a comparative study of numerical models for concrete cracking

8/20/2019 A COMPARATIVE STUDY OF NUMERICAL MODELS FOR CONCRETE CRACKING

1/19

European Congress on Computational Methods in Applied Sciences and EngineeringECCOMAS 2004

P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, and D. Knörzer (eds.)Jyväskylä, 24–28 July 2004

A COMPARATIVE STUDY OF NUMERICAL MODELS FORCONCRETE CRACKING

Christian Feist, Walter Kerber, Hermann Lehar and Günter Hofstetter

Institute for Structural Analysis and Strength of MaterialsTechnikerstrasse 13, University of Innsbruck, Austria

e-mail: [email protected], web page: http://ibft.uibk.ac.at

Key words: finite element method, concrete, crack, plasticity theory, smeared crackmodel, strong discontinuity approach

Abstract. The reliable numerical simulation of the development of cracks in plain con-crete plays an important role for the integrity assessment of concrete structures. To this end a large number of material models for the representation of cracks, based on different theories and different finite element methods, have been developed in recent years. It is the objective of a task group of the thematic network ”Integrity Assessment of Large Con-crete Dams” to conduct a systematic comparison of different nonlinear material models

for cracking of plain concrete. To this end, a data base is set up, which contains selected laboratory tests for plain concrete, subjected to 2D and 3D stress states. The present contribution focuses on the numerical simulation of tests, contained in the data base, by means of three different material models for concrete, consisting of (i) a concrete model based on the smeared crack approach and formulated within the framework of the theory of plasticity, (ii) a plastic-damage model, which is available in the commercial finite element program ABAQUS and (iii) a crack model, based on the strong discontinuity approach. In

particular, the crack models are applied to the numerical simulation of (i) L-shaped panel tests, (ii) wedge splitting tests, (iii) tests on single edge notched beams and (iv) mixed mode fracture tests on double edge notched panels. On the basis of the numerical results the advantages and shortcomings of the investigated models for cracking of concrete are outlined.

1


2/19


1 INTRODUCTION

The reliable numerical simulation of the development of cracks in plain concrete playsan important role for the integrity assessment of concrete structures. To this end a largenumber of material models for the representation of cracks, based on different theoriesand different finite element methods, have been developed in recent years.

It is the objective of a task group of the thematic network ”Integrity Assessment of Large Concrete Dams” (project acronym IALAD), funded by the European Community,to conduct a systematic comparison of different nonlinear material models for cracking of plain concrete. To this end, a data base is set up, which contains selected laboratory testsfor plain concrete, subjected to 2D and 3D stress states. The data base can be accessedby the internet (http://nw-ialad.uibk.ac.at/Wp2/Tg2/). For each test the test setup, the

geometry, the material parameters and the loading can be retrieved from the data base.In addition, experimental results, such as stress-strain diagrams, load-displacement curvesand crack patterns are provided and the numerical results, expected to be delivered byinterested participants, are stated.

Everybody is invited to make use of the data base for the validation of concrete mod-els and to contribute numerical results with a short description of the employed model.Suggestions, which further tests should be included in the data base, are welcome.

The present contribution focuses on the numerical simulation of tests, contained in thedata base, by means of three different material models for concrete, consisting of

• a concrete model for 2D stress states, based on the smeared crack approach and

formulated within the framework of the theory of plasticity (model A),• a plastic-damage model, which is available in the commercial finite element program

ABAQUS (model B) and

• a crack model, based on the strong discontinuity approach and formulated withinthe framework of plasticity theory (model C).

In particular, the models are applied to the numerical simulation of

• the L-shaped panel tests, conducted by Winkler [2],

• the wedge splitting tests, conducted by Trunk [10],

• tests on single edge notched beams, conducted by Feist [8] and

• mixed mode fracture tests on double edge notched panels, conducted by Nooru-Mohamed [11].

