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Computers 1 S~rur~ures Vol. 27. No. 3. pp. 417425. 1987 0045.7949187 f3.00 + 0.00
Printed I” circa: kitam. Q 1987 Perpmon Journals Ltd.
SUBSTRUCTURING TECHNIQUE IN NONLINEAR
ANALYSIS OF BRICK MASONRY SUBJECTED TO
CONCENTRATED LOAD
S. ALL I. D. MOOREand A. W. PAGE
Department of Civil Engineering and Surveying, University of Newcastle, New South Wales 2308,
Australia
(Received 5 February 1987)
Abstract-A multi-level substructuring technique and a mesh grading scheme are used in the nonlinear
finite element analysis of brick masonry subjected to in-plane concentrated loads. Masonry structures are
ideally suited for solution using these techniques since masonry consists of a regular assemblage of bricks
and joints in a repetitive pattern. Large wall panels can therefore be modelled without the need for
excessive computer storage requirements. It is shown that the dual application of these techniques is highly
efficient and leads to significant savings in costs. The possibility of modelling the elastic region of brick
masonry as an isotropic continuum using similar techniques is also considered.
INTRODUCTION in-plane concentrated loads. In the analysis, the part
of the structure which is known (by physical consider-
Since the introduction of the substructuring tech- ations or experience) to remain elastic during the
nique in the early sixties for the numerical analysis of deformation process is defined as one substructure
aerospace structures featuring many degrees of free- and is treated as a collection of elastic bricks and
dom, a number of refinements and variations of the mortar joints. The remainder of the masonry wall
technique have been reported in the literature [l-3]. which undergoes plastic deformation and fracturing
The primary concern of these studies has been the is defined as another substructure. The present model
reduction of the number of variables that must be allows for nonlinear material characteristics and
retained in the comptuer memory at a given time progressive local cracking for the inelastic region.
during the solution process. This permits the analysis The load is applied incrementally so that the response
of very large structural systems with computers hav- of the wall from first crack through to final failure is
ing small to medium in-core capacity. Another attrac- determined. The possibility of modelling the elastic
tive feature of substructuring techniques is that they region of brick masonry as an isotropic homogeneous
often lead to significant reductions in overall com- material is also considered. Example problems are
putational effort. included to show how the procedure operates and to
-The efficiency of a finite element analysis will also demonstrate the significant savings which can be
be improved by refining the finite element mesh where achieved through the use of substructuring and mesh-
accurate modelling of displacement and stress fields refinement techniques.
is most important, and by having a coarser discretis-
ation elsewhere. A number of investigators have
IN
NONLINEARAPPLICATIONS MASONRY
addressed this problem, and efficient mesh-refinement STRUCTURES
schemes [4,5] have been developed.
The finite element analysis of masonry is formid- Nonlinear analysis has become an important
able given the composite nature of the material, since aspect of research into the behaviour of masonry
masonry consists of brick units set in a mortar structures, particularly if ultimate loads are to be
matrix, each with differing deformation and strength predicted. Nonlinear behaviour in masonry struc-
characteristics. However, a masonry wall is a regular tures is produced both by the nonlinear deformation
assemblage of these identical structural units, and is characteristics of its component materials (particu-
therefore an ideal candidate for solution using sub- larly the mortar joints), and the progressive local
structuring techniques. For many problems the calcu- cracking that occurs in both bricks and joints. The
lation of brick masonry strength requires an bulk of this cracking is a result of the low bond
elastic-plastic fracture analysis, so that the use of a strength that typically exists between the
mesh refinement scheme is also advantageous when brick-mortar interface. For many cases of in-plane
the detailed modelling of brick and joint is necessary loading, such as walls subjected to concentrated
for crack propagation analysis. loads, shear walls or frames with masonry infill,
This paper reports how both substructuring and failure usually occurs by the formation of a few
mesh-refinement techniques have been successfully dominant cracks with very little failure elsewhere.
