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Historical Constructions, P.B. Loureno, P. Roca (Eds.), Guimares, 2001 685

ABSTRACT: With the construction of the North-South line metro link in Amsterdam, historicalmasonry facades face the risk of being damaged due to settlements. The situation may be criticalbecause of the combination of very soft soil, timber pile foundations and brittle unreinforced

masonry. In this paper, some aspects of structural modelling to simulate settlement damage inhistorical masonry are studied. The models are applied at large, structural scale to facades of

say, 20 meters height. This scale differs a factor 100 from the scale of the notched laboratoryspecimens that are usually taken as validation targets for numerical models. It will bedemonstrated that this large scale in combination with the relatively brittle masonrycharacteristics leads to a very jumpy snap-back behaviour, which is quite different from thebehaviour usually observed in laboratory-size specimens. The main focus is on a new

sequentially linear continuum model for fracture in quasi-brittle materials. This model

approximates the tensile softening stress-strain curve as a saw-tooth diagram. After a linearanalysis, the critical element, i.e. the element for which the stress is most close to the currentpeak in the saw-tooth diagram, is traced. Next, the stiffness of that element is reduced and theprocess is repeated. With this sequentially linear model, the very sharp snap-backs associatedwith brittle fracture of a historical masonry facade automatically come out, whereas nonlinear

analysis of the facade using smeared or discrete softening crack models shows substantialdifficulties.

Figure 1 : Amsterdam Central Station, by Cuijpers, 1890. Historical masonry, undergoing the risk of

settlement damage due to subsurface building activities.

The role of structural modelling in preserving Amsterdamarchitectural city heritage

J.G. RotsFaculty of Architecture, Delft University of Technology, P.O. Box 5043, 2600 GA Delft, Netherlands;

also TNO Building and Construction Research, Delft, Netherlands

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686 Historical Constructions

1 INTRODUCTION

In Amsterdam, a new North-South metro link is being planned underneath the urban historicalcity centre. This implies that historical masonry facades and monuments like the Amsterdam

Central Station, build around 1890 by the well-known architect Cuijpers, face the risk of beingdamaged due to the settlements imposed by the subsurface building activities. Fig. 1 shows thebuilding which consists mainly of masonry on fragile timber pile foundations, being sensitive to

imposed displacements. Fig. 2 shows the planned construction underneath the existing station,with very deep excavation walls and a table structure on top of it, including possibilities forjacking. These activities will definitely lead to settlements. The total system of soft soil, pilefoundation and masonry building should be looked upon in order to be able to provide a

settlement risk assessment. Large societal and financial risks are involved with a view to claimsof building owners. Here, experience and existing simplified design approaches can be enriched

by research using computational crack models. The research contributes to culturalsustainability or cultural durability as it helps to protect city heritage.

Figure 2 : Subsurface building activities planned underneath Amsterdam Central Station, leading to

settlements and thus risks of cracking and damage in the existing masonry.

2 REVIEW OF CRACK MODELS

Accepted ways to model cracking in quasi-brittle materials are the so-called smeared and

discrete crack methods with tensile softening. Smeared models consider the cracked material as

a continuum, whereby the formulation can be in terms of e.g. decomposed strain frameworks

(e.g. Rots 1993), Rankine plasticity based frameworks (e.g. Feenstra and de Borst 1995 for

isotropic concrete, Lourenco 1996 for orthotropic masonry), total-strain based concepts (e.g.

Feenstra et al. 1998) or concepts of damage mechanics. A possibility for discrete cracking is to

use interface elements that are predefined in the mesh based on engineering judgement (e.g.

Rots et al. 1994). The advantage of the latter approach is that the behaviour of the bulk material

in between the cracks is totally separated from the softening crack behaviour. Extensions of the

models to include visco-elasticity to simulate creep, have been made by Van Zijl (2000).

