Pena 2017a - Revision history

Rimni at 14:06, 13 July 2021

2021-07-13T14:06:14Z

Rimni at 12:34, 13 July 2021

2021-07-13T12:34:27Z

Rimni at 12:32, 13 July 2021

2021-07-13T12:32:17Z

Rimni: /* 4. Mechanical characteristics of the interfaces */

2021-07-13T12:31:41Z

‎4. Mechanical characteristics of the interfaces

Rimni: /* 4. Mechanical characteristics of the interfaces */

2021-07-13T12:20:49Z

‎4. Mechanical characteristics of the interfaces

Rimni at 12:11, 13 July 2021

2021-07-13T12:11:41Z

Rimni at 11:37, 13 July 2021

2021-07-13T11:37:03Z

Rimni at 11:33, 13 July 2021

2021-07-13T11:33:28Z

Rimni at 11:32, 13 July 2021

2021-07-13T11:32:01Z

Rimni at 11:30, 13 July 2021

2021-07-13T11:30:12Z

@@ Line 1: / Line 1: @@
 ==Abstract==
@@ Line 5: / Line 4: @@
 '''Keywords: '''Gravity dams, Semi-discrete model, non-linear behaviour, rigid body spring model, damage
 ==1. Introduction==
@@ Line 12: / Line 9: @@
 Gravity dams are massive structures, which their stability depends on the gravity loads applied to the structure [1]. Mainly, the structural assessment of gravity dams is performed by means of a gravity approach, where the resultant of all forces acting on the dam must lie in the third middle of the base. Generally, the hypothesis of no tension material is assumed [2]. Sometimes, this approach is too conservative and mostly does not reflect the real structural behaviour of the dam [2-4].
-The gravity approach is based on the calculation of the [1]: a) Position of the resultant force, where the resultant force must lie in the middle third of the base; b) Inclination of the resultant force, in order to evaluate the shear forces and the possible sliding of the dam; and c) Compressive stresses, in order to avoid the crushing of the material. [[#img-1|Figure 1]] depicts the forces acting on the dam; where <math display="inline"> P_H </math> is the hydraulic pressure (v=vertical, h=horizontal and e=downstream), <math display="inline"> P_p </math> is the own weight, <math display="inline"> P_s </math> is the uplift pressure, <math display="inline"> P_A </math> is the earth pressure, ''S'' is the seismic loads (<math display="inline">V=</math>vertical, <math display="inline">H=</math>horizontal) and <math display="inline"> P_O </math> is the hydrodynamic pressure.
+The gravity approach is based on the calculation of the [1]: a) Position of the resultant force, where the resultant force must lie in the middle third of the base; b) Inclination of the resultant force, in order to evaluate the shear forces and the possible sliding of the dam; and c) Compressive stresses, in order to avoid the crushing of the material. [[#img-1|Figure 1]] depicts the forces acting on the dam; where <math display="inline"> P_H </math> is the hydraulic pressure (v=vertical, h=horizontal and e=downstream), <math display="inline"> P_p </math> is the own weight, <math display="inline"> P_s </math> is the uplift pressure, <math display="inline"> P_A </math> is the earth pressure, <math display="inline">S</math> is the seismic loads (<math display="inline">V=</math>vertical, <math display="inline">H=</math>horizontal) and <math display="inline"> P_O </math> is the hydrodynamic pressure.
 <div id='img-1'></div>
@@ Line 376: / Line 373: @@
-The proposed model was validated by using the discrete element model of the Guil1hofrei dam (Portugal) performed by Bretas et al. [15]. This is a masonry gravity dam, built in 1938, with a maximum height of 39 m and a total length of 190 m. The soil foundation of the dam is a granitic rock mass, of good quality ([[#img-8|Figure 8]]). [[#tab-1|Table 1]] shows the mechanical properties of the masonry and the soil foundation. For details of the dam characteristics please refer to [15, 20]. [[#img-9|Figure 9]] shows the SDEM and the DEM meshes for the studied dam.
+The proposed model was validated by using the discrete element model of the Guil1hofrei dam (Portugal) performed by Bretas et al. [15]. This is a masonry gravity dam, built in 1938, with a maximum height of 39 m and a total length of 190 m. The soil foundation of the dam is a granitic rock mass, of good quality ([[#img-8|Figure 8]]). [[#tab-1|Table 1]] shows the mechanical properties of the masonry and the soil foundation. For details of the dam characteristics please refer to [15,20]. [[#img-9|Figure 9]] shows the SDEM and the DEM meshes for the studied dam.
 <div id='img-8'></div>

