Experimental insight into spalling behavior of concrete tunnel linings under fire loading

Abstract

New experimental insight into the spalling behavior of concrete in fire conditions is presented in this paper. Spalling was recorded by a high-speed camera. The slow-motion sequences allow us to determine the size, shape, and velocity of the spalled-off pieces. With this information at hand, the released energy associated with every spalling event is computed and compared to the energies associated with pore-pressure and thermal-stress spalling. This comparison provides new insight into the impact of the various thermal, mechanical, and hydral processes controlling concrete spalling.

Appendices

Appendix 1: Determination of pressurized pore volume V ₀

For determination of the pore volume right before spalling, V ₀ [m³]—containing water vapor at pressure p ₀ [Pa]—the ratio between pore volume V _p [m³] and total concrete volume (i.e., the porosity ϕ [–]) is assumed to be equal to the area ratio of an arbitrary plane section cut through the porous medium, giving

$$ {\frac{V_p}{V}} = \phi = {\frac{A_p}{A}}, $$

(14)

with A _p [m²] as the cumulative area of the pore sections cut by this plane. In addition, the following is assumed:

• 1. Pores cut by an arbitrary plane section have different diameters, with the distribution of these diameters following the pore-size distribution obtained from, e.g., mercury-intrusion porosimetry (MIP) and/or image analysis. According to [12, 13, 33], a combination of the two mentioned techniques is appropriate for identification of the pore structure of concrete. For the underlying evaluation, the real pore-size distribution is approximated by a straight line in the log (D)-V _p -diagram (see Fig. 21a).

• 2. Assuming spherical pores, an arbitrary section through concrete does not cut all pores at mid section but rather cuts them in a distributed manner (see Figs. 9, 22). Hence, pores of equal diameter contribute differently to the total area A _p . This is taken into account by evenly distributing the location of the intersecting plane over the sphere diameter (see Fig. 22).

Fig. 21

Illustration of a approximation of the pore-size distribution by [−k log (D/D _max)] [33] and b division of employed pore-size distribution into sub-pore ranges

Fig. 22

Illustration of different contribution of spheres with equal diameter to the total area of cut pores

Based on Assumption (1), the employed pore-size distribution is divided into a finite number of sub-pore ranges (see Fig. 21b) and the number of pores corresponding to the i-th sub-pore range, N _i [–], is determined from A _p,i [m²] and D _i [m]. Subsequently, the corresponding sub-pore volume, V _0,i [m³], and the total corresponding pore volume right before spalling, V ₀ [m³], are determined as

$$ V_{0,i} = N_i V_i = N_i {\frac{\pi}{6}} D_i^3 \quad \hbox{giving} \quad V_0 = \sum_i V_{0,i}. $$

(15)

Appendix 2: Experimental determination of specific fracture energy of concrete by three-point bending tests

The fracture energy of concrete can be determined by (i) direct tension or (ii) bending tests (see Fig. 23). Regarding the latter, the fracture energy may be determined according to [36]. Hereby, (i) weight-compensated tests (where the self weight of the beam is eliminated by a counter-weight system) and (ii) tests without weight compensation are distinguished. In case of no self-weight compensation, the fracture energy G _F [J/m²] is given by (see Fig. 23)

$$ G_F = {\frac{W_0 + W_1 + W_2 + W_3}{A_{\rm lig}}}, $$

(16)

with A _lig [m²] as the area of the ligament (with A _lig = b(h−a), where b [m] is the beam width, h [m] is the beam height, and a [m] is the notch height). In Eq. (16),

$$ W_0 = \int\limits_0^{\delta_0} P_{\rm exp} d\delta $$

(17)

is the external work W ₀ [J] (area under the experimentally obtained load-deflection curve) and

$$ W_1 = \left( {\frac{m_1}{2}} + m_2 \right) g \delta_0 $$

(18)

is the work performed by the mass of the beam between the supports, m ₁ [kg], and the mass of the part of the loading device not attached to the machine, m ₂ [kg] (following the beam until failure), g = 9.81 m/s is the gravity acceleration, and δ₀ [m] is the mid-span beam deflection at failure. According to [19, 32], W ₃ can be neglected.

Fig. 23

Three-point bending test: a test setup and b load-deflection curve in case of no weight compensation [19]

According to [14, 15, 18, 34], the so-obtained fracture energy changes with sample size which is attributed to the following characteristics of the experimental setup:

• 1. At the supports, friction ⁷ between support and beam leads to an overestimation of the fracture energy by 2–5% [18].

• 2. Dissipation of energy in the bulk material results in an overestimation of the fracture energy by 5–10% due to damage at central support and 1–2% due to damage in regions of high tensile stresses, respectively [34].

• 3. W₂ is determined by assuming rigid-body motion of the two parts of the beam [14, 15], giving

  $$M = b \int\limits_0^{z_c} \sigma\left[w(z)\right] z dz = {\frac{b} {\theta^2}} \int\limits_0^{w(z_c)} \sigma\left(w\right) w dw = {\frac{\zeta b} {\theta^2}}, $$

  (19)

  where θ [rad] is the opening angle and z was substituted by w/θ (see Fig. 23a). Inserting

  $$ M = \left[ P_{\rm exp} + \left( {\frac{m_1} {2}} + m_2 \right) g \right] {\frac{l}{4}} = {\frac{P l}{4}} \quad \hbox{and} \quad \theta = {\frac{4 \delta}{l}} $$

  (20)

  into Eq. (19) leads to [14, 15]

  $$ {\frac{M}{b}} = {\frac{1}{\theta^2}} \zeta \quad \hbox{and} \quad P = {\frac{\zeta b l}{4 \delta^2}}, $$

  (21)

  allowing extrapolation of the experimental P−δ curve as indicated in Fig. 23b. Hereby, the unknown parameter ζ [N] (introduced in Eq. (19)) is obtained from linear regression of the experimental results (see Fig. 24).

Fig. 24

Determination of parameter ζ from linear regression of the part of the bending experiment close to failure of the beam, i.e., for large values of θ [14]

Accordingly, the fracture energy G _F , determined from application of Eqs. (16) and (21) to the results of the three-point bending experiments, was reduced by 10%, accounting for the aforementioned dissipative processes. Moreover, aging of concrete was considered by the empirical relation ⁸ [7, 28]

$$ G_F(28\,\hbox{days}) = G_F(t) {\frac{1}{1+0.277\cdot \log (t/28)}}. $$

(22)

Concerning the temperature dependence of the fracture energy, contradictory experimental results are reported in the open literature:

• In [7], the fracture energy of concrete was determined at elevated temperatures up to 200°C, showing a decrease of G _F with temperature.

• In [29, 43], the residual fracture energy continuously increased up to a temperature of 300–400°C and decreased thereafter. The fracture energy obtained on hot concrete specimens, on the other hand, showed a decreasing behavior up to a temperature of 150°C followed by a continuous increase. It is, however, stated in [29, 43] that transient effects at temperatures up to 150°C may have altered the experimental results for G _F at the respective temperatures.

• In [4, 16], no clear trend for the residual fracture energy was obtained and it was therefore concluded that G _F may be assumed to be independent of temperature.

Considering these contradictory conclusions regarding the temperature-dependence of the fracture energy of concrete, G _F was assumed to be temperature-independent, with a mean value for the fracture energy obtained from 46 experiments given by G _F  = 90 J/m² (see Table 2).

Table 2 Adjusting the experimental result for G _F [J/m²]