2


3/19


2 BRIEF DESCRIPTION OF THE EMPLOYED CRACK MODELS

2.1 Smeared crack model based on plasticity theory (model A)

The material model, formulated for 2D stress states within the framework of the theoryof plasticity is based on the concrete model proposed in [1]. It is characterized by acomposite yield surface, consisting of the Rankine criterion to limit the tensile stress anda Drucker-Prager yield function to describe the compressive regime. The plastic strainsare computed by means of an associated flow rule. Suitable softening laws are employed todescribe tensile and compressive failure. Cracking is represented in a smeared manner bydistributing the crack opening over the width of the respective finite element. Objectivityof the numerical results with respect to the employed mesh size is obtained by employingthe specific fracture energy for tensile failure of concrete and an equivalent length. The

latter is simply taken as the root of the area of the respective finite element. Crushingfailure is treated in a similar manner. Damage due to tensile stresses is coupled withdamage due to compressive stresses for mixed tension-compression loading in order todescribe the decreasing compressive strength with increasing tensile stresses in lateraldirection. Unloading and reloading is modelled by the introduction of an isotropic scalardamage model. A detailed description of the model, which also includes the representationof tension-stiffening of reinforced concrete, can be found in [2, 3]. However, for thenumerical analyses described in the present paper primarily those parts of the modelreferring to concrete cracking are essential.

2.2 Plastic damage model (model B)

The plastic damage model for concrete, which is available in the commercial FE-program ABAQUS [4], is based on the concrete model proposed in [5]. It allows theconsideration of cyclic and dynamic loading by modelling the stiffness degradation due tounloading and reloading. To this end, the yield function is formulated in terms of effectivestresses rather than total stresses. The plastic strains are computed by means of a non-associative flow rule in order to control dilatancy. Hardening and softening behaviour isrepresented in terms of two hardening variables, which represent equivalent plastic strainsin tension and compression, respectively. They control the evolution of the yield surfaceand can be related to the specific fracture energies in uniaxial tension and compression inorder to ensure objective results with respect to the employed mesh size. The evolution of

stiffness degradation due to cyclic loading is decoupled from the outlined material modelfor the elastic-plastic response. The degradation of the elastic stiffness during unloadingand reloading is modelled by means of two independent damage variables, one for tensileunloading and reloading and one for compressive unloading and reloading, which also de-pend on the mentioned two hardening variables. Moreover, recovery of the elastic stiffnessdue to crack closure can be modelled. For the numerical analyses described in the presentpaper primarily those parts of the model referring to concrete cracking due to monotonicloading are essential.

3


4/19


2.3 Crack model based on the strong discontinuity approach (model C)

The crack model is based on the concept of finite elements with embedded discontinu-ities. The strong discontinuity approach, characterized by the representation of cracks bydiscontinuities in the displacement field, is formulated within the framework of plasticitytheory [6, 7]. The yield condition provides a relation between the normal component of thetraction vector and the crack opening and the flow rule yields the enhanced strains, whichapproximate the effect of the discontinuity. Being distributed over the width of those fi-nite elements, which are crossed by the discontinuity, the enhanced strains complementthe regular strains of the continuous displacement field. Discontinuities are allowed tocross elements in arbitrary ways and continuity of the crack path is enforced by means of a partial domain crack tracking strategy [8]. The model is based on the fixed crack con-

cept assuming the geometrical representation of the discontinuity surface to be fixed withrespect to time. The crack direction upon incipient cracking is predicted on the basis of the direction of the maximum principal nonlocal strain, which is obtained from a nonlocalaveraging procedure of the strains by means of a bell-shaped distribution function withina particular interaction radius. A detailed description is given in [9].

3 NUMERICAL SIMULATIONS OF SELECTED TESTS

3.1 L-shaped panel test

The L-shaped panel has become a popular benchmark test for the validation of com-putational models for the numerical simulation of cracking of plain concrete. In order

to provide experimental data, tests on L-shaped structural members were performed atthe University of Innsbruck [2]. The test setup with the geometric properties and theboundary conditions is shown in Fig. 1(a).