@@ Line 166: / Line 166: @@
 {| style="text-align: center; margin:auto;"
 |-
-| style="text-align: center;" | <math> \{ \sigma \}^T  = \{\sigma^P_1, \sigma^Q_1, \sigma^R_1, \sigma^P_2, \sigma^Q_2, \sigma^R_2, \cdots , \sigma^P_r, \sigma^Q_r, \sigma^R_r\} </math>
+| style="text-align: center;" | <math> \{ \sigma \}^T  = \left\{ \sigma^P_1, \sigma^Q_1, \sigma^R_1, \sigma^P_2, \sigma^Q_2, \sigma^R_2, \cdots , \sigma^P_r, \sigma^Q_r, \sigma^R_r \right\} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (8)

← Older revision		Revision as of 12:32, 13 July 2021
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	For the own weight analysis, only the volumetric weight of the dam was taken into account. [[#tab-2\|Table 2]] shows the results obtained by DEM [15] and the proposed model. It can be observed that the results are practically the same for both models. The own weight and the compression stress have error percentages are less than 10%. Although the error for displacements are around 15%, the displacements are in mm. Thus, small variations give great errors.		For the own weight analysis, only the volumetric weight of the dam was taken into account. [[#tab-2\|Table 2]] shows the results obtained by DEM [15] and the proposed model. It can be observed that the results are practically the same for both models. The own weight and the compression stress have error percentages are less than 10%. Although the error for displacements are around 15%, the displacements are in mm. Thus, small variations give great errors.
−

	<div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">		<div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">

@@ Line 275: / Line 275: @@
 {| style="text-align: center; margin:auto;"
 |-
-|  style="text-align: center;" | <math>\{H\}^T = \{Η_{11}, Η_{22}, Η_{12}\} </math>
+|  style="text-align: center;" | <math>\{H \}^T = \{ H_{11}, H_{22}, H_{12}\} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (15)

@@ Line 275: / Line 275: @@
 {| style="text-align: center; margin:auto;"
 |-
-|  style="text-align: center;" | {''H''}<sup>''T''</sup> = {''Η''<sub>11</sub>, ''Η''<sub>22</sub>, ''Η''<sub>12</sub>}
+|  style="text-align: center;" | <math>\{H\}^T = \{Η_{11}, Η_{22}, Η_{12}\} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (15)

