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INTERNATIONAL JOURNA L
FOR
N U M E R I C A L M E T H O D S I N E N G I N E E R I N G , V O L.
22 ,75 1-767
(1986)
SNAP-THROUGH AND SNAP-BACK RESPONSE IN
CONCRETE STRUCTURES AND THE DANGERS
OF UNDER-INTEGRATION
M. A. CRISFIE LD
Transport and Road Research Labo ratory , Department of Transport Cronthorne, Berkshire.
U . K
S U M M A R Y
Snap-through and snap-back responses are usually associated with the buckling of shells. However, it is
shown in this paper that they can also be expected with the cracking
of
concrete structures. I t is also
demonstrated that,
for
such structures, the common use of 2
x
2 Gauss ian integrat ion with the 8-noded
isoparametric element can lead to spurious responses associated with ‘trun cated hour-glass modes’.
I N T R O D U C T I O N
The phenomena that will be described are largely associated with the softening response that
follows cracking. Fo r two-dimensional c ontin uum analyses, this softening can be related to the
fracture energy.’-’’ In a n attemp t to avoid mesh-dependency,I3 this energy can be used to
provide a degree-of-softening that is inversely proportional to some ‘characteristic length’.
‘ - I ’
The present work adopt s a smeared concrete m ~ d e l , ’ ~ . ~ ’
o
that this length should be associated
w ith an in te gra tio n sta tio n o r G a us s p ~ i n t . ~ - ’ ~o an extent, this procedure produces a
‘non-local’ stress/strain relationship. Th is conc ept has been taken further by Bazant,’ who
advocates overlapping or ‘imbricated’ elements.
Fo r beam or slab analyses, involving plane-sections tha t remain plane, an empirical softening
is usually provided. This phenomenon is called ‘tension ~tiffening”~-’*ecause tensile stresses
are generated in the concrete beyond a crack a s
a
result of the transfer, via shear and bond, of
stress from the reinforcement. The Gauss point models both the crack (or cracks) and the
adjacent concrete and consequently its response should be stiffer than
i t
would be for a purely
brittle failure.
Softening m aterials ar e know n to induce ‘strain-localization’,’-’
2 ,1 9 p 2
in which a local region
softens (or cracks) while the adjoining material unloads elastically. These localizations may be
accompanied by dy namic ‘snap-throughs’ or ‘snap-backs’. Th e former ph eno me non involves a
dynam ic jum p to a new displacement state a t a fixed load level, while the latter involves a
dynamic jump to a new load level under a fixed displacement state (Figure 1). Such snap-backs can
only be obtained experimentally
if
a very stiff testing frame is available while, un de r load control,
a snap-through might appear as a local load plateau. Traditional static nonlinear analysis
techniques have considerable difficulties with such phenomena and this may account for the
convergence difficulties that are often encountered with concrete problems.
As
a result, analysts
often a do pt a displacement convergence criterion which c an allow very significant out-of-balance
forces.23
0029-5981/86/060751-17 08.50
0
986 by John Wiley
&
Sons, Ltd.
Received J u l y 1985
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7 5 2 M .
A.
CRISFIELD
Deflection
p
Figure 1. ‘Snap buckling’
Short
of
adopting a dynamic analysis, special pseduo-static solution procedures are required.
The author adopts the spherical arc-length m e t h ~ d ~ ~ ~ ~ jnd couples the technique with ‘line
~ e a r c h e s ’ . ~ ~ . ~ ~ith these procedures, it is hoped to trace the equilibrium response beyond the
maximum load and hence to establish the cause of collapse. Tradit ional, load-controlled analyses
equate failure of the structure with the failure of the iterative solution technique. As a consequence,
in a brittle environment, they can fail to establish the particular cracks or mechanism that
initiates the collapse because there is no converged equilibrium state to study.
S IM P LE M O D E L S FOR STRAIN SOFTENING AND LOCALIZATION
I t can be shown that the cracking of concrete is a ductile phenomenon that can be measured
if
a sufficiently stiff testing machine is a ~ a i l a b l e . ~ , ~ ~ , ~ ~he fracture energy may then be considered
as a fundamental material property, G ( 100N/m).1,8926,27 his energy (per unit area) can be
related to a simple softening stress-strain relationship (Figure
2)
via
G = O. ~ CCI CJ . E
1 )
E
E
a c t
Figure 2. Assumed stress-strain relationship for concrete in tension
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R E SPON SE I N C O N C R E T E ST R U C T U R E S
7 5 3
(a )
Simple t ie-bar
b) t r uc tu r a l response
Figure
3.
