A stress‐strain model is developed for concrete subjected to uniaxial compressive loading and confined by transverse reinforcement. The concrete section may contain any general type of confining steel: either spiral or circular hoops; or rectangular hoops with or without supplementary cross ties. These cross ties can have either equal or unequal confining stresses along each of the transverse axes. A single equation is used for the stress‐strain equation. The model allows for cyclic loading and includes the effect of strain rate. The influence of various types of confinement is taken into account by defining an effective lateral confining stress, which is dependent on the configuration of the transverse and longitudinal reinforcement. An energy balance approach is used to predict the longitudinal compressive strain in the concrete corresponding to first fracture of the transverse reinforcement by equating the strain energy capacity of the transverse reinforcement to the strain energy stored in the concret...

/pdf/theoretical-stress-strain-model-for-confined-concrete-3eemamm28s.pdf

Theoretical stress strain model for confined concrete

http://mech.spbstu.ru/images/b/bd/Potyondy_Cundall_2004_A_bonded-particle_model_for_rock.pdf

A bonded-particle model for rock

A fracture theory for a heterogenous aggregate material which exhibits a gradual strain-softening due to microcracking and contains aggregate pieces that are not necessarily small compared to structural dimensions is developed. Only Mode I is considered. The fracture is modeled as a blunt smeard crack band, which is justified by the random nature of the microstructure. Simple triaxial stress-strain relations which model the strain-softening and describe the effect of gradual microcracking in the crack band are derived. It is shown that it is easier to use compliance rather than stiffness matrices and that it suffices to adjust a single diagonal term of the complicance matrix. The limiting case of this matrix for complete (continuous) cracking is shown to be identical to the inverse of the well-known stiffness matrix for a perfectly cracked material. The material fracture properties are characterized by only three parameters—fracture energy, uniaxial strength limit and width of the crack band (fracture process zone), while the strain-softening modulus is a function of these parameters. A method of determining the fracture energy from measured complete stres-strain relations is also given. Triaxial stress effects on fracture can be taken into account. The theory is verified by comparisons with numerous experimental data from the literature. Satisfactory fits of maximum load data as well as resistance curves are achieved and values of the three material parameters involved, namely the fracture energy, the strength, and the width of crack band front, are determined from test data. The optimum value of the latter width is found to be about 3 aggregate sizes, which is also justified as the minimum acceptable for a homogeneous continuum modeling. The method of implementing the theory in a finite element code is also indicated, and rules for achieving objectivity of results with regard to the analyst's choice of element size are given. Finally, a simple formula is derived to predict from the tensile strength and aggregate size the fracture energy, as well as the strain-softening modulus. A statistical analysis of the errors reveals a drastic improvement compared to the linear fracture theory as well as the strength theory. The applicability of fracture mechanics to concrete is thus solidly established.

/pdf/crack-band-theory-for-fracture-of-concrete-4hm1hrkirl.pdf

Crack band theory for fracture of concrete

A plastic-damage model for concrete

National Institute of Standards and Technology における超伝導研究及び生活

During the last two decades, researches on quasibrittlc failure have led to major advances in the understanding and modeling of the energetic and statistical size effects in the mean statistical sense •2• Computational approaches and simple design code formulas giving better mean predictions have becn developed. It now appears, however, that the existing design codes• and standard practice for concrete and other quasibrittle structures also necessitate major revisions with regard to the effect of structure size and, more generally, degrec of brittleness, which depend not only on the size of the structure but also on its structure geometry.

Reliability of fracturing concrete structures and challenges of stochastic finite element modeling

Model B3.1 for multi-decade concrete creep and shrinkage: Calibration by combined laboratory and bridge data

Normal and refractory concretes for lmfbr applications, volume 1, 2.

Arecalibration of the B3 model for creep and shrinkage has become possible with the availability of a new expanded statistical database on creep and shrinkage at Northwestern University that now also covers high performance concretes with admixtures. The recently collected evidence on excessive multi-decade deflections of numerous pre-stressed long-span box-girder bridges shows that this recalibration is indeed necessary. 
Data containing precise information of concrete materials as well as curing and testing conditions are needed in
order to update the model parameters.To accurately capture the long-termbehavior of a structure, the model must
also be calibrated with long-term data.Yet, precise laboratory tests pertain mostly to durations <6 years and are
not available on the time scale of many decades that engineers need to predict the behavior with desired lifetimes
>50 years and often>100 years.To performthe required recalibration, two databases are used simultaneously—
one for laboratory test data with accurate material information but unknown long-term behavior and another
for long-term bridge deflections with unknown or uncertain material information and structural characteristics.
A Bayesian approach to this problem is presented. First the parameters of the B3 model are recalibrated with
the enlarged laboratory test database, which is regarded as the Bayesian prior information, see Figure 1. Then
the model is updated in the Bayesian sense using the incomplete bridge deflection data to obtain the posterior
information. The resulting model predicts the long-term structural behavior more accurately and provides the
associated measures of uncertainty based on both sets of data.

Zdenek P. Bazant

Papers

Reliability of fracturing concrete structures and challenges of stochastic finite element modeling

Model B3.1 for multi-decade concrete creep and shrinkage: Calibration by combined laboratory and bridge data

Normal and refractory concretes for lmfbr applications, volume 1, 2.

Recalibration and uncertainty quantification of the B3 creep model for long term estimates using Bayesian methods

Probabilistic Prediction of Creep and Shrinkage in Concrete Structures: Combined Sampling and Spectral Approach