Closed-Form Moment-Curvature Expressions for Homogenized Fiber-Reinforced Concrete

A closed form solution is presented for the moment-curvature response of cement-based composites with homogeneously distributed reinforcement. The derivation is based on parametric representation of uniaxial material constitutive response using piece-wise linear and quadratic segments. Effects of tensile and compressive constitutive relations on moment-curvature response were studied and it was observed that the tensile stiffness from the first cracking to the ultimate tensile strength and the ultimate tensile strain were the most important parameters. The moment-curvature relation was combined with crack localization rules to simulate the flexural load-deformation response of a beam under four-point loading conditions. Model simulations indicate that the direct use of uniaxial tension stress-strain response underpredicts the flexural results. This is attributed to the differences in the effective volume of the material subjected to critical stress. By applying a single scaling factor to material models, the model simulations can match the experimental data.

Keywords: bending moment; composite concrete flexural members; ferrocement; fiber-reinforced concrete; tension.

(ProQuest-CSA LLC: ... denotes formulae omitted.)

INTRODUCTION

Cement-based composite systems with improved tensile capacity, fracture toughness, strain hardening behavior, and reduced crack width are increasingly being used. In steel fiber-reinforced concrete (SFRC)1 fiber addition V^sub f^ = 0.5 to 2.0% only marginally increases properties such as stiffness and tensile strength through crack bridging.1 Composites with higher fiber volume fractions, such as slurry-infiltrated fiber concrete (SIFCON),2 V^sub f^ = 3 to 15% and ultra-high performance cement composites with V^sub f^ = 2 to 5% fibers exhibit significant improvements in tensile strength.3,4 Continuous fiber and mesh systems such as ferrocement,5 and more recently textile-reinforced cements (TRCs)6 have been shown to improve tension capacity and strain hardening behavior by exhibiting distributed cracking mechanisms.7 Due to the nonlinear material response, the elastically equivalent flexural strength in these composites is as much as 70 to 100% higher than the uniaxial tensile strength. The current challenge in this general area is threefold: 1) development of a unified procedure to correlate the tensile and flexural responses; 2) procedures for back calculation of fundamental tensile properties from routinely conducted flexural tests; and 3) development of simplified design guides for flexural members.

This paper presents a parameterized uniaxial constitutive response of cement-based composites. Instead of a conventional iterative strain distribution approach, the moment curvature response is analytically derived as a function of the extreme fiber tensile strain. Results are expressed in normalized form to eliminate the effect of specimen size and strength at first cracking. This momentcurvature relationship can be used with crack localization rules and moment area method to predict the flexural response of a four-point bending test.

RESEARCH SIGNIFICANCE

This approach provides computational efficiency by explicitly using closed form solutions to determine location of neutral axis, internal moment, and curvature in a flexural member as a function of the bottom tensile strain and nonlinear material parameters. An inverse analysis algorithm is presented to back-calculate material properties from the experimental load deflection data and correlate the tensile and flexural response of nonlinear fiberreinforced concrete (FRC) systems. The effect of material characteristics for improving flexural performance can be studied. Furthermore, the generated moment curvature relationship can be used as a sectional property of a beam element in nonlinear finite element method for analysis and design of structural members with FRC materials.

CLOSED FORM SOLUTIONS FOR MOMENT-CURVATURE RESPONSE

Figure 1(a) and (b) show the homogenized material model consisting of a parabolic curve for compression and a trilinear curve for tension. The compression model is expressed with three parameters: initial modulus E^sub c^0, rate of softening κ, and the ultimate compressive strain ε^sub cu^

σ^sub c^ = E^sub c0^ (ε^sub c^ - κε^sub c^^sup 2^) 0 ≤ ε^sub c^ ≤ ε^sub cu^ (1)

The trilinear tension model is expressed with five independent parameters of initial tensile modulus E^sub t0^, cracked tensile modulus E^sub t1^, first cracking strain ε^sub t0^, strain at the maximum tensile strength ε^sub t1^, and the termination strain ε^sub t2^. Using the tensile strain at the bottom fiber ε^sub tbot^ as the independent variable, the model consists of three straight lines: the linear elastic range up to the first cracking stress defined as the bend over point (BOP)8 (0

... (2)

The material model parameters defined in Eq. (1) and (2) are replaced by the two intrinsic parameters, E^sub t0^ and ε^sub t0^, and five normalized parameters, α^sub 1^, α^sub 2^, β, η, and γ, defined as