Strain localization
where
c
defines a 'characteristic length'. '-' F o r a sme ared mod el, this length should be associated
with
a Gauss point. To this end, the author uses the intersection of the principle tensile stress,
at first cracking, with a skewed ellipse derived from the Ja cob ian at the G au ss point.
The concept of strain localization can be simply illustrated by reference to Figure 3. In Figure
3(a), the elements each contain a single Ga uss point an d a re assumed to follow the stress-strain
curve of Figure 2 (with a fixed
I
value). As a consequence, four alternative equilibrium paths
are possible (Figure 3b). Th e shallowest (path 1) involves all four elements softening down the
line
A C
(Figure
2),
while the snap-back of path 4 is associated with only one softening element
while the other three elements unload elastically down
A 0
(Figure
1).
Only this last response
(path 4) is stable, in a material sense, because a small perturbation in the tensile strength would
invalidate the other paths.
If such a perturbation were provided and the softening parameter
CI
(Figure
2)
was made
inversely propo rtiona l to the element length (using equatio n l), the structura l response would be
independent of the mesh. This independence ha s been dem on strated in a nu mb er of more general
finite element an alys es.6-s , '0. '1 However, these analyses hav e usually involved a single crack ev en
where a smeared approach is adopted. Real structures are more complicated and also involve
reinforcement. The simple model of Figure 3 can be extended t o such situations (Fig ure 4) an d
can produ ce the complicated load/deflection response show n in Figure 4(c). Assum ing a small
perturbation had been applied to the tensile strength, the weakest element would crack at point
A ,
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754 M. A. CRISFIELD
I I
I I
1 P
(a) St i f fened t ie -bar
P A
-
( b ) So l u t i o n w i t h O >cYcrit,
( c ) S o l ut i o n w i t h fca cri,.
Figure
4
Strain localization
for
stiffened tie-bar
while the strongest would crack at point G and a t point H , the last remaining uncracked concrete
element would reach the bottom of its softening ‘curve’ (point
C,
Figure 2).
At
this stage, the
structure would respond as a steel bar on its own. The falling lines A B ,
C D ,
E F and G H
all
involve
one localized softening element and three elastically unloading elements. However, for line A B ,
these three elements would all be reinforced concrete, while for G H they would all consist of
reinforcement on its own. The steepness of the peaks in Figure 4(c) will be increased as a is reduced
and flattened as
ct
is increased. Finally, with ct greater than
a
critical value, [ l +
E , A , / E , y A , ) ] ,
he
response will be of the form indicated in Figure 4(b) and only one element will have cracked. These
simple models have illustrated the complexities that can be encountered once a softening model is
introduced.
I t
will
be shown later that similar phenomena can be encountered in more large-
scaled finite element analyses.
FINITE ELEMENT AND MATERIAL
M O D E L L I N G
The finite element and material modelling is fairly standard15 and is detailed in Reference 12. In
particular, perfect bond is assumed while a softening-hardening plastic model
is
adopted for the
compressive regime. However, little compressive nonlinearity was encountered in the examples
that will be presented. As already indicated, the adoption of a softening model is bound to
introduce ‘material unloading’ away from the localized zones. Physically, this unloading involves
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RESPONSE IN CONCRETE STRUCTURES 755
the closing of cracks. For the present analyses, the falling line
BO
of Figure
2
has been adopted.
A specific weakness of the present model, which is shared by most other procedures, relates to the
provision of fixed orthogonal cracks.
As
shear is transmitted across the first primary crack, via the
shear retention factor,I5 new principle tensile stresses will build up and can exceed the tensile
strength. Unless this new cracking stress is orthogonal to the original crack, no new crack or
weakness is generated. A model to overcome this deficiency has recently been proposed by
de Borst et ~ 1 . ~ 3 ’ ~
T h e arc-length method
General reviews of the literature on the arc-length m e t h ~ d ’ ~ . ~ ’re contained in References
23
and 31, while the special details of the author’s technique are described in References 22-24. For
the present we will merely outline the main concepts with emphasis on those feature that
relate to the analysis of concrete structures.
