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SSGG 7 - Continuous and Discontinuous Modelling of Fracture in Concrete Using FEM

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This research addresses the modeling of fracture in concrete using Finite Element Method (FEM), analyzing both continuous and discontinuous approaches. The study highlights the necessity of incorporating microstructural characteristics into classical constitutive laws to accurately represent strain localization and size effects. Both monotonic and cyclic loading conditions are simulated, utilizing various continuum models and formulations to capture macro-crack formation and mesh-objective results.

Figures (410)

Fig. 2.4 Relationship between compressive strength of concrete prisms and compressive strength of concrete cubes @ against slenderness ratio of concrete specimen h/d or h/b (Leonhard 1973)
Fig. 2.4 Relationship between compressive strength of concrete prisms and compressive strength of concrete cubes @ against slenderness ratio of concrete specimen h/d or h/b (Leonhard 1973)
Fig. 2.12 Maximum stresses in plane stress state in biaxial tests by Kupfer et al. (1969) (Klisinski and Mréz 1988)
Fig. 2.12 Maximum stresses in plane stress state in biaxial tests by Kupfer et al. (1969) (Klisinski and Mréz 1988)
Fig. 2.13 Volume changes during biaxial tests by Kupfer et al. (1969) (Klisinski and Mr6z 1988)
Fig. 2.13 Volume changes during biaxial tests by Kupfer et al. (1969) (Klisinski and Mr6z 1988)
Fig. 2.16 Stress-volume strain during hydrostatic compression (Klisinski and Mréz 1998)
Fig. 2.16 Stress-volume strain during hydrostatic compression (Klisinski and Mréz 1998)
Fig. 2.19 Failure surface for concrete in space of principal stresses o; (a) and in deviatoric plane (@- Lode angle) (Klisinski and Mréz 1988, Pivonka et al. 2003)
Fig. 2.19 Failure surface for concrete in space of principal stresses o; (a) and in deviatoric plane (@- Lode angle) (Klisinski and Mréz 1988, Pivonka et al. 2003)
Fig. 2.22 Stress-strain diagram for reinforcement steel during cyclic loading (Klisinski and Mroz 1988)
Fig. 2.22 Stress-strain diagram for reinforcement steel during cyclic loading (Klisinski and Mroz 1988)
Fig. 2.23 Peak stresses and corresponding strains of the normal strength concrete during uniaxial compression specimens depending upon friction along horizontal edges (van Vliet and van Mier 1996)
Fig. 2.23 Peak stresses and corresponding strains of the normal strength concrete during uniaxial compression specimens depending upon friction along horizontal edges (van Vliet and van Mier 1996)
Fig. 2.26 Post-peak deformations of normal strength concrete specimens with differe1 boundary conditions along horizontal edges (van Vliet and van Mier 1996)
Fig. 2.26 Post-peak deformations of normal strength concrete specimens with differe1 boundary conditions along horizontal edges (van Vliet and van Mier 1996)
Fig. 2.27 Axial strain against lateral strain for normal strength concrete cubes with: a) high friction and b) low friction along horizontal edges (van Vliet and van Mier 1996)
Fig. 2.27 Axial strain against lateral strain for normal strength concrete cubes with: a) high friction and b) low friction along horizontal edges (van Vliet and van Mier 1996)
‘ig. 2.30 Specimens for size effect tests during uniaxial tension (van Vliet and van Mier 2000)
‘ig. 2.30 Specimens for size effect tests during uniaxial tension (van Vliet and van Mier 2000)
Fig. 2.35 Experimental results of size effect in three-point bending tests (points — experimental results, line — size effect law by Bazant (Bazant and Planas 1998), D — specimen size, D, — characteristic size, B — parameter, f, — tensile strength) (Le Bellego et al. 2003)
Fig. 2.35 Experimental results of size effect in three-point bending tests (points — experimental results, line — size effect law by Bazant (Bazant and Planas 1998), D — specimen size, D, — characteristic size, B — parameter, f, — tensile strength) (Le Bellego et al. 2003)
Fig. 2.39 Short reinforced concrete beams (Walraven and Lehwalter 1994)
Fig. 2.39 Short reinforced concrete beams (Walraven and Lehwalter 1994)
Fig. 2.42 Size effect for short reinforced concrete beams of normal concrete with shear reinforcement ratio p,,: nominal shear strength versus effective height d during cracking and failure (V, - ultimate vertical force, b - beam width, f. - compressive strength) (Walraven and Lehwalter 1994)
Fig. 2.42 Size effect for short reinforced concrete beams of normal concrete with shear reinforcement ratio p,,: nominal shear strength versus effective height d during cracking and failure (V, - ultimate vertical force, b - beam width, f. - compressive strength) (Walraven and Lehwalter 1994)
Fig. 2.43 Test reinforced concrete beams for group I (a) and group II (b) (Belgin and Sener 2008)
Fig. 2.43 Test reinforced concrete beams for group I (a) and group II (b) (Belgin and Sener 2008)
Fig. 2.44 Test reinforced concrete beams for group III (a) and group IV (b) (Belgin and Sener 2008)
Fig. 2.44 Test reinforced concrete beams for group III (a) and group IV (b) (Belgin and Sener 2008)
Fig. 2.50 Distribution of shear stresses tin contact zone between concrete and rein- forcement for increasing pull-out force P (Klisinski and Mréz 1988)
Fig. 2.50 Distribution of shear stresses tin contact zone between concrete and rein- forcement for increasing pull-out force P (Klisinski and Mréz 1988)
Fig. 2.53 Pull-out force versus slip for two bars with different diameter: 10 mm (HA10) and 12 mm (HA12) (Dahou et al. 2009)
Fig. 2.53 Pull-out force versus slip for two bars with different diameter: 10 mm (HA10) and 12 mm (HA12) (Dahou et al. 2009)
Fig. 2.57 Influence of a relatively thick cover on load-deflection curve in reinforced concrete beams (BaZant and Planas 1998)
Fig. 2.57 Influence of a relatively thick cover on load-deflection curve in reinforced concrete beams (BaZant and Planas 1998)
Fig. 3.1 Drucker-Prager criterion in the space of principal stresses (a) and on the plane g-p (b)
Fig. 3.1 Drucker-Prager criterion in the space of principal stresses (a) and on the plane g-p (b)
Continuum damage models initiated by the pioneering work of Katchanov (1986) describe a progressive loss of the material integrity due to the propagation and coalescence of micro-cracks and micro-voids. Continuous damage models (Simo and Ju 1987, Lemaitre and Chaboche 1990) are constitutive relations in which the mechanical effect of cracking and void growth is introduced with internal state variables which act on the degradation of the elastic stiffness of the material. They
Continuum damage models initiated by the pioneering work of Katchanov (1986) describe a progressive loss of the material integrity due to the propagation and coalescence of micro-cracks and micro-voids. Continuous damage models (Simo and Ju 1987, Lemaitre and Chaboche 1990) are constitutive relations in which the mechanical effect of cracking and void growth is introduced with internal state variables which act on the degradation of the elastic stiffness of the material. They
where the tangent modulus C represents the relation between the normal crack strain increment and normal stress increment during loading, S$ is the secant modulus of the unloading branch (Fig. 3.6), G is the elastic shear modulus and / is the shear stiffness reduction factor (the term {G account for effects like aggregate interlock). In addition, a threshold angle is introduced which allows new cracks to
where the tangent modulus C represents the relation between the normal crack strain increment and normal stress increment during loading, S$ is the secant modulus of the unloading branch (Fig. 3.6), G is the elastic shear modulus and / is the shear stiffness reduction factor (the term {G account for effects like aggregate interlock). In addition, a threshold angle is introduced which allows new cracks to
Fig. 3.6 Relationships between normal crack stress versus normal crack strain in softening range
Fig. 3.6 Relationships between normal crack stress versus normal crack strain in softening range
where the parameter /, is a characteristic length of micro-structure and r is a distance between two material points. The averaging in Eq. 3.93 is restricted to a small representative area around each material point (the influence of points at the distance of r=3/, is only of 0.01%) (Fig. 3.7). A characteristic length is usually related to the micro-structure of the material (e.g. maximum aggregate size and crack spacing in concrete, pore and grain size in granulates, crystal size in metals). It is determined with an inverse identification process of experimental data (Geers et al. 1996, Mahnken and Kuhl 1999, Le Bellego et al. 2003). However, the determination of a representative characteristic length of micro- structure J. is very complex in concrete since strain localization can include a mixed failure mode (cracks and shear zones) and a characteristic length (which is a scalar value) is related to the fracture process zone with a certain volume which changes during a deformation (the width of the fracture process zone increases according to e.g. Pijaudier-Cabot et al. 2004, but decreases after e.g. Simone and Sluys 2004). In turn, other researchers conclude that the characteristic length is not a constant, and it depends on the type of the boundary value problem and the current level of damage (Ferrara and di Prisco 2001). Thus, a determination of /, requires further numerical analyses and measurements, e.g. using a Digital Image Correlation (DIC) technique (Bhandari and Inoue 2005). FE simulations of tests with measured load-displacement curves and widths of fracture process zones for different boundary value problems and specimen sizes are of importance. According to Pijaudier-Cabot and BaZant (1987), Bazant and Oh (1983), it is in concrete approximately 3xd,""", where d,'"“ is the maximum aggregate size. Other representations can be also used for the function @ (Ozbolt 1995, Akkermann 2000, Jiraések 2004a, di Prisco et al. 2002, Bazant and Jirdsek 2002); e.g. the polynomial bell-shaped function reads
where the parameter /, is a characteristic length of micro-structure and r is a distance between two material points. The averaging in Eq. 3.93 is restricted to a small representative area around each material point (the influence of points at the distance of r=3/, is only of 0.01%) (Fig. 3.7). A characteristic length is usually related to the micro-structure of the material (e.g. maximum aggregate size and crack spacing in concrete, pore and grain size in granulates, crystal size in metals). It is determined with an inverse identification process of experimental data (Geers et al. 1996, Mahnken and Kuhl 1999, Le Bellego et al. 2003). However, the determination of a representative characteristic length of micro- structure J. is very complex in concrete since strain localization can include a mixed failure mode (cracks and shear zones) and a characteristic length (which is a scalar value) is related to the fracture process zone with a certain volume which changes during a deformation (the width of the fracture process zone increases according to e.g. Pijaudier-Cabot et al. 2004, but decreases after e.g. Simone and Sluys 2004). In turn, other researchers conclude that the characteristic length is not a constant, and it depends on the type of the boundary value problem and the current level of damage (Ferrara and di Prisco 2001). Thus, a determination of /, requires further numerical analyses and measurements, e.g. using a Digital Image Correlation (DIC) technique (Bhandari and Inoue 2005). FE simulations of tests with measured load-displacement curves and widths of fracture process zones for different boundary value problems and specimen sizes are of importance. According to Pijaudier-Cabot and BaZant (1987), Bazant and Oh (1983), it is in concrete approximately 3xd,""", where d,'"“ is the maximum aggregate size. Other representations can be also used for the function @ (Ozbolt 1995, Akkermann 2000, Jiraések 2004a, di Prisco et al. 2002, Bazant and Jirdsek 2002); e.g. the polynomial bell-shaped function reads
where the lower subscript at the variable « denotes the number of the triangular element in the specified quadrilateral (Fig. 3.8), and dx and dy are the distances between the triangle centres in the neighbouring quadrilaterals in a horizontal and vertical direction, respectively. The calculations of the second derivatives of the variable «in other triangles are similar. Thus, the effect of neighbouring elements near each element is taken into account. In FE calculations, the mixed derivative (Eq. 3.113) was neglected to reproduce the Laplacian of the variable xonly. The advantage of a gradient approach is that it is suitable (as a non-local approach) for both shear and tension (decohesion) dominated applications. The explicit second-gradient strain isotropic damage approach (Eqs. 3.111 and 3.112)
where the lower subscript at the variable « denotes the number of the triangular element in the specified quadrilateral (Fig. 3.8), and dx and dy are the distances between the triangle centres in the neighbouring quadrilaterals in a horizontal and vertical direction, respectively. The calculations of the second derivatives of the variable «in other triangles are similar. Thus, the effect of neighbouring elements near each element is taken into account. In FE calculations, the mixed derivative (Eq. 3.113) was neglected to reproduce the Laplacian of the variable xonly. The advantage of a gradient approach is that it is suitable (as a non-local approach) for both shear and tension (decohesion) dominated applications. The explicit second-gradient strain isotropic damage approach (Eqs. 3.111 and 3.112)
Fig. 3.8 Diagram for determination of the gradient of the constitutive variable « in triangular account non-local variables and a non-linear finite element equation is solved. Between odd and even iterations, the same element configuration is imposed. In the second method, only one mesh is used which contains user’s elements (defined by the UEL procedure). During odd iterations, the information about the elements is stored, and during even iterations, the stresses within a non-local theory are determined. As compared to the first method, the second one consumes less time. However, it is less comfortable for the user due to the need of the definition of the stiffness matrix and out-of-balance load vector in finite elements.
Fig. 3.8 Diagram for determination of the gradient of the constitutive variable « in triangular account non-local variables and a non-linear finite element equation is solved. Between odd and even iterations, the same element configuration is imposed. In the second method, only one mesh is used which contains user’s elements (defined by the UEL procedure). During odd iterations, the information about the elements is stored, and during even iterations, the stresses within a non-local theory are determined. As compared to the first method, the second one consumes less time. However, it is less comfortable for the user due to the need of the definition of the stiffness matrix and out-of-balance load vector in finite elements.
Fig. 3.11 Bond-slip relationship between concrete and reinforcement proposed by Haskett et al. (2008) where u;=1.5 mm is slip corresponding to the peak.
Fig. 3.11 Bond-slip relationship between concrete and reinforcement proposed by Haskett et al. (2008) where u;=1.5 mm is slip corresponding to the peak.
Fig. 4.4 Minimal distances between crack segment and triangle vertices/sides With the same values of Vinin and Jnin, the first condition is stronger, since it takes into account also the distance between the vertex and segment. Finally, a new scheme for calculating strains, stresses, internal forces and stiffness in a cracked
Fig. 4.4 Minimal distances between crack segment and triangle vertices/sides With the same values of Vinin and Jnin, the first condition is stronger, since it takes into account also the distance between the vertex and segment. Finally, a new scheme for calculating strains, stresses, internal forces and stiffness in a cracked
where s; are the components of the deviatoric principle stress tensor, J denotes the second invariant of the deviatoric stress tensor and Vis the Poisson’s ratio. During compression, two shear zones are simultaneously created expandin; outward from the weak element on the left side (Fig. 5.7). They occur directly befor the peak of the resultant vertical force on the top. After the peak, and up to the enc only one shear zone dominates. The complete shear zone is noticeable shortly afte the peak. The shear zone is wider than one finite element (Fig. 5.4). The thickness o a shear zone is approximately t,=2.4 cm (3.2x/) and does not depend upon the mes! size. The inclination of the shear zone against the bottom is equal to 0=44.8°, 44.8 and 46.8° with a coarse, medium and fine mesh, respectively. These values are in. good agreement with an inclination of a shear zone (@=46°) obtained from a1 analytical formula based on a bifurcation theory (Sluys 1992)
where s; are the components of the deviatoric principle stress tensor, J denotes the second invariant of the deviatoric stress tensor and Vis the Poisson’s ratio. During compression, two shear zones are simultaneously created expandin; outward from the weak element on the left side (Fig. 5.7). They occur directly befor the peak of the resultant vertical force on the top. After the peak, and up to the enc only one shear zone dominates. The complete shear zone is noticeable shortly afte the peak. The shear zone is wider than one finite element (Fig. 5.4). The thickness o a shear zone is approximately t,=2.4 cm (3.2x/) and does not depend upon the mes! size. The inclination of the shear zone against the bottom is equal to 0=44.8°, 44.8 and 46.8° with a coarse, medium and fine mesh, respectively. These values are in. good agreement with an inclination of a shear zone (@=46°) obtained from a1 analytical formula based on a bifurcation theory (Sluys 1992)
ECecitiot tie UVUUUIIEL, The evolution of the vertical force along the top before and after the peak is the same for a various discretization (Fig. 5.5). The maximum vertical force, 1.95 MN (corresponding to the compressive strength 49 MPa), occurs at the vertical displacement of 0.4 mm. The distribution of non-local equivalent plastic strains in a section perpendicular to the shear zone is fully uniform (Fig. 5.6). Due to that a non-locality parameter is larger than 1, the difference between the non-local and local plastic strains results in a constant value of the plastic strain (Jirasek and Rolshoven 2003). Separately, the non-local and local plastic strains have an exponential shape. The FE-calculations were also performed with a mesh alignment lower than 45°. The width of the shear zone was equal to t,=2.4 cm for all meshes except for the coarsest one (8x56) where the localization zone was equal to t,=3.6 cm. The inclination of the shear zone was: 44.0° (8x56, G=26.6°), 43.9° (12x56, 0=36.9°), 47.6° (16x84, @=33.7°) and 45.2° (20x84, @=39.8°). The load-displacement diagrams were similar using all mesh discretizations. In the FE-studies assuming a mesh inclination greater than 45°, the width of the shear zone was again 2.4 cm with the meshes 20x56 and 24x56, and 2.9 cm with the meshes 12x28 and 16x28.
