Qiang Yu

Excessive Long-Time Deflections of Prestressed Box Girders. I: Record-Span Bridge in Palau and Other Paradigms

The segmental prestressed concrete box girder of Koror-Babeldaob (KB) Bridge in Palau, which had a record span of 241 m (791 ft), presents a striking paradigm of serviceability loss because of excessive multidecade deflections. The data required for analysis have recently been released and are here exploited to show how the analysis and design could be improved. Erected segmentally in 1977, this girder developed a midspan deflection of 1.61 m (5.3 ft) compared with the design camber after 18 years, and it collapsed in 1996 as a consequence of remedial prestressing, after a 3-month delay. Compared with three-dimensional analysis, the traditional beam-type analysis of box girder deflections is found to have errors up to 20%, although greater errors are likely for bridges with higher box-width-to-span ratios than the KB Bridge. However, even three-dimensional finite-element analysis with step-by-step time integration cannot explain the observed deflections when the current American Concrete Institute, Japan Society of Civil Engineers, Comite Euro-International du Beton (or Comite Euro- International du Beton—Federation internationale de la precontrainte), and Gardner and Lockman prediction models for creep and shrinkage are used. These models give 18-year deflection estimates that are 50-77% lower than measured and yield unrealistic shapes of the deflection history. They also predict the 18-year prestress loss to be 46-56% lower than the measured mean prestress loss, which was 50%. Model B3, which is the only theoretically based model, underestimates the 18-year deflection by 42% and gives a prestress loss of 40% when the default parameter values are used. However, in Model B3, several input parameters are adjustable and if they are adjusted according to the long-time laboratory tests of Brooks, a close fit of all the measurements is obtained. For early deflections and their extrapolation, it is important that Model B3 can capture realistically the differences in the rates of shrinkage and drying creep caused by the differences in the thickness of the walls of the cross section. The differences in temperature and possible cracking of the top slab also need to be taken into account. Other paradigms on which data have recently been released are four bridges in Japan and one in the Czech Republic. Their excessive deflections can also be explained. The detailed method of analysis and the lessons learned are presented in Part II. DOI: 10.1061/(ASCE)ST.1943-541X .0000487.

Choice of standard fracture test for concrete and its statistical evaluation

The main characteristics of the cohesive (or fictitious) crack model, which is now generally accepted as the best simple fracture model for concrete, are (aside from tensile strength) the fracture energies GF and Gf corresponding to the areas under the complete softening stress-separation curve and under the initial tangent of this curve. Although these are two independent fracture characteristics which both should be measured, the basic (level I) standard test is supposed to measure only one. First, it is argued that the level I test should measure Gf, for statistical reasons and because of relevance to prediction of maximum loads of structures. Second, various methods for measuring Gf (or the corresponding fracture toughness), including the size effect method, the Jenq-Shah method (TPFM), and the Guinea et al. method, are discussed. The last is clearly the most robust and optimal because: (1) it is based on the exact solution of the bilinear cohesive crack model and (2) necessitates nothing more than measurement of the maximum loads of notched specimens of one size, supplemented by tensile strength measurements. Since the identification of material fracture parameters from test data involves two random variables, f′t (tensile strength) and Gf, statistical regression should be applied. But regression is not feasible in the original Guinea et al.s method. The present study proposes an improved version of Guinea et al.s method which reduces the statistical problem to linear regression thanks to exploiting the systematic trend of size effect. This is made possible by noting that, according to the cohesive (or fictitious) crack model, the zero-size limit σN0 of nominal strength σN of a notched specimen is independent of Ff and thus can be easily calculated from the measured f′t. Then, the values of σN0 obtained from the measured f′t values, together with the measured σN-values of notched specimens, are used in statistical regression based on the exact size effect curve calculated in advance from the cohesive crack model for the chosen specimen geometry. This has several advantages: (1) the linear regression is the most robust statistical approach; (2) the difficult question of statistical correlation between measured f′t and the nominal strength of notched specimens is bypassed, by virtue of knowing the size effect trend; (3) the resulting coefficient of variation of mean Gf is very different and much more realistic than in the original version; (4) the coefficient of variation of the deviations of individual data from the regression line is very different from the coefficient of variation of individual notched test data and represents a much more realistic measure of scatter; and (5) possible accuracy improvements through the testing of notched specimens with different notch lengths and the same size, or notched specimens of different sizes, are in the regression setting straightforward. For engineering purposes where high accuracy is not needed, the simplest approach is the previously proposed zero-brittleness method, which can be regarded as a simplification of Guinea et al. method. Finally, the errors of TPFM due to random variability of unloading-reloading properties from one concrete to another are quantitatively estimated.