In order
to
prevent divergence of the Newton or modified Newton iterative techniques,3z the
load level Is treated as a variable, while a constraint
ApTAp = A P
2)
is used to limit the magnitude
of
the incremental displacement vector, A p. The scalar A in equation
2)
is a prescribed ‘length increment’ which varies from increment to increment in order to reflect
the degree of nonlinearity.22723 he first, trial, incremental solution, A p , , is based on the tangential
displacement vector, ti so that
3 )
where K , is the tangent stiffness matrix and q is a fixed load vector. Both equat ion (3)and - A i d ,
satisfy equation
2)
if
Ap,
=
AAtiT
=
AAKG’q
A number of different procedures can be used to choose the sign. For example, both Bergan’s
current stiffness parameter33and the sign of the determinant of the tangent stiffness matrix have
been used. In Reference 25, the author switched from the latter technique to the former so as to
prevent material instabilities, which are associated with materially unstable equilibrium states,
being mistaken for limit points. However, the use of the current stiffness parameter will lead to
failure
if
sharp snap-backs are involved and consequently the author has reverted lo thc system
whereby the sign in equation
(4)
follows the sign of the determinant of the tangent stiffness matrix,
K,.. The latter can be obtained from the pivots, D, resulting from the Crout, LDLT, actorization.
Material instabilities can often be overcome by adding a small perturbation to the tensile
In any case, it is unwise to simply ignore the negative pivots which indicate an
unstable equilibrium state.
A s the iterations are applied, the load level is varied by 62, where
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756
M . A. C R I SFI E L D
where
g
is the residual or out-of-balance force vector and
Apo
is the ‘old’ increm ental d isplacem ent.
Equation
(5)
ensure that the new incremental displacement vector,
Ap,
*
Apn= Ap,
+6 + 63-6,
8)
also satisfies the constrain t
of
equat ion (2). If the modified New ton-R aphso n m eth od is used,
K,
in
equa tion (7) is fixed as the tangen t stiffness ma trix ?t the beginning of the increm ent (as in equa tion
3),
so
that the
6,’s
in
(6)
are also fixed and only 6 need be re-computed at each iteration.
Of the two roots to eq uatio n (5), the chosen value ensures
a
positive
Ap;fAp,.
This algorithm will
solve many problems but will sometimes fail, especially when strain-localizations accompany
material softening.23
2 5
Various improvements can be made. In particular, line searches can be
added23
2 5 so
that
A p n = A p , + u ] ~ + 6 ~ 6 , )
(9)
where u] is the step-length param eter which can be chosen to m ake the energy (at the new lo ad level)
stationary . ‘Slack line searches’ have been found to be invaluab le for the disp lacem ent- con trolled
analysis of concrete str~cture.’~nfortunately, for the arc-length me thod , they introduce q in to
the coefficients of (6)
so
that
I
and u] are coupled and a more complex simultaneous solution
is
required. H owever, the difficulties can be o verco me.2 4 O th er imp rovem ents involve the p eriodic
updating of K, using th e N ew to n r at he r t ha n th e m odified N ew to n m e t h ~ d . ’ ~ - ~ ~
The problem of the alternative roots in equation (5) can be ove rcome by a do ptin g
a
form
of
g eneralized d i s p l a c e m e n t - ~ o n t r o l , ~ ~ ~ ~n which certain displacement mo des are prescribed. Th is
technique should be very effective when th e analyst ca n estimate the bu ckling o r sna ppin g modes.
However, for the present concrete structures, these modes ca n be both local and unpredictable.
Consequently, the author has con tinued to use the arc-length method. However, it is true tha t for
these structures, where multiple equilibrium states may exist, there m ay b e difficulties in know ing
which equilibrium paths are being traced an d some times ‘false’ pa ths c an be tem porarily followed.
This phenomenon will be illustrated in the following example.
6
-
Z
400
-
.A
Y
X
-
200
0
----
- - - - - - -
- El
Experimental collapse load G
-
‘ . G u n l o a d i n grc length procedure aused
by
15.2 m 4.3
m
0 50 100
350 400
450
Deflection I m m )
Figure
5.