ECecitiot tie UVUUUIIEL, The evolution of the vertical force along the top before and after the peak is the same for a various discretization (Fig. 5.5). The maximum vertical force, 1.95 MN (corresponding to the compressive strength 49 MPa), occurs at the vertical displacement of 0.4 mm. The distribution of non-local equivalent plastic strains in a section perpendicular to the shear zone is fully uniform (Fig. 5.6). Due to that a non-locality parameter is larger than 1, the difference between the non-local and local plastic strains results in a constant value of the plastic strain (Jirasek and Rolshoven 2003). Separately, the non-local and local plastic strains have an exponential shape. The FE-calculations were also performed with a mesh alignment lower than 45°. The width of the shear zone was equal to t,=2.4 cm for all meshes except for the coarsest one (8x56) where the localization zone was equal to t,=3.6 cm. The inclination of the shear zone was: 44.0° (8x56, G=26.6°), 43.9° (12x56, 0=36.9°), 47.6° (16x84, @=33.7°) and 45.2° (20x84, @=39.8°). The load-displacement diagrams were similar using all mesh discretizations. In the FE-studies assuming a mesh inclination greater than 45°, the width of the shear zone was again 2.4 cm with the meshes 20x56 and 24x56, and 2.9 cm with the meshes 12x28 and 16x28.
Figure 5.14 presents the results with one strong element in the middle of the left side of the specimen. The location of the shear zone is not connected with the position of the imperfection. The shear zone (reflected from the rigid bottom) is created in the lower part of the specimen. Created 1 We loOWer Part Of We specimen. The evolution of non-local plastic strains in the specimen with three initi weak elements distributed at the same distance along the left side is demonstrate in Fig. 5.15. Before the peak, two shear zones are created at each week elemen The shear zones propagate towards both the top and bottom of the specimen. Aft the peak, only one shear zone dominates in the whole specimen. The location « this shear zone is, however, different as compared to the results with one wea element at mid-height. The other calculations showed that the size and number « weak elements and distance between them did not influence the thickness an inclination of the shear zone, and the load-displacement curve. Only the locatio of the shear zone was affected. This result is in contrast to FE-calculations by SI and Chang (2003) wherein the thickness of a shear zone was influenced by th imperfection spacing.
Figure 5.14 presents the results with one strong element in the middle of the left side of the specimen. The location of the shear zone is not connected with the position of the imperfection. The shear zone (reflected from the rigid bottom) is created in the lower part of the specimen. Created 1 We loOWer Part Of We specimen. The evolution of non-local plastic strains in the specimen with three initi weak elements distributed at the same distance along the left side is demonstrate in Fig. 5.15. Before the peak, two shear zones are created at each week elemen The shear zones propagate towards both the top and bottom of the specimen. Aft the peak, only one shear zone dominates in the whole specimen. The location « this shear zone is, however, different as compared to the results with one wea element at mid-height. The other calculations showed that the size and number « weak elements and distance between them did not influence the thickness an inclination of the shear zone, and the load-displacement curve. Only the locatio of the shear zone was affected. This result is in contrast to FE-calculations by SI and Chang (2003) wherein the thickness of a shear zone was influenced by th imperfection spacing.
Fig. 5.14 Evolution of non-local equivalent plastic strains in the specimen during uniaxial
Fig. 5.14 Evolution of non-local equivalent plastic strains in the specimen during uniaxial
Finally, the effect of the boundary roughness was investigated in concrete elements (a weak element was inserted at the lower left corner) (Fig. 5.18). In this case, the horizontal displacements along horizontal boundaries were fixed. With smooth boundaries, the strength is always the same. When the boundaries are very rough, the strength increases with increasing characteristic length. The brittleness increases as usually with decreasing /. independently of the boundary roughness. Next, the problem of a symmetric double notched concrete specimen under uniaxial tension was numerically studied with a Rankine’a constitutive model (Eqs. 3.32, 3.93 and 3.97) (Bobinski and Tejchman 2006). Experimentally it was investigated by Hordijk (1991). The geometry of the concrete specimen (width b=60 mm, height h=125 mm, thickness in the out-of-plane direction =50 mm) and
Finally, the effect of the boundary roughness was investigated in concrete elements (a weak element was inserted at the lower left corner) (Fig. 5.18). In this case, the horizontal displacements along horizontal boundaries were fixed. With smooth boundaries, the strength is always the same. When the boundaries are very rough, the strength increases with increasing characteristic length. The brittleness increases as usually with decreasing /. independently of the boundary roughness. Next, the problem of a symmetric double notched concrete specimen under uniaxial tension was numerically studied with a Rankine’a constitutive model (Eqs. 3.32, 3.93 and 3.97) (Bobinski and Tejchman 2006). Experimentally it was investigated by Hordijk (1991). The geometry of the concrete specimen (width b=60 mm, height h=125 mm, thickness in the out-of-plane direction =50 mm) and
Fig. 5.22 Calculated stress—elongation diagrams for a specimen under uniaxial tension with different FE-meshes compared to the experimental diagram by Hordijk (1991): a) elasto- plastic model with non-local linear softening (softening modulus H=1300 MPa), b) elasto- plastic model with non-local linear softening (H;=2300 MPa), c) elasto-plastic model with non-local exponential softening, d) damage model with non-local softening (Eq. 3.38), e) damage model with non-local softening (Eq. 3.35) (Bobinski and Tejchman 2006)
Fig. 5.22 Calculated stress—elongation diagrams for a specimen under uniaxial tension with different FE-meshes compared to the experimental diagram by Hordijk (1991): a) elasto- plastic model with non-local linear softening (softening modulus H=1300 MPa), b) elasto- plastic model with non-local linear softening (H;=2300 MPa), c) elasto-plastic model with non-local exponential softening, d) damage model with non-local softening (Eq. 3.38), e) damage model with non-local softening (Eq. 3.35) (Bobinski and Tejchman 2006)
Fig. 5.24 Calculated contours of the non-local parameter A2 in a specimen under uniaxial tension for fine mesh within elasto-plasticity with non-local softening: a) linear softening (H,=1300 MPa), b) linear softening (H,=2300 MPa), c) curvilinear softening by Eq. 3.55 (Bobinski and Tejchman 2006) Fig. 5.23 Calculated contours of the non-local parameter «2 in a specimen under uniaxial tension for a) coarse, b) medium and c) fine mesh: A) elasto-plastic model with non-local softening (softening modulus H=1300 MPa), B) damage model with non-local softening (Eq. 3.38), C) damage model with non-local softening (Eq. 3.35) (Bobinski and Tejchman 2006)
Fig. 5.24 Calculated contours of the non-local parameter A2 in a specimen under uniaxial tension for fine mesh within elasto-plasticity with non-local softening: a) linear softening (H,=1300 MPa), b) linear softening (H,=2300 MPa), c) curvilinear softening by Eq. 3.55 (Bobinski and Tejchman 2006) Fig. 5.23 Calculated contours of the non-local parameter «2 in a specimen under uniaxial tension for a) coarse, b) medium and c) fine mesh: A) elasto-plastic model with non-local softening (softening modulus H=1300 MPa), B) damage model with non-local softening (Eq. 3.38), C) damage model with non-local softening (Eq. 3.35) (Bobinski and Tejchman 2006)
parameter called transitional size. To find the parameters B and D, from FE- analyses, a linear regression was used. Figure 5.27 presents a comparison between FE-result and the size effect law by Bazant (2003). A good agreement was obtained. The normalized strength decreases almost linearly with increasing size ratio h/I, in the considered range. sey ge a — = as - x = wae = ae ae SAU PY Ue AEE UI VU Ee ARM EVs In addition, similar elasto-plastic calculations (using the Rankine’s failure criterion, Eq. 3.32) were carried out with a concrete specimen under uniaxial tension possessing one non-symmetric notch (Fig. 5.28). The tensile strength was equal to f=3 MPa, the softening modulus in tension H=1.0 GPa, the non-local parameter m=2 and the characteristic length /=15 mm. Figure 5.29 presents a normalized load-displacement diagrams for 3 different geometrically similar
parameter called transitional size. To find the parameters B and D, from FE- analyses, a linear regression was used. Figure 5.27 presents a comparison between FE-result and the size effect law by Bazant (2003). A good agreement was obtained. The normalized strength decreases almost linearly with increasing size ratio h/I, in the considered range. sey ge a — = as - x = wae = ae ae SAU PY Ue AEE UI VU Ee ARM EVs In addition, similar elasto-plastic calculations (using the Rankine’s failure criterion, Eq. 3.32) were carried out with a concrete specimen under uniaxial tension possessing one non-symmetric notch (Fig. 5.28). The tensile strength was equal to f=3 MPa, the softening modulus in tension H=1.0 GPa, the non-local parameter m=2 and the characteristic length /=15 mm. Figure 5.29 presents a normalized load-displacement diagrams for 3 different geometrically similar
Fig. 5.27 Relationship between calculated normalized loads: P(Ex,bt) and PAf,bt) during uniaxial tension (with /=5 mm) and ratio h/], as compared to size effect law by Bazant (2003) within: a) damage mechanics, b) elasto-plasticity (Bobinski and Tejchman 2006)
Fig. 5.27 Relationship between calculated normalized loads: P(Ex,bt) and PAf,bt) during uniaxial tension (with /=5 mm) and ratio h/], as compared to size effect law by Bazant (2003) within: a) damage mechanics, b) elasto-plasticity (Bobinski and Tejchman 2006)
Fig. 5.32 Calculated force—displacement curves during uniaxial tension (using XFEM of Chapter 4.2) (Bobinski and Tejchman 2011)
Fig. 5.32 Calculated force—displacement curves during uniaxial tension (using XFEM of Chapter 4.2) (Bobinski and Tejchman 2011)
Fig. 5.34 Geometry of notched beam and boundary conditions in laboratory tests by Le Bellego et al. (2003)
Fig. 5.34 Geometry of notched beam and boundary conditions in laboratory tests by Le Bellego et al. (2003)
In the elasto-plastic simulations, the modulus of elasticity was taken as E=38.5 GPa and the Poisson ratio as v=0.2. In the tensile regime of the elasto-plastic model, the Rankine criterion with the exponential curve by Hordijk (1991) was used (Eqs. 3.32, 3.55, 3.93 and 3.97). Two different internal lengths were chosen ir the FE-analysis: .=5 mm and /.=10 mm. Due to two different /., two sets of the material parameters were also chosen in Eq. 3.55 (with m=2): f=3.6 MPa, «,=0.005 for 1.=5 mm, and f=3.3 MPa, «=0.003 for /.=10 mm to obtain the best agreement between the load-displacement diagrams from FE-analyses and laboratory tests (Le Bellego et al. 2003). The internal friction angle was equal to g=10° (Eq. 3.27) and the dilatancy angle y&5° (Eq. 3.30). The compressive strength was equal to f.=40 MPa. A linear softening modulus in compression was H,.=0.8 MPa. The effect of material parameters in compression was found to be insignificant on the FE-results. In the case of a damage model with the Rankine criterion (Eqs. 3.35, 3.40, 3.93 and 3.99), two sets of material parameters were again chosen for two different /,-: =7-10°, a=0.99, f=600 for J.=5 mm, and K=6.25-10°, @&=0.99, B=1000 for /.=10 mm. Figure 5.36 presents load-displacement curves for all beams obtained from FE- calculations using the characteristic length /=5 mm and /.=10 mm, respectively (compared to experiments). A satisfactory agreement with experiments was achieved. The FE-results overestimate slightly the load bearing capacity of the small and medium beam and underestimate the maximum load for the large beam. The ratio between the specimen strength of the large and medium beam and of the medium and small beam is similar. The same numerical results were obtained by
In the elasto-plastic simulations, the modulus of elasticity was taken as E=38.5 GPa and the Poisson ratio as v=0.2. In the tensile regime of the elasto-plastic model, the Rankine criterion with the exponential curve by Hordijk (1991) was used (Eqs. 3.32, 3.55, 3.93 and 3.97). Two different internal lengths were chosen ir the FE-analysis: .=5 mm and /.=10 mm. Due to two different /., two sets of the material parameters were also chosen in Eq. 3.55 (with m=2): f=3.6 MPa, «,=0.005 for 1.=5 mm, and f=3.3 MPa, «=0.003 for /.=10 mm to obtain the best agreement between the load-displacement diagrams from FE-analyses and laboratory tests (Le Bellego et al. 2003). The internal friction angle was equal to g=10° (Eq. 3.27) and the dilatancy angle y&5° (Eq. 3.30). The compressive strength was equal to f.=40 MPa. A linear softening modulus in compression was H,.=0.8 MPa. The effect of material parameters in compression was found to be insignificant on the FE-results. In the case of a damage model with the Rankine criterion (Eqs. 3.35, 3.40, 3.93 and 3.99), two sets of material parameters were again chosen for two different /,-: =7-10°, a=0.99, f=600 for J.=5 mm, and K=6.25-10°, @&=0.99, B=1000 for /.=10 mm. Figure 5.36 presents load-displacement curves for all beams obtained from FE- calculations using the characteristic length /=5 mm and /.=10 mm, respectively (compared to experiments). A satisfactory agreement with experiments was achieved. The FE-results overestimate slightly the load bearing capacity of the small and medium beam and underestimate the maximum load for the large beam. The ratio between the specimen strength of the large and medium beam and of the medium and small beam is similar. The same numerical results were obtained by
Fig. 5.38 Calculated contours of the non-local parameter A2 along the beam height at the left side of notch: a) /=5 mm, b) /.=10 mm, A) /=80 mm, B) h=160 mm, C) h=320 mm (damage model with non-local softening, Eqs. 3.35 and 3.40) (Bobinski and Tejchman 2006)
Fig. 5.38 Calculated contours of the non-local parameter A2 along the beam height at the left side of notch: a) /=5 mm, b) /.=10 mm, A) /=80 mm, B) h=160 mm, C) h=320 mm (damage model with non-local softening, Eqs. 3.35 and 3.40) (Bobinski and Tejchman 2006)