The long and the short edges of the L-shaped panel are given as 500 and 250 mm,respectively; its thickness is 100 mm. The lower horizontal edge of the vertical leg isfixed. A vertical load F v, acting uniformly across the thickness opposite to the directionof gravity, is applied at the lower horizontal surface of the horizontal leg at a distanceof 30 mm from the vertical end face. Shortly before reaching the maximum load theexperiment is switched from load-control to displacement control.

Young’s modulus, Poisson’s ratio, the cylindrical compressive strength, the uniaxialtensile strength and the specific fracture energy are given as E = 25850 N/mm2, ν = 0.18,

f c = 31.0 N/mm2, f t = 2.70 N/mm2 and Gf = 0.09 Nmm/mm2.Fig. 2 depicts the relationship between the applied load F v and the vertical displacement

v at the point of load application. The grey shaded area shows the scatter of the exper-imental results from three tests on identical specimens, whereas the load-displacementcurves refer to the numerically predicted behaviour employing the three crack models andthe four FE-meshes, shown in Fig. 3. The mesh data of the employed FE-meshes are givenin Table 1. The meshes LSP-600, LSP-1151 and LPS-2910 consist of linear triangular el-ements (CST-elements), whereas the FE-mesh LSP-2246 contains bilinear quadrilateral

4


5/19


500

250 250

500

250

250

all lengths in [mm]

F v , v

30

t = 100 mm

(a) (b)

Figure 1: L-shaped panel test: (a) Test setup, (b) scatter of observed crack paths

l o a

d

[ k N ]

0.0

exp. spectrum

model A

model B

model C

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

(a) LSP-600 (b) LSP-1151

l o a

d

[ k N ]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

exp. spectrum

model A

model B

model C

l o a

d

[ k N ]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

(c) LSP-2910 (d) LSP-2246

l o a

d

[ k N ]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

11.0

exp. spectrum

model A

model B

exp. spectrum

model A

model B

model C

vertical displacement [mm] vertical displacement [mm]

vertical displacement [mm] vertical displacement [mm]

Figure 2: L-shaped panel test: Comparison of measured and computed load-displacement curves

5


6/19


model A model B model C

F E - m e s h L S P - 6 0 0

F E - m e s h L S P - 1 1 5 1

F E - m e s h L S

P - 2 9 1 0

F E - m e

s h L S P - 2 2 4 6

Figure 3: L-shaped panel test: FE-meshes and computed crack paths

6


7/19


element type CST CPS4

mesh identifier LSP-600 LSP-1151 LSP-2910 LSP-2246number of elements 600 1151 2910 2246number of nodes 341 630 1539 2316number of dofs 682 1260 3078 4632

Table 1: L-shaped panel test: Properties of the employed FE-meshes

elements (CPS4-elements). Since model C, which is based on the strong discontinuityapproach, is only implemented for CST-elements, for the latter model numerical resultsare presented only for the meshes with CST-elements.

Apart from the FE-meshes, Fig. 3 also shows the computed crack paths by markingthose elements, which are crossed by the crack, by grey shading. The computed crackpaths can be compared with the respective crack paths observed in the tests, which areshown in Fig. 1(b).

3.2 Wedge splitting test

A series of wedge splitting tests on dam concrete was performed at the Institute forBuilding Materials of the Swiss Federal Institute of Technology, ETH Zürich, in order toinvestigate the size dependence of fracture mechanics parameters. The extensive experi-mental program is documented in [10]. In the present context only the material data forthe tests on dam concrete, denoted as CP250, are described. A schematic diagram of the

wedge splitting specimens is shown in Figure 4.The dimensions H and B of tested specimens ranged from 400 mm to 3200 mm. In the

present context only the specimen, characterized by H = B = 1600 mm and a thicknessof t = 400 mm is considered. The remaining dimensions are given as k = 100 mm,s = 100 mm and a0 = 775 mm. In the vertical plane of symmetry the specimen has anotch of depth (a0 − s/2). The specimen rests on two supports, which are located belowthe centre of gravity of each half of the specimen. Young’s modulus, Poisson’s ratio,the uniaxial compressive strength, the uniaxial tensile strength and the specific fractureenergy are given as E = 28300 N/mm2, ν = 0.18, f c = 44.7 N/mm

2, f t = 2.11 N/mm2

and Gf = 0.482 Nmm/mm2.

