CMN 2017

CMN 2017 Congress on Numerical Methods in Engineering July 3 - 5, Valencia, Spain Edited by: Irene Arias, Jesús María Blanco, Stephane Clain, Paulo Flores, Paulo Lourenço, Juan José Ródenas and Manuel Tur

Congress on Numerical Methods in Engineering

CMN 2017

July 3 - 5 Valencia, Spain

A publication of: International Center for Numerical Methods in Engineering (CIMNE) Barcelona, Spain

Printed by: Artes Gráficas Torres S.L., Huelva 9, 08940 Cornellà de Llobregat, Spain

ISBN: 978-84-947311-0-5

Rigid body spring model for the structural assessment of old masonry dams

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Fernando Pe˜na and Laura Robles

1

INTRODUCTION

Old masonry dams are massive structures, which their stability depends on the gravity loads applied in the structure. Mainly, the structural assessment of old masonry dams is performed by means of a static approach, where the resultant of all forces acting on the dam must lie in the third middle of the base. Figure 1 depicts the forces acting on the dam; where PH is the hydraulic pressure (v=vertical, h=horizontal and e=downstream), Pp is the own weight, Ps is the water underpressure, PA is the earth pressure and S is the seismic loads (V=vertical, H=horizontal).

Figure 1: Forces acting on the dam.

The static analysis of the masonry dams must calculate the [1]: - Position of the resultant force. Where the resultant force must lie in the middle third of the base. - Inclination of the resultant force. In order to evaluate the shear forces and the possible sliding of the dam. - Compressive stresses. In order to avoid the crushing of the masonry. However, this approach is too conservative and mostly does not reflect the real structural behavior of the dam. Mainly, because the tensile strength of the material is neglected. In this context, there is the need of models that are simplified enough to allow a simple and fast parametric analyses, but they should also take into account the peculiar behavior of the masonry. Thus, in this paper a Rigid Body Spring Model (RBSM) for the structural assessment of old masonry dams is presented. 2

RIGID BODY SPRING MODEL

The model follows the philosophy of the Rigid Element Model (REM) proposed in [2, 3, 4]. This philosophy is based on: - There is a limited number of damage mechanisms. Thus, zones, types and directions of the possible damage are possible to be known a priori. - Damage is not concentrated in a single point of the structure.

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- There are zones with similar behaviour of a rigid body, between which there are damaged zones. In this study only the in–plane deformations are considered. The dam is idealized as an assemblage of rigid elements. Damage and deformation are concentrated in the contact sides between adjacent elements. These elements are quadrilateral and have the kinematics of rigid bodies with two linear displacements and one rotation. Three devices (springs) connect the common side between two rigid elements or the restrained side. These devices are two axial springs, separated by a distance 2b that take into account the bending moment, and one shear spring at the middle of the side. The masonry material is considered deformable but this deformation is concentrated in the connection points, while the element is not deformable. Each connecting device is independent of the behavior of the other connecting devices and depends only on the Lagrangian displacements. In other words, the connection points represent the elastic and post–elastic mechanical characteristics of the masonry material and, at the same time, represent the capacity of the model to take into account the separation or the sliding between elements. The proposed model was developed as a semi–discrete model. Therefore, the RBSM can detect separation and sliding of the elements. However, relative motion between two adjacent elements can occur. Initial contacts do not change during the analysis and a relative continuity between elements exists, in order to simplify the computational effort. Thus, separation and sliding between two adjacent elements can occur. The semi–discrete model can be thought as an analysis technique that combines the advantages of the discrete analysis techniques (e.g. it considers the relative motion among elements) with the computational advantages of the continuous analysis technique (e.g. no new contacts update is necessary). 3

FORMULATION

The masonry dam is considered as a two–dimensional plane solid body Ω, partitioned into m quadrilateral elements ωi such that no vertex of one quadrilateral lies on the edge of another quadrilateral. A local reference frame {oi , ξi , ηi }, whose axes are initially parallel to the ground reference frame {O, x, y} is fixed in each element’s barycentre oi . These elements are rigid, so the displaced configuration of the discrete model is described by the position of these local reference frames, as shown in Figure 2. Given the local coordinate of a point (η, ξ), the

Figure 2: Reference frame, forces and displacements of a rigid element.

