UNDERSTANDING THE BEHAVIOUR OF SHEAR WALLS: A

UNDERSTANDING THE BEHAVIOUR OF SHEAR WALLS:
A NUMERICAL REVIEW
Paulo B. Lourenço!' 2 and Jan G. Rots!
1. ABSTRACT
A new interface model that includes all failure mechanisms found in masonry structures
was recently developed by the authors . The model is used to analyse solid and hollow
masonry shear-walls and is capable of predicting the experimental collapse load and
behaviour accurately. Detailed comparisons between experimental and numerical results
perrnit a clear understanding of the walls structural behaviour, fiow of internal forces
and redistribution of stresses both in the pre- and post-peak regime . The parameters
necessary to charaeterize the rnodel are well-defined from available rnicro-experiments .
2. INTRODUCTION
The behaviour of rnasonry shear-walls observed in experirnents is difficult to reproduce
numerieally. The solid clay shear-walls described in this paper present ali the basic
types of mechanisms that characterize masonry: (a) sliding along a bed or head joint at
low values of normal stress, (b) crack.ing of the masonry units in direct tension, (c) diagonal tension crack.ing of masonry units at values of normal stress sufficient to develop
frietion in joints and (d) splining of units in tension as a result of mortar dilatancy at
high values of normal stress. Numerieal models developed for masonry structures
include these mechanisms, in general, only partially (see [I] for references). To the
knowledge of the authors previous numerical rnicro-analyses (and also most of the
experimental results) are lirnited to the structural pre-peak regime. Computations
beyond the lirnit load down to a possibly lower residual load are, however, needed to
assess the safety of the structure. This point becomes much more relevant due to the
extremely low dilatancy of masonry joints, i.e., the need to consider non-associated
plasticity [1] .
A rnicro-rnodel that includes alI the above mechanisrns and the post-peak behaviour at
eonstitutive levei was reeently developed by the authors [2]. Such model will be used to
analyse as well as discuss the behaviour of full-scale solidlhol1ow shear-walls.
Keywords: Masonry ; Walls ; Shear; Micro; Modelling
(I)
Res. Engrs., DelfI University ofTechnology, Faculty of Civil Engineering, P.O. Box 5048, 2600 GA
Delft. The Netherlands and TNO Building and Construction Research. The Netherlands.
(2)
On leave from School of Engineering, University of Minho. Guimarães. Portugal.
11
3. MODELLING STRATEGY
Bricks are modelled with continuum elements and joints are modelled with interface
elements. Interface elements aIlow discontinuities in the displacement field and establish the direct relation: tractions (o-, r) versus relative displacements across the interface
(uo ' us)' The authors tried before (see [I]) to model shear and tensile failure of the joints
in the interface element and compressive failure of masonry in the continuum element.
It was then concIuded that without modelling of cracks in the units an overstiff postpeak response is obtained, leading to a limit load much higher than the one observed in
the experiments. IncIusion of a smeared crack approach for the bricks leads to numerical
difficulties and, presently, does not seem to be a solution. The approach followed here is
to concentrate all the damage in the relativeIy weak joints and in potential pure tension
cracks in the units placed vertically in the middle of each brick (see Fig. I). The joint
interface yield surface has then to include three mechanisms: tensile and shear faiIure of
the joint and compressive faiIure of masonry. Inclusion of the first two mechanisms has
been discussed before (for example, as in Fig. 2) whereas the latter is nove!. By limiting
the compressionlshear stress combinations the compressive damage can be included in
the mode!. This means that a compression cap must be added to the model shown in Fig.
2. Such an approach can certainly be questioned and a discussion is given later in this
papeL
1
~l
í'~
'·' ' ·' ·' ·' ' ' ·'·:."I
· ·
t ~: . . l
fb·.'
oJ
IC
r ol . o~
........ ..
Potential
brick crack
Jl
V
0 0.
.. .. .... ..
Interface
Elements
o-
zerothi~kn~
Continuum Eleme~
Fig. 2 - Interface failure criterion [1]
Fig. 1 - Modelling strategy
4. INTERFACE CAP MODEL
The new interface cap model consists of a tension cut-off, the CouIomb friction model
and an elliptical cap (see Fig. 3). This composite yield function is similar to the experimental results obtained at macro-leveI in shear-waIls [3] and paneIs [4].
