDAS is providing 

homogenized nonlinear masonry model for this kind of analysis. Tensile cracks in mortar and brick can be 

traced with simply defined nonlinear masonry material model. It should be noted that the nonlinear behavior 

of masonry structure is very sensitive to the material properties such as tensile strength and reduced 

stiffness after cracking. So, proper material properties should be carefully defined by thorough investigation 

and experimental consideration. 

It is widely recognized that the satisfactory behavior of masonry structure is retained only when the outof-plane 

damage is well prevented and the structure shows in-plane reaction as a whole. Although these two 

types of damage take place simultaneously, the separate detailed analyses are conducted in practical 


easons. 

Fig.4 Example of in-plane damage mechanism 


5. Importance of nonlinear analysis of masonry model 

To appreciate the importance of nonlinear masonry model, as shown in figure 5, a two story masonry wall 

is analyzed linearly and nonlinearly. As suggested by Magenes8, the wall model with openings is subjected 

to in-plane simple pushover loadings. The model has 6m-width and 6.5m-height and is meshed by eight 

node solid elements. 

Firstly the model is analyzed linearly, which means the stiffness reduction factor is set to be unit value, ‘1’. 

And then, for nonlinear behavior, the stiffness reduction factor is reduced to have very small value of ‘1.e-10’, 

which leads elasto-plastic behavior. The horizontal forces are loaded incrementally with 10 steps, and the 

cracked deformed shape at the step 8 is presented in figure 6. The marked points are representing crack 

points and the contour results are based on effective stress results. 

Fig.5 Two story masonry wall model 

8 Guido Magenes, Masonry Building Design in Seismic Areas: Recent Experiences and Prospects from a European Standpoint, First 

European Conference on Earthquake Engineering and Seismology, Paper Number: Keynote Address K9, Geneva, Switzerland, 3-8 

September, 2006. 


Fig.6 Cracked and deformed shape at step 8 

In both cases two models have the same homogenization procedure. The only difference is the reduced 

stiffness of a constituent at which the crack took place. The force-displacement result shown in figure 7 gives 

that the analytic behavior is significantly dependent on the stiffness of the masonry constituents after crack. 

The deformation is extracted from the nodal results of the right top point. 

Nonlinear vs. Linear Analysis of Masonry 

[ Both cases have the same homogenization ] 

9 

8 

7 

Loading Steps 

6 

5 

4 

3 

2 

1 

Dx[m] 

0 

0.0E+00 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03 3.0E-03 

Nonlinear 

Linear 

Fig.7 Force-deformation result 


In the reference of figure 8, the significance of nonlinearity is more convincing if we consider the resultant 

base shear results. In the figure 8, the horizontal axis is showing the pier positions and the vertical axis is 

representing the resultant shear of each pier divided by the total shear force. In the left pier, the base shear 

result of nonlinear masonry model is almost half of that of linear masonry model. Contrarily, in the right pier, 

the resultant base shear of nonlinear is almost twice that of linear masonry model. Also, overall shear force 

distribution is quite different. The linear masonry model shows symmetric force distribution about the mid pier. 

In the nonlinear masonry model, however, the right pier has the largest shear force results. From this 

consideration, it should be noted that the shear forces after crack are shared not by elastic stiffness but by 

the strength capacity as suggested by Magenes(2006). 

Nonlinear vs. Linear Analysis of Masonry 

[ Both cases have the same homogenization ] 

V/Vtotal [%] 

55 

50 

45 

40 

35 

30 

25 

20 

15 

10 

5 

0 

Left Pier Mid Pier Right Pier 

Nonlinear 

Linear 

Fig.8 Base shear force distribution 


APPENDIX I 

ORTHOTROPIC PROPERTIES OF MASONRY BASED ON STRAIN ENERGY RULE 

Orthotropic material properties of masonry can be derived employing a strain energy concept and the details 

are given in the following. It is noted that homogenization is performed between brick and bed joint first. 

Similar details can also be obtained when brick and head joint are homogenized first. 

