Nonlinear finite element modelling of the tension softening of
conventional and fibrous cementitious composites
R. Y. Xiao and C. S. Chin
Civil & Computational Engineering Centre
School of Engineering
University of Wales Swansea
It is quite clear that nonlinear finite element analysis incorporating comprehensive pre-
cracking and post-cracking characteristics of cementitious composites has become more
important for predicting the stress-deformation response and stress-strain distribution
accurately. Attempts have been made in the recent research where two types of tension
softening models (i.e. Tension Softening Material and Enhanced Multilinear Isotropic
Softening Models) were developed by employing ANSYS finite element software. The
intention of this paper is thereby to discuss the proposed softening models to validate
the complete stress-deformation response of prismatic specimens subjected to uniaxial
tension. The outcomes from the verification of both modelling techniques have shown
to provide good agreement over the experimental results obtained from literatures 
2. THE CONSTITUITIVE TENSION SOFTENING MODELS
It’s generally acknowledged that there is a growing interest in utilizing commercial
finite element software for modelling cementitious material either in conducting
research or industrial applications. The conventional smeared cracking model has been
used by previous researchers   for modelling the material or structural
characteristics of cementitious composites. However, the conventional smeared
cracking approach always attempts convergence problems whereas approaching failure,
the subsequent equilibrium point was found to be unstable which caused divergence
leading to the unfeasibility to simulate the post-cracking response. For this reason,
efforts have been made to develop new modelling technique which enables the
determination of tension softening characteristics of both conventional and fibrous
concrete in tension. In the preliminary study, prismatic specimen subjected to direct
tension as presented in the previous chapter has been analyzed by adopting the proposed
TSM and EMIS models. The TSM model is generally characterized by the nonlinear
isotropic behaviour described by four Voce’s material constants as specified in an
σ = k + Ro ε pl + R∞ 1 − e −bε
σ (stress, N/mm2); k (elastic limit, N/mm2); Ro (threshold stress, N/mm2); εpl(equivalent
plastic strain); R∞ (asymptotic stress, N/mm2); b(Voce’s parameter),
On the other hand, The EMIS model utilizes the multilinear isotropic assumption where
the uniaxial tensile behaviour was described by stress-strain curve. The uniaxial tensile
stress-strain curve was defined with reference to  where the ascending and
descending branches were defined by equation 2 and 3 correspondingly.
= 1.20 − 0.20 ε (2) = p (3)
f ε ε f β
t p p t ε
α − 1 + ε
σ (stress, N/mm2); ε(tensile strain); εp(tensile strain corresponds to peak stress); ft
(ultimate tensile strength, N/mm2); α (coefficient); β (coefficient)
3. VERIFICATION TEST CASES
Since the proposed modeling technique has shown its ability in capturing the complete
stress-deformation response of prismatic specimen subjected to uniaxial tensile load.
This has been extended to the present research in verifying different test cases intended
to provide more confidence in the solution capabilities of TSM model proposed. An
attempt has been made to employ three types of specimens with distinctive material and
physical properties in the present verification tests. Alternatively, some of the
specimens were also employed for validating the EMIS±SR model. The geometry of
throated specimen subjected to direct tension was modelled by adopting solid elements.
The boundary conditions were modelled where the constraints were fixed at the end
surface of the specimen in the direction of loading. Additionally, the loads applied the
opposite end of the specimen were iterated by adopting Newton-Raphson method.
2HB-1 2HB-2 Dramix RF4000 PFRC SFRC
     
Compressive 31.41 31.41 46.3 46.3
Strength, fcu N/mm2 N/mm2 N/mm2 N/mm2
Tensile 2.819 2.295 4.31 4.50 3.93 4.58
Strength, ft N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 N/mm2
100 x 100 x
100 x 100 x
350 x 350 x 330 x
Specimen 100 x 100 x 330 x 127 x
100 x 20 100 x 20 127 x 28
Size 210/70 210/70 28 mm
mm mm mm
x 70 x 70
Fibre Type Fibrillated Hooked
(Volume - - Polypropylene Steel
(2 %) (2 %)
Fraction) (0.5 %) (0.5%)
TABLE 1. Specifications of various test cases.
a) The throated
c) The throated
FIGURE 1. Comparison of complete stress-deformation curves between experimental
specimens and numerical models.
The stress-deformation response of the softening model has evidently shown that, the
pre-cracking behaviour was initiated by nearly linearly elastic response followed by
non-linear plastic deformation until the ultimate stress was reached. The uniqueness of
the present model is where the tension softening response could be achieved without
neither post-cracking instability nor convergence problem. The output from the present
numerical analysis has indicated that the peak stresses were developed at the tip of the
prism (it’s always believed that cracks would occur within the regions which govern the
highest stresses) where this finding is similar to those obtained experimentally.
4. CONCLUDING REMARKS
From all validated test cases, the tension softening material (TSM) model proposed
herein has been proved to its ability in determining the complete stress-deformation
response for specimen of various geometries. For stress-strain distribution prediction, it
is also capable to give reasonably accurate results. Subsequently, it has given a great
significance on fibre reinforced concrete research. Further research is required to have
more modelling validations to amplify the confidence level of using the TSM model for
wide range of finite element analysis. Above and beyond, the EMIS model, despite the
fact that it’s not as powerful as TSM model, it could be preserved as an alternative to
determine to softening response of either conventional of fibrous cementitious
 Guo Zhen-hai and Zhang Xiu-qin, “Investigation of Complete Stress-
Deformation Curves for Concrete in Tension”, ACI Materials Journal / July-
August 1987, 84-M29, pg. 278-285.
 Zongjin Li, Faming Li, Tse-Yung Paul Chang, and Yiu-Wing Mai, “Uniaxial
Tensile Behaviour of Concrete Reinforced with Randomly Distributed Short
Fibres”, ACI Materials Journal / September-October 1998, 95-M54, pg. 564-574.
 Z. Li, S. M. Kulkarni, and S. P. Shah, “New Test Method for Obtaining
Softening Response of Unnotched Concrete Specimen Under Uniaxial Tension”,
Experimental Mechanics, September 1993, pg. 181-188.
 Thomas W. D. Lee, “Numerical Simulation and testing of High Performance
Polymer Concrete”, MSc Thesis, C/Msc/395/02, Department of Civil
Engineering, University of Wales Swansea, 2002.
 R. Y. Xiao and T. O’Flaherty, “Finite Element Analysis of Tested Reinforced
Concrete Connections”, Journal of Computer & Structures, Vol.78, 2000, pg.
 R. Y. Xiao and C. S. Chin, “Non-linear Finite Element Modelling of Tension
Stiffening of High Performance Fibrous Concrete”, ANSYS/ICEM UK
Conference 2003, Coventry, November 2003, 15.
 R. Y. Xiao and C. S. Chin, “Fracture and Tension Softening of High
Performance Fibrous Concrete”, in Proceedings of The Seventh International
Conference on Computational Structures Technology, B. H. V. Topping and C.
A. Mota Soares, (Editors), Civil-Comp Press, Lisbon, paper 209, 2004.