Coulomb
Friction
Mode
Irl
Cap
Mode
--~----------------~~~----~--,c~~--
Initial yield surface
Residual yield surface
o-
+ Tension
+ Mode
Fig. 3 - Proposed cap model for interfaces
The model is fully described in [2] but its features will be briefty reviewed here. The
model is established in modem plasticity concepts including non-associated plasticity,
12
correct treatment of the corners and consistent tangent operators. Isotropic coupling is
assumed between shear and tension modes, rneaning that the percentage of softening on
the cohesion ç is the same as on the tensile strength ft. No coupling is assumed between
cap and shear modes .
4.1 Model behaviour in tension
Exponential softening on the tensile
strength is assumed. This fits the experimental results well [5J . The area under the
curve is the Mode I fracture energy G},
defined as the amount of energy to create
one unit area of a crack in the brick or
along the brick-mortar interface.
Oi
)
Experimental
Numerical
--Crack displacement - U n [mm)
FigA - Tensile behaviour
4.2 Model behaviour in shear
't[MPa]
2.0 r-;------------~
Exponential softening on the cohesion is
assumed . This fits well the experimental
results [6J . The area defined by the curve
and the residual dry friction levei is the
Mode II fracture energy G~. Friction, measured by tantP , where tP is the internal friction angle, is assumed to vary between an
inÍtial value (tantPo) and an ultimate value
(tantPu) proportionally to the variation on
the cohesion ç (see [2]).
-- --- Experimental
Numerical
1.5
--
1.0
cr = -0.5 MP.
0.5
0.0
L_==::::':::====:=::":"'_
0 .0
0.2
0.4
0.6
0.8
1.0
Shear slip - U s [mm)
Fig.5 - Shear behaviour
4.3 Model behaviour in compression
o
Im
hardenA
parabolic/exponential
inglsoftening law is assumed on the
masonry compressive strength Im . The
necessary input data shown in Fig. 6 can
be obtained from the statistical database
[7J but, in the pre-peak regime, the
adopted law is in agreement with available
experimental results [5) . Note that experimental work on concrete in compression
[8J showed that this failure is also a localized phenomenon and therefore an energy
based diagram should be adopted .
------ Experimental
- - Numerical
r---' -:~:-::-::;-
)
1:! -~~~-://~E e1ast
Fig.6 - Compressive behaviour
13
5. NUMERICAL EXAMPLES
Tests on solid/hollow walls were carried out by Venneltfoort and Raijmakers [9, 10].
The specimen consists of a pier with a width/height ratio of one (990 x 1000 rnrn 2 ),
built up with 18 la~ers (16 layers unrestrained) of Joosten solid clay bricks (dimensions
204 x 98 x 50 rnrn ) and 10 rnrn thick mortar (1:2:9, cement:lime:sand by volume). The
piers were subjected to different vertical precompression forces P before a horizontal
load was monotonically increased under top displacement control until failure. Fig. 7
shows the failure patterns of the solid specimens. lts behaviour is characterized by initial horizontal tension cracks that develop at the bottom and top of the wall. Tlús is followed by a diagonal shear crack that under increasing defonnation leads finally to the
collapse of the wall simultaneously with cracks in the bricks and masonry crushing in
the compressed toes. The existence of a hole in the wall produces a rearrangement of
the internal forces. The diagonal compressive strut is forced to spread through both
sides of the hole leading to the more distributed cracking pattern illustrated in Fig. 8.
º
a) P = 30 kN
b)P=120kN
c)P=21OkN
Fig. 7 - Failure patterns for different confining pressures, solid walls
Fig. 8 - Failure patterns for confining pressure of 30 kN, hollow walls
For the numerical analysis, bricks are represented by plane stress continuum elements
(8-noded) wlúle line interface elements (6-noded) are adopted for joints and for the
potential vertical cracks in the middle of each brick. Each brick is modelled with 4x2
elements. For the joints, the composite yield function of Fig. 3 is adopted and, for the
potential cracks in the bricks, a simple Mode I cracking model with exponential tensile
softening and irnrnediate drop to zero of the shear stress after initiation of the crack is
assumed. The material data obtained from the micro tension, compression and shear
tests as well as from samples collected for each wall are (P = 30 kN) : for the brick continuum, Eb = 16700 MPa and vb = O. 15; for the brick mode I crack, fI = 2. 4 MPa,
kn = k, = 1. O x 106 N/rnrn 3 and
= O. 1 N/rnrn; for the joint, kn = 82 N/mrn3 ,
G:
14
kt = 36 N/mm 3 , ft = 0. 25 MPa, G} = 0.018 N/mm, c = 1. 4ft' tan ~o = tan ~n = 0.75,
tan If/ = O. O, G}I = 0.125 N/mm, fc = 10. 5 MPa.