For the Amsterdam Central Station, studies using the standard smeared crack or interface

element approach were recently carried out, with a preliminary report made up by t Hart, Rots

and Schat (2001). These studies contributed to insight in the complex stress redistributions and

damage processes induced by the intervention, and helped in designing mitigating measures like

jacking schemes for the table structure. Although the inclusion of creep and options of stagedconstruction sequence are important considerations, it appears that the major problem with these

analyses is the explosive nature of cracking. Initially, unimportant cracks occur at window

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corners, but suddenly, at the top of the faade a catastrophic crack occurs, displaying a sharp

snap-back in the settlement versus crack width relation.The importance of capturing explosive cracking also emerged from a related study that

considered a representative Amsterdam masonry faade of three house units, underpassed by a

bored tunnel, again connected to the plans for the North-South metro link. This case is lesscomplex than the station and will be considered in more detail. Figs. 3-5 summarize results byRots (2000). The facade represents three house units with a length of 20.4 m and height 15.5 m.

Please note that these dimensions are no less than 150 to 200 times larger than the dimensionsof popular notched laboratory beams that usually have a height of approximately 100 mm. First,the facade is subjected to dead load, which leads to vertical pre-compression in the masonry,while horizontal stresses partially crept out over the long term. Subsequently, a settlement

trough in the form of a Gauss curve is imposed which is incremented until and beyond fracture.The settlement trough leads to bending and shear and superposes tensile stresses to the existing

pre-compression, which may lead to cracking. The aim is to predict (post-peak) crack widths asthis is the main damage process that governs the risk. Fig. 3 shows the results for a displacementcontrolled smeared crack analysis and for an arc-length controlled discrete crack analysis, interms of the angular distortion, which is related to the magnitude of the settlement trough,

versus the maximum crack width in the facade. Fig. 5 shows the deformed mesh for the smearedcrack analysis. For the discrete crack analysis, interface elements were predefined at the verticalline where localization was found in the smeared crack analysis. For details of all parametersetc., the reader is referred to Rots (2000). Here, we only display the results, because it is a goodillustration of the fact that the behavior is extremely brittle. The large scale in combination withthe relatively low fracture energy of masonry leads to a very sharp snap-back in the angulardistortion versus crack width diagram of Fig. 3. Only with the discrete crack analysis, this snap-

back could be followed partially using special arc-length schemes. However, the crack has tosnap four times, from window to window in the vertical direction. This saw-tooth four-step

nested path could not be traced completely, and sometimes luckily a rigorous jump to another

tooth was found, see the zoomed-in graph in Fig. 4. After the difficult regime, the discrete crackanalysis produced a rising curve again, corresponding to a fully broken facade.

This second branch makes it possible to find the post-peak crack width, which suddenly jumpsfrom approximately 0.5 mm at peak to 18 mm after fracture completion. The discrete crack path

was converged. With the smeared crack analysis, no convergence could be found near and

Figure 3 : Angular distortion vs. maximum

crack width in masonry faade subjected tosettlement trough. Smeared (dashed) anddiscrete crack (drawn) analysis.

Figure 4 : Zoom-in of snap-back path in Fig. 3.

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beyond peak. Use of a tangent stiffness scheme could not be maintained, while arc-length

methods failed, independent whether a decomposed strain based crack model, a total strainbased crack model or a Rankine plasticity based crack model was adopted. The dashed line in

Figs. 3 and 4 corresponds to a rough displacement controlled 'push-through' analysis whereby

the secondary branch shows unconverged states. The very brittle response is jumped over andsignificant stress-locking was observed.

The irregular snap-regime of Fig. 4 gave rise to the idea of sequentially linear analysis where

the local peaks emerge automatically rather than delicately trying to pass them using arc-length

procedures in nonlinear analysis. Furthermore, inspiration came from lattice models, e.g.

