Resistance and Sub-Resistances of RC Beams Subjected to Multiple Failure Modes
Geometric and mechanical properties all influence the
resistance of RC structures and may, in certain combination of
property values, increase the risk of a brittle failure of the whole
This paper presents a statistical and probabilistic investigation on
the resistance of RC beams designed according to Eurocodes 2 and 8,
and subjected to multiple failure modes, under both the natural
variation of material properties and the uncertainty associated with
cross-section and transverse reinforcement geometry. A full
probabilistic model based on JCSS Probabilistic Model Code is
derived. Different beams are studied through material nonlinear
analysis via Monte Carlo simulations. The resistance model is
consistent with Eurocode 2. Both a multivariate statistical evaluation
and the data clustering analysis of outcomes are then performed.
Results show that the ultimate load behaviour of RC beams
subjected to flexural and shear failure modes seems to be mainly
influenced by the combination of the mechanical properties of both
longitudinal reinforcement and stirrups, and the tensile strength of
concrete, of which the latter appears to affect the overall response of
the system in a nonlinear way. The model uncertainty of the
resistance model used in the analysis plays undoubtedly an important
role in interpreting results.
 W.J. Krefeld, C.W. Thurston, “Contribution of Longitudinal Steel to
Shear Resistance of Reinforced Concrete Beams,” ACI Journal 63, No.
3, 1966, pp. 325-344.
 W.J. Krefeld, C.W. Thurston, “Studies of the Shear and Diagonal
Tension Strength of Simply Supported Reinforced Concrete Beams,”
ACI Journal 63, No. 4, 1966, pp. 451-476.
 R. Park, T. Paulay, “Reinforced Concrete Structures,” New York:
 B.G. Ang, M.J.N. Priestley, T. Paulay, “Seismic Shear Strength of
Circular Reinforced Concrete Columns,” ACI Struct. J. 86, No. 1, 1989,
 M.J.N. Priestley, V. Ravindra, X. Yan, “Seismic Shear Strength of
Reinforced Concrete Columns,” ASCE Journal of Structural
Engineering 120, No. 8, 1994, pp. 2310-2329.
 F. Sangiorgio, J. Silfwerbrand, G. Mancini, “Statistical Investigation on
the Ultimate Load Behaviour of RC Beams Subjected to Multiple
Failure Modes,” Nordic Concrete Research 50, 2014, pp. 457-460.
 F. Sangiorgio, J. Silfwerbrand, G. Mancini, “Probabilistic Investigation
of the Ultimate Load Behaviour of RC Structures Designed According
to EN 1992-1-1 and Subjected to Multiple Failure Modes,” Engineering
Structures, submitted for publication.
 Eurocode 2 (EN 1992-1-1), “Design of concrete structures. Part 1-1:
general rules and rules for buildings,” 2004.
 Eurocode 8 (EN 1998-1), “Design of structures for earthquake
 G. Bertagnoli, L. Giordano, G. Mancini, “Safety Format for the
Nonlinear Analysis of Concrete Structures. Studi e ricerche,” Politecnico
di Milano, Scuola di specializzazione in costruzioni in cemento armato
25, 2004, pp. 31-56.
 V. Červenka, “Global Safety Format for Nonlinear Calculation of
Reinforced Concrete,” Beton und Stahlbetonbau 103, 2008, pp. 37-42.
 H. Schulne, M. Plos, K. Gylltoft, “Safety Formats for Nonlinear
Analysis Tested on Concrete Beams Subjected to Shear Forces and
Bending Moments,” Eng. Struct. 33, No. 8, 2011, pp. 2350-2356.
 M. Sykora, M. Holicky, “Safety Format for Non-Linear Analysis in the
Model Code - Verification of Reliability Level,” Fib Symposium on
Concrete engineering for excellence and efficiency, Prague, Czech
Concrete Society, 2011, pp. 943-946.
 Joint Committee on Structural Safety, “Probabilistic Model Code 2000,”
 T.Vrouwenvelder, “Reliability Based Code Calibration. The Use of the
JCSS Probabilistic Model Code,” Joint Committee of Structural Safety
Workshop on Code Calibration, 2002, March 21/22, Zurich.
 J.K. Kim, T.G. Lee, “Nonlinear Analysis of Reinforced Concrete Beams
with Softening,” Computer and structures 44, No. 3, 1992, pp. 567-573.
 H.A.S. Rasheed, K.S. Dinno, “An Efficient Nonlinear Analysis of RC
Sections,” Computers and Structures 53, No. 3, 1994, pp. 613-623.
 H.A.S. Rasheed, K.S. Dinno, “An Improved Nonlinear Analysis of
Reinforced Concrete Frames,” Computers and Structures 53, No. 3,
1994, pp. 625-636.
 H.G. Kwak, S.P. Kim, “Nonlinear analysis of RC beams based on
moment-curvature relation,” Computer and structures 80, 2002, pp. 615-
 H.R. Valipour, S.J. Foster, “A Total Secant Flexibility-Based
Formulation for Frame Elements with Physical and Geometrical
Nonlinearities,” Finite Elements in Analysis and Design 46, 2010, pp.
 M. Sargin, “Stress-Strain Relationships for Concrete and the Analysis of
Structural Concrete Sections,” University of Waterloo, Solid Mechanics
Division, SM Study 4, 1971, pp. 23-46.
 B. Massicotte, A.E. Elwi, J.G. MacGregor, “Tension-Stiffening Model
for Planar Reinforced Concrete Members,” ASCE J. Struct. Eng. 116,
vol. 11, 1990, pp. 3039-3058.
 E.L. Droguett, A. Mosleh, “Bayesian Methodology for Model
Uncertainty Using Model Performance Data,” Risk Analysis 28, No. 5,
2008, pp. 1457-1476.
 J.B. MacQueen, “Some Methods for Classification and Analysis of
Multivariate Observations,” Proceedings of the 5th Berkeley Symposium
on Mathematical Statistics and Probability, 1967, pp. 281-297.
 M.R. Anderberg, “Cluster Analysis for Applications,” Academic Press,
 A.K. Jain, R.C. Dubes, “Algorithms for Clustering Data,” Prentice Hall,
 L. Kaufman, P.J. Rousseeuw, “Finding Groups in Data - An Introduction
to Cluster Analysis,” Wiley, 1990.
 P.N. Tan, M. Steinbach, V. Kumar, “Introduction to Data Mining,”
 P.J. Rousseeuw, “Silhouttes: A Graphical Aid to the Interpretation and
Validation of a Cluster Analysis,” J Comput Applied Math 20, 1987, pp.
 G.W. Snedecor, W.G. Cochran, “Statistical Methods,” 6th ed. Iowa State
Univ. Press, Ames, 1967.
 E. Garcia, “A Tutorial on Correlation Coefficients,”
http://www.miislita.com, 2010 - simmons.edu.
 G.E. Dallal, “Correlation Coefficients,”