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displacements (∆x, ∆y) in the x - y plane are evaluated as follows:   � � � �  ue  ∆x 1 0 η v = ∆y 0 1 −ξ  e  ψe

(1)

The translation components ui , vi and the rotation angle ψi associated with each element, are collected into the vector of Lagrangian coordinates {u}. The loads are condensed into three resultants associated with each element: the forces pi and qi applied to the element centroid considering the initial undeformed geometry, and the couple µi . They are assembled into the vector of external loads {fe } which is conjugated in virtual work with {u} as follows: {u}T = {u1 , v1 , ψ1 , u2 , v2 , ψ2 , ..., um , vm , ψm } T

{fe } = {p1 , q1 , µ1 , p2 , q2 , µ2 , ..., pm , qm , µm }

(2) (3)

The elements are interconnected by line springs placed along each side, in correspondence of three points named P , Q and R, as shown in Figure 3. Three average strain measures are associated with these connecting devices: the axial strains, ǫP and ǫR are associated with the volumes of pertinence V P and V R , while the shear strain ǫQ is associated with the volume V Q = V P +V R . Considering a discrete model with r sides which connect all the elements (interfaces), the vector of generalized strain {ǫ} and the diagonal matrix of volumes of pertinence [V ] (Fig. 4) are defined as follows:

Figure 3: Assembly of rigid elements.

R P Q R P Q R {ǫ}T = {ǫP1 , ǫQ 1 , ǫ1 , ǫ2 , ǫ2 , ǫ2 , . . . , ǫ r , ǫr , ǫr }

(4)

[V ] = Diag{V1P , V1Q , V1R , V2P , V2Q , V2R , ..., VrP , VrQ , VrR }

(5)

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a)

b)

c)

d)

Figure 4: Volume of pertinence: a) Tensile volume, b) compressive volume, c) shear volume, d) sliding volume.

Under a small rotation assumption, the strain–displacement relation can be expressed by considering a 3r x 3m matrix [B] as follows: {ǫ} = [B]{u}

(6)

1 (7) hi + hj  [sin(αi − ϑ)di + b] − cos αj − sin αj −[sin(αi − ϑ)di + b] [cos(αi − ϑ)di ] sin αj − cos αj −[cos(αi − ϑ)di ]  [sin(αi − ϑ)di − b] − cos αj − sin αj −[sin(αi − ϑ)di − b] [B] =



cos αi sin αi − sin αi cos αi cos αi sin αi

Where αi is the angle of the connection side of element i referring to ξ-axis and ϑ is called distortion angle. A measure of stress, work–conjugated to the strain, is assigned to each connecting device, and is assembled into the vector {σ} as follows: {σ}T = {σ1P , σ1Q , σ1R , σ2P , σ2Q , σ2R , . . . , σrP , σrQ , σrR }

(8)

Where σ P and σ R are the axial stresses in the connection point P and R, and σ Q is the shear stress in Q. Forces and stresses are related by: {fe } = [B]T {σ} (9)

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The constitutive law correlates the strains and stresses: {σ} = [D]{ǫ}

(10)

Where [D] is the tangential stiffness matrix of the connection side: [D] = Diag[k P , k Q , k R ]

(11)

Replacing Equation 6 in Equation 10 and this in Equation 9, it obtains: {fe } = [B]T [D][B]{u} = [K]{u}

(12)

Being [K] the global stiffness matrix. 4 4.1

MECHANICAL CHARACTERISTICS OF THE INTERFACES Elastic properties

The elastic characteristics of the connecting devices are assigned with the criterion of approximating the strain energy of the corresponding volumes of pertinence in the cases of simple deformation. For an orthotropic material in plane deformation, the matrix of elasticity is given by:   A11 A12 0 [A] = A21 A22 0  (13) 0 0 A33

νE E E Where A11 =A22 = (1−ν 2 ) , A12 =A21 = 1−ν 2 , A33 =2G= 1+ν 2 ; E is the Young modulus, ν is the Poisson’s coefficient and G is the shear modulus. On the other hand, the stress Σ and the strain H vectors are:

{Σ}T = {Σ11 , Σ22 , Σ12 } T

{H} = {H11 , H22 , H12 }

(14) (15)

The stiffness of the elastic devices is obtained by equating the strain energy of the masonry material Πm and the strain energy of the connections Πc : 1 1 Πm = {ǫ}T [A]{ǫ}V ol = {q}T [k]{q} = Πc 2 2

