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The Elastic Modulus Variation During the Shotcrete Curing Jointly Investigated by the Convergence-Confinement and the Hyperstatic Reaction Methods

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Abstract

Induced stresses in sprayed concrete (or shotcrete) are quite complex to evaluate and depend on many factors such as the size and depth of the tunnel, the geomechanical characteristics of the surrounding ground in which the tunnel is excavated, the type of shotcrete, the evolution of its mechanical parameters over time and the excavation face advance rate. In particular, the evolution of the mechanical properties of the shotcrete is crucial regarding the interaction with the tunnel wall and the development of the bending moments and the normal forces which occur along the circumference of the lining. In this research, a new calculation procedure based on the combined use of two calculation methods, the convergence confinement and the hyperstatic reaction methods, is presented. Thanks to this procedure, it is possible to progressively apply the load acting on the lining as the curing phase of the concrete progresses and therefore with the evolution of its mechanical parameters. This procedure has been applied to several examples of calculation, obtaining useful considerations regarding the mechanical behavior of the shotcrete lining when some fundamental parameters of the calculation change. It is possible to achieve bending moments and forces in the lining with the progress of the load steps. It is also possible to determine the trend of the lining safety factor over time and at the end of the loading phase, allowing a proper design of the support, with particular attention to the type of shotcrete and the thickness of the lining.

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Fig. 1

(picture courtesy Roland Mayr, BASF)

Fig. 2

(modified after Oreste 2009)

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Abbreviations

A :

Area of the lining section

\(E_{rm}\) :

Elastic modulus of the rock mass

\(E_{,mean }\) :

Mean value of the elastic modulus of shotcrete

\(E_{, t}\) :

Elastic modulus of shotcrete at the time \(t\)

\(E_{, 0}\) :

Value of the asymptotic elastic modulus of the shotcrete, for \(t = \infty\)

\(\left\{ F \right\}\) :

Nodal forces applied to the numerical model

\(J_{z}\) :

Moment of inertia of the lining section

\(K\) :

Global stiffness matrix

\(k_{i }\) :

Local stiffness matrix of the element i

\(K_{n}\) :

Normal stiffness of the interaction spring in the node of the model

\(K_{0}\) :

Lateral earth pressure

\(K_{s}\) :

Shear stiffness of the interaction spring in the node of the model

l :

Length of the one-dimensional element

M :

Bending moment

N :

Normal force

\(p\) :

Internal tunnel pressure

\(p_{cr}\) :

Critical pressure at the limit between the elastic and the plastic behavior

\(p_{fict}\) :

Fictitious internal tunnel pressure

\(p_{0}\) :

Lithostatic pressure

\(R\) :

Tunnel radius

\(r\) :

Generic radial coordinate

\(\left\{ S \right\}\) :

Vector of nodal displacements

T :

Shear force

\(t_{0}\) :

Final installation time of the support

\(u\) :

Tunnel wall radial displacement

v :

Poisson’s ratio

\(\alpha\) :

Time constants \((t^{ - 1} )\) of the curing equation for the elastic modulus

\(\alpha_{i}\) :

Angle of inclination of the element ith with respect to the horizontal

\(\phi\) :

Rotation of the element in correspondence to the nodes

\(\delta\) :

Advance step

\(\delta n\) :

Nodal normal displacement between the structure and the rock mass

\(\delta s\) :

Shear displacement between the structure and the rock mass

\(\sigma_{c,t}\) :

Unconfined compressive strength for the shotcrete at the time t

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Author notes

  1. Cesar Alejandro Luna Ramos

    Present address: Faculty of Engineering, Universidad Mariana, Calle 18 No. 34-104, Pasto, Colombia

Authors and Affiliations

  1. Department of Environmental, Land and Infrastructural Engineering, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129, Turin, Italy

    Pierpaolo Oreste & Cesar Alejandro Luna Ramos

  2. BASF Construction Solutions GmbH, Dr.-Albert-Frank-Straße 32, 83308, Trostberg, Germany

    Giovanni Spagnoli

Authors
  1. Pierpaolo Oreste
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  2. Giovanni Spagnoli
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  3. Cesar Alejandro Luna Ramos
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Correspondence to Giovanni Spagnoli.

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Oreste, P., Spagnoli, G. & Luna Ramos, C.A. The Elastic Modulus Variation During the Shotcrete Curing Jointly Investigated by the Convergence-Confinement and the Hyperstatic Reaction Methods. Geotech Geol Eng 37, 1435–1452 (2019). https://doi.org/10.1007/s10706-018-0698-1

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