Acta Mechanica

, Volume 229, Issue 2, pp 849–866 | Cite as

Bearing capacity of concrete hinges subjected to eccentric compression: multiscale structural analysis of experiments

  • Johannes Kalliauer
  • Thomas Schlappal
  • Markus Vill
  • Herbert Mang
  • Bernhard PichlerEmail author
Open Access
Original Paper


Existing design guidelines for concrete hinges are focusing on serviceability limit states. Lack of knowledge about ultimate limit states was the motivation for this work. Experimental data are taken from a testing series on reinforced concrete hinges subjected to eccentric compression up to their bearing capacity. These tests are simulated using the finite element (FE) software Atena science and a material model for concrete implemented therein. The first simulation is based on default input derived from measured values of Young’s modulus and of the cube compressive strength of the concrete. The numerical results overestimate the initial stiffness and the bearing capacity of the tested concrete hinges. Therefore, it is concluded that concrete was damaged already before the tests. A multiscale model for tensile failure of concrete is used to correlate the preexisting damage to corresponding values of Young’s modulus, the tensile strength, and the fracture energy of concrete. This allows for identifying the preexisting damage in the context of correlated structural sensitivity analyses, such that the simulated initial stiffness agrees well with experimental data. In order to simulate the bearing capacity adequately, the triaxial compressive strength of concrete is reduced to a level that is consistent with regulations according to Eurocode 2. Corresponding FE simulations suggest that the ductile structural failure of concrete hinges results from the ductile material failure of concrete at the surface of the compressed lateral notch. Finally, Eurocode-inspired interaction envelopes for concrete hinges subjected to compression and bending are derived. They agree well with the experimental data.

List of symbols


Neck width


Maximum aggregate size


Width of the partially loaded area \(A_{c0}\)


Width of the partially loaded area \(A_{c1}\)


Softening parameter

\(c_1, c_2\)

Coefficients appearing in the crack opening law


Initial value of c


Depth of the partially loaded area \(A_{c0}\)


Depth of the partially loaded area \(A_{c1}\)


Eccentricity of the normal force

\(e_\sigma \)

Parameter influencing the Menétrey–Willam failure surface in the deviatoric plane


base vectors in x, y, and z-direction


Uniaxial compressive strength


Initial elastic limit under uniaxial compression


Evolving elastic limit under uniaxial compression


Design value of the uniaxial compressive strength


Characteristic value of the uniaxial compressive strength


Mean value of the cube compressive strength


Characteristic value of the cube compressive strength


Uniaxial tensile strength of the Rankine failure surface


Uniaxial tensile strength of the Menétrey–Willam failure surface


Uniaxial tensile strength of damaged concrete


Parameter influencing the shape of the Menétrey–Willam failure surface


Reduction of uniaxial compressive strength, due to cracks with crack-plane normal vectors orthogonal to the loading direction


Crack opening displacement


Value of w corresponding to the vanishing cohesive stress


Critical compression displacement


Value of w due to preexisting damage


Cartesian coordinates


Partially loaded area


Distribution area with a similar shape to \(A_{c0}\)


Young’s modulus


Young’s modulus of uncracked concrete


Young’s modulus of damaged concrete


Failure function of the Menétrey and Willam criterion


Maximum design compressive force


Flag for modeling of crack rotation: \(Fixed=1\) ... no crack rotation


Fracture energy


Fracture energy of damaged concrete


Fracture toughness


Bending moment


Normal force

\(\beta \)

Flag for modeling the direction of the plastic flow: \(\beta =0~\dots \) purely deviatoric plastic flow

\(\varDelta G_f\)

Reduction of fracture energy due to preexisting damage

\(\varDelta \varphi \)

Rotation angle

\(\epsilon _c^p\)

Plastic strain at uniaxial compressive strength

\(\vartheta \)

Lode angle

\(\lambda _t\)

Auxiliary-to-actual uniaxial tensile strength ratio

\(\nu \)

Poisson’s ratio

\(\nu _c\)

Poisson’s ratio of uncracked concrete

\(\xi \)

Hydrostatic stress invariant

\(\rho \)

Deviatoric stress invariant

\(\sigma \)

Softening tensile strength of smeared crack model

\({\varvec{\sigma }}\)

Cauchy stress tensor

\(\sigma _1,\sigma _2,\sigma _3\)

Principal stresses

\(\sigma _\ell \)

Principal normal stress in the loading direction

\(\sigma _{\ell u}\)

Maximum normal stress in the loading direction

\(\sigma _y\)

von Mises yield stress of steel

\(\omega \)

Crack density parameter



Open access funding provided by Austrian Science Fund (FWF). Financial support by the Austrian Ministry for Transport and Technology (bmvit), the Austrian Research Promotion Agency (FFG), ÖBB-Infrastruktur AG, ASFINAG Bau Management GmbH, provided within the VIF-project 845681 “Optimierte Bemessungsregeln für dauerhafte bewehrte Betongelenke” and corresponding discussions with Michael Schweigler (TUWien), Susanne Gmainer and Martin Peyerl (Smart Minerals GmbH), Alfred Hüngsberg (ÖBB-Infrastruktur AG), Erwin Pilch and Michael Kleiser (ASFINAG Bau Management GmbH) are gratefully acknowledged. Additional interesting discussions regarding the use of concrete hinges inmechanized tunneling, carried out within theAustrian Science Fund (FWF) project P 281 31-N32 “Bridging theGap by means ofMultiscale Structural Analysis” with Yong Yuan (Tongji University) and Jiaolong Zhang (TU Wien/Tongji University) are also gratefully acknowledged.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Johannes Kalliauer
    • 1
  • Thomas Schlappal
    • 1
  • Markus Vill
    • 2
  • Herbert Mang
    • 1
    • 3
  • Bernhard Pichler
    • 1
    Email author
  1. 1.Institute for Mechanics of Materials and StructuresTU Wien – Vienna University of TechnologyViennaAustria
  2. 2.Vill Ziviltechniker GmbHViennaAustria
  3. 3.College of Civil EngineeringTongji UniversityShanghaiChina

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