5.1 Discussion of the numerical approach
Three aspects of the approach suggested in this paper may be questioned. First, an
alternative to the simplified approach used here would be to model bricks and mortar
with continuum e!ements with additional interface elements for the two mortar/brick
interfaces at either side of the joint. This approach would require a very fine and sophisticated finite element mesh. For the rei ative dimensions of mortar and brick in the tests
above (thickness ratio 1 to 5), it is not clear whether our simplified approach is acceptable, or whether full brick continuum - brickJmortar interface - mortar continuum modelling is necessary. This comparison will be given in a near future.
A second simplification is that tensile cracking of the brick is modelled via a predefined
discrete crack approach. Cracks are constrained to occur vertically, in the rrúddle of
each brick. This approach was adopted because an unconstrained smeared crack
approach for the bricks turned out to produce numerical bifurcations when used in combination with interface softening for the joints [2].
A third simplification is that compressive failure of masonry is modelled via the interface elements. In preceding analyses [1] the compressive failure was modelled via Von
Mises plasticity for the continuum elements. The idea of modelling compressive failure
of masonry in the interface arose as a process to model diagonal cracking in the brick by
lirrúting the shear stresses in the interface. Experiences in rock joints [lI] for higher
confining stresses also show some deviation from the usual linear Coulomb friction
mode! to a parabolic Mohr failure envelope. Therefore the cap model suggested can easily include ali these phenomena. A comparison between modelling the compressive failure in the brick (von Mises) and in the interface (cap model) is given in Fig. 9a for different cap fittings . Both models are of the type hardening/ideal plasticity and cracks in
the bricks were not inc1uded. Fig. 9b shows the caps used in comparison with the ultimate Mohr-circle in compression (15: controls the shape of the ellipse in
a 2 + K ,,2 = f~), which should be applicable if masonry was a homogeneous material.
In this case (K = 4) the result of both types of modelling is sirrúlar.
150.0
r--~-----------
- - Experimental
- - VonMises
ê
s-;;
----- Cap madel
---
100.0
co
.F:
o
:r:
K=O.\
K=4
K=9
- - Yield surface
----- Mohr-circle
50.0
--,
----.,
K=4
K=9
0.0
0.0
L-_~
_____
2.0
4.0
~
6.0
_ _ _ _--.J
8.0
10.0
a
Horizontal displacement [mm]
a) Load-displacement diagram
b) Yield surface in compression
Fig. 9 - Comparison between von Mises (for the continuum) and interface cap model
5.2 Numerical results of solid walls
The comparison between numerical and experimental load-displacement diagrams is
shown in Fig. 10. The experimental behaviour is satisfactorily reproduced and the colIapse load can be estimated within a 15% range of the experimental values. The sudden
15
load drops are due to the opening of each complete crack across one brick or local
unloading in a single joint integration point. In the latter case a mesh refinement is likely
to smooth the curve.
150.0
'~"
100.0
l
-;::;
co
o
:r:
l
I
.2
.~
Experimental
Numerical
,
z
=.
I
i
50.0
,
~
0.0
0.0
~----
p= 210kN
,
p= 120kN
'---------------, --:'-",",,-~:_--_-_-_-_~
__
-I
1.0
2.0
3.0
Horizontal displacement lrnrnl
p = 30 kN
4.0
Fig. 10 - Load-displacement curves for solid walls
The shear-wall with a confining vertical force of 30 kN was chosen for further discussion because two specimens were built. The numerical simulation is ilIustrated at the
final stage (d = 4.0 mm) in Fig. 11 . A diagonal crack through head joints, bed joints and
bricks occurred followed by failure of the compressed toes.
_
..1
---
••
•
1
..1
1
I
_I
..1 l~
I
1
I
I
,
,
Cap
I---' !._~ -r
-r --Ln
I
a) Total displacements at final stage
_
Shear + Tension
I
I
...I.
