For tunnels with low overburden, the shotcrete lining is the main element of ground support, which is why shotcrete behaviour is of prime importance. Following its application, shotcrete not only gains strength and stiffness with time but – if loaded at an early stage – is also subject to creep, which in turn leads to stress relaxation. For two-dimensional FE calculations, a computational modulus of elasticity of shotcrete is to be used. Based on parameter studies it was found that the modulus of elasticity of shotcrete depends on both shotcrete mix and ground behaviour. While the Young’s Modulus used for young shotcrete may range from 1,000MPa to 7,000MPa, the Young’s Modulus used for hardened shotcrete is best represented by a value of 15,000MPa.

Structural analyses for the initial lining of tunnels are often performed using finite element, finite difference or beam element models. This article illustrates how shotcrete, as a main support measure for tunnels with low overburden, is modelled in numerical analyses.

Rokahr and Lux 1989(5) were the first to describe shotcrete behaviour determined by creep and relaxation. Pöttler(4) then used these findings in connection with structural analyses for tunnels and defined the so-called hypothetical Young’s Modulus of young shotcrete. In the following years, various authors were to use different values for the hypothetical Young’s Modulus of shotcrete(7). For a realistic derivation of this value, an extensive study was conducted considering the following aspects:

  • Once applied, the shotcrete gradually hardens gaining both strength and stiffness

  • The time dependency of shotcrete is governed by creep and relaxation depending on the shotcrete stress level and age

  • While hardening, the shotcrete strains increase due to deformations of the cavity

  • The stress-strain behaviour of the shotcrete is nonlinear

    Behaviour of young shotcrete

    The time-dependent development of the shotcrete’s Young’s Modulus is taken into account using Weber’s equations for concrete(6) (Figure 1).

    [Equation 1]

    aREF, cREF, n = material properties to derive from tests

    E28, = Young’s Modulus after 28 days

    The time-dependent development of Young’s Modulus of steel-fibre reinforced shotcrete was investigated by Ding(2). In view of the fact that different steel-fibre contents (20, 40, 60kg/m³) have no substantial effect on this development, the constants of Weber’s equation are to the greatest possible extent adjusted to the mean value of the three tests. The constants aREF = 1.14, cREF = -0.353, E28 = 32,720MN/m², n = 1.17 in Weber’s equation harmonise well with the results obtained for steel-fibre reinforced shotcrete (Figure 2).

    It was found that steel-fibre reinforced shotcrete hardens considerably faster than unreinforced shotcrete, in other words, the steel fibres enhance the shotcrete’s stiffness (Figure 3).

    With the Young’s Modulus of shotcrete being time-dependent, it is necessary to determine stress changes based upon strain changes.

    [Equation 2]

    The tangent modulus of shotcrete ET is determined from the following constitutive relation:

    [Equation 3]

    EC = Tangent modulus at zero strain

    ECl = Secant modulus at strain εcl

    εcl = Strain at maximum concrete compressive stress fcm

    fcm = Mean concrete compressive strength

    The tangent modulus is determined as follows by the derivation of the above equation for εel.

    [Equation 4]

    The tangent modulus thus depends on the mean concrete compressive strength fcm. The time-dependent development of the compressive strength is derived from the time-dependent development of the Young’s Modulus, using the relation included in MC 90(1) in the following way:

    [Equation 5]

    ßcc(t) = Relation to describe the time-dependent compressive strength development acc. to MC 90

    ßE(t) = Relation to describe the time-dependent Young’s Modulus development acc. to MC 90

    Figure 4 demonstrates the stress-strain relationships for 1, 2, 4, 7 and 28-day-old concrete.

    Creep and relaxation

    The creep behaviour of young shotcrete is – in the present study – considered by applying Zachow’s relation(8), which represents a further development of Petersen’s(3) creep model. The creep rate is determined as follows:

    [Equation 6]

    Differences in the magnitude of creep are reflected by different constants in Equation 6. These constants are determined by a comparison of test and computation results, adopting the method of least squares.

    The evaluation of previous creep tests conducted on different types of shotcrete revealed that the shotcrete mix design considerably influences the creep behaviour. As a result, a differentiation is made between slight, moderate, and pronounced creep potential.

    The constants for pronounced creep are derived from Petersen’s tests. In the course of these tests, the shotcrete was – up to an age of 280 hours – subjected to constant compressive stresses of 5, 7 and 9MPa.

    The parameters for moderate and slight creep are derived from Ding’s tests(2). The results from tests performed on steel-fibre reinforced shotcrete characterised by a fast strength development are used to establish the slight creep parameters, whilst the results achieved by tests with unreinforced concrete are used to determine the moderate creep parameters.

    Rock-mass behaviour

    Depending on rock mass type, stress redistribution occurs at different rates. This is why two scenarios are investigated – a fast redistribution representing brittle behaviour, completed after 5 days and a slow redistribution representing ductile behaviour completed after 15 days. The time-dependent development of deformations which reflects stress redistribution is assumed to follow a parabolic curve with a horizontal tangent at the end. Deformations are assumed to start after about 5 hours (approximately corresponding to the first round of advance following shotcrete application).