In the tests, the relationship between the horizontal splitting force F , acting uniformlyalong the inner vertical faces of the specimen, and the change of the distance ∆s of thepoints of load application, denoted as crack mouth opening displacement, was measured.Fig. 5 depicts the measured and the computed relationships between F and ∆s, the latteron the basis of the three different crack models employing the four FE-meshes, shown inFig. 6.

The FE-meshes, denoted as WST-1512 and WST-2472 are irregular meshes consistingof CST-elements without taking any advantage of symmetry, whereas the FE-meshes

7


8/19


B

H

a 0

s /2

s /2

detail

detail k

s, F

Figure 4: Wedge splitting test: Test setup

l o a

d

[ k N ]

0.0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.0

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

110.0

l o a

d

[ k N ]

0.0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.0

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

110.0

l o a

d

[ k N ]

0.0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.0

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

0.0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.0

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

l o a

d

[ k N ]

100.0

110.0

100.0

(a) FE-mesh WST-1512

crack mouth opening displacement [mm]

(b) FE-mesh WST-2472


(c) FE-mesh WST-2042


(d) FE-mesh WST-1224


exp. curve

model A

model B

model C

exp. curve

model A

model B

model C

exp. curve

model A

model B

exp. curve

model A

model B

Figure 5: Wedge splitting test: Comparison of measured and computed load-displacement diagrams

8


9/19



F E - m e s h W S T - 1 5 1 2

F E - m e s h W S T - 2 4 7 2

F E - m e s h W S T - 2 0 4 2

F E - m

e s h W S T - 1 2 2 4

Figure 6: Wedge splitting test: FE-meshes and computed crack paths

9


10/19



mesh identifier WST-1512 WST-2472 WST-2042 WST-1224number of elements 1512 2472 2042 1224number of nodes 816 1301 2158 1287number of dofs 1632 2602 4316 2574

Table 2: Wedge splitting test: Properties of the employed FE-meshes

WST-2042 and WST-1224 are symmetric with respect to the theoretical crack path. Thelatter is a straight line, extending in the vertical plane of symmetry from the notch to thebottom of the specimen. The mesh properties are summarized in Table 2. The predictedcrack paths, computed on the basis of the three crack models and the different FE-meshesare depicted in Fig. 6.

3.3 Single edge notched beam test

Beam-shaped specimens were tested by Feist [8] in order to investigate the crack devel-opment in plain concrete subjected to 2D and 3D stress states. Two series of experimentswere carried out with five identical specimens for each series. In the first series a 2Dstress state was generated to obtain reference data on curved crack paths for 2D stressstates, whereas in the second series a 3D stress state was generated to obtain data oncurved crack surfaces for the validation of material models for 3D stress states. Thepresent contribution only focuses on the 2D tests. The dimensions of the specimens for

the 2D tests are 600 × 180× 100 mm with a span of 500 mm (Fig. 7). Young’s modulus,Poisson’s ratio, the uniaxial compressive strength, determined on cubic specimens, theuniaxial tensile strength and the specific fracture energy are given as E = 34760 N/mm2,ν = 0.21, f c = 50.4 N/mm

2, f t = 3.4 N/mm2 and Gf = 0.071 Nmm/mm

2.At a horizontal distance of 175 mm from the left support a uniform line load is applied

along the entire width of the beam. A notch of 30 mm depth and 5 mm width is locatedin the plane of symmetry perpendicular to the beam axis at the tensile face of the beam.The notch serves as the location of crack initiation. Both, the notch width and the notchdepth are constant over the width of the beam.