Load/deflection response for beam-and-slab bridge with
90
per cent initial prestress (to
allow
for creep and
shrinkage)
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R E SPON SE I N C O N C R E T E ST R U C T U R E S
757
A
beam-and-slab bridge and a prestressed T-beam
A
numb er of numerical solutions for beam an d slab problems, th at have involved local maxim a
or snap-throughs, have been described in References 12,
21
and 23-25. The present solution
relates to a prestressed concrete beam -and-slab bridge tha t was tested to destruction at Otta wa ,
Illinois in 1961.36,37 ull details of the finite elemen t mo delling will be given in
a
separate paper.
Fo r the present, we will concentrate on the co mp uted load/deflection response (Figure 5 ) which
was obtained for
a
softening parameter
a
(Figure
2) of
15. This param eter is meant to account
for the tension stiffening.
A first local maximum (point
A ,
Figure 5 ) was obtained at a load of 440 kN which is only
two-thirds of the computed maximum load. This local maximum was detected when an
equilibrium point just beyond the maximum load dro ppe d, the various cracks in th e vicinity of
the three axles (Figure 5 ) localized
so
that only one set of transverse cracks opened while the
others closed. A t point
B
(Figure 5), the tensile stresses in th e lowest set of G au ss points at th e
cracking section reached the fully open position (point C in Figure 2) and consequently the
structure stiffened, the load increased a nd the loca lization vanished. Th is procedu re was repeated
for the two subsequent local maxima
(C
and
D)
nd related to adjacent sets of Gauss points.
Detailed descriptions of a similar set of localizations, that were computed for
a
prestressed
T - b e a ~ ~ , ~ ~re given in References 21 and
23 .
As with the beam-and-slab bridge (Figure 5) , the
compu tations for the T-beam (Figure
6)
produced
a
first local limit load tha t was only two-thirds
re nforcement
Overall length
of beam :
10.75m
Test span
: 9.91
m
I .
t
Bearing
earing
Figure 6. A prestressed concrete
T-beam
with in
si tu
slab
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M . A.
CRISFIELD
58
3
200
-
z
1 0 0
-
L
= ’localised
crack ’
inite
element
0 I I I I 1
0 1 2 3
4
5 6 7
8
9 10
Average curvature
over
’constant
m o m e n t z o n e ’
x
1 0 6 - m m - 1
Figure 7. Response for prestressed T-beam
of the final collapse load. Prior to this stage, a set of local max ims w ere com pu ted with a localiscd
crack (L-Figure
7)
moving from the centre, where the dead load produced the maximum
bending moment, towards the load point like a ‘shock wave’. In Figure
7,
the symbol
C
means
‘cracking’ (line AC -Figure 2), the symbol
S
mea ns ‘shutting’ (line BO -Figure
2)
and the symbol
0
means ‘open’ (line CD-Figure
2) ,
while the symbol
D ,
on the second row of Gauss points
from the bottom, relates to the prestressing steel an d defines the par ticu lar par t of the stress strain
~ u r v e . ~ ’ . ~ ~he total absence
of
any symbol means that the response is linear elastic (either
loading or unloading b ut not ‘shutting’). As with the simple model of Figure 4, the localizations
only occur as the load drops.
Returning to the beam-and-slab bridge of Figure
5,
the local limit points
A , C
and
D
were
later followed by a further local limit point at
E .
At this load level, the tangent stiffness matrix
gave a negative pivott and consequently the negative sign was ado pted in equa tion (4) in ord er
to define the next load increment. Th e spurious eq uilibrium point
F
was then obtained a nd all
the pivots were positive
so
that the load was automatically reversed and proceeded, via the
intermediate equilibrium points to a position th at was very close to th e original poin t
E
(Figure
5).
A
negative pivot was again encountered but,
at
this stage, the correct path EG was followed
and, finally, the comp utation was stopped when
a
prestressing cable reached its ma xim um s train.
This occurred at a load that was very close to the experimental collapse which was also caused
by rupturing of the cables.
’More recent
work
has indicated that this negative pivot
may
have been associated with a ‘non-localised‘
materially-unstable equilibrium path.21
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RESPONSE IN C O N C R E T E ST R U C T U R E S
759
I -
- 1
2286rnrn
Figure
8.
Dimensions and finite element mesh for reinforced concrete beam
A plane-stress analysis
The previous computations invoked the Kirchhoff hypothesis so that plane sections were
forced to remain plane. In contrast, the beam of Figure 8 was analysed using eight-noded
plane-stress elements. The modelling was related to a beam (OA2) tested by Bresler and
S~ordelis .~’he following properties were adopted:
Breadth of beam =
305
mm.