-L- L
_'
I
I
I
I
. ' ::=L
I
-~
I
I
I
-
-..-L_~
_'--,_
I
I
-j
1
,I
!
i
i
I
,
..1
I
:::L.
;
I
b) Damage at final stage
Fig. 11 - Numerical results for P = 30 kN
In the experiment a1so the vertical reaction was measured. This allows additional compari sons and further assessments of the performance of the model. Fig. 12 demonstrates
that the dilatant behaviour is reproduced well. Also the redistribution of vertical
stresses is dosely followed as illustrated in Fig. 13 by the position of the vertical reaction. This reaction moves initially from the middle of the specimen in the direction of
the compresses toes. Due to plastification/crushing of the compressed zone and cracking
of the jointslbricks dose to the compressed toes the vertical reaction drifts back in the
direction of the center of the specimen in a later stage (see Fig. 14). These phenomena
can be easily understood from the force ftow shown in Fig. 15.
16
120.0
/-------,
Z
=.
c
.2
100.0
-,----
80.0
Ü
~
(1.)
.....
"@
u
60.0
'E
(1.)
>
Experimental
Numerical
40.0
20.0
0.0
5.0
4.0
l.0
2.0
3.0
Horizontal displacement [mm]
Fig. 12 - Vertical reaction
-1-'-'- - JI- - - - -.- - - -Ir -.- - -
;
0.0
E
S
,
x
100.0
-----
~-
I>~ - ......
1.0
~
'\
~
:a
3.0
.~
4.0
o
:r:
Experimental
N umenca
' 1
" ' .....
- .... ,
8 2.0
~o
x [mm']
40ü.0
500.0
300.0
0.0 I-~;;;;;;;;;;e;:=============='i
~i=
~
'=4
f
200.0
-·-·-·t·
\
\"
\
\"''5.0
L-_~
__
~
__
~
__
~
_ _...J
Fig. 13 - Position of the vertical reaction
a
a
[MPaJ
2.0
[MPaJ
2.0
r
o.o~
~s:=-~'"='>~
0.0 1r-rrTT1:l21'~TTT-nrr="'-====
-2.0
-2.0 ,
-4 .0
-4.0
-6.0
______ d = 0.5
l
mm
d = 1.5 mm
d =2.0 mm
d =2.5 mm
-6.0
- - d= 1.0mm
-8.0 r
-8.0
-10.0 .
-10.0
l
Fig. 14 - Diagram of vertical stresses dose to support (cont.)
17
(J
[MPa]
2.0 '
f"'WlJlllV
I'
'\i'
-200
O -,
.,,;
-4 O
"
-.-.-.- d = 3 Ornm
------ d = 3.5 rnm
- - d = 4.0 rnm
-6.0
-8.0
-10.0
l
Fig. 14 - Diagram of vertical stresses dose to support
o
'"
'l'
,...
'<t
«
'"
00
'"
'"
'"'"
-;'
'"CO!-;'
'" a) d'" = 1.0 rnrn
'<t
9
~
o
I
[MPa]
b)d=4.0rnrn
Fig. 15 - Minimum principal stresses (confined bricks ignored)
5.3 Numerical results of hollow walls
For this simulation potential cracks in the bricks were not modelled as the experimental
results, in general, indicated no visible cracks. Micro-cracking is likely to occur but it
was assumed irrelevant for this analysis. The comparison between the numerical and
experimentalload-displacement diagrams is shown in Fig. 16. The results agree reasonably welL It seems that the numerical results are, in a first phase, slightly toa stiff (probably because the cracks in the bricks are ignored aml/or the cap parameter}5: is toa low).
AIso the post-peak behaviour is slightly toa brittle (probably due to a low compressive
fracture energy and/or the initial stiff behaviour). Remarkably it is possible to trace the
complete path of the test without numerical difficulties. Note also that the snap backs
and snap throughs found in the numerical analysis are converged states due to local
unloadings probably caused by the coarse mesh used. The behaviour of the wall is illustrated in Fig. I7 (compare with Fig. 8).
6. CONCLUSIONS
An interface cap model that includes alI the possible failure mechanisms of masonry
structures was deveIoped. Application of this mode! to experiments on shear-walls
shows a very good agreement and provides additional knowledge about the behaviour of
such structures. The model is able to reproduce the complete path of the structures until
total degradation without numerical difficulties . Further applications must include mesh
18
50.0
....
Z
=O)
...u
..,./ \\. .....