FE results with XFEM The geometry of the specimen during three-point bending was taken from Haussler-Combe (2002). The beam was deformed by prescribing a vertical displacement at the upper edge of the beam at its mid-span up to the final value u=0.4 mm (Fig. 5.53). The starting point of the crack was defined in the middle of the bottom edge. The Young modulus was equal to E=30 GPa and the Poisson’s ratio was v=0.2. The tensile strength was taken as f=3 MPa. The fracture energy G,=120 N/m with exponential softening was defined. The fixed vertical crack was assumed (Chapter 4.2). The simulations were performed with three FE meshes with 410, 1620 and 3630 3-node triangles (Bobinski and Tejchman 2011). A plane stress state was assumed. The odd number of finite elements between supports was defined to locate the crack starting point in the middle of the elements’ edge. Figure 5.54 shows the calculated force — displacement curves which are similar. For a coarse mesh, several drops in the force evolution were obtained which are caused by an edge-like crack propagation (crack tip can be placed at finite element edge only). With decreasing element size this phenomenon disappears. The obtained deformed coarse mesh is shown in Fig. 5.55 (displacements were scaled 20 times). It should be noted here that the use of the crack direction propagation criterion based on the maximum principal stress resulted in a sudden unrealistic change of the crack direction at the certain stage of deformation. The reason was an isotropic stress stage (biaxial tension) attained at the front of the crack tip. Next, the vertical tensile stresses become larger than horizontal ones and the crack turned by 90 degrees in the left (or right) direction. A beam with a notch under three-point bending was also analyzed according to
FE results with XFEM The geometry of the specimen during three-point bending was taken from Haussler-Combe (2002). The beam was deformed by prescribing a vertical displacement at the upper edge of the beam at its mid-span up to the final value u=0.4 mm (Fig. 5.53). The starting point of the crack was defined in the middle of the bottom edge. The Young modulus was equal to E=30 GPa and the Poisson’s ratio was v=0.2. The tensile strength was taken as f=3 MPa. The fracture energy G,=120 N/m with exponential softening was defined. The fixed vertical crack was assumed (Chapter 4.2). The simulations were performed with three FE meshes with 410, 1620 and 3630 3-node triangles (Bobinski and Tejchman 2011). A plane stress state was assumed. The odd number of finite elements between supports was defined to locate the crack starting point in the middle of the elements’ edge. Figure 5.54 shows the calculated force — displacement curves which are similar. For a coarse mesh, several drops in the force evolution were obtained which are caused by an edge-like crack propagation (crack tip can be placed at finite element edge only). With decreasing element size this phenomenon disappears. The obtained deformed coarse mesh is shown in Fig. 5.55 (displacements were scaled 20 times). It should be noted here that the use of the crack direction propagation criterion based on the maximum principal stress resulted in a sudden unrealistic change of the crack direction at the certain stage of deformation. The reason was an isotropic stress stage (biaxial tension) attained at the front of the crack tip. Next, the vertical tensile stresses become larger than horizontal ones and the crack turned by 90 degrees in the left (or right) direction. A beam with a notch under three-point bending was also analyzed according to
Fig. 5.58 Geometry of DEN specimen (dimensions in mm) and experimental curved cracks during loading path ‘4c’ (Nooru—Mohamed 1992)
Fig. 5.58 Geometry of DEN specimen (dimensions in mm) and experimental curved cracks during loading path ‘4c’ (Nooru—Mohamed 1992)
Fig. 5.66 Calculated force-displacement curves and cracks for shear force P,=10 KN using cohesive elements of Chapter 4.1 (Bobinski and Tejchman 2010)
Fig. 5.66 Calculated force-displacement curves and cracks for shear force P,=10 KN using cohesive elements of Chapter 4.1 (Bobinski and Tejchman 2010)
seilin®! “<eacciadieiieicinaltas ened 4 4e° Yel Tide The results of our FE simulations within continuum mechanics of the concrete behaviour under mixed mode conditions have shown that a proper choice of a constitutive law is a very important issue. The models show a different capability to capture the localized zone phenomenon. In general, an elasto-plastic model with the Rankine failure criterion is the most effective among continuous models. The usefulness of an isotropic damage model depends on the definition of the equivalent strain measure. The influence of the material description in the tensile- compression regime had to be taken into account (e.g. by a non-realistic increase compression regime had to be taken into account (e.g. by a non-realistic increase
seilin®! “<eacciadieiieicinaltas ened 4 4e° Yel Tide The results of our FE simulations within continuum mechanics of the concrete behaviour under mixed mode conditions have shown that a proper choice of a constitutive law is a very important issue. The models show a different capability to capture the localized zone phenomenon. In general, an elasto-plastic model with the Rankine failure criterion is the most effective among continuous models. The usefulness of an isotropic damage model depends on the definition of the equivalent strain measure. The influence of the material description in the tensile- compression regime had to be taken into account (e.g. by a non-realistic increase compression regime had to be taken into account (e.g. by a non-realistic increase
Fig. 6.1 Geometry of dog-bone shaped specimen (Vliet and van Mier 2000)
Fig. 6.1 Geometry of dog-bone shaped specimen (Vliet and van Mier 2000)
Fig. 6.7 Experimental and calculated force-displacement curves using 4 different coupled elasto-plastic-damage models with non-local softening during quasi-static four-point cyclic bending under tensile failure (Hordijk 1991): a) model ‘1’ (damage based on total strains), b) model ‘2’, c) model ‘3’ and d) model ‘4’ (damage based on elastic strains) (Marzec and Tejchman 2009, 2010)
Fig. 6.7 Experimental and calculated force-displacement curves using 4 different coupled elasto-plastic-damage models with non-local softening during quasi-static four-point cyclic bending under tensile failure (Hordijk 1991): a) model ‘1’ (damage based on total strains), b) model ‘2’, c) model ‘3’ and d) model ‘4’ (damage based on elastic strains) (Marzec and Tejchman 2009, 2010)
Fig. 6.8 Calculated contours of localized zone near notch in a beam under four-point bending with 4 different coupled elasto-plastic-damage models with non-local softening during four-point bending: a) model ‘1’, b) model ‘2’, c) model ‘3’ and d) model ‘4’ (at deflection u=0.15 mm) (Marzec and Tejchman 2010) zone above the notch is different due to the material stiffness in a softening regime induced by the material formulation. The shape of a localized zone in the models ‘1’ and ‘4’ is the same due to a similar model formulation, and is typical for other solutions within damage mechanics (e.g. Peerlings 1999, Pamin and de Borst 1999).
Fig. 6.8 Calculated contours of localized zone near notch in a beam under four-point bending with 4 different coupled elasto-plastic-damage models with non-local softening during four-point bending: a) model ‘1’, b) model ‘2’, c) model ‘3’ and d) model ‘4’ (at deflection u=0.15 mm) (Marzec and Tejchman 2010) zone above the notch is different due to the material stiffness in a softening regime induced by the material formulation. The shape of a localized zone in the models ‘1’ and ‘4’ is the same due to a similar model formulation, and is typical for other solutions within damage mechanics (e.g. Peerlings 1999, Pamin and de Borst 1999).
Fig. 6.10 Geometry and boundary conditions of a notched beam under three-point cyclic bending (Perdikaris and Romeo 1995) Table 6.2 Beam dimensions in cyclic tests by Perdikaris and Romeo (1995)
Fig. 6.10 Geometry and boundary conditions of a notched beam under three-point cyclic bending (Perdikaris and Romeo 1995) Table 6.2 Beam dimensions in cyclic tests by Perdikaris and Romeo (1995)
Fig. 6.17 Response of coupled elasto-plastic-damage model ‘4’ for concrete specimen under uniaxial cyclic compression from FE calculations (with damage scale factors a=0.0 and a.=1.0): a) deformed FE mesh, b) contours of calculated non-local parameter, c) calculated and experimental stress-strain curve by Karsan and Jirsa (1969), d) calculated stress-strain stress-strain curve, e) experimental stress-strain curve by Karsan and Jirsa (1969)
Fig. 6.17 Response of coupled elasto-plastic-damage model ‘4’ for concrete specimen under uniaxial cyclic compression from FE calculations (with damage scale factors a=0.0 and a.=1.0): a) deformed FE mesh, b) contours of calculated non-local parameter, c) calculated and experimental stress-strain curve by Karsan and Jirsa (1969), d) calculated stress-strain stress-strain curve, e) experimental stress-strain curve by Karsan and Jirsa (1969)
Fig. 7.1 Reinforced concrete bar in pure tension: a) schematically (units in mm), b) FE- mesh (Widulinski et al. 2009)
Fig. 7.1 Reinforced concrete bar in pure tension: a) schematically (units in mm), b) FE- mesh (Widulinski et al. 2009)
iil iaineiniaiaiiiiniiiiniiaas = Sauer” ial The horizontal tensile force increases with increasing characteristic length. The width of localized zones w, increases with increasing /, and is about w.=5x/.=25 mm for /.=5 mm, w,=4x/,=40 mm for /=10 mm and w,=3.5x/,=52 mm for /=15 mm. In turn, the spacing s, of localized zones also increases and is s=11X/=55 mm for /.=5 mm, s.=10x/.=100 mm for /=10 mm and s,=9x/.=137 mm for /.=15 mm.
iil iaineiniaiaiiiiniiiiniiaas = Sauer” ial The horizontal tensile force increases with increasing characteristic length. The width of localized zones w, increases with increasing /, and is about w.=5x/.=25 mm for /.=5 mm, w,=4x/,=40 mm for /=10 mm and w,=3.5x/,=52 mm for /=15 mm. In turn, the spacing s, of localized zones also increases and is s=11X/=55 mm for /.=5 mm, s.=10x/.=100 mm for /=10 mm and s,=9x/.=137 mm for /.=15 mm.
Fig. 7.7 Load-displacement curves for different non-local parameters m (/.=5 mm, p=2.0%, Uu,=0.06 mm): a) m=2, b) m=3 (Widulinski et al. 2009)
Fig. 7.7 Load-displacement curves for different non-local parameters m (/.=5 mm, p=2.0%, Uu,=0.06 mm): a) m=2, b) m=3 (Widulinski et al. 2009)
Fig. 7.8 Load-displacement curves for different tensile strength distributions (=5 mm, p=2.0%, U =0.06 mm): a) uniform, b) stochastic with standard deviation sy=0.05 MPa, c) stochastic with standard deviation sy=0.03 MPa, d) stochastic with standard deviation sg=0.01 MPa (Widulinski et al. 2009)
Fig. 7.8 Load-displacement curves for different tensile strength distributions (=5 mm, p=2.0%, U =0.06 mm): a) uniform, b) stochastic with standard deviation sy=0.05 MPa, c) stochastic with standard deviation sy=0.03 MPa, d) stochastic with standard deviation sg=0.01 MPa (Widulinski et al. 2009)
Fig. 7.11 Load-displacement curves and distribution of non-local softening parameter for different initial stiffness of bond slip law by Dorr (1980) at u=1 mm (/.=5 mm, P=2.0%): a) curve ‘a’ of Fig. 7.10 (u,=0.03 mm), b) curve ‘b’ of Fig. 7.10 (u,=0.06 mm), c) curve ‘c’ of Fig. 7.10 (u,=0.12 mm), d) curve ‘d’ of Fig. 7.10 (u,=0.24 mm), e) curve ‘e’ of Fig. 7.10 (u,=0.48 mm) (Widulinski et al. 2009)
Fig. 7.11 Load-displacement curves and distribution of non-local softening parameter for different initial stiffness of bond slip law by Dorr (1980) at u=1 mm (/.=5 mm, P=2.0%): a) curve ‘a’ of Fig. 7.10 (u,=0.03 mm), b) curve ‘b’ of Fig. 7.10 (u,=0.06 mm), c) curve ‘c’ of Fig. 7.10 (u,=0.12 mm), d) curve ‘d’ of Fig. 7.10 (u,=0.24 mm), e) curve ‘e’ of Fig. 7.10 (u,=0.48 mm) (Widulinski et al. 2009)
Fig. 7.13 Reinforced concrete bar with stirrups in pure tension (units in mm) (Widulinski et al. 2009)
Fig. 7.13 Reinforced concrete bar with stirrups in pure tension (units in mm) (Widulinski et al. 2009)
Fig. 7.15 FE-meshes used for calculations: a) small-size beam 1, b) medium-size beam 2, c large-size beam 3 (Marzec et al. 2007)
Fig. 7.15 FE-meshes used for calculations: a) small-size beam 1, b) medium-size beam 2, c large-size beam 3 (Marzec et al. 2007)
Fig. 7.21 Calculated load-displacement curves for large-size beam (h=750 mm, bond-slip by Dérr (1980), p=0.75%, a/d=3, k=0.003) as compared to experiments by Walraven (1978) (P — vertical resultant force, u — vertical displacement, ‘e’ — experiment by Walraven, 1978): a) /.=10 mm, b) /.=20 mm, c) experiment (Marzec et al. 2007)
Fig. 7.21 Calculated load-displacement curves for large-size beam (h=750 mm, bond-slip by Dérr (1980), p=0.75%, a/d=3, k=0.003) as compared to experiments by Walraven (1978) (P — vertical resultant force, u — vertical displacement, ‘e’ — experiment by Walraven, 1978): a) /.=10 mm, b) /.=20 mm, c) experiment (Marzec et al. 2007)
Fig. 7.27 Calculated and experimental load-displacement curves for medium-size beam (a/d=3, p=0.75%, «,=0.003, 1.=20 mm): a) perfect bond, b) bond-slip (Marzec et al. 2007)
Fig. 7.27 Calculated and experimental load-displacement curves for medium-size beam (a/d=3, p=0.75%, «,=0.003, 1.=20 mm): a) perfect bond, b) bond-slip (Marzec et al. 2007)
Fig. 7.31 Calculated and experimental load-displacement curve in reinforced concrete beam for small-size beam (A) and medium-size beam (B): /.=5 mm (a), /.=10 mm (b), /.=12.5 mm (c) using explicit second-gradient strain isotropic damage approach (Matecki and Tejchman 2009)
Fig. 7.31 Calculated and experimental load-displacement curve in reinforced concrete beam for small-size beam (A) and medium-size beam (B): /.=5 mm (a), /.=10 mm (b), /.=12.5 mm (c) using explicit second-gradient strain isotropic damage approach (Matecki and Tejchman 2009)
The calculations were carried out with bond-slip using a relationship between the bond shear stress t, and slip u according to Dérr (1980) (Eqs. 3.114 and 3.115) due to the fact that bond traction values were far from the limiting value (thus, the shape of the bond law after the peak was unimportant). To investigate the effect of the bond stiffness, several numerical tests were carried out with a different value of uo changing from 0.06 mm (D6rr 1980) up to 1.0 mm (Haskett et al. 2008). Table 7.1 Specimen properties and failure loads (Skarzynski et al. 2010)
The calculations were carried out with bond-slip using a relationship between the bond shear stress t, and slip u according to Dérr (1980) (Eqs. 3.114 and 3.115) due to the fact that bond traction values were far from the limiting value (thus, the shape of the bond law after the peak was unimportant). To investigate the effect of the bond stiffness, several numerical tests were carried out with a different value of uo changing from 0.06 mm (D6rr 1980) up to 1.0 mm (Haskett et al. 2008). Table 7.1 Specimen properties and failure loads (Skarzynski et al. 2010)
Fig. 7.33 Geometry of reinforced concrete beams used in laboratory tests (Walraven and Lehwalter 1994)
Fig. 7.33 Geometry of reinforced concrete beams used in laboratory tests (Walraven and Lehwalter 1994)
experimental crack pattern (Fig. 7.38). Figure 7.35 shows the calculated force-displacement curves (P — vertical force at the mid-point of the beam top, u — vertical displacement of this mid-point) for the beams of h=200-800 mm. The distribution of a non-local tensile and compressive softening parameter is depicted in Figs. 7.36 and 7.37 at the beam failure. In addition, the distribution of non-local tensile softening parameter is shown at the normalized vertical force of Vbdf.)=0.10 as compared to the experimental crack pattern (Fig. 7.38).
experimental crack pattern (Fig. 7.38). Figure 7.35 shows the calculated force-displacement curves (P — vertical force at the mid-point of the beam top, u — vertical displacement of this mid-point) for the beams of h=200-800 mm. The distribution of a non-local tensile and compressive softening parameter is depicted in Figs. 7.36 and 7.37 at the beam failure. In addition, the distribution of non-local tensile softening parameter is shown at the normalized vertical force of Vbdf.)=0.10 as compared to the experimental crack pattern (Fig. 7.38).
