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New Soft Rock Pillar Strength Formula Derived Through Parametric FEA Using a Critical State Plasticity Model

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Abstract

Many abandoned room and pillar mines have been excavated not far from the surface of large areas of important European cities. In Rome, these excavations took place at shallow depths (3–15 m below the ground surface) in weak pyroclastic soft rocks. Many of these cavities have collapsed; others appear to be in a stable condition, although an appreciable percentage of their structural components (pillars, roofs, etc.) have shown increasing signs of distress from both the morphological and mechanical points of view. In this study, the stress–strain behaviour of soft rock pillars sustaining systems of cavities under vertical loads was numerically simulated, starting from the in situ initial conditions due to excavation of the cavities. The mechanical behaviour of the constituent material of the pillar was modelled according to the Modified Cam-Clay constitutive law (elasto-plastic with strain hardening). The influence of the pillar geometry (cross-section area, shape, and height) and mechanical parameters of the soft rock on the ultimate compressive strength of the pillar as a whole was parametrically investigated first. Based on the numerical results, an original relationship for pillar strength assessment was developed. Finally, the estimated pillar strengths according to the proposed formula and well-known formulations in the literature were compared.

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Abbreviations

\(\chi \) :

Pillar safety factor

\(\sigma _{\mathrm{r}}\) :

Pillar compressive strength

\(\sigma _{\mathrm{p}}\) :

Average pillar stress

\(\sigma _{\mathrm{p}}^\gamma \) :

Average pillar stress contribution due to the weight of coating soils

\(\sigma _{\mathrm{p}}^{\mathrm{y}}\) :

Pillar yield stress on the (\(\sigma _{\mathrm{p}},\varepsilon _{\mathrm{p}}\)) curve

\(\varepsilon _{\mathrm{p}}\) :

Average pillar strain

\(\varepsilon _{\mathrm{p}}^{\mathrm{y}}\) :

Average pillar strain at yielding

\(\delta v_{\mathrm{gs}}\) :

Imposed vertical ground surface displacements

\(A\) :

Pillar cross-section area

\(b\) :

Small side of pillar cross-section

\(l\) :

Large side of pillar cross-section

\(d\) :

Pillar diameter

\(h\) :

Pillar height

\(b_{\mathrm{eq}}\) :

Side of equivalent square section pillar

\({b}/{h}\) :

Pillar width-to-height ratio

\({b_{\mathrm{eq}}}/{h}\) :

Pillar average width-to-height ratio

\(\sigma _1\) :

Material uniaxial compressive strength

\(\sigma _{\mathrm{u}}\) :

Unconfined compressive strength of a cubical pillar specimen

\(k_{\mathrm{lp}}\) :

Pillar shape term (Lunder and Pakalnis’s formulation)

\(c_{\mathrm{pav}}\) :

Pillar average confinement (Lunder and Pakalnis’s formulation)

\(p\) :

Equivalent pressure

\(q\) :

Deviator stress

\(f(p,q)\) :

Yield function

\(g(p,q)\) :

Plastic potential

\({\varepsilon }^{pl({\mathrm el})}_{ij}\) :

Plastic (elastic) strain tensor

\(\varepsilon ^{pl}_{\mathrm{mag}}\) :

Plastic strain magnitude

\(\varepsilon _{p}^{\mathrm{pl(el)}}\) :

Plastic (elastic) volumetric strain

\(\varepsilon _{q}^{\mathrm{pl(el)}}\) :

Plastic (elastic) deviator strain

\(p_{\mathrm{c}}^*\) :

Current size of the yield surface

\(J^{\mathrm{pl}}\) :

Plastic volume change

\(p_{\mathrm{c}}\) :

Preconsolidation pressure

\(M\) :

Slope of the critical state line on the (\(p,q\))-plane

\(\lambda \) :

Logarithmic hardening constant in pure compression

\(k\) :

Logarithmic bulk modulus in pure compression

\(E\) :

Young’s modulus

\(\phi \) :

Shear angle

\(c\) :

Cohesion

\(\nu \) :

Poisson’s ratio

\(G\) :

Elastic shear modulus

\(e\) :

Void ratio

\(V\) :

Specific volume

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Authors and Affiliations

  1. LMT (ENS Cachan, CNRS, Université Paris Saclay), 61 Avenue du Président Wilson, 94235, Cachan Cedex, France

    Giuseppe Rastiello

  2. Department of Civil Engineering and Computer Science, University of Rome “Tor Vergata”, Via del Politecnico 1, 00133, Rome, Italy

    Francesco Federico & Silvio Screpanti

Authors
  1. Giuseppe Rastiello
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  2. Francesco Federico
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  3. Silvio Screpanti
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Correspondence to Giuseppe Rastiello.

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Rastiello, G., Federico, F. & Screpanti, S. New Soft Rock Pillar Strength Formula Derived Through Parametric FEA Using a Critical State Plasticity Model. Rock Mech Rock Eng 48, 2077–2091 (2015). https://doi.org/10.1007/s00603-014-0693-7

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