used in the analysis of masonry walls subjected to The computing cost of nonlinear analysis is often
418 S. ALI ei al.
an order of magnitude grcatcr than for a linear then cqn (6)
can bc rewritten as
analysis of the same structure. Fortunately, nonlinear
behaviour usually occurs in isolated regions in many rPp= K,AU,. (9)
masonry structures, especially for the case of a wall
subjected to concentrated loads. Considerable advan- in which
tages can therefore be gained by subdividing the
structure into elastic and inelastic regions, and using
the substructuring technique to determine the
stiffness of the boundary of the elastic region. Addi- and
tional simplifications can be achieved by modelling
the elastic region as an equivalent isotropic homoge- R,= K~~-K:. (11)
neous material, thus avoiding the need to consider
bricks and joints separately. The terms AP: and K:, defined in eqns (7) and (8)
are respectively the condensed incremental load and
the condensed stiffness matrix for the elastic sub-
THE SUBSTRlJClWRING CONCEPT FOR
ELASTO-PLASTIC FRACIWRE ANALYSIS structure. After the problem has been solved for the
nodal displacements in the plastic region (AUp) using
Considering the usual incremental/iterative process eqn (9), the nodal displacements in the elastic region
for the solution of elasto-plastic fracture problems, (AU,) are obtained by substitution of AL$ into eqn
the application of the substructuring technique can be (5).
summarised as follows.
The relationship between the stiffness matrix K, the USING SUBSTRUCTURES TO MODEL THE ELASTIC
incremental nodal forces AP, and the incremental REGION OF BRICK MASONRY
nodal displacements AU, of a structure is given by
Perhaps the most obvious feature of any masonry
KAU = AP. structure is its composition-it is a multitude of
(1)
identical structural units joined together in a repeti-
Equation (1) can be partitioned into elastic and tive pattern. It is by definition a natural system of
inelastic parts and written in matrix form as substructures, and the regular nature of its construc-
tion means that most masonry structures can be
subdivided into units of various scales. This charac-
[:; :] [$I = [:;I. @) teristic can be used to improve the efficiency of
structural modelling considerably by the calculation
where subscripts e and p denote elastic and plastic of the boundary stiffness of the elastic region.
terms respectively. Equation (2), when expanded, Non-homogeneous model
leads to
The small brick masonry panel shown in Figs 1 and
2 illustrates how the multi-level substructuring con-
(3)
cept can be employed within a region of elastic brick
and masonry. A detailed finite element analysis of this
region based on conventional modelling techniques
without substructuring (Fig. 1) requries a large num-
APp = K,AU, + KppAUp. (4)
ber of nodes and elements if the bricks and joints are
Solving eqn (3) for AUc: to be modelled separately. The analysis of a storey
height wall modelled in a similar fashion would thus
ALI, = K,‘(AP, - KcflAUp), be extremely expensive. However, the substructuring
(5)
technique shown in Fig. 2 can be used to simplify the
which can be substituted into eqn (4) to yield an problem. The substructuring is based on the repeated
expression for AP, in terms of only the degrees of masonry unit shown in Fig. 2(a) which incorporates
freedom in the plastic region AU,,, viz.: several bricks and joints.
The substructuring technique used in this study can
AP,-K,K,'AP,=(K,,- KwKly'KII))AUp. (6) be summarised as follows.
Step 1. The masonry unit shown in Fig. 2(a) com-
Now let posed of conventional finite elements is
selected to form the sub-superelement
AP:= K,K,'AP, (7) (Fig. 2(b)). The bricks and joints are
modelled separately. The size of the unit is
and the choice of the analyst. At this level, a
reasonably coarse finite element mesh can
K:= K,K,'K,,; (8) normally be used.
Nonlinear analysis of brick masonry subjected to concentrated load 419
Loading Plate
I II
t i ii i iiiiiiiiil
Detail A
I
4
Fig. 1. Conventional finite element modelling.
(a) Repeated Masonry (b) Sub-Superelement (c) Finite Element Mesh <
Unit Using Sub-Superelements
Slave Nodes
Inelastic Region
Master Nodes
A ]
Elastic Region
!
i
(d) Elastic Superelement and (e) Inelastic Region Separated
Inelastic Region from Elastic Boundary
Fig. 2. Multiple level of substructuring for the nonlinear analysis of brick masonry.