“U70 Uldt We CAPCIUMeNtal ONCs, The calculated geometry of localized zones within a smeared crack approach is imilar as this within elasto-plasticity except of beams with h>400 mm where the ocalized zones are more diffuse. The effect of crack type assumed in the model fixed or rotating crack model) was insignificant. In turn, large discrepancies yccur in the distribution of localized zones when using the damage model. The nclined localized zones were not obtained in FE analyses (only one vertical at ottom mid-point of a tensile type).
“U70 Uldt We CAPCIUMeNtal ONCs, The calculated geometry of localized zones within a smeared crack approach is imilar as this within elasto-plasticity except of beams with h>400 mm where the ocalized zones are more diffuse. The effect of crack type assumed in the model fixed or rotating crack model) was insignificant. In turn, large discrepancies yccur in the distribution of localized zones when using the damage model. The nclined localized zones were not obtained in FE analyses (only one vertical at ottom mid-point of a tensile type).
Fig. 7.36 Distribution of calculated non-local tensile softening parameter K, within elasto- plasticity in beams at failure for different beams: a) h=200 mm, b) h=400 mm, c) h=600 mm, d) h=800 mm (the beams are not proportionally scaled) (Skarzynski et al. 2010) Fig. 7.36 Distribution of calculated non-local tensile softening parameter K, within elasto-
Fig. 7.36 Distribution of calculated non-local tensile softening parameter K, within elasto- plasticity in beams at failure for different beams: a) h=200 mm, b) h=400 mm, c) h=600 mm, d) h=800 mm (the beams are not proportionally scaled) (Skarzynski et al. 2010) Fig. 7.36 Distribution of calculated non-local tensile softening parameter K, within elasto-
Fig. 7.38 Comparison of distribution of non-local tensile softening parameter K, within
Fig. 7.38 Comparison of distribution of non-local tensile softening parameter K, within
Fig. 7.43 Calculated size effect in reinforced concrete beams from FE-analyses compared to experiments (Walraven and Lehwalter 1994) and to the size effect law by Bazant (Bazant and Planas 1998) (b — beam width, d — effective beam height, f. — compressive strength of concrete, V,, — ultimate vertical force): a) experiments, b) FE-calculations (elasto-plasticity), c) FE-calculations (smeared crack model, Chapter 3.1.3), d) FE-calculations (damage mechanics, Eqs. 3.35 and 3.40), e) size effect law by Bazant (Skarzynski et al. 2010)
Fig. 7.43 Calculated size effect in reinforced concrete beams from FE-analyses compared to experiments (Walraven and Lehwalter 1994) and to the size effect law by Bazant (Bazant and Planas 1998) (b — beam width, d — effective beam height, f. — compressive strength of concrete, V,, — ultimate vertical force): a) experiments, b) FE-calculations (elasto-plasticity), c) FE-calculations (smeared crack model, Chapter 3.1.3), d) FE-calculations (damage mechanics, Eqs. 3.35 and 3.40), e) size effect law by Bazant (Skarzynski et al. 2010)
was induced by increments of the vertical displacement prescribed to both end: through steel plates with a large stiffness. In 2D plane strain calculations, the vertical reinforcement was assumed mainly in the form of 2D 3-node triangulat elements. The width of each steel column composed of 2D elements was assumec to be 2.0 mm and their length in the direction perpendicular to the deformatior plane was 32 mm for p=1.98% and 64 mm for p=3.95%. In 2D calculations, one simulation was also performed with vertical 2-node truss elements (describing reinforcement) with a diameter of 9.0 mm. In turn, in 3D calculations, one usec always vertical 2-node truss elements with a circular cross-section and diameter 0: 6 mm. The vertical reinforcement was fixed to the horizontal edge steel plates The stirrups were not taken into account since the FE-study concerned mainly the column behaviour up to peak (their influence is very important in the post-pealk regime). A bond-slip law by Dorr (1980) was assumed (Eqs. 3.114 and 3.115).
was induced by increments of the vertical displacement prescribed to both end: through steel plates with a large stiffness. In 2D plane strain calculations, the vertical reinforcement was assumed mainly in the form of 2D 3-node triangulat elements. The width of each steel column composed of 2D elements was assumec to be 2.0 mm and their length in the direction perpendicular to the deformatior plane was 32 mm for p=1.98% and 64 mm for p=3.95%. In 2D calculations, one simulation was also performed with vertical 2-node truss elements (describing reinforcement) with a diameter of 9.0 mm. In turn, in 3D calculations, one usec always vertical 2-node truss elements with a circular cross-section and diameter 0: 6 mm. The vertical reinforcement was fixed to the horizontal edge steel plates The stirrups were not taken into account since the FE-study concerned mainly the column behaviour up to peak (their influence is very important in the post-pealk regime). A bond-slip law by Dorr (1980) was assumed (Eqs. 3.114 and 3.115).
pb — perfect bond, bs — bond slip, bs-J - u,=0.06 mm, bs-2 - u,=0.30 mm Table 7.7 Spacing of localized zones from FE-calculations and analytical formulae (Majewski et al. 2008)
pb — perfect bond, bs — bond slip, bs-J - u,=0.06 mm, bs-2 - u,=0.30 mm Table 7.7 Spacing of localized zones from FE-calculations and analytical formulae (Majewski et al. 2008)
Table 7.7 summarizes some numerical and analytical results with respect to the spacing of localized zones.
Table 7.7 summarizes some numerical and analytical results with respect to the spacing of localized zones.
[1] Kim and Lee (2000), [2] Hsu (1988), [3] Lloyd and Rangan (1996), [4] Bazant and Kwon (1994).
[1] Kim and Lee (2000), [2] Hsu (1988), [3] Lloyd and Rangan (1996), [4] Bazant and Kwon (1994).
Fig. 7.71 Forces in corbel with horizontal stirrups using strut-and tie model by Hagberg (1983) results (Bobinski and Tejchman 2004). The elastic constants for concrete were: E=21.9 GPa and v.=0.20. The compressive and tensile strengths were f,=22.6 MPa and f;=2.26 MPa, respectively. The internal friction angle was g=20 and the dilatancy angle was y=10.. In the case of tension, the parameter «"=0.0017 was assumed (Fig. 7.72Aa). Thus, the tensile fracture energy was Gy =100 N/m. It was calculated as Gj=g/xwy,;, gy — area under the entire softening function (with w;~ 2/, — width of tensile localized zones with /,=5-10 mm). In the case of compression (Fig. 7.72Ab), we assumed x;"=0.0085. Thus, the compressive fracture energy G, was 5000 N/m. The non-locality parameter was chosen as m=2 on the basis of other calculations (Bobinski and Tejchman 2004).
Fig. 7.71 Forces in corbel with horizontal stirrups using strut-and tie model by Hagberg (1983) results (Bobinski and Tejchman 2004). The elastic constants for concrete were: E=21.9 GPa and v.=0.20. The compressive and tensile strengths were f,=22.6 MPa and f;=2.26 MPa, respectively. The internal friction angle was g=20 and the dilatancy angle was y=10.. In the case of tension, the parameter «"=0.0017 was assumed (Fig. 7.72Aa). Thus, the tensile fracture energy was Gy =100 N/m. It was calculated as Gj=g/xwy,;, gy — area under the entire softening function (with w;~ 2/, — width of tensile localized zones with /,=5-10 mm). In the case of compression (Fig. 7.72Ab), we assumed x;"=0.0085. Thus, the compressive fracture energy G, was 5000 N/m. The non-locality parameter was chosen as m=2 on the basis of other calculations (Bobinski and Tejchman 2004).
The horizontal bars and stirrups were assumed as 1D elements. The steel behaviour was described by the elasto-plastic law following Gonzales-Vidosa et al. (1988) (Fig. 7.72c) with E,=200 GPa, o,=488 MPa for bars and 0,=445 MPa for stirrups. Either perfect bond or bond-slip according to CEB-FIP (1990) or Dorr (1980) (Eq. 3.114-3.119) was assumed between bars and concrete. In calculations with slip-bond, the same displacements were enforced between reinforcement (truss) and concrete (solid) elements in nodes at the specimen surfaces.
The horizontal bars and stirrups were assumed as 1D elements. The steel behaviour was described by the elasto-plastic law following Gonzales-Vidosa et al. (1988) (Fig. 7.72c) with E,=200 GPa, o,=488 MPa for bars and 0,=445 MPa for stirrups. Either perfect bond or bond-slip according to CEB-FIP (1990) or Dorr (1980) (Eq. 3.114-3.119) was assumed between bars and concrete. In calculations with slip-bond, the same displacements were enforced between reinforcement (truss) and concrete (solid) elements in nodes at the specimen surfaces.
Fig. 7.75 Load-displacement curves from our 2D calculations using elasto-plastic model (a) compared with experimental result by Fattuhi (1990) (b) and numerical results by Strauss et al. (2006) (c) (Syroka et al. 2011)
Fig. 7.75 Load-displacement curves from our 2D calculations using elasto-plastic model (a) compared with experimental result by Fattuhi (1990) (b) and numerical results by Strauss et al. (2006) (c) (Syroka et al. 2011)
Fig. 7.77 Load-displacement curves for different characteristic lengths and various compressive and tensile fracture energy from 2D calculations (solid lines) compared to the experiments by Campione et al. (2005): a) .=5 mm, G=100 N/m, G.=10000 N/m, b) /.=10 mm, G=100 N/m, G,=10000 N/m, c) /.=5 mm, G=100 N/m, G,=5000 N/m, d) /.=5 mm, G=50 N/m, G,=5000 N/m (bond-slip law by CEB-FIP 1990) (Syroka et al. 2011)
Fig. 7.77 Load-displacement curves for different characteristic lengths and various compressive and tensile fracture energy from 2D calculations (solid lines) compared to the experiments by Campione et al. (2005): a) .=5 mm, G=100 N/m, G.=10000 N/m, b) /.=10 mm, G=100 N/m, G,=10000 N/m, c) /.=5 mm, G=100 N/m, G,=5000 N/m, d) /.=5 mm, G=50 N/m, G,=5000 N/m (bond-slip law by CEB-FIP 1990) (Syroka et al. 2011)
connected with the vertical one. The width of the vertical strain localization in the first stage is similar in both cases and larger by 10% than for the elasto-plastic constitutive model. The calculated load bearing capacities are 20% higher than in the experiment. Both curves exhibit similar softening as in the experiment. The smeared crack approach is also able to properly capture the propagation of i ee ae the experiment. Both curves exhibit similar softening as as in the experiment. The smeared crack approach is also able to properly capture the propagation of localized zones (Fig. 7.85). The sequence of localized zones (first, the vertical and next, the inclined one) is similar to the one observed in the experiment. As compared with the elasto-plastic model, the width of vertical zone is much larger for the smeared crack approach, moreover it has a different slope. The calculated maximum vertical force (Fig. 7.85) is significantly too high as compared to the experiment (by 45%). A similar outcome in the form of a high calculated vertical failure force with the smeared crack model was also obtained by Souza (2010).
connected with the vertical one. The width of the vertical strain localization in the first stage is similar in both cases and larger by 10% than for the elasto-plastic constitutive model. The calculated load bearing capacities are 20% higher than in the experiment. Both curves exhibit similar softening as in the experiment. The smeared crack approach is also able to properly capture the propagation of i ee ae the experiment. Both curves exhibit similar softening as as in the experiment. The smeared crack approach is also able to properly capture the propagation of localized zones (Fig. 7.85). The sequence of localized zones (first, the vertical and next, the inclined one) is similar to the one observed in the experiment. As compared with the elasto-plastic model, the width of vertical zone is much larger for the smeared crack approach, moreover it has a different slope. The calculated maximum vertical force (Fig. 7.85) is significantly too high as compared to the experiment (by 45%). A similar outcome in the form of a high calculated vertical failure force with the smeared crack model was also obtained by Souza (2010).
Fig. 7.83 Load-displacement curves from 2D calculations for various concrete models ‘solid lines) with non-local softening compared to the experimental curve (dashed line): a) elasto-plastic model, b) damage model with modified von Misses equivalent strain measure Eq. 3.38), c) damage mechanics with equivalent strain measure by Haussler-Combe Eq. 3.39), d) smeared fixed crack approach (for tests of Campione et al. (2005) (Syroka et il. 2011)
Fig. 7.83 Load-displacement curves from 2D calculations for various concrete models ‘solid lines) with non-local softening compared to the experimental curve (dashed line): a) elasto-plastic model, b) damage model with modified von Misses equivalent strain measure Eq. 3.38), c) damage mechanics with equivalent strain measure by Haussler-Combe Eq. 3.39), d) smeared fixed crack approach (for tests of Campione et al. (2005) (Syroka et il. 2011)
Fig. 7.86 Reinforcement detail of wall bottom corner (Bobinski and Tejchman 2009) aD In an Opposite direction to simpliry reinrtorcement Works. During the virgin water filling of one tank (at this moment, the tank walls were not covered with soil from the outside), an excessive horizontal displacement of vertical walls occurred. The maximum displacement of the exterior wall at the top was 75 mm at the water height of 5.7 m. The shape of the displacement curve indicated a wall rotation against the bottom. The maximum settlement of tanks was about 30 mm. The filling process was stopped and the tank was emptied. After tank emptying, a maximum permanent horizontal wall displacement at the op reduced to 20 mm.
Fig. 7.86 Reinforcement detail of wall bottom corner (Bobinski and Tejchman 2009) aD In an Opposite direction to simpliry reinrtorcement Works. During the virgin water filling of one tank (at this moment, the tank walls were not covered with soil from the outside), an excessive horizontal displacement of vertical walls occurred. The maximum displacement of the exterior wall at the top was 75 mm at the water height of 5.7 m. The shape of the displacement curve indicated a wall rotation against the bottom. The maximum settlement of tanks was about 30 mm. The filling process was stopped and the tank was emptied. After tank emptying, a maximum permanent horizontal wall displacement at the op reduced to 20 mm.
Fig. 7.87 Tank fragment assumed for plane strain FE calculations (Bobinski and Tejchman 2009) a > eas ie The behaviour of the structure was analyzed during initial tank filling for 4 different variants of reinforcement in the wall corner subjected to a positive (opening) bending moment (Fig. 7.89). The bearing capacity of the structure and wall displacements were calculated. Next, water loading was increased in the calculations up to the structure failure (assumed as a begin of material softening). Depending upon the reinforcement variant, a different bearing capacity was obtained (Tab. 7.12).
Fig. 7.87 Tank fragment assumed for plane strain FE calculations (Bobinski and Tejchman 2009) a > eas ie The behaviour of the structure was analyzed during initial tank filling for 4 different variants of reinforcement in the wall corner subjected to a positive (opening) bending moment (Fig. 7.89). The bearing capacity of the structure and wall displacements were calculated. Next, water loading was increased in the calculations up to the structure failure (assumed as a begin of material softening). Depending upon the reinforcement variant, a different bearing capacity was obtained (Tab. 7.12).