@@ Line 143: / Line 143: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\left[ B\right] =\frac{1}{{h}_{i}+{h}_{j}}\left[ \begin{matrix}\mathrm{cos}\,{\alpha }_{i}&\mathrm{sin}\,{\alpha }_{i}&\left[ \mathrm{sin}\,\left( {\alpha }_{i}-\vartheta \right) {d}_{i}+b\right] \quad \\-\mathrm{sin}\,{\alpha '''}_{i}&\mathrm{cos}\,{\alpha }_{i}&\left[ \mathrm{cos}\,\left( {\alpha }_{i}-\vartheta \right) {d}_{i}\right] \\\mathrm{cos}\,{\alpha }_{i}&\mathrm{sin}\,{\alpha }_{i}&\left[ \mathrm{sin}\,\left( {\alpha }_{i}-\vartheta \right) {d}_{i}-b\right] \end{matrix}\begin{matrix}-\mathrm{cos}\,{\alpha }_{j}&-\mathrm{sin}\,{\alpha }_{ij}&-\left[ \mathrm{sin}\,\left( {\alpha }_{j}-\vartheta \right) {d}_{j}+b\right] \\\mathrm{sin}\,{\alpha }_{j}&-\mathrm{cos}\,{\alpha }_{j}&-\left[ \mathrm{cos}\,\left( {\alpha }_{j}-\vartheta \right) {d}_{j}\right] \\-\mathrm{cos}\,{\alpha }_{j}&-\mathrm{sin}\,{\alpha }_{j}&-\left[ \mathrm{sin}\,\left( {\alpha }_{j}-\vartheta \right) {d}_{j}-b\right] \end{matrix}\right]</math>
+| <math>\left[ B\right] =\frac{1}{{h}_{i}+{h}_{j}}\left[ \begin{matrix}\mathrm{cos}\,{\alpha }_{i}&\mathrm{sin}\,{\alpha }_{i}&\left[ \mathrm{sin}\,\left( {\alpha }_{i}-\vartheta \right) {d}_{i}+b\right] \quad \\-\mathrm{sin}\,{\alpha '''}_{i}&\mathrm{cos}\,{\alpha }_{i}&\left[ \mathrm{cos}\,\left( {\alpha }_{i}-\vartheta \right) {d}_{i}\right] \\\mathrm{cos}\,{\alpha }_{i}&\mathrm{sin}\,{\alpha }_{i}&\left[ \mathrm{sin}\,\left( {\alpha }_{i}-\vartheta \right) {d}_{i}-b\right] \end{matrix}\begin{matrix}-\mathrm{cos}\,{\alpha }_{j}&-\mathrm{sin}\,{\alpha }_{ij}&-\left[ \mathrm{sin}\,\left( {\alpha }_{j}-\vartheta \right) {d}_{j}+b\right] \\\mathrm{sin}\,{\alpha }_{j}&-\mathrm{cos}\,{\alpha }_{j}&-\left[ \mathrm{cos}\,\left( {\alpha }_{j}-\vartheta \right) {d}_{j}\right] \\-\mathrm{cos}\,{\alpha }_{j}&-\mathrm{sin}\,{\alpha }_{j}&-\left[ \mathrm{sin}\,\left( {\alpha }_{j}-\vartheta \right) {d}_{j}-b\right] \end{matrix}\right]</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (7)
 |}
-where <math>\alpha_i</math> is the angle of the connection side of element <math>i</math> referring to <math>\xi</math>–axis and <math>\vartheta</math> ''ϑ'' is called distortion angle ([[#img-5|Figure 5]]). <math>d_i</math> is the distance from the baricenter of element i to the center of the interface <math>i-j</math> and <math>h_i</math> is the minimum distance between the baricenter of element and the interface <math>i-j</math>.
+where <math>\alpha_i</math> is the angle of the connection side of element <math>i</math> referring to <math>\xi</math>–axis and <math>\vartheta</math> is called distortion angle ([[#img-5|Figure 5]]). <math>d_i</math> is the distance from the baricenter of element i to the center of the interface <math>i-j</math> and <math>h_i</math> is the minimum distance between the baricenter of element and the interface <math>i-j</math>.
 <div id='img-5'></div>
@@ Line 159: / Line 159: @@
-A measure of stress, work–conjugated to the strain, is assigned to each connecting device, and is assembled into the vector {''σ''} as follows:
+A measure of stress, work–conjugated to the strain, is assigned to each connecting device, and is assembled into the vector <math>\{ \sigma \}</math> as follows:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 166: / Line 166: @@
 {| style="text-align: center; margin:auto;"
 |-
-|<div style="text-align: right; direction: ltr; margin-left: 1em;">
+| style="text-align: center;" | <math> \{ \sigma \}^T  = \{\sigma^P_1, \sigma^Q_1, \sigma^R_1, \sigma^P_2, \sigma^Q_2, \sigma^R_2, \cdots , \sigma^P_r, \sigma^Q_r, \sigma^R_r\} </math>
-{'' σ'' }''<sup>T</sup>'' = {'' σ<sup> P</sup><sub>1</sub>'', ''σ<sup> Q</sup><sub>1</sub>'', ''σ<sup> R</sup><sub>1</sub>'', ''σ<sup> P</sup><sub>2</sub>'', ''σ<sup> Q</sup><sub>2</sub>'', ''σ<sup> R</sup><sub>2</sub>'', . . . , ''σ<sup>P</sup><sub>r</sub>'' , ''σ<sup>Q</sup><sub>r</sub>'', ''σ<sup> R</sup><sub>r</sub>''} </div>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (8)
 |}
-where ''σ<sup>P</sup>'' and ''σ<sup>R</sup>'' are the axial stresses in the connection point P and R, and ''σ<sup>Q</sup>'' is the shear stress in Q. Forces are related by:
+where <math>\sigma^P</math> and <math>\sigma^R</math> are the axial stresses in the connection point <math display="inline"> P </math> and <math display="inline"> R </math>, and <math>\sigma^Q</math> is the shear stress in <math display="inline"> Q </math>. Forces are related by:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 179: / Line 178: @@
 {| style="text-align: center; margin:auto;"
 |-
-|<div style="text-align: right; direction: ltr; margin-left: 1em;">
+|style="text-align: center;" | <math> \{f_e\} = [B]^T \{\sigma \} </math>
-{''f<sub>e</sub>''} = [''B'']''<sup>T</sup>''{''σ''} </div>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (9)
@@ Line 192: / Line 190: @@
 {| style="text-align: center; margin:auto;"
 |-
-|<div style="text-align: right; direction: ltr; margin-left: 1em;">
+|style="text-align: center;" | <math> \{ \sigma \} = [D]\{\varepsilon \} </math>
-{''σ''} = [''D'']{''ε''} </div>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (10)
 |}
-where [''D''] is the tangential stiffness matrix of the connection side:
+where <math>[D]</math> is the tangential stiffness matrix of the connection side:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 205: / Line 202: @@
 {| style="text-align: center; margin:auto;"
 |-
-|<div style="text-align: right; direction: ltr; margin-left: 1em;">
+|style="text-align: center;" | <math> [D]  =Diag [k^P, k^Q, k^R] </math>
-[''D''] =''Diag'' [''k<sup>P</sup>, k<sup>Q</sup>, k<sup>R</sup>''] </div>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (11)
@@ Line 218: / Line 214: @@
 {| style="text-align: center; margin:auto;"
 |-
-|<div style="text-align: right; direction: ltr; margin-left: 1em;">
+|style="text-align: center;" | <math> \{f_e\} = [B]^T [D][B]\{u\}=[K]\{u\} </math>
-{''f<sub>e</sub>''} = [''B'']''<sup>T</sup>''[''D''][''B'']{''u''}=[''K'']{''u''} </div>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (12)