420 S. ALI e/ al.
Step 2. The stiffness of the sub-superelement of the wall being relatively lightly stressed. Conse-
(Fig. 2(b)) is evaluated using static condens- quently a fine mesh is required near the load to model
ation of the internal degrees of freedom. the high stress gradients present, with a coarser mesh
The order of this sub-superelement depends elsewhere. The use of this fine mesh near the loading
on the size of the unit chosen in step 1. point in conjunction with a coarse mesh in the elastic
Step 3. The stiffness of a section of the elastic part of the brick masonry can be facilitated by using
masonry panel composed of a number the mesh-refinement scheme of Gupta [5]. Slave (or
of the sub-superelements from step 2 dependent) nodes are introduced between master
(Fig. 2(c)) is evaluated using conventional nodes when the transition from one discretisation
assembly methods. level to another is required (see Fig. 2(e)). In this
study the interface point concept proposed by Anand
Steps 2 and 3 can be repeated as many times as
and Shaw [7] for the constant strain triangular ele-
necessary. For an ‘infinitely’ (or ‘very’) long wall the
ment has been applied to the linear quadrilateral
process can be repeated until interaction between
element.
each end of the structure is negligible [6]. This is
manifest when terms in the superelement stiffness
matrix which relate degrees of freedom at opposite MODELLINC OF THE INELASTlC REGION
ends of the structure become negligible. Otherwise
Once the elastic boundary stiffness is known, the
the process ceases when the whole of the elastic
inelastic region together with this boundary stiffness
masonry panel has been modelled. For an analysis
can be analysed using the techniques normally re-
featuring a section of the structure which responds
quired for any nonlinear fracture problems. The
inelastically (part ‘B’ in Fig. 2(d)), only the elastic
analysis of this region requires a representative ma-
stiffness of the boundary (I-2-3) surrounding the
terial model. Only a brief description of the material
inelastic region is required at this level.
model used for this region will be presented in this
A result of the substructuring process as described
section, since the model is described in detail else-
is that the modelling of the displacements and stress
where [8].
fields is the same as that for a conventional solution
A complete material model requires the definition
based on the smallest element unit. However, huge
of constitutive relations before and after failure and
reductions in the number of equations to be manipu-
a suitable failure criterion. In this study the wall has
lated have resulted, leading to significant savings in
been assumed to be in a state of plane stress. Previous
computation and storage.
work has shown that this is a reasonable assumption
Homogeneous isotropic model
provided the concentrated load is applied across a
significant portion of the wall thickness [9]. Previous
For concentrated load problems, where nonlinear elastic analysis has shown that the stress state be-
behaviour is localised near the loading point, it may neath the concentrated load is predominantly one of
be possible to model the region of elastic brick biaxial tension+ompression. The main cause of non-
masonry as a homogeneous material (rather than linear behaviour in this case will therefore be
modelling bricks and joints separately). This would progressive local cracking rather than material non-
enable a coarser mesh to be used for the repeated linearity. Hence, relatively simple constitutive re-
masonry unit to be condensed as a sub-superelement. lations for the material are justified.
The substructuring and mesh-refinement schemes de- In the experimental study which was carried out in
scribed above will still be employed and these tech- parallel with the analytical investigation, all masonry
niques will be advantageous if large areas are to be was constructed from solid concrete bricks and a
modelled. compatible mortar. Tests were performed on both
The equivalent homogeneous material properties small samples of the masonry and its components as
were calculated by examining the behaviour of small well as on larger wall panels [8, lo].
masonry panels under uniform prescribed displace-
ment. For the brick/mortar combination considered Deformation characteristics
in this study, the stiffness in directions normal and The deformation characteristics of the bricks were
parallel to the bed joint were found to be similar. This found to be nonlinear in nature. In the analytical
simplified model was also used to analyse the behav- formulations, incremental stress-strain relations were
iour of a storey height wall subjected to concentrated used with a tangent modulus of elasticity appropriate
load, and the results compared to that obtained from to the stress level [8]. Poisson’s ratio was found to be
the more refined analysis (modelling elastic region as approximately constant up to a stress level corre-
non-homogeneous material) previously described. sponding to 75% of the brick strength. Since brick
stresses rarely reached this level in the analysis, a
Use of slave nodes to control mesh-refinement constant value of 0.16 was adopted.