Schlangen & Van Mier (1992) and Beranek & Hobbelman (1995). In their publications, muchemphasis is placed on the imitation of the continuum by a lattice. However, in the author's

opinion the charm of lattice models is probably not so much the replacement of the continuum

by a lattice, but rather the fact that lattice models adopt sequentially linear techniques. After a

linear analysis, the most critical beam element of a lattice is traced and subsequently removed,

after which the next linear analysis is carried out and the process is repeated. The question is

whether the charm of sequentially linear analysis can also be gained directly for the continuum,

so that the notions of elasticity, strength and fracture energy can be preserved at macro-level.

Together with the motivation related to sharp snap-backs, this marked the starting point for the

study below.

3 SEQUENTIALLY LINEAR PROCEDURE WITH SAW-TOOTH SOFTENING

3.1 Global procedure

The structure is discretized using standard elastic continuum elements. Youngs modulus,Poissons ratio and a tensile strength are assigned to the elements. Subsequently, the followingsteps are sequentially carried out:

Add the external load (e.g. the prescribed settlement trough) as a unit load. Perform a linear elastic analysis. Extract the critical element from the results. The critical element is the element for which

the principal tensile stress is most close to its current strength. This principal tensile stress

criterion is widely accepted in mode-I fracture mechanics of quasi-brittle materials.

Calculate the critical global load as the unit load times the current strength divided by stressof the critical element.

Figure 5 : Deformed mesh of masonry faade subjected to

settlement trough, smeared crack analysis.

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Extract also a corresponding global displacement measure, so that later an overall load-displacement curve can be constructed.

Reduce the stiffness and strength, i.e. Youngs modulusEand tensile strengthftof thecritical element, according to a saw-tooth tensile softening stress strain curve as described in

the next section. Repeat the previous steps for the new configuration, i.e. re-run a linear analysis for the

structure in whichE andftof the previous critical element are reduced.

. Repeat again, etc. .. Construct the overall load-displacement curve by connecting all load-displacement sets

consecutively found in the above steps.

Plot deformed meshes. These plots reveal the fracture localization because the series ofcritical weakened elements will display the largest strains, representing crack width.

3.2 Saw-tooth softening model via stepwise reduction of Youngs modulus

The outcome of the above scheme depends on the way in which the stiffness and strength of thecritical elements are progressively reduced. This constitutes the essence of the model. A very

rough method would be to reduce E to zero immediately after the first, initial strength is

reached. This elastic perfectly brittle approach, however, is likely to be mesh-dependent as it

will not yield the correct energy consumption upon mesh refinement (Bazant and Cedolin

1979). In this study, the consecutive strength and stiffness reduction is based upon the concept

of tensile strain softening, which is fairly accepted in the field of fracture mechanics of concrete

(Bazant and Oh, 1983).

The tensile softening stress-strain curve is defined by Youngs modulusE, the tensile strength

ft , the shape of the diagram, e.g. a linear or exponential diagram, and the area under the

diagram. The area under the diagram represents the fracture energy Gfdivided by the crack band

width h, which is a discretisation parameter associated with the size, orientation and integration

scheme of the finite element. Although there is some size-dependence, the fracture energy canbe considered to be a material property. This softening model usually governs nonlinear

constitutive behaviour in an incremental-iterative strategy. Please note that here we adopt the

curve only as a 'mother' or envelope curve that determines the consecutive strength reduction in

sequentially linear analysis. In the present study, attention is confined to a linear softening

diagram, but extension to any other shape of the diagram is possible. For a linear softening

diagram, the ultimate strain u of the diagram reads:

u = ( 2 Gf ) / (ft h) (1)

In a sequentially linear strategy, the softening diagram can be imitated by consecutively

reducing Youngs modulus as well as the strength. Youngs modulus can e.g. be reducedaccording to:

Ei =Ei-1 / a for i= 1 to n (2)

with idenoting the current stage in the saw-tooth diagram, i-1 denoting the previous stage in the

saw-tooth diagram and abeing a constant. When ais taken as 2, Youngs modulus of a critical

element is reduced by a factor 2 compared to the previous state. n denotes the amount of

reductions that is applied in total for an element. When an element has been critical ntimes, it is

removed completely in the next step. This complete removal can be done explicitly so that a

hole in the mesh occurs for full cracks, or it can be approximated by maintaining the element

but giving it a very low residual Youngs modulus for reasons of computational convenience

(e.g. 10-6 times the initial Youngs modulus). The reduced strength fti corresponding to the

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reduced Youngs modulus Ei is taken in accordance with the envelope softening stress-strain

curve:

fti = uEi (D / (Ei + D ) ) (3)

with

Ei =E / ( ai (4)

and

D =ft / ( u ( ft / E ) ) (5)

being the tangent to the tensile stress-strain softening curve. Note that this is the softening curve

in terms of stress versus totalstrain, i.e. the sum of elastic strain and crack strain of an imaginedcracked continuum. The diagram includes the initial rising branch, which is steep compared tothe downward slope in case of small-scale elements (small crack band width) and/or highfracture energy. As an example, Fig. 6 shows the envelope softening curve and thecorresponding saw-tooth curve for an init ial Youngs modulusE of 38000 N/mm

2, initial tensile

strengthftof 3 N/mm2, fracture energy Gfof 0.06 N/mm, crack band width h of 5 mm, factor a

equal to 2 and number of reductions n equal to 10.Strictly speaking, the sequentially linear approach corresponds to a set of unconnected lines of

different slope, starting from the origin up to the current strength. For reasons of presentation,the lines have been connected into one discontinuous curve, by letting the stress drop verticallyfrom the peaks to a residual level at which the new rising branch of reduced slope passes. The

model is simple. It always provides a solution, as ill-conditioning or divergence does not appearin sequentially linear analysis. A physical explanation to the model is that fracture is a gradual

separation process whereby the net cross section that connects material, and thus the stiffness, isgradually reduced. An advantage of the model is that the regular notions of fracture mechanics,

like the principal tensile stress criterion, the envelope strength and fracture energy aremaintained which helps in reaching realistic energy consumption and toughness as observed in

experiments.

The model has been elaborated here for continuum crack methods, using softening stress-

strain curves. Sequentially linear saw-tooth approaches can also be employed for discrete crack

methods with stress-relative displacement curves in interface elements or embedded

representations. Unloading of softening zones is possible and inherently of the secant type,

because of the reduction of elasticity.

3.3 Alternative definition of saw-tooth diagramIn the previous section, the starting point for defining the saw-tooth diagram was a consecutive

reduction of Youngs modulus via the factor a while the corresponding reduced strength was

subsequently determined from the envelope curve according to (3). In first trials of the model,

the starting point was taken alternatively by reducing the strength from ft to zero in n steps,

while the corresponding reduced Youngs modulus was subsequently computed from the

envelope curve. The decreasing line of the envelope curve is in fact split into n equidistant

portions. When n is assumed to be 10 and when the element is critical for the first time, the

strength is reduced to 90% of the original strength. When it is critical for the second time, the

strength is reduced to 80% of the original strength, etc. The resulting saw-tooth diagram for this

alternative approach is depicted in Fig. 7, using the same parameters as subject to Fig. 6. Other

alternative ways of stepwise reduction might be used as well.

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4 ANALYSIS OF NOTCHED BEAM

4.1 Geometry and meshes

To test the model, a symmetric notched beam of total length 500 mm, span 450 mm, height 100

mm, thickness 50 mm and notch depth 10 mm was selected for analysis. The distance between

the loading points in the symmetric four-point loading scheme is 150 mm. In an earlier study

(Rots 1993) this beam served the purpose of illustrating stress-locking for smeared crack models

and prevention thereof via isotropic damage formulations. In the present study, on purpose a

regular straight mesh was adopted, so that the solution is not affected by any disturbing effect

due to zig-zag crack band paths. First, the coarse mesh of Fig. 8 was used, with a symmetric

center crack band of 20 mm. The element height for the central elements was taken the same as

the element width, so that the amount of center elements over the depth of the beam amounts to