--- - - Experimental
\ - - Numerical
---,
40.0
1
.... _~\".\-:-, ----I
//
30.0
..2
'--- '--_ .... _,' '.....
,\
\. __
-;;
Õ
, ,
,
,,
1
20.0
-,
1
.~
5
:c
.
1
"
11_____
" 1--
,
1
10.0
0.0
1
0.0
10.0
30.0
20.0
Horizontal displacement [mml
Fig. 16 - Load-displacement curves for hollow wall
_
_
Shear + Tension
I
I
I
I
I
I
I
I
I
I
I
~
I
..I
I
I
I
I
I
J
I
I
I
I
.1
.J
.J
I
I
I
I
I
_
I
I
I
I
I
.-
t
I
I
I
.I
c) Total displacements (d = 25.0 mm)
I
I
I
I
I
I
I
I
I
I
...
I
I
d) Damage (d
Fig. 17 - Numerical results for hollow wall
19
•
I
I
.J
I
r
I
I
I
I
I
I
I
I
I
I
I
I
I
i
I
I
I
_
Shear + Tension
I
I
I
= 2.5 mm)
b) Damage (d
a) Total displacements (d = 2.5 mm)
.J
I
I
I
I
Cap
I
,
I
I
Cap
J
I
T
I
I
I
I
I
I
I
I
I
I
T
I
=25.0 mm)
I
I
retinement, analysis of input data sensitivity and other structures. However, the importance of numerical modelling for supporting safe and econoIlÚc designs of masonry
structures is already evident from the present paper.
7. ACKNOWLEDGEMENTS
The calculations have been carried out with DIANA tinite element code of TNO Building and Construction Research. The research is supported tinancially by the Netherlands
Technology Foundation (STW) under grant DCT 33.3052 and the Netherlands brick
industries via CUR comrrúttees C96 and A33 on Structural Masonry.
8. REFERENCES
1.
Lourenço, P.B. and Rots, J.G., "On the use of IlÚcro-modelling for the analysis
of masonry shear-walls", Proc. 2nd Int. Symp. on Comp. Meth. in Struct.
Masonry, Swansea, U.K., 1993
2.
Lourenço, P.B., Rots, J.G. and Blaauwendraad, l ., "Implementation of an interface cap model for the analysis of masonry structures", Proc. Euro-C Conf. on
Comp. ModeI. of Concrete Struct., Innsbruck, Austria, 1994, Eds. R. de Borst
and N. Bicanic, Pineridge Press, D.K.
3.
Mann, W. and Müller, H., Failure of shear-stressed masonry - an enlarged theory,
tests and application to shear walls, Proc. British Ceramic Society, VoI. 30, pp.
223-235, 1982
4.
Dialer, c., "Some remarks on the strength and deformation behaviour of shear
stressed masonry panels under static monotonic loading", Proc. of the 9th
IB2Mac, Berlin, Germany, 1991, pp. 276-283
5. Vermeltfoort, A.Th. and Pluijm, R.v.d. , Deformation controlled tensile and compressive tests on units, mortar and masonry, report B-91-0561, TNO-BOUWrru
Eindhoven, Building and Construction Research, The Netherlands, 1991 (in
Dutch)
6.
Pluijm, R.v.d., Deformation controlled IlÚCro shear tests on masonry samples,
Report BI-92-104, TNO-BOUW, Building and Construction Research, The
Netherlands, 1992 (in Dutch)
7. Atkinson, R.H. and Yan, G.G., Results of a statistical study of masonry
deformability, The Masonry Society ]oumal, VoI. 9( 1), 1990
8. Vonk, R.A., Softening of concrete loaded in compression, Dissertation, Eindhoven University ofTechnology, The Netherlands, 1992
9.
Raijmakers, T.M.J. and Vermeltfoort, A.Th., Deformation controlled meso shear
tests on masonry piers, Report B-92-1156, TNO-BOUWrrU Eindhoven, Building and Construction Research, The Netherlands, 1992 (in Dutch)
10. Vermeltfoort, A.Th. and Raijmakers, T.M.J., Deformation controlled meso shear
tests on masonry piers, Part 2, Draft report, TU Eindhoven, The Netherlands,
1993 (in Dutch)
11.
Hoek, E., Strength of jointed rock masses, Géotechnique, VoI. 33(3), 187-223,
1983
20