When a masonry wall is subjected to a concen- A relatively simple nonlinear stress-strain formu-
trated load, very high stresses are developed in the lation was adopted for the mortar joints. A model
region beneath the loading plate, with the remainder similar to that proposed by Dhanasekar et al. (111
Nonlinear analysis of brick masonry subjected to concentrated load 421
was chosen. The mortar stress-strain curves were (3) Crushing of the brick or mortar under a stress
derived indirectly from prism tests (for normal and state of biaxial compression-compression.
parallel strains), and from brick masonry couplets To predict joint bond failure (Type (1) above), a
with sloping bed joints (for shear strains). The tests three dimensional failure surface in terms of the
adopted and the derived relationships have been normal, parallel and shear stresses on the joint (a,, oP
presented elsewhere [8]. and ‘t) was used (see Fia. 3). Details of the experi-
mental derivation bf this-surface have been reported
Failure criteria elsewhere [IO]. To predict a cracking type of failure,
the failure surface shown in Fig. 4(a) was used for
Three types of failure are possible for brick ma-
both brick and mortar with the appropriate tensile
sonry in a state of plane stress:
and compressive strength values being substituted.
(I) Bond failure at the interface of the bricks and the For a crushing type of failure, a Von Mises crushing
mortar. This often occurs when the stress normal surface (in terms of principal strains) was used (see
to the interface is tensile. Fig. 4(b)). The use of this simplified surface was
(2) Cracking of either the brick or the mortar under considered justified since previous studies had indi-
a stress state of either biaxial tension-com- cated that a local crushing failure was unlikely for
pression or biaxial tension-tension. concentrated load analyses (81.
-T
Fig. 3. Bond failure surface for mortar joints.
Uncrushed
1:: Present
1 :; Criterlon
-.
\-
f \; ’
Crushed
(a) Cracking Surface (b) Crushing Surface
Fig. 4. Typical biaxial failure surfaces for concrete.
422 S. ALI er al.
When failure was indicated, the effects of the turing region whose height is about one half the
failure were ‘smeared’ over the full width of the height of the wall with the material in the bottom half
element. The stiffness coefficients appropriate to the of the structure being modelled elastically. For this
failure mode were reduced to their appropriate value, study the depth of the inelastic region was taken to
and the problem resolved, to allow stress redis- be 60% of the wall height.
tribution to occur. The assumed width of the inelastic region varies
The material model described above was originally depending on the size of the loading plate. In this
incorporated into a conventional finite element model study the width of the region was selected in such a
without substructuring, and its validity demonstrated way that there was at least one potential vertical
by comparing the predicted and observed per- plane of weakness outside the edge of the loading
formance of small concrete masonry panels subjected plate (the planes of weakness correspond to the lines
to concentrated loads [8]. The present study focuses of the vertical joints).
on the incorporation of substructuring and mesh-
refinement techniques into the finite element model to NUMERICAL EXAMPLES
allow the analysis of more realistic storey height walls
subjected to concentrated loads. Examples of the use In order to demonstrate the application and
of these techniques follow. effectiveness of the numerical techniques described,
two brick masonry walls were selected for analysis.