Area of reinforcing steel
=
3227 mm2.
Young’s modulus of the reinforcing steel = 21
8,000
N/mm2.
Young’s modulus of the concrete = 24,000N/mm2.
Softening factor, a = 10.0 which, for the adopted mesh (Figure 8) gives a fracture energy,
G
-
100N/m.
Constant shear retention factor,
N o details are given on the effective stress-strain relationship in compression because only
the linear part of the adopted curve was used for the results that will be discussed.
I t
can be
seen from Figure
8
that no overhang was provided beyond the support in order to anchor the
reinforcing bar. In addition, ‘spreader plates’ should have been introduced to distribute both
the load and the reaction. Consequently, the idealization is inadequate even for a coarse mesh.
However, interesting results can stem from mistakes and consequently the solutions will be
described, although it should be borne in mind that they relate to a beam with reinforcement
that is improperly anchored.
Standard displacement control was adopted for the first analysis which proceeded satisfactorily
until about load point A on the load/deflection plot of Figure
9.
At this stage, a negative pivot
was encountered. By ignoring this phenomenon, i t was possible to proceed with the analysis.
However, in order to investigate the cause, an eigenvalue analysis was performed on the s tructure
of Figure
10
and the eigenmode corresponding to the negative eigenvalue was as shown in
Figure 1 1 and implies a materially unstable state because both points A and B have cracked. A
later analysis, with slightly different increment sizes, produced a stable checkerboard cracking
state below the reinforcement with point B remaining uncracked. A similar phenomenon was
encountered by Dodds et aL4’ Both the ‘stable’ and ‘unstable’ solutions produced load/deflection
plots that were similar to the curve O A B in Figure 9. However, no matter how small the
displacement-increments were made in the vicinity of point B, no converged equilibrium state
could be achieved beyond this displacement level (compared with point
G
in Figure 4).
In an attempt to overcome the problem, the beam was reanalysed using the arc-length method
and it is this solution that is actually plotted in Figure 9. A t the maximum load (point
B ) ,
the
= 0.2
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760
M.
A . C R I SFI E L D
160
140
120
100
-
a 80
’D,
_I
60
40
20
0
0.00 0.80 1.60
2.40
3.20
4.00 4.80
Deflection
under
load
(mm)
Figure 9. Relationship between load a n central deflection
for
reinforced concrete beam (plan e stress)
c
Figure 10. ‘Unstable’ deformation and cracks (near point A , Figure 9) (magnification factor for deformation =
100)
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,, ,, I
I I
I1
1
,, ,
1
, 1
- -
76 1
a
, 1 ( ,
1
I , (1 4 ,
I 4 1
a 4 ,
-
I
e
(a) A t local maximum (point
B
-
F i g
9
e
b) With reduc ing load (poin t C -
F i g
9
(c) With final increasing load (point E
-
F i g . 9 )
Figure 12. Deformations an d crack-pattern s for reinforced concrete beam (pane stress) (deform ationsmagnified by a factor
of 100). Cracks:-opening;---'closing'
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762
M .
A .
C R I SFI E L D
computed crack p attern was as show in Figure 12(a) with the crack A , along the reinforcing
bar, having only just occurred. Without the arc-length method, this crack could not have been
detected.
O n factorizing the tang ent stiffness ma trix for the next increm ent, one negative pivot was
encountered and consequently the sign of the next load increment was automatically reversed.
The load/deflection response then proceeded down the curve
BCD
of Figure
9
with the crack
patterns being as shown in Figure 12(b). Clearly, a localized failure is occurring a bo ve the
support while all the other cracks are ‘closing’. (In fact, the computer program merely indicated
whether the stress state was on A C (Figure 2) o r
O B
(Figure 2), with the terminology ‘closing’
being applied t o the latter, althou gh strictly the stress cou ld be m oving from
0
to B ) .