(16)

Thus, the axial and shear stiffness are: k P = k R = A11

(17)

k Q = A33

(18)

In addition, the two axial devices are separated from the middle point of the side by a length b in order to take into account the bending moment, where b= 2√l 3 . 6

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4.2

Strength and Plastic properties

The monotonic constitutive laws are assigned to the connecting devices adopting a phenomenological approach. These laws are based on experimental monotonic tests currently available in literature. Different rules are assumed for the axial devices and for the shear device, as sketched in Figure 5. For the axial spring, the skeleton curve under compression is given by:

a)

b) Figure 5: Constitutive laws: a) axial, b) shear.

σ = E0 exp



−ǫ ǫc



(19)

Where E0 is the initial elastic modulus and ǫc is the strain at the peak compression strength σc . Along this skeleton curve, the spring stiffness (k P , k R ) for compression loading is:     −ǫ −ǫ P R exp (20) k = k = E0 1 − ǫc ǫc The tensile axial response is defined by a tri–linear skeleton curve identified by the couples of points (σt ,ǫt ) and (σr ,ǫt ) which correspond to the peak and residual strengths. The plastic response of each axial connection is independent from the behavior of any other connection. Symmetric stiffness and strength have been attributed to the shear connections. The skeleton curve is tri-linear,defined by four parameters: the initial shear stiffness G, the softening shear stiffness Gr , the maximum shear strength τ and the residual shear strength τr . The shear strength is related to the stresses of the axial connections according with Mohr–Coulomb criterion [2]: τ = c − σ tan(φ)

(21)

Where c is the cohesion, σ is the axial stress and φ is the internal friction angle. 5

VALIDATION

The proposed model was validated by comparing it with a discrete element model of the Guilhofrei dam (Portugal) taken from the literature [5]. This dam was built in 1938 with a height of 49 m and 190 of longitude (Fig. 6). The mechanical properties of the materials (Table 1) were obtained from [5]. The soil foundation of the dam was also considered. Figure 7 shows the RBSM and the DEM meshes for the studied dam. Four different load cases were considered: 7

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Figure 6: Guilhofrei dam [5].

Mechanical Property Volumetric Weight Elastic modulus Coefficient of Poisson Shear modulus Cohesion Friction angle Compressive Strength Tensile Strength

Dam 24 kN/m3 10 GPa 0.2 4 GPa 1.58 MPa 55o 10 MPa 1 MPa

Foundation 25 kN/m3 10 GPa 0.2 4 GPa Elastic Elastic Elastic Elastic

Table 1: Mechanical properties of the materials.

Figure 7: Numerical models: a) Rigid Body Spring Model, b) Discrete Element Model [5].

- Load Case 1: Own weight (PoPo) - Load Case 2: Hydrostatic pressure (PH) - Load Case 3: Own weight plus hydrostatic pressure (PoPo + PH) - Load Case 4: Own weight plus hydrostatic pressure plus underpressure (PoPo + PH + Ps) 5.1

Load Case 1: Own weight

For the own weight analysis, only the volumetric weight of the curtain was taken into account. Table 2 shows the results obtained by DEM [5] and the proposed model. It can be observed 8

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that the results are practically the same for both models, since for the own weight and the compression stress the error percentages are less than 10% Result DEM RBSM Ownd weight (kN) 9,700 9,500 Maximum compressive stress (MPa) 0.94 0.87 Maximum horizontal displacement (mm) 2.5 2.1

Error (%) 2.0 7.4 14.5

Table 2: Results of the own weight analysis.

Figure 8 shows the deformation produced by the curtain’s own weight, in which it can be observed that it deforms slightly upstream. This coincides with the real phenomenon, since most of the mass is on this side of the curtain. So that when the hydrostatic pressure will apply, the forces remain in equilibrium.

a)

b)

Figure 8: Own weight deformed: a) Rigid Body Spring Model, b) Discrete Element Model [5].

