Snap‐Through and Snap‐Back Response in Concrete Structures and the Dangers of Under‐Integration - Crisfield

8/19/2019 Snap‐Through and Snap‐Back Response in Concrete Structures and the Dangers of Under‐Integration - Crisfield

1/17

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


2/17

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


3/17

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 ,


4/17

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


6/17

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


8/17

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


9/17

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


10/17

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)


11/17

,, ,, 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'


12/17

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‘


13/17

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)


14/17

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)


15/17

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.


16/17

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.

1.

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19

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Documents

Snap‐Through and Snap‐Back Response in Concrete Structures and the Dangers of Under‐Integration - Crisfield