Excessive Long-Time Deflections of Prestressed Box Girders. II: Numerical Analysis and Lessons Learned

As a sequel to Part I, which clarified the causes of the unexpectedly large deflections of the Koror-Babeldaob Bridge in the Pacific island nation of Palau, Part II presents the numerical procedure and reviews the lessons learned. The box girder represents a thick shell that is discretized by eight-node, three-dimensional (3D) finite elements. Except for corrections due to cracking, concrete creep is assumed to follow aging linear viscoelasticity and is modeled by a rate-type law based on the Kelvin chain, the properties of which are adjusted for humidity conditions and temperature. In each time step and at each integration point, Widder’s formula is used to convert the aging compliance function to a continuous retardation spectrum for the current age of concrete, and discretization of the spectrum yields the current elastic moduli of the Kelvin units. The shrinkage strains depend on the environmental humidity and the thickness of each plate in the cross section. The computations proceed according to Bazant’s exponential algorithm, which is unconditionally stable and reduces the problem to a sequence of elasticity problems with an orthotropic effective stiffness of material and nonisotropic inelastic strains, different for each integration point in each time step. These problems are solved by commercial software ABAQUS. The segmental construction sequence is also modeled. The computer results reported in Part I explain the excessive deflections and compare the performance of various material models for creep and shrinkage. Part II formulates the lessons learned and makes recommendations for implementation.

Size-Effect Testing of Cohesive Fracture Parameters and Nonuniqueness of Work-of-Fracture Method

The cohesive crack model has been widely accepted as the best compromise for the analysis of fracture of concrete and other quasibrittle materials. The softening stress-separation law of this model is now believed to be best described as a bilinear curve characterized by four parameters: the initial and total fracture energies Gf and GF, the tensile strength ft′, and the knee-point ordinate σ1. The classical work-of-fracture test of a notched beam of one size can deliver a clear result only for GF. Here it is shown computationally that the same complete load-deflection curve can be closely approximated with stress-separation curves in which the ft′ values differ by 77% and Gf values by 68%. It follows that the work-of-fracture test alone cannot provide an unambiguous basis for quasibrittle fracture analysis. It is found, however, that if this test is supplemented by size-effect testing, all four cohesive crack model parameters can be precisely identified and the fracture analysis of structures becomes una...

Can Stirrups Suppress Size Effect on Shear Strength of RC Beams

This paper demonstrates the size effect on the shear strength of reinforced concrete (RC) beams with stirrups and does so in two separate and independent ways: (1) by fracture mechanics, based on finite-element analysis calibrated by a large beam test; and (2) by purely statistical analysis in which a newly assembled database of 234 tests is filtered to eliminate spurious size effects caused by nonuniformity of secondary influencing parameters. Both ways show that stirrups, whether minimum or heavier, cannot suppress the size effect completely, although they can mitigate it significantly for beam depth d 1 m (39.4 in.), the size effect cannot be neglected. DOI: 10.1061/(ASCE)ST.1943-541X.0000295.

Problems with Hu-Duan Boundary Effect Model and Its Comparison to Size-Shape Effect Law for Quasi-Brittle Fracture

Recent disagreements on the mathematical modeling of fracture and size effect in concrete and other quasi-brittle materials are obstacles to improvements in design practice, and especially in design codes for concrete structures. In an attempt to overcome this impediment to progress, this paper compares the Hu-Duan boundary effect model BEM expounded since 2000 to the size-shape effect law SEL proposed at Northwestern University in 1984 and extended to the geometry or shape effects in 1990. It is found that within a rather limited part of the range of sizes and shapes, the fracture energy values identified by BEM and SEL from the test data on maximum loads are nearly the same. But in other parts of the range the BEM is either inferior or inapplicable. The material tensile strength values identified by BEM have a much larger error than those obtained from the SEL after calibration by the cohesive crack model. From the theoretical viewpoint, several hypotheses of BEM are shown to be unrealistic. DOI: 10.1061/ASCEEM.1943-7889.89 CE Database subject headings: Cracking; Concrete; Structural failures; Data analysis; Size effect. Author keywords: Fracture scaling; Fracture energy; Concrete; Asymptotics of fracture; Cohesive cracks; Failure of structures; Evalu- ation of experimental data.