Referring to Fig. 1, volume fraction of brick and bed joint can be described as 

h 

tbj 

µ 

b 

= ; µ 

bj 

= 

h+ t h+ 

t 

bj 

bj 

A.1 

where subscript b and bj represent the brick and bed joint, respectively. If the 

brick and bed joint are homogenized in the beginning, the following stress/strain 

components in the sense of volume averaging can be established; 

{ xx, yy, zz 

, 

xy, yz, 

zx} 

T 


zx} 

σ = σ σ σ τ τ τ 


T 

A.2 

where, 

σ 

ε 

2 

1 

= ∑∫ σ dV 

A.3 

xx xxi i 

V 

Vi 

i= 

1 

2 

1 

= ∑∫ ε dV 

A.4 

xx xxi i 

V 

Vi 

i= 

1 

and i=1 for brick, i=2 for bed joint. For each strain component, 


1 

ε = σ −νσ −νσ 

E 

( ) 

xxi xxi i yyi i zzi 

i 

1 


E 

( ) 

yyi yyi i xxi i zzi 

i 

1 


E 

γ 

γ 

γ 

( ) 

zzi zzi i xxi i yyi 

i 

xyi 

yzi 

xzi 

τ 

= 

G 

τ 

= 

G 

τ 

= 

G 

xyi 

xyi 

yzi 

yzi 

xzi 

xzi 

A.5 

Now the strain energy for each component and 1 layer prism can be denoted as 

U 

U 

2 

1 

= + + + + + 

∑ ∫ 

( σ ε σ ε σ ε τ γ τ γ τ γ ) 

re xxi xxi yyi yyi zzi zzi xyi xyi yzi yzi xzi xzi i 

1 2 

Vi 

i= 

1 

= 

2 

∫ + + + + + 

V 

( σ ε σ ε σ ε τ γ τ γ τ γ ) 

e xx xx yy yy zz zz xy xy yz yz xz xz 

dV 

dV 

A6 

where ‘re’ and ‘e’ represent the component and layer prism, respectively, and it is 

obvious that 

U 

re 

= U 

A7 

e 

Introduce auxiliary stresses/strains, 

σ 

σ 

σ 

τ 

τ 

τ 

xxi xx xxi 

yyi 

yy 

zzi zz zzi 

xyi 

yzi 

= σ + A 

= σ 

= σ + A 

= τ 

= τ 

xy 

yz 

= τ + A 

xzi xz xzi 

A.8 

and 


ε 

ε 

ε 

γ 

γ 

γ 

xxi 

xx 

yyi yy yyi 

zzi 

zz 

xyi xy xyi 

yzi yz yzi 

xzi 

= ε 

= ε + B 

= ε 

= γ + B 

= γ + B 

= γ 

xz 

A.9 

then, from eqs. (A.3) & (A.8), 

2 

∑ 

i= 

1 

2 

∑ 

i= 

1 

2 

∑ 

i= 

1 

µ A 

i 

µ A 

i 

µ A 

i 

xxi 

zzi 

xzi 

= 0 

= 0 

= 0 

A.10 

and 

2 

∑ 

i= 

1 

2 

∑ 

i= 

1 

2 

∑ 

i= 

1 

µ B 

i 

µ B 

i 

µ B 

i 

yyi 

xyi 

yzi 

= 0 

= 0 

= 0 

A.11 

where, µ 

1 

and µ 

2 

represent the volume fraction of brick and bed joint, respectively. 

From eqs. (A.5),(A.9) & (A.11), 


E 

x 

2 

ζ 

= − 

α α 

1 µ µ 

bj ⎛ν b 

zy ν ⎞ ⎛ν b 

zy 

ν ⎞ 

bj 

= + + 2χb⎜ − ⎟+ 2χbj 

− 

Ey Eb Ebj Ez E ⎜ b 

Ez E ⎟ 

⎝ ⎠ ⎝ bj ⎠ 

E 

z 

2 

ζ 

= α − α 

µ 

1 

G 

xy 

1 

G 

G 

ν 

ν 

ν 

yz 

xz 

xy 

xz 

zy 

= 

= 

= 

∑ 

i 

i 

∑ 

i 

∑ 

G 

i 

xyi 

µ 

i 

G 

yzi 

( µ G ) 

χζ 

= χ − 

α 

ζ 

= 

α 

χζ 

= χ − 

α 

xzi 

A.12 


where, 

µν 

b b 

χb 

= 

1 − ν 

2 2 

( 1− ) + E ( 1− 

) 

2 2 

( 1−νb 

)( 1−νbj) 

( 1 ) E ( 1 

2 2 

( 1−νb 

)( 1−νbj) 