    Computation method

    The computation adopts a time-stepping procedure during which the differential equation is solved as follows: At any time t, the shotcrete compressive strain ε(t) is determined and the viscous creep rate is calculated by use of Equation 6. The elastic strain increments are derived from the total compressive strain increments and the creep strain increments:

    [Equation 7]

    The elastic stress increments Δσ(t) are determined from the elastic strain increments Δεel(t) and the tangent modulus ET(σ,t). . The integration of the stress increments is achieved by summation. The ratio of stress to total strain leads to the computational value representing the Young’s Modulus of young shotcrete. This value reflects the time-dependent development of both shotcrete stiffness and creep.

    Determination of stiffness of Young Shotcrete

    The different shotcrete mix designs result in differences in material behaviour, which may be characterised as follows (Table 1):

  • Slow hardening and pronounced creep

  • Fast hardening and moderate creep

  • Fast hardening and slight creep

    Combining various material behaviours with different rock mass behaviours leads to six scenarios (Table 1).

    Figure 5, for example, presents the results of computation 2.2. It illustrates the development of deformations due to stress redistribution, the development of the computational modulus of elasticity and of stresses in the shotcrete, as well as the degree of utilisation of the shotcrete as ratio of stress to the respective concrete compressive strength. The computational modulus of elasticity is calculated.

    With a slow development of deformations due to stress redistribution, a fast shotcrete hardening, and a moderate creep, the computational value of the Young’s Modulus of young shotcrete reaches its maximum of 16,000MN/m² one day after initial loading. Subsequently it continuously decreases. The maximum stress in the shotcrete occurs after approx. 6 days at a computational value of the Young’s Modulus of 6,500MN/m². The degree of utilisation remains quite constant for up to 8 days, then gradually decreases. At the end of the stress redistribution, the computational value of the Young’s Modulus drops to 3,000MN/m².

    Table 2 lists the rounded computational values in the following sequence: maximum value/value at maximum degree of utilisation/value at maximum stress/value after completed stress redistribution. The results reveal the importance of choosing the appropriate computational value for the stiffness of young shotcrete. As the maximum value occurs at a very early stage, and amounts to a fraction of the total load, this is of no relevance. Bearing in mind that the stiffness at maximum degree of utilisation and maximum stress considerably exceeds the final value, it should be checked which value in Table 2 is to be used for the structural analyses.

    Computational Young’s Modulus of shotcrete

    For the shotcrete lining design, a recommendation shall be made below regarding the computational value of Young’s Modulus. Based on the results of this study, for young shotcrete, this is made considering:

    Reinforcement: Shotcrete reinforced by steel wire mesh has not been considered since the test results available are insufficient. Tests with steel-fibres reveal that the reinforcement influences the material behaviour. A differentiation is hence made between unreinforced or slightly reinforced shotcrete (with one layer), moderately reinforced shotcrete (with two layers), and heavily reinforced shotcrete (with two overlapping layers), equalling steel-fibre reinforced shotcrete.

    Material behaviour of shotcrete: Often, test results which allow a shotcrete characterisation distinguishing between slow/fast hardening and pronounced/ moderate/slow creep, are not available. In this case the following is recommended: Slow hardening/pronounced creep is allocated to shotcrete featuring a 1-day strength of <10MPa with light reinforcement; Fast hardening/ moderate creep is allocated to shotcrete featuring a 1-day strength of >10MPa with moderate reinforcement; Fast hardening/slight creep is allocated to shotcrete featuring a 1-day strength of >10MPa with heavy reinforcement or with steel-fibre reinforcement.

    It is therefore recommended to choose the respective Young’s Modulus of young shotcrete as presented in Table 3, considering given parameters. Also, the advance rate must be taken into account. With an advance rate of 6m/day of continuous operation the stress redistribution at a distance of 30m to the working face may (in most cases) be assumed to be completed after 5 days (fast redistribution). With a continuous advance rate of 2m/day, the 30m distance to the working face will only be achieved after 15 days (slow redistribution). Any tunnel advance interruption should be considered by modifying the computational Young’s Modulus of young shotcrete. Any interruption for several days induces shotcrete hardening and its relaxation capacity to decrease. For shotcrete, which has already hardened when the advance is resumed, a computational Young’s Modulus of 15,000MPa for hardened shotcrete should be used. The difference between the results from the initial computation and the results achieved with the increased Young’s Modulus allows the additional support measures by rockbolts or shotcrete to be determined.

    Related Files
    Fig 1 – Time dependent development of concrete Young’s Modulus according to Weber(6)
    Equation 3
    Equation 9
    Fig 2 – Time dependent development of Young’s Modulus for steel-fibre reinforced shotcrete according to Ding(2)
    Fig 4 – Time dependent stress-strain relationships for concrete
    Equation 4
    Equation 5
    Equation 7
    Equation 8
    Fig 5 – Results of computation 2.2 according to Table 1
    Fig 3 – Numerical assumption for time dependent development of Young’s Modulus
    Equation 1
    Equation 2
    Equation 6