Fig. 8 contains the measured and the computed relations between the applied loadF and the crack mouth opening displacement ∆s (see Fig. 7(a)). The latter have beenobtained on the basis of the three different crack models employing the four FE-meshes,which are partly shown in Fig. 9. The FE-meshes PCT2D-700, PCT2D-2190 and PCT2D-3590 consist of CST-elements, whereas the mesh PCT2D-2418 contains bilinear quadrilat-eral elements. The first mesh is an irregular mesh, whereas the second one is regular. Thethird mesh is characterized by a regular fine discretization in the vicinity of the expectedcrack, whereas the fourth mesh is characterized by an irregular fine discretization in thisregion. The properties of the employed FE-meshes are summarized in Table 3.

10


11/19


300.0 300.0

225.0 375.0

130.0

50.0

50.0

support sleeve support sleeve

F

notch

s

(a) (b)

SG

all lengths in [mm]

50.0

30.0

Figure 7: Notched beam test: (a) Test setup, (b) observed crack path

l o a

d

[ k N ]

0.0 0.1 0.2 0.3 0.4

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0.5

(a) FE-mesh PCT2D-700


l o a

d

[ k N ]

0.0 0.1 0.2 0.3 0.4

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0.5

(b) FE-mesh PCT2D-2190


l o a

d

[ k N ]

0.0 0.1 0.2 0.3 0.4

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0.5

(c) FE-mesh PCT2D-3590


l o a

d

[ k N ]

0.0 0.1 0.2 0.3 0.4

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0.5

(d) FE-mesh PCT2D-2418


exp. curve

model A

model B

exp. curve

model A

model B

model C

exp. curve

model A

model B

model C

exp. curve

model A

model B

model C

Figure 8: Notched beam test: Comparison of measured and computed load-displacement diagrams

11


12/19



F E

- m e s h P C T 2 D - 7 0 0

F E - m e s h P C T 2 D - 2 1 9 0

F E - m e s h P C T 2 D - 3 5 9 0

F E - m e s h P C

T 2 D - 2 4 1 8

Detail domain for the crack path plots

Figure 9: Notched beam test: FE-meshes and computed crack paths

12


13/19


14/19


3.4 Mixed mode fracture test

A series of mixed mode fracture tests was undertaken by Nooru-Mohamed and vanMier at Delft University of Technology. Results from seven series of tests with differentload paths were reported in Nooru-Mohamed’s Ph.D. thesis [11] and by van Mier [12].The square shaped, double edge notched specimens are characterized by dimensions of 200 × 200 × 50 mm, a notch depth of 25 mm and a notch width of 5 mm. A schematicdiagram of the specimens and of the testing arrangement, relevant to the results reportedhere, is shown in Figure 11(a).

200

200

v s , P s

25

5

u n , P n

25

t = 50 mm

all lengths in [mm]

(a) (b) (c)

front facerear face

front facerear face

Figure 11: Mixed mode fracture test: (a) Test setup, (b) observed crack paths for load-path 4a, (c)observed crack paths for load-path 4b

Here, only results from test series 4a and 4b are considered. In series 4a (4b) first ashear load was applied to the specimen in displacement control up to P s = 5 kN (10 kN),with the axial load maintained at zero. Then an axial tensile load P n was applied underdisplacement control, whilst the shear force was maintained at a constant level. In [11]the test designation for the test of series 4a is 48-03 and the two identical tests of series4b are denoted as 46-05 and 47-01.

The compressive strength, obtained from cubes with dimensions of 150 mm, and thesplitting tensile strength for the specimen 46-05 are given as f c = 49.66 N/mm

2, f s =3.76 N/mm2, for the specimen 47-01 as f c = 46.19 N/mm

2, f s = 3.78 N/mm2 and for

the specimen 48-03 as f c = 46.24 N/mm2, f s = 3.67 N/mm

2. From the splitting tensilestrength the uniaxial tensile strength was estimated for both tests as f t = 3.0 N/mm

2.Young’s modulus and Poisson’s ratio are not provided in the test data, however, in anumerical simulation, described in [11], they were chosen as E = 30000 N/mm2 andν = 0.2. The specific fracture energy is chosen as Gf = 0.110 Nmm/mm

2.