Size Effect on Strength of Laminate-Foam Sandwich Plates

Experiments on size effect on the failure loads of sandwich beams with PVC foam core and skins made of fiber-polymer composite are reported. Two test series use beams with notches at the ends cut in the foam near the top or bottom interface, and the third series uses beams without notches. The results demonstrate that there is a significant nonstatistical (energetic) size effect on the nominal strength of the beams, whether notched or unnotched. The observed size effect shows that the failure loads can be realistically predicted on the basis of neither the material strength concept nor linear elastic fracture mechanics (LEFM). It follows that nonlinear cohesive (quasi-brittle) fracture mechanics, or its approximation by equivalent LEFM, must be used to predict failure realistically. Based on analogy with the previous asymptotic analysis of energetic size effect in other quasibrittle materials, approximate formulas for the nominal strength of notched or unnotched sandwich beams are derived using the approximation by equivalent LEFM. Different formulas apply to beams with notches simulating pre-existing stress-free (fatigued) cracks, and to unnotched beams failing at crack initiation. Knowledge of these formulas makes it possible to identify from size effect experiments both the fracture energy and the effective size of the fracture process zone.

Minimizing Statistical Bias to Identify Size Effect from Beam Shear Database

A broad database can be used to statistically calibrate a design formula capturing the size effect on shear strength of reinforced concrete beams without stirrups. This database, however, can have a bias of two types: 1) most data points are crowded in the small size range; and 2) the means of the subsidiary influencing parameters, such as the steel ratio and shear-span ratio are very different within different intervals of beam size or beam depth. The database must be properly filtered to minimize the second type of bias. To this end, the size range is first subdivided into intervals of constant size ratio. Then, in each size interval, a computer program progressively restricts the range of influencing parameters both from above and from below, until the mean of the influencing parameter values remaining in that interval attains about the same value in all the size intervals. The centroids of the filtered shear strength data within the individual size interval are found to exhibit a rather systematic trend. Giving equal weight to each interval centroid overcomes the first bias. The centroids can be closely matched by bivariate least-square regression using Bazant’s size effect law. This purely statistical inference of minimized bias also supports the previous fracture-mechanics-based conclusion that, for large sizes, the bi-logarithmic size effect plot must terminate with the asymptotic slope of –1/2. Similar filtering of the database gives further evidence for the previous empirical observation that the shear strength of beams is approximately proportional to the 3/8-power of the longitudinal reinforcement ratio. The proposed statistical procedure can be used to improve the calibration of formulas in concrete design codes.

Scaling of Strength of Metal-Composite Joints-Part II: Interface Fracture Analysis

The effect of the size of hybrid metal-composite joint on its nominal strength, experimentally demonstrated in the preceding paper (part I), is modeled mathematically. Fracture initiation from a reentrant corner at the interface of a metallic bar and a fiber composite laminate sheet is analyzed. The fracture process zone (or cohesive zone) at the corner is approximated as an equivalent sharp crack according to the linear elastic fracture mechanics (LEFM). The asymptotic singular stress and displacement fields surrounding the corner tip and the tip of an interface crack emanating from the corner tip are calculated by means of complex potentials. The singularity exponents of both fields are generally complex. Since the real part of the stress singularity exponent for the corner tip is not -1/2, as required for finiteness of the energy flux into the tip, the interface crack propagation criterion is based on the singular field of the interface crack considered to be embedded in a more remote singular near-tip field of the corner from which, in turn, the boundaries are remote. The large-size asymptotic size effect on the nominal strength of the hybrid joint is derived from the LEFM considering the interface crack length to be much smaller than the structure size. The deviation from LEFM due to finiteness of the interface crack length, along with the small-size asymptotic condition of quasiplastic strength, allows an approximate general size effect law for hybrid joints to be derived via asymptotic matching. This law fits closely the experimental results reported in the preceding paper. Numerical validation according to the cohesive crack model is relegated to a forthcoming paper.

Size effect on compressive strength of sandwich panels with fracture of woven laminate facesheet

Prismatic sandwich specimens of various sizes, geometrically scaled in the ratio 1:2:4:8, are subjected to eccentric axial compression and tested to failure. The sandwich core consists of a closed-cell polyvinyl chloride foam, and the facesheets are woven glass-epoxy laminates, scaled by increasing the number of plies. The test results reveal a size effect on the mean nominal strength, which is strong enough to require consideration in design. The size effect observed is fitted with the size effect law of the energetic (deterministic) size effect theory. However, because of inevitable scatter and limited testing range, the precise form of the energetic size effect law to describe the test results is not unambiguous. The Weibull-type statistical size effect on the mean strength is ruled out because the specimens had small notches which caused the failure to occur in only one place in the specimen, and also because the observed failure mode was kink band propagation, previously shown to cause energetic size effect. Various fallacies in previous applications of Weibull theory to composites are also pointed out.