I t
might have been anticipated that this very sudden ‘brittle failure’ was the end of the
‘structure’, but the computer model provided a local minimum at
D
(Figure
9)
followed by the
new rising load/deflection relationship D E F . At the minimum, the two cracks abo ve the support
almost simultaneously reached the fully-open position
C
of Figure 2. Th e tang ent stiffness ma trix
was then positive definite, so that the solution procedure automatically increased the load and
the rising equilibrium curve D E F was produced: load could have been increased well beyond
point
F ,
but the solution was aba ndo ned as being unrealistic (Figure 12c). In p articular, a
truncated form of ‘hour-glass mode’41 app ears to have developed in the element ab ove the
support. Dodds et
d ”
ave also reported unusual results for similarly under-integrated elements
following the formation
of
a horizontal crack above the support . In order to improve the supp ort
conditions, stiff elastic elements A and B , Figure
13)
were added t o simulate the spreader an d
anchorage plates. An initial analysis with un iform
2
x 2 integration was very quickly in trouble
because the linear elastic element over the support developed an ‘hour-glass’ made. It is well
known that isolated elements are susceptible to such m odes an d c onsequently 3 x 3 integration
was used for the two linear elements A and B . However, difficulties were again encountered
and as illustrated in Figure 13; the deflected shape (with the displacements to an exagerrated
scale) indicated that ‘hour-glassing’ had infected the results.
A
simplc
full-out tesl
Although i t is well known that the eight-noded element co ntains a single ‘hour glass’ mech anism,
i t is argued4’ that the mode cannot propagate in an assembly of elements. Both the last example
E
Figure 13
Deformed mesh sho wing the effects of ‘hour-glassing‘
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R E S P O N S E I N C O N C R E T E S T R U C T U R E S
- _
763
P
i
1
m
A,
=
400mrnz
A’
E = 30 000N/mm2.
v c =
0.15
Ut
=
3Nlmm2,
rmax 5Nlmm2
Gf=GON/m.
E,
=
200 000N/mm 2
p =
0.1
Figure
14.
Idealized two-element pull-out test
and the previous work of Dodds
et
and de Borst et
~ 1 . ~ ~
ast doubts on the validity of
this assertion for cracked concrete elements. In order to clarify the matter, i t was decided to
analyse a very simple structure using both 2
x 2
and
3 x
3 integration.
A
two-element pull-out
test was therefore devised (Figure 14) and, by invoking symmetry, it became a single element
test. Although only one element is involved, the test should still provide useful data on the ‘hour
glass’ mode because the centre-line (AA’-Figure
14)
is forced to remain straight so that the
mode should not develop.
The computed static load/deflection relationships are shown in Figures
15
and
16
and would
24
a
. 16
U
8
0
0 00 0.16 0 . 32 0.48
Displacement (mm)
Figure IS. Load/deflection relationship for
pull-out test
(2 x 2
integration)
32
24
-
x
16
TI
m
8
0
1
0.00
0
80 1 6 0 2
40
Displacement
(inrnl
Figure 16. Load/deflection relationship for
pull-out test 3
x
3 integrat ion)
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764 M .
A.
C R I S F I E L D
clearly involve snap-throughs and snap-backs if by dynamics were allowed. By comparing the
crack patterns (Figures 17 and 18) with the equivalent load/deflection relationship (Figures I5
and 16), it can be seen that the descending equilibrium paths are associated with strain-
localizations so that the cracks close in regions away from the localized zone. This behaviour
is in line with the earlier observations including those relating to the simple model of Figure 4.
Figures 17(c) and 17(d) show that a truncated form of hour-glass mode has dominated the
response and that this mode has developed despite the elastic behaviour of the two Gauss points
on the right-hand side. A spurious mechanism is clearly involved. Comparing the load/deflection
relationship of Figure 15 with that of Figure 16 it can be seen that, as a result of this mechanism,
the
2
x
2
integration has produced a fare more flexible response than the
3
x
3
integration. The
plateau around point 4 in Figure 16 was induced when the specified maximum shear stress of
5 N/mm2 was reached across some of the cracks. A t this stage, the computer program allowed
no
further increase in these shears. This facility was input as a measure to partially compensate
for the limitations of the constant shear retention factor p). The limiting shear stress was only
applied
to
the out-of-balance forces,
g,
and was not incorporated in the tangent stiffness matrix.
I t was not reached when
2
x
2
integration was adopted.
Although the physical significance of these analyses may be questioned, it is worth briefly
examining the computed crack histories
if
only to emphasize the complexity of the solutions.