5. The ultimate strain u of the envelope input curve was adapted to h according to Eq. (1).

Four-node linear elements were used. These were integrated using a two by two Gaussian

scheme, except for the elements in the center band which were integrated using a single center-

point integration. This means that, if an integration point of an element in this center band

reaches a local peak in the saw-tooth curve, Youngs modulus of the entire element can be

reduced. The material parameters are the same as mentioned in section 3 for defining Fig. 6.

Poissons ratio was taken as 0.2. The model of Fig. 6 was adopted. The factor ain Eq. (2) was

taken as 2. The factor nwas taken as 10, i.e. the stiffness of an element can be reduced 10 timesaccording to Eq. (2), assuming the element becomes 10 times critical in a global sense. Beyond

that, the element is removed.

Figure 8 : Coarse and fine mesh

for mode-I fracture analysis of notched beam.

Figure 6 : Envelope softening stress-strain

diagram (dashed) and saw-tooth approximationdrawn .

Figure 7 : Envelope softening stress-strain

diagram (dashed) and alternative saw-toothapproximation (drawn).

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4.2 Results

Fig. 9 shows the result for the coarse mesh in terms of the total load (sum of the two loads)versus displacement at the loading points. The curve is constructed by connecting the criticalloads and the corresponding displacements for all linear analyses that have been executedsequentially. As a comparison, the reference curve (see also Rots 1993) from a nonlinearsoftening analysis with the same parameters is included in the graph.

Figure 9 : Load-displacement curve, coarse mesh. Dashed: nonlinear. Drawn: sequentially linear.

The load-displacement curve from sequentially linear analysis is irregular. This is because theprocess of elements becoming critical is discontinuous. If two elements have almost the same

ratio of stress versus current strength, still the most critical one is selected first, but in the nextstep the other one will fail very soon, maybe at a lower load and lower displacement than in the

previous step. The irregular saw-tooth response at global level can be interpreted as the globalpendant of the saw-tooth input at local level.

For the present mesh, the agreement between the sequentially linear result and the nonlinearreference result is very good. The envelope of the discontinuous curve matches the continuouscurve. Please note that the model is especially derived with a view towards large-scale brittle

fracture analysis, i.e. large element sizes or coarse meshes, which is thus promising. Theadvantage of the sequentially linear result is that no incremental-iterative solution technique is

required and negative tangent stiffness due to softening is absent.

4.3 Effect of mesh fineness

Mesh objectivity is a major issue in fracture analysis. With smeared softening models, the

ultimate strain of the softening diagram should be adjusted to the element size to achieve correctresults (Bazant & Oh 1983). With lattice models, the issue of mesh objectivity has not yet beenproperly addressed. For mesh-objectivity of the present sequentially linear continuum model,the reader is referred to Rots (2001). As an example, Fig. 10 shows the result for the four timesrefined mesh of Fig. 8, using the same parameters as for the coarse mesh. The fine meshproduces a smoother response (more peaks and less crude jumps) but leads to a lower peak load,which deviates from the correct pak load from the nonlinear reference analysis that is to be

mimicked. This demonstrates that sequentially linear models are in principle not objective withrespect to mesh refinement. This is because increasing mesh fineness leads to sharper stress

peaks at the crack band tip, so that the strength at the crack band tip is reached earlier than for acoarse mesh. In fact, a kind of zip-fastener effect is introduced. With a sudden full removal of

elements, i.e. a diagram with only one saw-tooth instead of the present ten saw-teeth, this effectwould be even more pronounced. That was described already by Bazant & Cedolin (1979), who

also indicated a method of strength enhancement to remedy this problem.