The first wall was 20 brick courses high and six bricks
SELECTION OF THE INELASTIC REGION
FOR THE CONCENTRATED LOAD ANALYSES wide (Fig. 5) and is small enough to permit analysis
by the conventional finite element method (without
Earlier investigations [8] indicated that cracks al- substructuring) for purposes of comparison. The
ways initiated from the vertical joints some distance second wall was 24 courses high and 12 bricks wide
away from the loading plate and then propagated (see Fig. 7). These dimensions are typical of those of
towards the loading plate and the base of the wall. a real wall. Because of the large number of bricks and
During the initial stages, the crack (or cracks) prop- joints involved, it cannot be readily analysed using
agated as the load was increased, indicating that the the conventional finite element method if the joints
crack propagation was stable. The response remained and the bricks are to be modelled separately.
stable until cracks reached a depth of approximately Two solutions were obtained for the first case, one
40-60% of the wall height below the loading plate. using the conventional finite element procedure and
After this, further crack propagation was accom- the other using a procedure incorporating the sub-
panied by a decrease in the applied load, representing structuring and mesh-refinement techniques. Only
an unstable condition. Both the stable and unstable half of the wall was analysed because of symmetry.
conditions can be analysed because the load is ap- A total of 1672 four-noded quadrilateral elements
plied in the form of a prescribed displacement. It is were used for the conventional solution since a fine
therefore reasonable to specify a plastic and frac- mesh was needed near the loading point (see Fig.
i Pult= 244.5 kN
(a) Conventional Finite (b) Mode of Failure
Element Discretisation
Fig. 5. Conventional nonlinear finite element analysis.
Nonlinear analysis or brick masonry subjcctcd to concentrated load 423
5(a)). For the second procedure the same finite substructuring technique for two different loading
element mesh was used in the inelastic region but conditions: a concentric load (Fig. 7(a)), and an edge
substructuring and mesh refinement techniques were loading (Fig. 7(b)). Loads were again applied in the
used in the modelling of the elastic region (see Fig. form of prescribed displacements with a loaded area
6(a)). equal to 10% of the wall area. Failure patterns and
Load was applied incrementally in the form of failure loads are shown in Figs 7(a) and 7(b). In both
prescribed displacement over 10% of the wall area. cases the cracks formed in vertical lines correspond-
The nonlinear analysis was subdivided into 25 load ing with the edge of the loading plate where both
increments, and an average of 12 iterations were shear and normal stresses are quite high. These
needed for each load increment. Before any local stresses initiated joint bond failure in vertical mortar
failures occurred, the initial stiffness method was joints with subsequent propagation of the crack in a
found to be the most effective in reproducing material vertical direction through alternate joints and bricks.
nonlinearities. However, once local failure occurred The ultimate load for the concentric case was consid-
in the masonry constituents, convergence became erably higher than the corresponding edge loading.
very slow using the above method. Hence, after the This is due to the higher local stresses in the region
initiation of the first crack, the solution procedure of the edge of the loading plate for the eccentric case
was changed to modified Newton-Raphson method. [91.
Using this method the stiffness matrix was updated at The concentric loading case shown in Fig. 7(a) was
the first iteration for each load increment. also analysed using the homogeneous isotropic ma-
The behaviour of the wall was traced from the terial model for the elastic region. The results of the
linear elastic condition into a plastic state, and even- analysis are shown in Fig. 8. It can be seen that the
tually to ultimate failure. The final failure patterns ultimate load and failure pattern are almost identical
and failure loads are shown in Figs 5(b) and 6(b). to those shown in Fig. 7(a) for the substructuring
Almost identical cracking patterns and ultimate loads analysis based on explicit brick/joint modelling for
were predicted using both analyses, with the ultimate the elastic region (the failure loads differed by only
loads differing by only 1.3%. For the conventional 1.14%). It is clear that for this particular problem it
solution, the CPU time using a VAX-11/780 com- is quite reasonable to use the homogeneous isotropic
puter was 3 hr 24 min 27 sec. For the second analysis material model for the elastic region. This then
the CPU time was only 47 min 22 set, less than 25% provides an even more efficient procedure for calcu-
of the former case. For larger walls the differences lating the stiffness of the elastic boundary, since a
would be even more significant, and the benefits to be coarser mesh can be employed. More work is needed
gained from the use of the techniques described to more generally identify the circumstances under
earlier are obvious. which the homogenous mode1 can be expected to
The larger masonry wall was analysed using the yield satisfactory results.