µ 

bEb ν 

bj 

µ 

bj bj 

νb 

α = 

µ ν E − ν + µ ν −ν 

ζ = 

2 2 

b b b bj bj bj bj b 

b 

b 

µν 

bj bj 

χbj 

= 

1 − ν 

bj 

χ = χ + χ 

bj 

) 

A.13 

and the relationship below can also be established; 

ν 

yx 

E 

y 

= ν 

xy 

A.14 

Ex 

For the system of masonry panel, the homogenization is applied to the layered material and head joint based 

on the assumption of continuous head joint. Now, volume fractions of the constituent materials are 

l 

thj 

µ 

eq 

= ; µ 

hj 

= A.15 

l + t l + t 

hj 

hj 

where, subscript eq and hj represent layered material and head joint, respectively. As in the previous case, 

the following stress/strain components in the sense of volume averaging can be established; 


zx} 

T 

{ xx, yy, zz 

, 

xy, yz 

, 

zx} 

σ = σ σ σ τ τ τ 


T 

A.16 


Introducing auxiliary stresses/strains, 

σ 

σ 

σ 

τ 

τ 

τ 

xxi 

xx 

yyi yy yyi 

zzi zz zzi 

xyi 

xy 

yzi yz yzi 

xzi 

= σ 

= σ + C 

= σ + C 

= τ 

= τ + C 

= τ 

xz 

A.17 

and 

ε 

ε 

ε 

γ 

γ 

γ 

xxi xx xxi 

yyi 

zzi 

yy 

zz 

xyi xy xyi 

yzi 

= ε + D 

= ε 

= ε 

= γ + D 

= γ 

yz 

= γ + D 

xzi xz xzi 

A.18 

where, i=1 & i=2 represent the layered material and head joint, respectively. Following the same procedure 

and defining the following coefficients, 

µ E µ E 

α = + 

1−ν ν 1 

eq y hj hj 

2 

yz zy 

−νhj 

µ E µ E 

β = + 

1−ν ν 1 

eq z hj hj 

2 

yz zy 

−νhj 

µν E µν E 

ζ = + 

1 1 

eq yz z hj hj hj 

2 

−ν yzνzy −νhj 

eq 

( + ) 

µ 

eq 

νzx ν 

yxνzy 

χeq 

= 

1−ν ν 

µν 

hj hj 

χhj 

= 

1 − ν 

hj 

χ = χ + χ 

hj 

yz 

zy 


eq 

( + ) 

µ 

eq 

ν 

yx 

ν 

yzνzx 

λeq 

= 

1−ν ν 

µν 

hj hj 

λhj 

= 

1 − ν 

hj 

λ = λ + λ 

hj 

yz 

zy 

A.19 

the orthotropic material properties of the masonry panel are finally derived; 

1 µ 

eq 

µ ⎛ 

hj 

ν 

yx 

ν ⎞ ⎛ 

xy 

ν 

yx 

ν ⎞ 

hj 

= + + λeq 

− + λhj 

− 

Ex Ex E ⎜ hj 

Ey E ⎟ ⎜ 

+ 

⎝ x ⎠ ⎝ Ey E ⎟ 

hj ⎠ 

E 

y 

2 

αβ −ζ 

Ez 

= 

α 

1 µ 

eq 

µ 

hj 

= + 

G G G 

1 µ 

eq 

µ 

hj 

= + 

G G G 

2 

xy xy hj 

G = µ G + µ G 

ν 

ν 

ν 

ν 

yz eq yz hj hj 

xz xz hj 

yx 

yz 

zx 

zy 

⎛νzx ν ⎞ ⎛ 

xz 

ν ν ⎞ 

zx hj 

χeq 

⎜ − ⎟+ χhj 

− 

Ez E ⎜ x 

Ez E ⎟ 

⎝ ⎠ ⎝ hj ⎠ 

αβ −ζ 

= 

β 

ζχ 

= λ − 

β 

= 

ζ 

β 

ζλ 

= χ − 

α 

ζ 

= 

α 

A.20 


APPENDIX II 

STRUCTURAL RELATIONSHIP OF MASONRY 

Structural relationship of each constituent materials with respect to overall masonry can be established 

through utilizing auxiliary stress/strain components introduced in Appendix I. Details of each relationship are 

now deduced. 