Fig. 7.89 4 different variants of reinforcement of wall corner assumed in FE calculations (Bobinski and Tejchman 2009)
Fig. 7.89 4 different variants of reinforcement of wall corner assumed in FE calculations (Bobinski and Tejchman 2009)
Fig. 7.93 Formation of strain localization in the wall corner during: a) initial water filling,
Fig. 7.93 Formation of strain localization in the wall corner during: a) initial water filling,
Fig. 7.92 Method for tank strengthening (Bobinski and Tejchman 2009)
Fig. 7.92 Method for tank strengthening (Bobinski and Tejchman 2009)
Fig. 8.1 Size effect models SEL by BaZant (2004) in logarithmic scale with oy - nominal strength, D — element size: a) type 1 (structures without notches and pre-existing large cracks), b) type 2 (notched structures) (material strength is bound for small sizes by plasticity limit whereas for large sizes, the material follows Linear Elastic Fracture Mechanics LEFM) where Oy is the nominal strength, D is the characteristic structure size, f,~, D, and r denote the positive constant representing unknown empirical parameters to be determined; f,~ represents the solution of the elastic-brittle strength reached as the nominal strength for large structures, 7 controls the curvature and shape of the law and D, is the deterministic characteristic length having the meaning of the thickness of the cracked layer (if D,=0, the behaviour is elastic-brittle, Eq. 8.1). In turn, in Eq. 8.2, f, denotes the tensile strength, B is the dimensionless geometry- dependent parameter (depending on the geometry of the structure and crack) and D, denotes the size-dependent parameter called transitional size (both unknown parameters to be determined).
Fig. 8.1 Size effect models SEL by BaZant (2004) in logarithmic scale with oy - nominal strength, D — element size: a) type 1 (structures without notches and pre-existing large cracks), b) type 2 (notched structures) (material strength is bound for small sizes by plasticity limit whereas for large sizes, the material follows Linear Elastic Fracture Mechanics LEFM) where Oy is the nominal strength, D is the characteristic structure size, f,~, D, and r denote the positive constant representing unknown empirical parameters to be determined; f,~ represents the solution of the elastic-brittle strength reached as the nominal strength for large structures, 7 controls the curvature and shape of the law and D, is the deterministic characteristic length having the meaning of the thickness of the cracked layer (if D,=0, the behaviour is elastic-brittle, Eq. 8.1). In turn, in Eq. 8.2, f, denotes the tensile strength, B is the dimensionless geometry- dependent parameter (depending on the geometry of the structure and crack) and D, denotes the size-dependent parameter called transitional size (both unknown parameters to be determined).
Fig. 8.2 Size effect by Carpintieri et al (1994): nominal strength oy versus specimen size D
Fig. 8.2 Size effect by Carpintieri et al (1994): nominal strength oy versus specimen size D
Deterministic and statistical calculations The two-dimensional FE-calculations (Bobinski et al. 2009) of free supportec 10tched beams with free ends under bending (assuming constant values of tensile strength f;) were performed with 4 different beam sizes of a similar geometry hxL, 3x32 cm? (called small-size beam), 16x64 cm? (called medium-size beam) 32x128 cm’ (called large-size beam) and 192x768 cm’ (called very large-size seam) (h — beam height, L, — total beam length). The span length L was equal tc 3h for all beams (Fig. 8.4). The size of the first 3 beams was similar as ir corresponding experiments carried out by Le Bellego et al. (2003). The quadrilateral elements divided into triangular elements were used to avoic volumetric locking. 7628 triangular (small-size beam), 14476 (medium-sizé yeam), 28092 (large size beam) and 104310 (large-size beam) triangular element: were used, respectively. The mesh was particularly very fine in the region of < 10tch (Fig. 8.5) to properly capture strain localization in concrete (where the finite >lement size was equal to 1/3x/,, 1=5 mm). The ratio between the width of thi: fine region and beam length was always the same.
Deterministic and statistical calculations The two-dimensional FE-calculations (Bobinski et al. 2009) of free supportec 10tched beams with free ends under bending (assuming constant values of tensile strength f;) were performed with 4 different beam sizes of a similar geometry hxL, 3x32 cm? (called small-size beam), 16x64 cm? (called medium-size beam) 32x128 cm’ (called large-size beam) and 192x768 cm’ (called very large-size seam) (h — beam height, L, — total beam length). The span length L was equal tc 3h for all beams (Fig. 8.4). The size of the first 3 beams was similar as ir corresponding experiments carried out by Le Bellego et al. (2003). The quadrilateral elements divided into triangular elements were used to avoic volumetric locking. 7628 triangular (small-size beam), 14476 (medium-sizé yeam), 28092 (large size beam) and 104310 (large-size beam) triangular element: were used, respectively. The mesh was particularly very fine in the region of < 10tch (Fig. 8.5) to properly capture strain localization in concrete (where the finite >lement size was equal to 1/3x/,, 1=5 mm). The ratio between the width of thi: fine region and beam length was always the same.
Fig. 8.5 FE mesh in the case of a medium-size beam (Bobinski et al. 2009) To describe the behaviour of concrete under tension during three-point bending, a Rankine criterion was used with a yield function with isotropic softening (Eq. 3.32). To model the concrete softening under tension, the exponential curve by Hordijk (1991) with the tensile strength of the concrete of f=3.6 MPa was assumed (x,=0.005 b\=3.0, bo>=6.93) (Eq. 3.55). The modulus of elasticity was assumed to be E=38.5 GPa and the Poisson ratio was v=0.24 (Le Bellego et al. 2003). The calculations were performed under plane strain conditions (the differences between the results obtained within Rankine plasticity under plane stress and plane strain conditions are insignificant). A large-displacement analysis available in the ABAQUS finite element code (1998) was used (although the influence of such analysis was negligible). In this method, the current configuration of the body was taken into account. The Cauchy stress was taken as the stress measure. The conjugate strain rate was the rate of deformation. The rotation of the stress and strain tensor was calculated with the Hughes-Winget method (1980). The non-local averaging was performed in the current pe peed s BCs Peas: A quasi-static deformation of a small, medium and large beam was imposed through a constant vertical displacement increment Aw prescribed at the upper mid-point of the beam top. To capture a snap-back behaviour in a very large size beam, the so-called arc-length technique was used. The actual load vector P was defined as APnax where A— multiplier and P,,,, — maximum constant load vector. In general, the determination of the length of the arc the P-u space (u — displacement vector) involves the displacements of all nodes (as e.g. the Riks method available in ABAQUS Standard 1998). However, for problems involving strain localization, it is more suitable to use an indirect displacement control method, where only selected nodal displacements are considered to formulate an additional condition in the P-w space. The horizontal distance between two nodes lying on the opposite sides of the notch was chosen as a control variable CVOD (crack mouth open displacement). The indirect displacement algorithm was implemented with the aid of two identical and independent FE-meshes and some additional node elements to exchange the information about the displacements between these meshes. The Monte Carlo method was used in statistical calculations. Application of
Fig. 8.5 FE mesh in the case of a medium-size beam (Bobinski et al. 2009) To describe the behaviour of concrete under tension during three-point bending, a Rankine criterion was used with a yield function with isotropic softening (Eq. 3.32). To model the concrete softening under tension, the exponential curve by Hordijk (1991) with the tensile strength of the concrete of f=3.6 MPa was assumed (x,=0.005 b\=3.0, bo>=6.93) (Eq. 3.55). The modulus of elasticity was assumed to be E=38.5 GPa and the Poisson ratio was v=0.24 (Le Bellego et al. 2003). The calculations were performed under plane strain conditions (the differences between the results obtained within Rankine plasticity under plane stress and plane strain conditions are insignificant). A large-displacement analysis available in the ABAQUS finite element code (1998) was used (although the influence of such analysis was negligible). In this method, the current configuration of the body was taken into account. The Cauchy stress was taken as the stress measure. The conjugate strain rate was the rate of deformation. The rotation of the stress and strain tensor was calculated with the Hughes-Winget method (1980). The non-local averaging was performed in the current pe peed s BCs Peas: A quasi-static deformation of a small, medium and large beam was imposed through a constant vertical displacement increment Aw prescribed at the upper mid-point of the beam top. To capture a snap-back behaviour in a very large size beam, the so-called arc-length technique was used. The actual load vector P was defined as APnax where A— multiplier and P,,,, — maximum constant load vector. In general, the determination of the length of the arc the P-u space (u — displacement vector) involves the displacements of all nodes (as e.g. the Riks method available in ABAQUS Standard 1998). However, for problems involving strain localization, it is more suitable to use an indirect displacement control method, where only selected nodal displacements are considered to formulate an additional condition in the P-w space. The horizontal distance between two nodes lying on the opposite sides of the notch was chosen as a control variable CVOD (crack mouth open displacement). The indirect displacement algorithm was implemented with the aid of two identical and independent FE-meshes and some additional node elements to exchange the information about the displacements between these meshes. The Monte Carlo method was used in statistical calculations. Application of
Fig. 8.7 The correlation distances for different coefficients 2 [1/m] (Bobinski et al. 2009)
Fig. 8.7 The correlation distances for different coefficients 2 [1/m] (Bobinski et al. 2009)
Fig. 8.10 Normalized force-displacement curves with constant values of tensile strength for 4 notched beams under three-point bending (Bobinski et al. 2009)
Fig. 8.10 Normalized force-displacement curves with constant values of tensile strength for 4 notched beams under three-point bending (Bobinski et al. 2009)
Fig. 8.9 Stochastic distribution of tensile strength f, close to the notch in small-size beam (strong correlation, small standard deviation) (Bobinski et al. 2009)
Fig. 8.9 Stochastic distribution of tensile strength f, close to the notch in small-size beam (strong correlation, small standard deviation) (Bobinski et al. 2009)
Fig. 8.12 Relationship between calculated normalized concrete strength Ino=In[PL/(f,h’t)| and ratio In(h/l,.) compared to size effect law by Bazant of Fig. 8.1b (Bazant and Planas 1998) for constant values of tensile strength (h- beam height, /, - characteristic length) (Bobinski et al. 2009) ee eee do D gies satiiaslaciveteaammaal™ sated! = ua — Seadeiatn The load-displacement curves for a very large beam are not smooth ir oftening regime when the tensile strength is distributed stochastically. The scatte: f the maximum vertical force around its mean value is similar for all beam size: Fig. 8.17). The deformation field above the notch is strongly non-symmetric Fig. 8.16). The mean width of the localized zone above the notch is slightly igher than the deterministic value, namely: w=16.56 mm (A=8 cm), w=18.88 mn h=32 cm) and w=19.67 mm (h=192 cm), Fig. 8.15.
Fig. 8.12 Relationship between calculated normalized concrete strength Ino=In[PL/(f,h’t)| and ratio In(h/l,.) compared to size effect law by Bazant of Fig. 8.1b (Bazant and Planas 1998) for constant values of tensile strength (h- beam height, /, - characteristic length) (Bobinski et al. 2009) ee eee do D gies satiiaslaciveteaammaal™ sated! = ua — Seadeiatn The load-displacement curves for a very large beam are not smooth ir oftening regime when the tensile strength is distributed stochastically. The scatte: f the maximum vertical force around its mean value is similar for all beam size: Fig. 8.17). The deformation field above the notch is strongly non-symmetric Fig. 8.16). The mean width of the localized zone above the notch is slightly igher than the deterministic value, namely: w=16.56 mm (A=8 cm), w=18.88 mn h=32 cm) and w=19.67 mm (h=192 cm), Fig. 8.15.
Fig. 8.13 The load-displacement curves from FE-calculations with constant values of tensile strength compared to the experiments by Le Bellego et al. (2003) with 3 notched concrete beams: h=8 cm (lower curves), =16 cm (medium curves) and h=32 cm (upper curves) (Bobinski et al. 2009)
Fig. 8.13 The load-displacement curves from FE-calculations with constant values of tensile strength compared to the experiments by Le Bellego et al. (2003) with 3 notched concrete beams: h=8 cm (lower curves), =16 cm (medium curves) and h=32 cm (upper curves) (Bobinski et al. 2009)
Fig. 8.15 Distribution of non-local softening parameter above the notch in the case of deterministic (red dashed lines) and random calculation (solid lines) for 3 notched beams under three-point bending: a) small-size beam (h=8 cm), b) large-size beam (h=32 cm), c) very large-size beam (h=192 cm) (A,)=1 I/m, A2=3 1/m, sy=0.424 MPa) (Bobirski et al. 2009)
Fig. 8.15 Distribution of non-local softening parameter above the notch in the case of deterministic (red dashed lines) and random calculation (solid lines) for 3 notched beams under three-point bending: a) small-size beam (h=8 cm), b) large-size beam (h=32 cm), c) very large-size beam (h=192 cm) (A,)=1 I/m, A2=3 1/m, sy=0.424 MPa) (Bobirski et al. 2009)
All specimens had again the constant uniformly distributed tensile strength jf=3.6 MPa. In order to properly capture strain localization in concrete, the mesh was very fine in the mid-part of the beam (Fig. 8.21) (where the element size was not greater than 3x/,). The width of this region s of Fig. 8.22 was determined with preliminary calculations: s=12 cm (D=8 cm), s=18cm (D=16 cm), s=24cm (D=32 cm) and s=192 cm (D=192 cm). A quasi-static deformation of a small and medium beam was imposed through a constant vertical displacement increment Au prescribed at the upper mid-point of the beam top.
All specimens had again the constant uniformly distributed tensile strength jf=3.6 MPa. In order to properly capture strain localization in concrete, the mesh was very fine in the mid-part of the beam (Fig. 8.21) (where the element size was not greater than 3x/,). The width of this region s of Fig. 8.22 was determined with preliminary calculations: s=12 cm (D=8 cm), s=18cm (D=16 cm), s=24cm (D=32 cm) and s=192 cm (D=192 cm). A quasi-static deformation of a small and medium beam was imposed through a constant vertical displacement increment Au prescribed at the upper mid-point of the beam top.