Considering, first, the analyses with 2 x 2 integration (Figures 15 and 17), the first nonlinearity
(point I Figure 15) was associated with the cracking of Gauss point
1 ,
Figure 17(a). This was
followed by the first limit point which occurred just beyond point
2,
Figure 15, when the second
Gauss point (point 2) in Figure 17(b) cracked. Beyond this stage, the load reduced while crack
Cracks openinq- closing - - -
(a ) Point 2
(b l
Point 3
c ) Point
5
F'igurc 17. Deformations and cracks for
pull-out test ( 2 x 2 integration)
dl Point
7
Cracks openi ng closing
- -
-
(b) Point 2a) Point 1
l c ) Point 3
d )
Point 4
Figure 18. D ef or mat ions and cracks for
pull-out test (3 x 3 integrat ion)
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RESPONSE
I N
CONCRETE STRUCTURES
765
1
closed and the stress in the adjacent Gauss point of the reinforcement (point A-Figure 17(b))
reduced.
The local minimum at point 4 was related to the attainment of the fully-open position (point
C Figure
2)
in crack
2
(Figure 17) and the rising line 4-5 (Figure 15) introduced increasing
strains at all of the Gauss points. The second maximum, at point
5
of Figure
15,
coincided
with
the formation of a third crack 3 (Figure 12c) which was orthogonal to crack 1 . Following this
cracking, the load reduced (line 5-6, Figure 15) with the cracks localizing so that only crack
3
was opening. The local minimum, near point 6-Figure 17, was associated with the attainment
of the fully-open position in this same crack. At this stage, there was no remaining load capacity
in the system, although the path 6-7 (Figure 15) was traced with crack 3 (Figure 17d) opening
while the other cracks ‘closed’. When 3
x
3 integration was adopted, the cracking history was
even more complicated. An indication can be obtained from Figures 16 and 18. The circled
numbers in the latter figure indicate the order of cracking. As in the earlier figures, the caption
‘closing’ relates to any stress state on line O B of Figure 2.
CONCLUSIONS AN D DISCUSSION
I t has been shown that the strain softening, introduced in many concrete models, can lead to
local maxima that, under load control, would lead to dynamic ‘snap-throughs’. In the
pseudo-static response that follows these maxima, the reducing load is accompanied by strain
localizations in which the adjacent material unloads semi-elastically. If the softening is steep
enough and their is sufficient adjacent material, snap-backs can also be encountered. In some
cases, snap-throughs can be traced pseudo-statically using displacement control although, for
concrete problems, line searches or other sophistications will be required.25 More generally, the
arc-length method can be used, but again added sophistications may be necessary.22p24
In part, the computed snapping phenomena are physical and in part they are numerical and
relate specifically to the adopted mesh. However, even for a smeared model, it is possible to
limit the mesh-dependency by making the degree-of-softening
E)
nversely proportional
to
a
characteristic length at a Gauss point. To this end, the fracture energy can be used. However,
severe difficulties remain for reinforced concrete in which a progression of localizing cracks may
occur. The accompanying load/deflection response is likely to be artificially jagged for coarse
meshes.
In many instances, the analyst will be uninterested in the finer details. However, he cannot
simply overcome the problem by adopting a smeared model because the localizations will still
occur. Cruder solution techniques with coarser convergence tolerances might avoid the high
cost of tracing the locally fluctuating equilibrium path. However, such a solution procedure
would also fail at, or near, the final collapse. Particularly for brittle failure, it would seem
important to achieve equilibrium states at or jus t beyond the maximum load so that the cause
of the collapse can be investigated.
A
possible solution involves a two-stage analysis with a
sophisticated final analysis being re-started from an earlier crude solution. However, especially
with the path dependency, there may be difficulties in recovering from an analysis that has
strayed too far from equilibrium. It may even be necessary to introduce dynamics or
p ~ e u d o - d y n a m i c s . ~ ~ , ~ ~
The present results coincide with those of other worker^^^,^ in casting doubt on the integrity
of the (2
x
2) under-integrated 8-noded membrane element for problems involving the cracking
of concrete. The ‘hour-glass’ mechanism that is normally prevented by the adjacent elements
can nevertheless form once sufficient cracking has occurred. The difficulties of ‘local mechanisms’
are always likely to be present with a softening model. In such circumstances,
i t
would seem
wise to avoid the use of under-integrated elements.
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766 M . A . C R I S F I E L D
A C K N O W L E D G E M E N T S
The work described in this paper forms part of the programme of the Transport and Road
Research Laboratory and is published by permission of the Director. The author would like to
acknowledge the help of his colleague, Mr. J. Wills.
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