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for a plane stress situation, with 0 referring to the preceding load set(s) and referring to theincrement for the current load set. This is more cumbersome than for the case of the initialstresses being zero. From the resulting equations for the principal stress according to Mohrs

circle, can be solved directly using e.g. Maple software, or numerically.

In the present study this was not yet implemented. The performance of the model can quite aswell be illustrated by using a fictitious, proportional load. Here, a vertical point load at the top

of the faade, slightly off-center, was taken as an arbitrary example. Fig. 12 shows the result interms of the vertical point load versus displacement. Fig. 13 shows the deformed mesh at one ofthe final steps of the sequentially linear procedure.

The result reveals the very sharp snap-back behavior, which is found in the sequentially linear

fashion without any numerical problems. In the post-peak behavior, we observe four nestedsnaps, which correspond to the subsequent jumps of the crack from window to window, startingat the bottom and ending up at the top of the faade. These snaps now automically come out.This is an adequate alternative to the incremental-iterative nonlinear analyses of section 2.

A minor aspect is that the present result showed partial stress locking. This was because themesh consisted of quadratic eight-node elements, integrated by a three by three Gaussian

scheme, while the stress at the fifth, central integration point was taken as an element average tobe compared with the current strength in the saw-tooth curve. This procedure can be made moreaccurate by using a mesh of lower order triangular elements, so that a single integration pointgoverns the total element behaviour and false stress transfer is impossible.

6 SUMMARY AND DISCUSSION

To protect historical buildings, there is a need for predictions of possible cracking damage, and

subsequently for prevention and mitigation thereof. To make the predictions, computational

models become more and more attractive. In this paper, it has been shown that a major problem

in this field is the relatively brittle behaviour of the unreinforced masonry in connection with the

relatively large scale of the buildings. This combination has been shown to lead to very jumpy

or snappy behaviour in terms of load (e.g. settlement trough) versus displacement (e.g. crack

width) characteristics, which gives significant convergence problems in case of incremental-

iterative softening analysis.

To circumvent these problems, a new, sequentially linear continuum model for large-scalefracture has been proposed. The model approximates the envelope softening stress-strain curveby a saw-tooth diagram. In each linear analysis, a critical element is traced by comparing

Figure 13. Deformed mesh for final stage in

Fig. 12.

Figure 12. Load-displacement curve for sequentially

linear analysis of masonry faade under point load.

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element stress with current element strength, i.e. with the current peak in the saw-tooth diagram.

Next the stiffness and strength of the critical element are reduced according to the subsequenttooth of the diagram, and the process is repeated.

The charm of this model is that words like iteration, ill conditioning and divergence do not

appear in the vocabulary. A notched beam analysis shows the ability of the method to reproducethe behavior of a nonlinear reference analysis on the same parameters. The advantage of themodel especially appears in case of brittle, complex snap-back behavior for large-scalestructures with coarse meshes. This was demonstrated via the example of a masonry facade of

three house units, representative for the Amsterdam situation with settlement due to boredtunnelling processes. The complex brittle behaviour automatically came out, whereas nonlinear

analysis failed in an early stage. Further extensions of the model are required, see also Rots(2001), e.g. with regard to strength enhancement for mesh size objectivity, and a non-localprincipal stress criterion for mesh directional objectivity. The present formulation of the modelis isotropic because Youngs modulus is reduced for all directions. Extension of the model to

fixed planes of orthotropy is possible, including memory of unloading/reloading and full crackclosure and re-opening. An extension of the model to non-proportional loadings has been

outlined, which is important in case of historical buildings as we always start with a verticaldead load which is subsequently superimposed by e.g. a settlement trough, wind load orearthquake load. The sequentially linear approach can also be used in combination with a no-tension instead of a softening assumption for the masonry, aiming at providing a safe lowerbound for the cracking scenario in historical buildings.

The models have also been applied to the front facade and back facade of the AmsterdamCentral Station. These results are not yet disclosed, but will be presented at the conference.