Correlated random fields describing a fluctuation of the tensile strength were used to capture a stochastic size effect. The distribution of this single random variable f, took the form of a truncated Gaussian function with the mean concrete tensile strength of 3.6 MPa (as in calculations with notched beams, Fig. 8.6). The concrete tensile strength values again changed between 1.6 MPa and 5.6 MPa ( f, =3.642.0MPa). The homogeneous correlation function by Eq. 8.4 was adopted (Bielewicz and Gérski 2002). We took again into account a stronger correlation of the tensile strength f, in a horizontal direction 1,,=1.0 1/m and a
Correlated random fields describing a fluctuation of the tensile strength were used to capture a stochastic size effect. The distribution of this single random variable f, took the form of a truncated Gaussian function with the mean concrete tensile strength of 3.6 MPa (as in calculations with notched beams, Fig. 8.6). The concrete tensile strength values again changed between 1.6 MPa and 5.6 MPa ( f, =3.642.0MPa). The homogeneous correlation function by Eq. 8.4 was adopted (Bielewicz and Gérski 2002). We took again into account a stronger correlation of the tensile strength f, in a horizontal direction 1,,=1.0 1/m and a
weaker correlation in a vertical directions 2,.=3.0 1/m in Eq. 8.4 (due to the way of the specimen preparation by means of layer-by-layer from the same concrete block). The dimension of the random field was identical as the finite element mesh. The same random values were assumed in four neighbouring triangular elements. To generate the random fields, the conditional-rejection method was again used. The selection was performed by the Latin sampling method (Fig. 8.23). The generated numbers formed the following 12 pairs: 1-4, 2-7, 3 -—3,4- 11,5 5,6-8,7-1,8-6,9-2, 10-9, 11 — 10 and 12 - 12 (Fig. 8.23). Figure 8.24 shows the distribution of the concrete tensile strength in a small-size (Fig. 8.24a) and very large-size concrete beam (Fig. 8.24b). Deterministic size effect
weaker correlation in a vertical directions 2,.=3.0 1/m in Eq. 8.4 (due to the way of the specimen preparation by means of layer-by-layer from the same concrete block). The dimension of the random field was identical as the finite element mesh. The same random values were assumed in four neighbouring triangular elements. To generate the random fields, the conditional-rejection method was again used. The selection was performed by the Latin sampling method (Fig. 8.23). The generated numbers formed the following 12 pairs: 1-4, 2-7, 3 -—3,4- 11,5 5,6-8,7-1,8-6,9-2, 10-9, 11 — 10 and 12 - 12 (Fig. 8.23). Figure 8.24 shows the distribution of the concrete tensile strength in a small-size (Fig. 8.24a) and very large-size concrete beam (Fig. 8.24b). Deterministic size effect
Fig. 8.25 Normalized horizontal normal (flexural) stress-deflection curves 1.5FL/f,D’t)=f(u/D) under 3-point bending with constant values of tensile strength for 4 different concrete beam heights: small D=8 cm (dashed line ‘a’), medium D=16 cm (dotted-dashed line ‘b’), large D=32 cm (dotted line ‘c’), very large D=192 cm (solid line ‘d’) (F - vertical force, L — beam length, D — beam height, t- beam thickness, f, - tensile strength) (Syroka et al. 2011)
Fig. 8.25 Normalized horizontal normal (flexural) stress-deflection curves 1.5FL/f,D’t)=f(u/D) under 3-point bending with constant values of tensile strength for 4 different concrete beam heights: small D=8 cm (dashed line ‘a’), medium D=16 cm (dotted-dashed line ‘b’), large D=32 cm (dotted line ‘c’), very large D=192 cm (solid line ‘d’) (F - vertical force, L — beam length, D — beam height, t- beam thickness, f, - tensile strength) (Syroka et al. 2011)
Fig. 8.28 Calculated normalized flexural tensile strength f/f=1.5FnaLAf,D°t) versus beam height D in unnotched concrete beams from deterministic FE calculations (red circles) versus beam height D compared with the deterministic size effect model by Bazant (blue dashed line by Eq. 8.1 with r=1, green dotted line by Eq. 8.1 with r=4 and red solid line by Eq. 8.7 with 7=1) (Syroka et al. 2011)
Fig. 8.28 Calculated normalized flexural tensile strength f/f=1.5FnaLAf,D°t) versus beam height D in unnotched concrete beams from deterministic FE calculations (red circles) versus beam height D compared with the deterministic size effect model by Bazant (blue dashed line by Eq. 8.1 with r=1, green dotted line by Eq. 8.1 with r=4 and red solid line by Eq. 8.7 with 7=1) (Syroka et al. 2011)
Fig. 8.34 Calculated normalized flexural tensile strength f/f=1.5Final/fD"i versus beam height D from deterministic (circles) and stochastic (triangles) F calculations compared with deterministic (line ‘a’, Eq. 8.7) and deterministic stochastic size effect law by BaZant (Eq. 8.10) for various Weibull moduli m an constant deterministic parameters (line “b’- m=48, line ‘c’- m=24, line ‘d’ - m=12 (Syroka et al. 2011) ssuming the Weibull modulus m=12-48. The stochastic outcomes indicate a ther decrease of the nominal strength with increasing element size while the sterministic ones reach their lower limit. Our deterministic-statistical results resent also a satisfactory agreement with the size effect law by Bazant by suming the recommended value of m=24 (L,=D,=16.95 mm, /,=0, r=1 and “=4.753 MPa). However, the Weibull modulus m=48 solely enables a transition om a pure deterministic to a coupled deterministic-statistical size effect. The alue m=12 underestimates the calculated deterministic-statistical flexural tensile renoth._
Fig. 8.34 Calculated normalized flexural tensile strength f/f=1.5Final/fD"i versus beam height D from deterministic (circles) and stochastic (triangles) F calculations compared with deterministic (line ‘a’, Eq. 8.7) and deterministic stochastic size effect law by BaZant (Eq. 8.10) for various Weibull moduli m an constant deterministic parameters (line “b’- m=48, line ‘c’- m=24, line ‘d’ - m=12 (Syroka et al. 2011) ssuming the Weibull modulus m=12-48. The stochastic outcomes indicate a ther decrease of the nominal strength with increasing element size while the sterministic ones reach their lower limit. Our deterministic-statistical results resent also a satisfactory agreement with the size effect law by Bazant by suming the recommended value of m=24 (L,=D,=16.95 mm, /,=0, r=1 and “=4.753 MPa). However, the Weibull modulus m=48 solely enables a transition om a pure deterministic to a coupled deterministic-statistical size effect. The alue m=12 underestimates the calculated deterministic-statistical flexural tensile renoth._
Fig. 8.35 Calculated normalized nominal (flexural) strength oy/f, (Oy=1.5FnaxLAD’t)) versus beam height D for: a) unnotched concrete beams from deterministic (red circles) and stochastic (blue triangles); b) for notched concrete beams from deterministic (green squares) and stochastic (green diamonds) FE calculations compared with deterministic size effect law by Bazant (Eq. 8.7) (red solid line), deterministic-stochastic size effect law by Bazant (Eq. 8.10) (blue dashed line) and deterministic size effect law by BaZant (Eq. 8.2) (green dotted-dashed line) (Syroka et al. 2011) stochastic (blue triangles); b) for notched concrete beams from deterministic (green
Fig. 8.35 Calculated normalized nominal (flexural) strength oy/f, (Oy=1.5FnaxLAD’t)) versus beam height D for: a) unnotched concrete beams from deterministic (red circles) and stochastic (blue triangles); b) for notched concrete beams from deterministic (green squares) and stochastic (green diamonds) FE calculations compared with deterministic size effect law by Bazant (Eq. 8.7) (red solid line), deterministic-stochastic size effect law by Bazant (Eq. 8.10) (blue dashed line) and deterministic size effect law by BaZant (Eq. 8.2) (green dotted-dashed line) (Syroka et al. 2011) stochastic (blue triangles); b) for notched concrete beams from deterministic (green
Fig. 8.37 Calculated normalized nominal (flexural) strength oj/f; (Oy=1.5 Final. D’t)) versus beam height D from coupled deterministic-stochastic FE calculations (circles) for notched beams (Bobinski et al. 2009) compared with two size effect laws: SEL by Bazant (Eq. 8.2) — solid line and MFSL by Carpinteri et al. (Eq. 8.3) — dashed line (Syroka et al. 2011) Fig. 8.36 Calculated normalized nominal (flexural) strength fAG=NS Finad/f,D't) versus beam height D from coupled deterministic-stochastic FE calculations (circles) for unnotched beams compared with two size effect laws: SEL by Bazant (Eq. 8.10) — solid line and MFSL by Carpinteri et al. (Eq. 8.3) — dashed line (Syroka et al. 2011)
Fig. 8.37 Calculated normalized nominal (flexural) strength oj/f; (Oy=1.5 Final. D’t)) versus beam height D from coupled deterministic-stochastic FE calculations (circles) for notched beams (Bobinski et al. 2009) compared with two size effect laws: SEL by Bazant (Eq. 8.2) — solid line and MFSL by Carpinteri et al. (Eq. 8.3) — dashed line (Syroka et al. 2011) Fig. 8.36 Calculated normalized nominal (flexural) strength fAG=NS Finad/f,D't) versus beam height D from coupled deterministic-stochastic FE calculations (circles) for unnotched beams compared with two size effect laws: SEL by Bazant (Eq. 8.10) — solid line and MFSL by Carpinteri et al. (Eq. 8.3) — dashed line (Syroka et al. 2011)
The calculations were carried out with periodic boundary conditions and material periodicity to avoid the effect of walls (van der Sluis 2001, Gitman 2006, Gitman et al. 2008). In the first case, the positions of nodes along corresponding specimen boundaries were the same before and after deformation. This is illustrated in Fig. 9.2, where an arbitrary periodically deformed unit cell under uniaxial extension conditions is shown. The deformation of each boundary pair is the same and the stresses are opposite in sign for each pair. The displacement boundary conditions are Table 9.1 Material parameters assumed for uniaxial tension (Skarzynski and Tejchman 2009)
The calculations were carried out with periodic boundary conditions and material periodicity to avoid the effect of walls (van der Sluis 2001, Gitman 2006, Gitman et al. 2008). In the first case, the positions of nodes along corresponding specimen boundaries were the same before and after deformation. This is illustrated in Fig. 9.2, where an arbitrary periodically deformed unit cell under uniaxial extension conditions is shown. The deformation of each boundary pair is the same and the stresses are opposite in sign for each pair. The displacement boundary conditions are Table 9.1 Material parameters assumed for uniaxial tension (Skarzynski and Tejchman 2009)
Fig. 9.1 Approximation of non-linear grading curve with discrete numbers of aggregate sizes (Skarzynski and Tejchman 2010)
Fig. 9.1 Approximation of non-linear grading curve with discrete numbers of aggregate sizes (Skarzynski and Tejchman 2010)
First, concrete specimens of five different sizes were investigated. The smallest and the largest unit cells were 10x10 mm” and 25x25 mm’, respectively (Fig. 9.6). For each specimen, five different stochastic realizations were performed (Fig. 9.7) with the aggregate density p kept constant (9=30%, p=45% and p=60%) (Fig. 9.8). Next, the calculations were carried out with a different characteristic length of micro-structure varying between /.=0.1 mm-2.0 mm. Later, the effect of an aggregate density (p=30%, p=45% and p=60%) on strain localization was investigated. In the final comparative calculations, non-locality was prescribed to the cement matrix only.
First, concrete specimens of five different sizes were investigated. The smallest and the largest unit cells were 10x10 mm” and 25x25 mm’, respectively (Fig. 9.6). For each specimen, five different stochastic realizations were performed (Fig. 9.7) with the aggregate density p kept constant (9=30%, p=45% and p=60%) (Fig. 9.8). Next, the calculations were carried out with a different characteristic length of micro-structure varying between /.=0.1 mm-2.0 mm. Later, the effect of an aggregate density (p=30%, p=45% and p=60%) on strain localization was investigated. In the final comparative calculations, non-locality was prescribed to the cement matrix only.
Fig. 9.7 Different stochastic distribution of aggregate for a concrete specimen of 15x15 mm’ (aggregate density p=30%) (Skarzynski and Tejchman 2009)
Fig. 9.7 Different stochastic distribution of aggregate for a concrete specimen of 15x15 mm’ (aggregate density p=30%) (Skarzynski and Tejchman 2009)
However, they are completely different in a softening regime after the peak is reached. An increase of the specimen size causes an increase of the material brittleness. The differences in the evolution of stress-strain curves in a softening regime are caused by strain localization contributing to a loss of the material homogeneity (Fig. 9.13). Strain localization in the form of a localized zone propagates between aggregates and can be strongly curved. The width of the calculated zone is about w.=(4x/,.)=2 mm (with /.=0.5 mm).
However, they are completely different in a softening regime after the peak is reached. An increase of the specimen size causes an increase of the material brittleness. The differences in the evolution of stress-strain curves in a softening regime are caused by strain localization contributing to a loss of the material homogeneity (Fig. 9.13). Strain localization in the form of a localized zone propagates between aggregates and can be strongly curved. The width of the calculated zone is about w.=(4x/,.)=2 mm (with /.=0.5 mm).
Fig. 9.14 Stress-strain curves for different characteristic lengths: a) /.=0.1 mm, b) /.=0.25 mm, c) /.=0.5 mm, d) /.=1.0 mm, e) /.=2.0 mm (specimen size 10x10 mm’, aggregate density p=30%) (Skarzynski and Tejchman 2009)
Fig. 9.14 Stress-strain curves for different characteristic lengths: a) /.=0.1 mm, b) /.=0.25 mm, c) /.=0.5 mm, d) /.=1.0 mm, e) /.=2.0 mm (specimen size 10x10 mm’, aggregate density p=30%) (Skarzynski and Tejchman 2009)
Fig. 9.20 Stress-strain curves for 2 different specimen sizes: A) 15x15 mm’, B) 25x25 mm? with a) non-locality prescribed to three phases and b) non-locality prescribed to cement matrix (aggregate density p=30%, characteristic length /.=0.5 mm) (Skarzynski and Tejchman 2009)
Fig. 9.20 Stress-strain curves for 2 different specimen sizes: A) 15x15 mm’, B) 25x25 mm? with a) non-locality prescribed to three phases and b) non-locality prescribed to cement matrix (aggregate density p=30%, characteristic length /.=0.5 mm) (Skarzynski and Tejchman 2009)
Fig. 9.21 Distribution of non-local softening strain measure for 2 different specimen sizes: A) 15x15 mm, B) 25x25 mm” with a) non-locality prescribed to three phases and b) non- locality prescribed to cement matrix (aggregate density p=30%, characteristic length /.=0.5 mm) (Skarzynski and Tejchman 2009)
Fig. 9.21 Distribution of non-local softening strain measure for 2 different specimen sizes: A) 15x15 mm, B) 25x25 mm” with a) non-locality prescribed to three phases and b) non- locality prescribed to cement matrix (aggregate density p=30%, characteristic length /.=0.5 mm) (Skarzynski and Tejchman 2009)
related computation time in FE calculations at meso-scale (two-dimensional analyses were carried out instead of three-dimensional ones) and b) to avoid differences in the hydration heat effects which are proportional to the thickness of the member (BaZant and Planas, 1998). The following concrete beams were used: a) small-size beams 80x320 mm”, b) medium-size beams 160x640 mm” and c) large-size beams 320x1280 mm” (Fig. 9.22a). The thickness of beams was always the same b=40 mm, and the beams’ span was equal to 3xD. A notch with a height of D/10 mm was located at the mid-span of the beam bottom. The beams were subjected to a vertical displacement in the mid-point at a very slow rate. Two different fine-grained concrete mixes were composed of ordinary Portland cement, water and fine sand (with a mean aggregate diameter dsj=0.5 mm and maximum aggregate diameter d,,,,=3.0 mm) or sand (d59=2.0 mm, d)_,=8.0 mm) (Fig. 9.22b). The width and shape of a localized zone above the notch on the surface of beams was determined with a Digital Image Correlation (DIC) method which is an optical way to visualize surface displacements by successive post-processing of
related computation time in FE calculations at meso-scale (two-dimensional analyses were carried out instead of three-dimensional ones) and b) to avoid differences in the hydration heat effects which are proportional to the thickness of the member (BaZant and Planas, 1998). The following concrete beams were used: a) small-size beams 80x320 mm”, b) medium-size beams 160x640 mm” and c) large-size beams 320x1280 mm” (Fig. 9.22a). The thickness of beams was always the same b=40 mm, and the beams’ span was equal to 3xD. A notch with a height of D/10 mm was located at the mid-span of the beam bottom. The beams were subjected to a vertical displacement in the mid-point at a very slow rate. Two different fine-grained concrete mixes were composed of ordinary Portland cement, water and fine sand (with a mean aggregate diameter dsj=0.5 mm and maximum aggregate diameter d,,,,=3.0 mm) or sand (d59=2.0 mm, d)_,=8.0 mm) (Fig. 9.22b). The width and shape of a localized zone above the notch on the surface of beams was determined with a Digital Image Correlation (DIC) method which is an optical way to visualize surface displacements by successive post-processing of
Fig. 9.23 Formation of localized zone with mean width of w,=3.5-4.0 mm directly above notch in 3 different experiments (‘a’, ‘b’ and ‘c’) with small-size notched fine-grained concrete beam 80x320x40 mm? using DIC (vertical and horizontal axes denote coordinates in [mm] and colour scales strain intensity) (Skarzynski et al. 2011) digital images taken at a constant time increment from a professional digital camera (based on displacements, strains can be calculated) (White et al. 2003). The experimental set-up and results were described in detail by Skarzynski et al. (2011). The beams of the same size were also used by Le Bellégo et al. (2003). Figure 9.23 shows the formation of a localized zone on one side of the surface of a fine-grained small-size concrete beam above the notch from laboratory tests
Fig. 9.23 Formation of localized zone with mean width of w,=3.5-4.0 mm directly above notch in 3 different experiments (‘a’, ‘b’ and ‘c’) with small-size notched fine-grained concrete beam 80x320x40 mm? using DIC (vertical and horizontal axes denote coordinates in [mm] and colour scales strain intensity) (Skarzynski et al. 2011) digital images taken at a constant time increment from a professional digital camera (based on displacements, strains can be calculated) (White et al. 2003). The experimental set-up and results were described in detail by Skarzynski et al. (2011). The beams of the same size were also used by Le Bellégo et al. (2003). Figure 9.23 shows the formation of a localized zone on one side of the surface of a fine-grained small-size concrete beam above the notch from laboratory tests
Fig. 9.24 Evolution of width w, (A) and height h, (B) of localized zone with deflection u directly above notch in experiments with small-size notched beam 80x320x40 mm’ of fine- grained concrete using DIC: a) aggregate dsj>=2 mm and d,,,,,=8 mm, b) aggregate ds9=0.5 mm and d,,,,=3 mm (x - maximum vertical force, ¢ - formation of macro-crack) (Skarzynski et al. 2011) using a DIC (Skarzynski et al. 2011). A localized zone occurred always before the peak on the force-deflection diagram and was strongly curved. In some cases, it branched. The measured width of a localized zone above the notch increased during deformation due to concrete dilatancy (Fig. 9.24A) up to w,=3.5-4.0 mm (Sdingx) in the range of the deflection u=0.01-0.04 mm until a macro-crack was created. The maximum height of a localized zone above the notch was about h.=50-55 mm at u=0.04 mm (Fig. 9.24B). The width of a localized zone did not depend upon the concrete mix type and beam size (Skarzynski et al. 2011).