14


15/19


l o a

d

[ k N ]

0.0 0.05 0.1 0.15 0.2

0.0

5.0

10.0

15.0

20.0

25.0

(a) load path 4a (FE-mesh NM-3474)

vertical displacement [mm]

(b) load path 4b (FE-mesh NM-3474)


(c) load path 4a (FE-mesh NM-10928)


(d) load path 4b (FE-mesh NM-10928)


l o a

d

[ k N ]

0.0 0.05 0.1 0.15 0.2

0.0

5.0

10.0

15.0

20.0

25.0

l o a

d

[ k N ]

0.0 0.05 0.1 0.15 0.2

0.0

5.0

10.0

15.0

20.0

25.0

l o a

d

[ k N ]

0.0 0.05 0.1 0.15 0.2

0.0

5.0

10.0

15.0

20.0

25.0

exp. curve

model A

model B

model C

exp. curve

model A

model B

model C

exp.

curve

model A

model B

exp.

curve

model A

model B

Figure 12: Mixed mode fracture test: Comparison of measured and computed load displacement diagrams

Fig. 12 shows the measured and computed relationships between the applied axialtensile load P n and the vertical displacement un. The load-displacement curves werecomputed employing the three different crack models and the two irregular FE-meshes,shown in Figs. 13 and 14. The first mesh, denoted as NM-3474, consists of CST-elements,whereas the second one, denoted as NM-10928, contains bilinear quadrilateral elements.The mesh data are summarized in Table 4. Fig. 13 also contains the computed crackpaths obtained by means of the three crack models employing the FE-mesh NM-3474.Fig. 14 shows the FE-mesh NM-10928 and the crack paths computed by means of modelA and model B (model C is not available for quadrilateral finite elements). Because of the very fine discretization the FE-mesh and the finite elements crossed by the cracksare shown separately in Fig. 14. The computed crack paths can be compared with theobserved crack paths, shown in Fig. 11(b) and (c).

15


16/19



F E - m e s h N M - 3 4 7 4

l o a d - p a t h 4 a

F E - m e s h N M

- 3 4 7 4

l o a d - p a t h

4 b

Figure 13: Mixed mode fracture test: Computed crack paths employing the FE-mesh NM-3474

FE-mesh NM-10928 model A model B

l o a d - p a t h

4 a

l o a d

- p a t h

4 b

Figure 14: Mixed mode fracture test: Computed crack paths employing the FE-mesh NM-10928

16


17/19



mesh identifier NM-3474 NM-10928number of elements 3474 10928number of nodes 1802 11078number of dofs 3564 22074

Table 4: Mixed mode fracture test: Properties of the employed FE-meshes.

4 CONCLUSIONS

The comparison of the numerical results, obtained for the four selected tests on plainconcrete specimens by means of the three numerical models for concrete cracking, leads

to the following conclusions:

• If regular meshes are employed, then the computed crack paths may be attractedby mesh lines. This shortcoming holds for the plastic damage model (model B) andto a somewhat lesser extent for the smeared crack model (model A). It results indiscrepancies between the experimental and the computed crack paths. This canbe seen for the L-shaped panel in Fig. 3 and for the single edge notched beam inFig. 9. The deviation of the computed crack path from the experimental one isalso reflected by the respective load-displacement diagrams (see Fig. 2(a) for theL-shaped panel and Fig. 8(b) and (c) for the single edge notched beam) by theoverestimation of the ultimate load.

• The shortcoming of computed crack paths following mesh lines by models A andB also holds for unstructured meshes. Crack paths then typically follow a zig-zagcourse, which can be seen for the L-shaped panel in Fig. 3 and for the wedge-splitting test in Fig. 6. Again, the deviation of the computed crack path from theexperimental one is reflected by the overestimation of the ultimate load (see Fig. 2(b)and (c) for the L-shaped panel and Fig. 5(b) for the wedge splitting test).