As in eq. (5), the structural matrix has the following form; 

[ S] 

⎡S S S 

⎢ 

⎢ 

S S S 

⎢S S S 

11 12 13 

21 22 23 

31 32 33 

= ⎢ 

⎢ 0 0 0 S44 

0 0 ⎥ 

⎢ 0 0 0 0 S55 

0 ⎥ 

⎢ 

⎥ 

⎢⎣ 

0 0 0 0 0 S66 

⎥⎦ 

0 0 0 ⎤ 

0 0 0 

⎥ 

⎥ 

0 0 0 ⎥ 

⎥ A.21 

Solving the auxiliary stress/strain components in eqs. (A.17) & (A.18), 

σ 

= σ + C 

yy, hj yy yy, 

hj 

2 

{ Ehjεyy νhjEhjεzz ( νhj νhj ) σxx} 

1 

= + + + 

1−νν 

hj 

hj 

⎧⎪ ν ⎛ν νν ⎞⎫⎪ ⎧⎪ ⎛ 1 νν ⎞⎫⎪ ⎧⎪ 

⎛ν ν 

= σxx ⎨ − η + + σ 

yy 

η − + σzz 

η − 

1 ν ⎜ E E ⎟⎬ ⎨ ⎜ ⎬ ⎨ 

E E ⎟ ⎜ 

⎪⎩ − ⎝ ⎠⎪⎭ ⎪⎩ ⎝ ⎠⎪⎭ ⎪⎩ 

⎝ E E 

hj yx hj zx hj zy hj yz 

hj y z y z z y 

⎞⎫ 

⎪ 

⎟⎬ ⎟ ⎠⎪ 

⎭ 

A.22 

where, 

E 

η = A.23 

1 − ν 

hj 

2 

hj 

Therefore, 


S 

S 

S 

hj xy hj zx 

hj,21 2 

1 ν ⎜ 

hj 

Ey Ez 

hj,22 

hj,23 

ν ⎛ν ν ν ⎞ 

= − η + 

− ⎟ 

⎝ ⎠ 

⎛ 1 νν ⎞ 

hj zy 

= η 

− 

⎜ 

Ey 

E ⎟ 

⎝ 

z ⎠ 

⎛νhj 

ν 

= η 

− 

⎜ 

⎝ Ez 

E 

yz 

y 

⎞ 

⎟ 

⎠ 

A.24 

The above equation can be rewritten as follows: 

S 

S 

S 

hj,21 

hj,22 

hj,23 

ν ⎛ 

hj 

ν 

yx 

νhjν 

⎞ 

zx 

= − ηhj 

+ 

1−ν 

⎜ 

hj 

Ey E ⎟ 

⎝ z ⎠ 

⎛ 1 νν ⎞ 

hj zy 

= ηhj 

− 

⎜ 

Ey 

E ⎟ 

⎝ 

z ⎠ 

⎛νhj 

ν 

= ηhj 

− 

⎜ 

⎝ Ez 

E 

yz 

y 

⎞ 

⎟ 

⎠ 

A.25 

where, 

η 

hj 

E 

= A.26 

1 − ν 

hj 

2 

hj 

Using same procedure, the remaining non-zero coefficients can also be derived; 

S 

S 

S 

S 

S 

S 

S 

hj,11 

hj,31 

hj,32 

hj,33 

hj,44 

hj,55 

hj,66 

= 1.0 

ν ⎧ 

hj ⎪νhjν yx ν ⎫ 

zx ⎪ 

= − ηhj 

⎨ + ⎬ 

1−ν 

hj ⎪⎩ 

Ey Ez 

⎪⎭ 

⎧⎪ν 

hj ν ⎫ 

zx ⎪ 

= ηhj 

⎨ − ⎬ 

⎪⎩Ey 

Ez 

⎪⎭ 

⎧⎪ 

1 νν ⎫ 

hj yz ⎪ 

= ηhj 

⎨ − ⎬ 

⎪⎩Ez 

Ey 

⎪⎭ 

= 1.0 

G 

= 

G 

hj 

yz 

= 1.0 

A.27 


Solving for the unknowns A, B, C and D in eqs. (A.8), (A.9), (A.17) and (A.18), the structural matrix for each 

component can be derived and the full details will be omitted.

THEORETICAL BACKGROUND OF MASONRY MODEL

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