Fig. 9.24 Evolution of width w, (A) and height h, (B) of localized zone with deflection u directly above notch in experiments with small-size notched beam 80x320x40 mm’ of fine- grained concrete using DIC: a) aggregate dsj>=2 mm and d,,,,,=8 mm, b) aggregate ds9=0.5 mm and d,,,,=3 mm (x - maximum vertical force, ¢ - formation of macro-crack) (Skarzynski et al. 2011) using a DIC (Skarzynski et al. 2011). A localized zone occurred always before the peak on the force-deflection diagram and was strongly curved. In some cases, it branched. The measured width of a localized zone above the notch increased during deformation due to concrete dilatancy (Fig. 9.24A) up to w,=3.5-4.0 mm (Sdingx) in the range of the deflection u=0.01-0.04 mm until a macro-crack was created. The maximum height of a localized zone above the notch was about h.=50-55 mm at u=0.04 mm (Fig. 9.24B). The width of a localized zone did not depend upon the concrete mix type and beam size (Skarzynski et al. 2011).
Table 9.2 Material parameters assumed for three-point bending (Skarzynski and Tejchman 2010) Four different fine-grained concrete mixes were numerically analysed (Fig. 9.25). To reduce the number of aggregate grains in calculations, the size of the smallest inclusions had to be limited (Fig. 9.25). The aggregate size varied between the minimum value d,,;,=2 mm and maximum value d,,,,,=8 mm with the mean value of dsj=2 mm (aggregate size distribution curve ‘a’ of Fig. 9.25 corresponding to the experimental one for sand concrete of Fig. 9.22b), dj,i,=2 mm and drg=l0 mm with dsj=4 mm (aggregate size distribution curve ‘b’ of Fig. 9.25), di»j,=2 mm and d,,,,=6 mm with d;s=4 mm (aggregate size distribution curve ‘c’ of Fig. 9.25) and d,,i,=0.5 mm and d,,,=3 mm with ds9=0.5 mm (aggregate size distribution curve ‘d’ of Fig. 9.25 corresponding to the experimental one for fine sand concrete of Fig. 9.22b). The width of ITZs was assumed to be t,=0-0.75 mm. The size of finite elements
Table 9.2 Material parameters assumed for three-point bending (Skarzynski and Tejchman 2010) Four different fine-grained concrete mixes were numerically analysed (Fig. 9.25). To reduce the number of aggregate grains in calculations, the size of the smallest inclusions had to be limited (Fig. 9.25). The aggregate size varied between the minimum value d,,;,=2 mm and maximum value d,,,,,=8 mm with the mean value of dsj=2 mm (aggregate size distribution curve ‘a’ of Fig. 9.25 corresponding to the experimental one for sand concrete of Fig. 9.22b), dj,i,=2 mm and drg=l0 mm with dsj=4 mm (aggregate size distribution curve ‘b’ of Fig. 9.25), di»j,=2 mm and d,,,,=6 mm with d;s=4 mm (aggregate size distribution curve ‘c’ of Fig. 9.25) and d,,i,=0.5 mm and d,,,=3 mm with ds9=0.5 mm (aggregate size distribution curve ‘d’ of Fig. 9.25 corresponding to the experimental one for fine sand concrete of Fig. 9.22b). The width of ITZs was assumed to be t,=0-0.75 mm. The size of finite elements
Fig. 9.25 Aggregate size distribution curves assumed for FE calculations (note that small aggregates were cut off to reduce the computation time)
Fig. 9.25 Aggregate size distribution curves assumed for FE calculations (note that small aggregates were cut off to reduce the computation time)
Fig. 9.26 Calculated and experimental nominal strength 1.5P//(bD’) versus normalised beam deflection u/D (u - beam deflection, D - beam height): A) FE-results, B) experiments by Le Bellégo et al. (2003), 1) small-size beam, 2) medium-size beam, 3) large-size beam (homogeneous one-phase material, /=2 mm) (Skarzynski and Tejchman 2010)
Fig. 9.26 Calculated and experimental nominal strength 1.5P//(bD’) versus normalised beam deflection u/D (u - beam deflection, D - beam height): A) FE-results, B) experiments by Le Bellégo et al. (2003), 1) small-size beam, 2) medium-size beam, 3) large-size beam (homogeneous one-phase material, /=2 mm) (Skarzynski and Tejchman 2010)
Fig. 9.28 Calculated force-deflection curves for two different random distributions of aggregate in small-size beam 80x320 mm” of sand concrete (d5g=2 mm, ding=8 mm, 1.=1.5 mm): a) entirely heterogeneous beam, b) partially heterogeneous beam with width of meso- scale section of b,,,=80 mm, c) partially heterogeneous beam with width of the meso-scale section of b,,;=40 mm (Skarzynski and Tejchman 2010) UlULUUUOH OL d DOM LOU! SULICHIII RS Pdl dele AVUVE Ule HOLT, The results show that the effect of the width of the meso-scale region on the results can be significant if b,,,;<D/2. However, if the width of a meso-scale region close to the notch equals D=80 mm, the results of forces and strains with an entirely and a partially heterogeneous beam are similar. In further calculations to save computational time, a representative meso-scale section was assumed to be always equal to the beam height D,,,=D (i.e. 80 mm for a small-size beam, 160 mm for a medium-size beam and 320 mm for a large-size beam).
Fig. 9.28 Calculated force-deflection curves for two different random distributions of aggregate in small-size beam 80x320 mm” of sand concrete (d5g=2 mm, ding=8 mm, 1.=1.5 mm): a) entirely heterogeneous beam, b) partially heterogeneous beam with width of meso- scale section of b,,,=80 mm, c) partially heterogeneous beam with width of the meso-scale section of b,,;=40 mm (Skarzynski and Tejchman 2010) UlULUUUOH OL d DOM LOU! SULICHIII RS Pdl dele AVUVE Ule HOLT, The results show that the effect of the width of the meso-scale region on the results can be significant if b,,,;<D/2. However, if the width of a meso-scale region close to the notch equals D=80 mm, the results of forces and strains with an entirely and a partially heterogeneous beam are similar. In further calculations to save computational time, a representative meso-scale section was assumed to be always equal to the beam height D,,,=D (i.e. 80 mm for a small-size beam, 160 mm for a medium-size beam and 320 mm for a large-size beam).
Fig. 9.30 Calculated distribution of non-local strain measure above notch (small size beam 80x320 mm’, .=1.5 mm) for gravel concrete (dsp=2 mm, dyygq=8 mm): a) entirely heterogeneous beam, b) partially heterogeneous beam with width of meso-scale section of bms=80 mm, c) partially heterogeneous beam with width of the meso-scale section of bys=40 mm (Skarzynski and Tejchman 2010)
Fig. 9.30 Calculated distribution of non-local strain measure above notch (small size beam 80x320 mm’, .=1.5 mm) for gravel concrete (dsp=2 mm, dyygq=8 mm): a) entirely heterogeneous beam, b) partially heterogeneous beam with width of meso-scale section of bms=80 mm, c) partially heterogeneous beam with width of the meso-scale section of bys=40 mm (Skarzynski and Tejchman 2010)
Fig. 9.34 Calculated force-deflection curves for different aggregate shape of Fig. 9.33: a) circular, b) octagonal, c) irregular (angular), d) rhomboidal (fine-grained concrete beam 80x320 mm’, /.=1.5 mm, p=60%, t,=0.25 mm) and different aggregate size distributions of Fig. 9.25: A) ds9=2 mm and djq,=8 mm (curve ‘a’), B) dsp=4 mm and ding,=10 mm (curve ‘b’), C) dsp=4 mm and d,,,,=6 mm (curve ‘c’)
Fig. 9.34 Calculated force-deflection curves for different aggregate shape of Fig. 9.33: a) circular, b) octagonal, c) irregular (angular), d) rhomboidal (fine-grained concrete beam 80x320 mm’, /.=1.5 mm, p=60%, t,=0.25 mm) and different aggregate size distributions of Fig. 9.25: A) ds9=2 mm and djq,=8 mm (curve ‘a’), B) dsp=4 mm and ding,=10 mm (curve ‘b’), C) dsp=4 mm and d,,,,=6 mm (curve ‘c’)
Fig. 9.35 Calculated force-deflection curves for different aggregate shape of Fig. 9.33: A) circular, B) octagonal, C) irregular (angular), D) rhomboidal (fine-grained concrete beam 80x320 mm’, /.=1.5 mm, p=60%, t,=0.25 mm) and different aggregate size distribution of Fig. 9.25: a) dsp5=2 mm and d,,q,=8 mm (curve ‘a’), b) dsp5=4 mm and d)q,=10 mm (curve ‘b’), c) dsg=4 mm and d,,q,=6 mm (curve ‘c’) The width of a localized zone equals approximately w.=3 mm for p=60% and is not influenced by the aggregate shape, aggregate distribution, mean and maximum srain size (Fig. 9.36). In turn, the form of a localized zone is strongly affected by the aggregate shape contributing thus to the different strength. The calculated width of a localized zone is in good agreement with our experiments with fine- srained concrete (Figs. 9.23 and 9.24A). Our outcome is in contrast to statements by Bazant and Pijauder-Cabot (1989), and Bazant and Oh (1983) wherein the width of a localized zone in usual concrete was estimated to be dependent upon dnax- It is also in contrast to experimental results by Mihashi and Nomura (1996) which showed that the width of a localized zone in usual concrete increased with increasing aggregate size. The differences between our and the experimental results (Bazant and Oh 1983, Mihashi and Nomura 1996) lie probably in a different concrete mix, specimen size and loading type. For instance, in our other tests with large reinforced concrete beams 6.0 m long without shear reinforcement under bending, the width of a localized zone in usual concrete was about 15 mm indicating that /=5 mm (Syroka and Tejchman 2011). This problem merits further experimental and numerical investigations.
Fig. 9.35 Calculated force-deflection curves for different aggregate shape of Fig. 9.33: A) circular, B) octagonal, C) irregular (angular), D) rhomboidal (fine-grained concrete beam 80x320 mm’, /.=1.5 mm, p=60%, t,=0.25 mm) and different aggregate size distribution of Fig. 9.25: a) dsp5=2 mm and d,,q,=8 mm (curve ‘a’), b) dsp5=4 mm and d)q,=10 mm (curve ‘b’), c) dsg=4 mm and d,,q,=6 mm (curve ‘c’) The width of a localized zone equals approximately w.=3 mm for p=60% and is not influenced by the aggregate shape, aggregate distribution, mean and maximum srain size (Fig. 9.36). In turn, the form of a localized zone is strongly affected by the aggregate shape contributing thus to the different strength. The calculated width of a localized zone is in good agreement with our experiments with fine- srained concrete (Figs. 9.23 and 9.24A). Our outcome is in contrast to statements by Bazant and Pijauder-Cabot (1989), and Bazant and Oh (1983) wherein the width of a localized zone in usual concrete was estimated to be dependent upon dnax- It is also in contrast to experimental results by Mihashi and Nomura (1996) which showed that the width of a localized zone in usual concrete increased with increasing aggregate size. The differences between our and the experimental results (Bazant and Oh 1983, Mihashi and Nomura 1996) lie probably in a different concrete mix, specimen size and loading type. For instance, in our other tests with large reinforced concrete beams 6.0 m long without shear reinforcement under bending, the width of a localized zone in usual concrete was about 15 mm indicating that /=5 mm (Syroka and Tejchman 2011). This problem merits further experimental and numerical investigations.
According to Kim and Abu Al-Rub (2011) the Young modulus linearly increases with increasing aggregate volume, and the tensile strength decreases with increasing aggregate density up to 9=40% and increases next from p=40% up to p=60%. The strain at the damage linearly decreases with increasing aggregate volume. He et al. (2009) and He (2010) concluded that concrete with a higher packing density of aggregate up to 50% has a decreasing tensile strength (due to a higher number of very weak interfacial transitional zones around aggregate). It seems that the property of ITZ (stiffness, strength and width) is essential for the global strength versus p.
According to Kim and Abu Al-Rub (2011) the Young modulus linearly increases with increasing aggregate volume, and the tensile strength decreases with increasing aggregate density up to 9=40% and increases next from p=40% up to p=60%. The strain at the damage linearly decreases with increasing aggregate volume. He et al. (2009) and He (2010) concluded that concrete with a higher packing density of aggregate up to 50% has a decreasing tensile strength (due to a higher number of very weak interfacial transitional zones around aggregate). It seems that the property of ITZ (stiffness, strength and width) is essential for the global strength versus p.
ITZ of smaller extent. This layer is highly heterogeneous and damaged and thus critical for the concrete behaviour (Srivener et al. 2004, Mondal et al. 2009). An accurate understanding of the properties and behaviour of ITZ is one of the most important issues in the meso-scale analysis because damage is initiated at the weakest region and ITZ is just this weakest link in concrete. We assumed that ITZs have the reduced stiffness and strength as compared to the cement matrix (Tabl.9.2). Figures 9.40 and 9.41 demonstrate the effect of the ITZ thickness in a fine- grained concrete beam of circular grains with the aggregate size distribution ‘a’ of Fig. 9.25 (ds9=2 mm, din_=8 mm) assuming the aggregate volume fraction p=45% and p=60% (/,=1.5 mm). Since there is very limited data on the thickness of ITZ, the thickness ¢, in our study was assumed to be 0 mm, 0.05mm (He et al. 2011, He 2010), 0.25 mm (Gitman et al. 2007) and 0.75 mm.