The numerical simulations help to design mitigating measures that have to be taken in order topreserve the historical buildings. They also serve to enrich existing analytical models like

equivalent tensile strain models (e.g. Burland and Wroth 1974) for settlement risk assessment.The analytical models have been carefully validated for the UK situation, but may not be

directly transformed to the Amsterdam situation, because the masonry, the building style, the(pile) foundations and the soil characteristics differ significantly. Here, numerical modelling canplay a supporting role, see e.g. Rots (2000).

ACKNOWLEDGEMENT

Financial support from Delft Cluster and the Netherlands Technology Foundation is

acknowledged. The research was carried out using an adapted version of DIANA.

REFERENCES

1. Bazant, Z.P. & Cedolin, L. 1979. Blunt crack band propagation in finite element analysis. ASCE J.Engineering Mechanics Division 105(2): 297-315.2. Bazant, Z.P. & Oh, B.H. 1983. Crack band theory for fracture of concrete. RILEM Materials and

Structures 16(93): 155-177.3. Beranek, W.J. & Hobbelman, G.J. 1985. 2D and 3D-Modelling of concrete as an assemblage of

spheres: revaluation of the failure criterion. In F.H. Wittmann (ed.), Fracture mechanics of concretestructures; Proc. FRAMCOS-2: 965-978. Freiburg: Aedificatio.

4. Burland, J.P. & Wroth, C.P. 1974. Settlement of buildings and associated damage. Proceedings ofConference on Settlement of Structures, Cambridge, pp. 611-654, 1974.

5. Feenstra, P.H., Rots, J.G., Arnesen, A., Teigen, J.G. & Hoiseth, K.V. 1998. A 3D constitutive modelfor concrete based on a co-rotational concept. In R. de Borst et al. (eds.), Computational modelling ofconcrete structures; Proc. EURO-C 1998: 13-22. Rotterdam: Balkema.

6. Feenstra, P.H. & Borst, R. de. 1995. A plasticity model and algorithm for mode-I cracking in concrete.Int. J. for Numerical Methods in Engineering 38: 2509-2530.

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7. t Hart, M., Rots, J.G. & Schat, B.J. 2001. Masonry Amsterdam Central Station Modelling andanalysis. Internal report TNO Building and Construction Research/TU Delft, on behalf of Arcadis

BouwInfra, 69 pp, Delft, Netherlands.8. Lourenco, P.B. 1996. Computational strategies for masonry structures. Dissertation, Delft University of

Technology, Delft, The Netherlands.9. Rots, J.G. 1993. The smeared crack model for localized mode-I tensile fracture. In F.H. Wittmann (ed.),Numerical models for fracture mechanics: 101-113. Rotterdam: Balkema.

10.Rots J.G. (Ed.) 1994. Structural Masonry An experimental/numerical basis for practical design rules.CUR-report 171 (in Dutch). Also published in English by Balkema, Rotterdam, 1997.

11.Rots, J.G. 2000. Settlement damage predictions for masonry. In L.G.W. Verhoef and F.H. Wittmann(eds.), Maintenance and restrengthening of materials and structures - Brick and brickwork, Proc. Int.

Workshop on Urban heritage and building maintenance: 47-62. Freiburg: Aedificatio.12.Rots, J.G. 2001. Sequentially linear continuum model for concrete fracture. In R. de Borst, J. Mazars,

G. Pijaudiet-Cabot and J.G.M. van Mier (eds.), Fracture Mechanics of Concrete Structures, Volume 2,

pp. 831-840, Lisse, The Netherlands: Balkema.13.Schlangen, E. & Van Mier, J.G.M. 1992. Experimental and numerical analysis of micro-mechanisms of

fracture of cement-based composites. Cement & Concrete Composites 14:105-118.

14.Van Zijl, G.P.A.G. 2000. Computational modelling of masonry creep and shrinkage. Dissertation, DelftUniversity of Technology, Delft,Netherlands.

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