← Older revision		Revision as of 11:37, 13 July 2021
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−	Under small rotation assumption, the strain-displacement relation can be expressed by considering a <math display="inline"> 3r\times m</math> matrix <math>[B]</math> as follows:	+	Under small rotation assumption, the strain-displacement relation can be expressed by considering a <math display="inline"> 3r\times 3m</math> matrix <math>[B]</math> as follows:

	{\| class="formulaSCP" style="width: 100%; text-align: center;"		{\| class="formulaSCP" style="width: 100%; text-align: center;"

← Older revision		Revision as of 11:33, 13 July 2021
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−	Under small rotation assumption, the strain-displacement relation can be expressed by considering a <math display="inline"> 3r\times x m</math> matrix <math>[B]</math> as follows:	+	Under small rotation assumption, the strain-displacement relation can be expressed by considering a <math display="inline"> 3r\times m</math> matrix <math>[B]</math> as follows:

	{\| class="formulaSCP" style="width: 100%; text-align: center;"		{\| class="formulaSCP" style="width: 100%; text-align: center;"

← Older revision		Revision as of 11:32, 13 July 2021
Line 124:		Line 124:


−	Under small rotation assumption, the strain-displacement relation can be expressed by considering a 3 <math display="inline"> r\times x </math> matrix <math>[B]</math> as follows:	+	Under small rotation assumption, the strain-displacement relation can be expressed by considering a <math display="inline"> 3r\times x m</math> matrix <math>[B]</math> as follows:

	{\| class="formulaSCP" style="width: 100%; text-align: center;"		{\| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 2: / Line 2: @@
 ==Abstract==
-<span id='_GoBack'></span>A new numerical model for the structural assessment of gravity dams by means of a semi-discrete approach is proposed. Gravity dams are massive structures, which their stability depends on the gravity loads applied to the structure. Mainly, its structural assessment is performed by means of a gravity approach. Sometimes, this approach is too conservative and mostly does not reflect the real structural behaviour of the dam. In this context, there is the need of models that are simplified enough to allow a simple and fast parametric analyses. The proposed model idealizes the dam as a set of rigid elements, where the damage and the deformation are concentrated in the contact sides between adjacent elements. Thus, the elements are rigid, but the material is considered as deformable. As the proposed model is semi-discrete, it can detect separation or sliding between elements. However, initial contacts do not change during the analyisis and a relative continuity among elements exists, in order to simplify the computational effort. The effective performance of the proposed model is demonstrated by numerical validation and by comparisons with some numerical models presented in the literature.
+A new numerical model for the structural assessment of gravity dams by means of a semi-discrete approach is proposed. Gravity dams are massive structures, which their stability depends on the gravity loads applied to the structure. Mainly, its structural assessment is performed by means of a gravity approach. Sometimes, this approach is too conservative and mostly does not reflect the real structural behaviour of the dam. In this context, there is the need of models that are simplified enough to allow a simple and fast parametric analyses. The proposed model idealizes the dam as a set of rigid elements, where the damage and the deformation are concentrated in the contact sides between adjacent elements. Thus, the elements are rigid, but the material is considered as deformable. As the proposed model is semi-discrete, it can detect separation or sliding between elements. However, initial contacts do not change during the analyisis and a relative continuity among elements exists, in order to simplify the computational effort. The effective performance of the proposed model is demonstrated by numerical validation and by comparisons with some numerical models presented in the literature.
 '''Keywords: '''Gravity dams, Semi-discrete model, non-linear behaviour, rigid body spring model, damage
@@ Line 124: / Line 124: @@
-Under small rotation assumption, the strain-displacement relation can be expressed by considering a 3 <math display="inline"> r\times x </math> m matrix <math>[B]</math> as follows:
+Under small rotation assumption, the strain-displacement relation can be expressed by considering a 3 <math display="inline"> r\times x </math> matrix <math>[B]</math> as follows:
 {| class="formulaSCP" style="width: 100%; text-align: center;"