ITZ of smaller extent. This layer is highly heterogeneous and damaged and thus critical for the concrete behaviour (Srivener et al. 2004, Mondal et al. 2009). An accurate understanding of the properties and behaviour of ITZ is one of the most important issues in the meso-scale analysis because damage is initiated at the weakest region and ITZ is just this weakest link in concrete. We assumed that ITZs have the reduced stiffness and strength as compared to the cement matrix (Tabl.9.2). Figures 9.40 and 9.41 demonstrate the effect of the ITZ thickness in a fine- grained concrete beam of circular grains with the aggregate size distribution ‘a’ of Fig. 9.25 (ds9=2 mm, din_=8 mm) assuming the aggregate volume fraction p=45% and p=60% (/,=1.5 mm). Since there is very limited data on the thickness of ITZ, the thickness ¢, in our study was assumed to be 0 mm, 0.05mm (He et al. 2011, He 2010), 0.25 mm (Gitman et al. 2007) and 0.75 mm.
The results show that the thickness and strength of ITZs strongly affect both the load-displacement response and shape of localized zone. Since ITZ is the weakest phase, the ultimate beam strength decreases with increasing bond thickness (Fig. 9.40). This result is in agreement with those by He et al. (2009), He (2010) and Kim and Abu Al-Rub (2011). However, the residual strength rather increases with increasing bond thickness as in calculations by Kim and Abu Al-Rub (2011). The width of a localized zone is w=4.5 mm (p=45%) and w.=3 mm (p=60%) and is not affected by the ITZ size t, (Fig. 9.41).
The results show that the thickness and strength of ITZs strongly affect both the load-displacement response and shape of localized zone. Since ITZ is the weakest phase, the ultimate beam strength decreases with increasing bond thickness (Fig. 9.40). This result is in agreement with those by He et al. (2009), He (2010) and Kim and Abu Al-Rub (2011). However, the residual strength rather increases with increasing bond thickness as in calculations by Kim and Abu Al-Rub (2011). The width of a localized zone is w=4.5 mm (p=45%) and w.=3 mm (p=60%) and is not affected by the ITZ size t, (Fig. 9.41).
Fig. 9.42 Numerical effect of notch size on force-deflection curve for two different aggregate densities: a) 0x0 mm”, b) 3x3 mm? and c) 6x3 mm”, A) p=30%, B) p=60% (fine- grained concrete beam 80x320 mm’, /.=1.5 mm, circular aggregate distribution ‘a’ of Fig. 9.25 with ds9=2 mm and d,,,,=8 mm)
Fig. 9.42 Numerical effect of notch size on force-deflection curve for two different aggregate densities: a) 0x0 mm”, b) 3x3 mm? and c) 6x3 mm”, A) p=30%, B) p=60% (fine- grained concrete beam 80x320 mm’, /.=1.5 mm, circular aggregate distribution ‘a’ of Fig. 9.25 with ds9=2 mm and d,,,,=8 mm)
Fig. 9.44 Effect of aggregate stiffness on force-deflection curve and distribution of non- local strain measure close to beam notch: a) strong circular aggregate, b) weak circular aggregate (fine-grained concrete beam 80x320 mm’, /.=1.5 mm, circular aggregate distribution ‘c’ of Fig. 9.25 with ds>=4 mm and d,,,,=10 mm, p=60%)
Fig. 9.44 Effect of aggregate stiffness on force-deflection curve and distribution of non- local strain measure close to beam notch: a) strong circular aggregate, b) weak circular aggregate (fine-grained concrete beam 80x320 mm’, /.=1.5 mm, circular aggregate distribution ‘c’ of Fig. 9.25 with ds>=4 mm and d,,,,=10 mm, p=60%)
Fig. 9.45 Calculated load-deflection curves for different characteristic lengths /,: a) 1.=0.5 mm, b) /,=1.5 mm, c) /,=2.5 mm and d) J.=5 mm (concrete beam 80x320 mm’, ITZ thickness t,=0.25 mm), A) volume fraction of circular aggregate p=30% (concrete mix ‘d’ of Fig. 9.25 with ds9=0.5 mm and d,,,,,=3 mm), B) volume fraction of circular aggregate p=45% (concrete mix ‘a’ of Fig. 9.25 with ds9=2 mm and d,,,,=8 mm), C) volume fraction of angular aggregate p=60% (concrete mix ‘b’ of Fig. 9.25 with ds9=4 mm and d,,,,=10 mm)
Fig. 9.45 Calculated load-deflection curves for different characteristic lengths /,: a) 1.=0.5 mm, b) /,=1.5 mm, c) /,=2.5 mm and d) J.=5 mm (concrete beam 80x320 mm’, ITZ thickness t,=0.25 mm), A) volume fraction of circular aggregate p=30% (concrete mix ‘d’ of Fig. 9.25 with ds9=0.5 mm and d,,,,,=3 mm), B) volume fraction of circular aggregate p=45% (concrete mix ‘a’ of Fig. 9.25 with ds9=2 mm and d,,,,=8 mm), C) volume fraction of angular aggregate p=60% (concrete mix ‘b’ of Fig. 9.25 with ds9=4 mm and d,,,,=10 mm)
Fig. 9.47 The calculated evolution of width (A) w, and height h, (B) of localized zone versus beam deflection u: a) concrete mix ‘a’ of Fig. 9.25 with dsp=2 mm and d,,,,=8 mm, irregular aggregate, p=60%, /.=1.5 mm, b) concrete mix ‘b’ of Fig. 9.25 with dso>=4 mm and dinax=10 mm, octagonal aggregate, p=60%, /.=1.5 mm, c) concrete mix ‘c’ of Fig. 9.25 with dsg=4 mm and d,,,,=6 mm, circular aggregate, p=60%, /,=1.5 mm, d) concrete mix ‘a’ of Fig. 9.25 with dsy>=2 mm and d,,,,=8 mm, circular aggregate, p=60%, beam without notch, /.=1.5 mm (¢ - maximum vertical force)
Fig. 9.47 The calculated evolution of width (A) w, and height h, (B) of localized zone versus beam deflection u: a) concrete mix ‘a’ of Fig. 9.25 with dsp=2 mm and d,,,,=8 mm, irregular aggregate, p=60%, /.=1.5 mm, b) concrete mix ‘b’ of Fig. 9.25 with dso>=4 mm and dinax=10 mm, octagonal aggregate, p=60%, /.=1.5 mm, c) concrete mix ‘c’ of Fig. 9.25 with dsg=4 mm and d,,,,=6 mm, circular aggregate, p=60%, /,=1.5 mm, d) concrete mix ‘a’ of Fig. 9.25 with dsy>=2 mm and d,,,,=8 mm, circular aggregate, p=60%, beam without notch, /.=1.5 mm (¢ - maximum vertical force)
Fig. 9.49 Calculated nominal strength 1.5P//(bD*) versus normalised beam deflection u/D (u - beam deflection, D - beam height): A) FE-results, B) experiments by Le Bellégo et al. (2003): 1) small-size beam, (2) medium-size beam, (3) large-size beam (three-phase random heterogeneous material close to notch, b,,,=D) (Skarzynski and Tejchman 2010)
Fig. 9.49 Calculated nominal strength 1.5P//(bD*) versus normalised beam deflection u/D (u - beam deflection, D - beam height): A) FE-results, B) experiments by Le Bellégo et al. (2003): 1) small-size beam, (2) medium-size beam, (3) large-size beam (three-phase random heterogeneous material close to notch, b,,,=D) (Skarzynski and Tejchman 2010)
Fig. 9.51 Calculated and measured size effect in nominal strength 1.5P1/(bD*) versus beam height D for concrete beams of a similar geometry (small-, medium- and large-size beam): a) our laboratory experiments, b) our FE-calculations (homogeneous one-phase material), c) our FE-calculations (heterogeneous material close to notch, b,,,=D), d) size effect law by Bazant (2004), Eq. 5.5, e) experiments by Le Bellégo et al. (2003) (Skarzynski and Tejchman 2010)
Fig. 9.51 Calculated and measured size effect in nominal strength 1.5P1/(bD*) versus beam height D for concrete beams of a similar geometry (small-, medium- and large-size beam): a) our laboratory experiments, b) our FE-calculations (homogeneous one-phase material), c) our FE-calculations (heterogeneous material close to notch, b,,,=D), d) size effect law by Bazant (2004), Eq. 5.5, e) experiments by Le Bellégo et al. (2003) (Skarzynski and Tejchman 2010)
Fig. 9.52 Deformed three-phase concrete specimen with periodicity of boundary conditions and material periodicity (Skarzynski and Tejchman 2012)
Fig. 9.52 Deformed three-phase concrete specimen with periodicity of boundary conditions and material periodicity (Skarzynski and Tejchman 2012)
Fig. 9.53 Stress-strain curves for various sizes of concrete specimens and two different random distributions of aggregate (a) and (b) using standard averaging procedure (characteristic length /.=1.5 mm, aggregate density p=30%) (Skarzynski and Tejchman 2012) case it was at random in different unit cells. The results show that the stress-strain curves are the same solely in an elastic regime independently of the specimen size, aggregate density and aggregate distribution. However, they are completely different at the peak and in a softening regime. An increase of the specimen size causes a strength decrease and an increase of material brittleness (softening rate). The differences in the evolution of stress-strain curves in a softening regime are caused by strain localization (in the form of a curved localized zone propagating between aggregates, Figs. 9.54 and 9.55) contributing to a loss of material homogeneity (due to the fact that strain localization is not scaled with increasing specimen size). The width of a calculated localized zone is approximately w.=3 mm=2x/,=12xs,, (unit cell 5x5 mm°*), w.=5 mm=3.33x/,=20xS,,, (unit cell 10x10 mm’) and w.=6 mm=4x/,=24xs,,, (unit cells larger than 10x10 mm”). Figure 9.56 presents the expectation value and standard deviation of the tensile fracture energy Gy versus the specimen height h for 3 different realizations. The fracture energy Gy was calculated as the area under the strain-stress curves gy
Fig. 9.53 Stress-strain curves for various sizes of concrete specimens and two different random distributions of aggregate (a) and (b) using standard averaging procedure (characteristic length /.=1.5 mm, aggregate density p=30%) (Skarzynski and Tejchman 2012) case it was at random in different unit cells. The results show that the stress-strain curves are the same solely in an elastic regime independently of the specimen size, aggregate density and aggregate distribution. However, they are completely different at the peak and in a softening regime. An increase of the specimen size causes a strength decrease and an increase of material brittleness (softening rate). The differences in the evolution of stress-strain curves in a softening regime are caused by strain localization (in the form of a curved localized zone propagating between aggregates, Figs. 9.54 and 9.55) contributing to a loss of material homogeneity (due to the fact that strain localization is not scaled with increasing specimen size). The width of a calculated localized zone is approximately w.=3 mm=2x/,=12xs,, (unit cell 5x5 mm°*), w.=5 mm=3.33x/,=20xS,,, (unit cell 10x10 mm’) and w.=6 mm=4x/,=24xs,,, (unit cells larger than 10x10 mm”). Figure 9.56 presents the expectation value and standard deviation of the tensile fracture energy Gy versus the specimen height h for 3 different realizations. The fracture energy Gy was calculated as the area under the strain-stress curves gy
Fig. 9.57 Stress-strain curves for various sizes of concrete specimens and two different random distributions of aggregate (a) and (b) using localized zone averaging procedure (characteristic length /.=1.5 mm, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.57 Stress-strain curves for various sizes of concrete specimens and two different random distributions of aggregate (a) and (b) using localized zone averaging procedure (characteristic length /.=1.5 mm, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.59 Stress-strain curves for various sizes of concrete specimens and two different random distributions of aggregate (a) and (b) using varying characteristic length approach (reference unit size 15x15 mm”, characteristic length according to Eq. 9.6, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.59 Stress-strain curves for various sizes of concrete specimens and two different random distributions of aggregate (a) and (b) using varying characteristic length approach (reference unit size 15x15 mm”, characteristic length according to Eq. 9.6, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.60 Stress-strain curves for various sizes of concrete specimens and two different random distributions of aggregate (a) and (b) using varying characteristic length approach (reference unit size 30x30 mm”, characteristic length according to Eq. 9.7, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.60 Stress-strain curves for various sizes of concrete specimens and two different random distributions of aggregate (a) and (b) using varying characteristic length approach (reference unit size 30x30 mm”, characteristic length according to Eq. 9.7, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.61 Distribution of non-local softening strain measure for various specimen sizes and stress-strain curves from Fig. 9.59a using varying characteristic length approach (reference unit size 15x15 mm”, characteristic length according to Eq. 9.6, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.61 Distribution of non-local softening strain measure for various specimen sizes and stress-strain curves from Fig. 9.59a using varying characteristic length approach (reference unit size 15x15 mm”, characteristic length according to Eq. 9.6, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.62 Distribution of non-local softening strain measure for various specimen sizes and stress-strain curves far Fig. 9.59b using varying characteristic length approach (reference unit size 15x15 mm’, characteristic length according to Eq. 9.6, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.62 Distribution of non-local softening strain measure for various specimen sizes and stress-strain curves far Fig. 9.59b using varying characteristic length approach (reference unit size 15x15 mm’, characteristic length according to Eq. 9.6, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.63 Distribution of non-local softening strain measure for various specimen sizes and stress-strain curves iret Fig. 9.60a using varying characteristic length approach (reference unit size 30x30 mm”, characteristic length according to Eq. 9.7, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.63 Distribution of non-local softening strain measure for various specimen sizes and stress-strain curves iret Fig. 9.60a using varying characteristic length approach (reference unit size 30x30 mm”, characteristic length according to Eq. 9.7, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.64 Distribution of non-local softening strain measure for various specimen sizes and stress-strain curves from Fig. 9.60b using varying characteristic length approach (reference unit size 30x30 mm”, characteristic length according to Eq. 9.7, aggregate density p=30%) (Skarzynski and Tejchman 2012)
Fig. 9.64 Distribution of non-local softening strain measure for various specimen sizes and stress-strain curves from Fig. 9.60b using varying characteristic length approach (reference unit size 30x30 mm”, characteristic length according to Eq. 9.7, aggregate density p=30%) (Skarzynski and Tejchman 2012)

Key takeaways

  • The differences concern the height and shape of localized zones, • the FE-results are similar within two different continuum crack models, although a slightly better agreement with experiments was achieved with a damage model, • the beam strength increases mainly with increasing reinforcement ratio, characteristic length, tensile strength, fracture energy and decreasing beam size and shear span ratio.
  • The width is about (3-4)×l c , • the spacing of localized zones increases with increasing characteristic length l c , tensile softening modulus and decreasing fracture energy, reinforcement ratio and initial bond stiffness.
  • Thus, the isotropic damage model needs improvements to describe localized shear zones in reinforced concrete elements under shear-tension failure, • the calculated spacing of localized tensile zones increased with increasing characteristic length, softening rate and beam height and decreasing fracture energy and bond stiffness.
  • To model the concrete softening under tension, the exponential curve by with the tensile strength of the concrete of f t =3.6 MPa was assumed (κ u =0.005 b 1 =3.0, b 2 =6.93) (Eq.
  • For instance, in our other tests with large reinforced concrete beams 6.0 m long without shear reinforcement under bending, the width of a localized zone in usual concrete was about 15 mm indicating that l c =5 mm .

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