Figure 9 shows the stress maps for shear and normal (vertical and horizontal) stresses. It can be seen that the maximum compressive stress is located upstream at the foot of the curtain. 5.2

Load Case 2: Hydrostatic Pressure

In this analysis, only the hydrostatic pressure was considered without taking into account the curtain’s own weight. In this case, the stability of the dam is due to the tensile strength of the material. The resulting hydrostatic pressure was approximately 5000 kN/m. Table 3 shows the results obtained, it can be observed that the results are acceptable with a minimum error. Figure 10 shows the deformation produced by hydrostatic pressure. It can be seen that the Result DEM Maximum tensile stress (MPa) 0.84 Maximum compressive stress (MPa) 0.77 Maximum horizontal displacement (mm) 6.5

RBSM Error (%) 0.89 6.0 0.76 1.0 5.7 12.3

Table 3: Results of the hydrostatic pressure analysis.

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Figure 9: Stress maps due to own weight: a) Normal (horizontal) RBsM, b) Normal (vertical) RBSM, c) Shear RBSM, d) Discrete Element Model [5].

curtain rotates slightly downstream. There is a slightly tensile damage in the base of the curtain upstream (Fig. 10c), since the own weight of the dam is not considered.

Figure 10: Deformed mesh due to hydrostatic pressure: a) Rigid Body Spring Model, b) Discrete Element Model [5], c) Damage map of RBSM.

5.3

Load Case 3: Own weight plus hydrostatic pressure

In this analysis, the own weight of the dam is first applied and then the hydrostatic pressure load, since the analysis is non–linear. Table 4 shows the results obtained. It can see that no tensile stresses are in the dam due to the own weight load. In this context, the maximum horizontal displacement is lesser than when the own weight is not considered (see Fig. 11). This means that the own weight contributes to the stability of the curtain. The error percentages are around 15%, which can be considered acceptable.

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Result DEM Maximum stress upstream (MPa) 0.29 Maximum stress downstream (MPa) 0.96 Maximum horizontal displacement (mm) 4.0

RBSM Error (%) 0.33 12.1 1.10 14.5 3.4 15.0

Table 4: Results of the load combination 3.

Figure 11: Deformed mesh due to load case 3: a) Rigid Body Spring Model, b) Discrete Element Model [5].

5.4

Load Case 4: Own weight plus hydrostatic pressure plus underpressure

This load case considers additionally to the own weight and the hydrostatic pressure, the underpressure and a flood of 5 m over the crown of the curtain (failure load). The resultant of the underpressure load was equal to 1,015 kN. This type of combination loads are similar to the failure loads of the dam. As the previous load case, the own weight of the dam is first applied and then the hydrostatic pressure and underpressure loads are applied, since the analysis is non–linear. The maximum compressive stress at the base of the curtain downstream was equal to 1.81 MPa for the DEM [5] and 2.1 for the RBSM (16% of error). Figure 12 shows the deformed mesh and the failure mechanism of the dam. The curtain overturns downstream, since the tensile stresses at the base of the dam are overpassed. 6

CONCLUSIONS

In this paper, a Rigid Body Spring Model for the structural assessment of old masonry dams is presented. The main conclusions are: - The proposed model is a semi–discrete model. - The model can detect sliding, separation, overturning, crushing, tensile and shear damage. - It was validated by comparing with a discrete element model of a dam. - The validation of the model was taking into account different load cases. - The tensile strength of the masonry is an important parameter in the structural assessment of old masonry dams. - The proposed model can detect the different collapse mechanism of the dams, mainly: overturning and sliding. 11

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Figure 12: Deformed mesh and failure mechanism due to load case 4: a) deformed mesh of RBSM, b) Failure mechanism RBSM, c) Deformed mesh DEM [5], d) Failure mechanism DEM [5].

REFERENCES [1] W. Creager, Engineering for masonry dams, John Wiley & Sons, Inc., New York, (1917). [2] F. Pe˜na, Rigid element model for dynamic analysis of in–plane masonry structures, Ph.D. Thesis, Politecnico di Milano, Italy, (2002). [3] S. Casolo, “Modelling in-plane micro-structure of masonry walls by rigid elements”, International Journal of Solids and Structures, Vol. 13, pp. 3625-3641, (2004). [4] S. Casolo and F. Pe˜na, “Rigid element model for in-plane dynamics of masonry walls considering hysteretic behaviour and damage”, Earthquake Engineering and Structural Dynamics, Vol. 36, pp. 1029-1048, (2007). [5] E. Bretas, Desenvolvimento de um modelo de elementos discretos para o estudo de barragens gravidade em alvanaria, Ph.D. Thesis, Universidade do Minho, Portugal, (2012).

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