• Except for the mixed mode fracture test, for which all three models for concretecracking overestimate the ultimate load (Fig. 12), only the crack model based on thestrong discontinuity approach (model C) yields reliable predictions of the ultimate

load irrespective of the employed mesh.• The smeared crack model (model A) and the plastic damage model (model B) only

allow the prediction of elements crossed by cracks. In Figs. 3, 6, 9, 13 and 14 therespective elements were marked by grey shading. In contrast to models A and B, thecrack model based on the strong discontinuity approach (model C) allows to predictthe discrete crack path by a discontinuity line, which is continuous across adjacentelements. Hence, the illustrations of the ”cracked” elements by grey shading inFigs. 3, 6, 9, 13 and 14, obtained on the basis of model C, are complemented in

17


18/19


Fig. 15 by plots of the predicted discrete cracks paths. The latter can be compared

with the experimentally obtained crack paths shown in Figs. 1(b), 7(b) and 11(b)and (c). This comparison shows that model C provides an objective resolution of the macroscopic cracks irrespective of the employed mesh.

(a) L-shaped panel testFE-mesh LSP-2910

(b) Wedge splitting testFE-mesh WST-2472

(c) Single edge notched beam testFE-mesh PCT2D-2190

(d) Mixed mode fracture testload-path 4aFE-mesh NM-3474

(e) Mixed mode fracture testload-path 4bFE-mesh NM-3474

Figure 15: Crack paths predicted by model C, based on the strong discontinuity approach

18


19/19


REFERENCES

[1] P.H. Feenstra and R. de Borst. A composite plasticity model for concrete. Interna-tional Journal of Solids and Structures , 33, 707–730, 1996.

[2] B. Winkler. Traglastuntersuchungen von unbewehrten und bewehrten Betonstrukturen auf der Grundlage eines objektiven Werkstoffgesetzes f¨ ur Beton. Dissertation, Uni-versity of Innsbruck, Austria, 2001.

[3] B. Winkler, G. Hofstetter and H. Lehar. Application of a constitutive model forconcrete to the analysis of a precast segmental tunnel lining. Numerical and Analytical Methods in Geomechanics , in print.

[4] ABAQUS Standard User’s Manual, Version 6.3, Rhode Island, USA, 2003.[5] J. Lee and G. L. Fenves. Plastic-damage model for cyclic loading of concrete struc-

tures. Journal of Engineering Mechanics , ASCE, 124, 892 – 900, 1998.

[6] J. Oliver, M.Cervera, and O.Manzoli. Strong discontinuities and continuum plasticitymodels: The strong discontinuity approach. International Journal of Plasticity , 15,319–351, 1999.

[7] J. Mosler and G. Meschke. 3D modeling of strong discontinuities in elastoplasticsolids: Fixed and rotating localization formulations. International Journal for Nu-merical Methods in Engineering , 57, 1553–1576, 2003.

[8] C. Feist and G. Hofstetter. Mesh-insensitive strong discontinuity approach for frac-ture simulations of concrete. Proceedings of the 9th International Conference on Nu-merical Methods in Continuum Mechanics (NMCM 2003), CD-ROM, Zilina, Slo-vakia, 2003, 22p.

[9] C. Feist. Numerical modelling of cracking of plain concrete based on the strongdiscontinuity approach and experimental investigations. Dissertation. University of Innsbruck, 2004.

[10] B. Trunk. Einfluss der Bauteilgrösse auf die Bruchenergie von Beton. AEDIFICATIO

Publishers, Immentalstrasse 34, D-79104, Freiburg, 2000.[11] M.B. Nooru-Mohamed. Mixed-mode fracture of concrete: An experimental approach .

Ph.D. Thesis, Delft University of Technology, Delft, 1992.

[12] J.G.M. van Mier. Fracture Processes of Concrete . Series: New Directions in CivilEngineering, Vol. 12, CRC Press, 1997.

19

a comparative study of numerical models for concrete cracking

Documents