G P,,,*- 247.6 kN
Inelastic Region --, , , ,,
Elastic Region
(a) Finite Element Discretisation (b) Mode of Failure
Using Substructuring
Technique
Fig. 6. Nonlinear finite element analysis using substructuring technique.
424 S. ALI cl al.
(a) Failure Pattern for Concentric Concentrated Load
351.6 kN
(b) Failure Pattern for Eccentric Concentrated Load
Fig. 7. Nonlinear finite element analysis of storey height walls subjected to concentrated loads
CONCLUSIONS mesh-refinement schemes with condensation dras-
tically reduce the number of nodes involved in the
The substructuring technique in conjunction with a nonlinear solution, and ultimately save from 60 to
rn~h-refinement scheme has been successfully used 80% of CPU time. Because of this significant saving,
for the nonlinear fracture analysis of brick masonry the behaviour of full wah panels subjected to concen-
subjected to a concentrated load. The combined use trated loads can be studied.
of substructuring and mesh-refinement schemes per- The separation of the inelastic region from the
mits the analyst to select an extremely small element elastic region for the brick masonry enables the
site in regions of high stress gradients, and a much analyst to diyretise both regions inde~ndently, thus
coarser mesh elsewhere, thus achieving a balance also simplifying data preparation. In general, elastic
between realistic modelling of the actual behaviour substructures should be modelled by considering
and computational effort. For the concentrated load bricks and joints separately, each with differing prop-
problem, where nonlinea~ty primarily occurs in the erties. However, for the analysis of walls subjected to
vicinity of the loading point, substructuring and concentrated loads, it appears that the elastic region
Nonlinear analysis of brick masonry subjected to concentrated load 425
457.2 k8
Fig. 8. Nonlinear finite element analysis using homogeneous isotropic model for the elastic region.
can be modelled as an isotropic, homogeneous mate- element computations. Compul. Merh. appl. Mech.
rial, thus leading to further efficiencies. Engng 7, 93-105 (1976).
5. A. K. Gupta, A finite element for transition from a fine
mesh to a coarse mesh. Int. J. Numer. Meth. Engng 12,
Acknowledgem~~nrs-This work has been carried out in the 35-45 (1978).
Department of Civil Engineering and Surveying, The Uni- 6. J. R. Booker and J. C. Small, Finite element analysis of
versity of Newcastle. Part of the work has been funded by problems with infinitely distant boundaries. Inr. J.
the Australian Research Grants Scheme. Bricks were do- Numer. Anaiyt. Merh. Geomech. 5, 345-368 (1981).
nated by the Concrete Masonry Association of Australia.
.7. S. C. Anand and R. H. Shaw, Mesh-refinement and
substructuring technique in elastic-plastic finite element
analysis. Compur. Struct. 11, 13-21 (1980).
REFERENCES 8. S. Ali, A. W. Page and P. W. Kleeman, Nonlinear finite
element model for concrete masonry with particular
I. A. K. Noor, A. H. Kamel and R. F. Fulton, reference to concentrated loads. Proc. 4th Can. Mar.
Substructuring techniques-status and projections. Symp. I, pp. 137-148 (1986).
Compur. Swucr. 8, 621-632 (1978). 9. S. Ali and A. W. Page, An elastic analysis of concen-
2. C. S. Gurujee and V. L. Deshpande, An improved trated loads on brickwork. Mas. Inrl 6, 9-21 (1985).
method of substructure analysis. Compuf. Sfruct. 8, IO. S. Ali and A. W. Page, A failure criterion for mortar
147-152 (1978). joints in brickwork subjected to combined shear and
3. R. H. Dodds, Jr and L. A. Lopez, Substructuring in tension. Mar. Inrl 9, 43-54 (1986).
linear and nonlinear analysis. Inr. J. Numer. Mesh. I I. M. Dhanasekar, P. W. Kleeman and A. W. Page,
Engng 15, 583-597 (1980). Biaxial stress-strain relations for brick masonry. J.
4. G. F. Carey, A mesh-refinement scheme for finite Slrucf. Diu., ASCE 111, 1085-1100 (1985).
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