Reliability of Concrete Elements Designed for Alternative Load Combinations Provided in Eurocodes

The basic European standard for design of buildings and other engineering works, EN 1990 "Basis of structural design", provides alternative design procedures, for which national choice is allowed. One of the most important questions concerns three fundamental combinations of actions for persistent and transient design situations in the Ultimate limit states. Simple examples of reinforced concrete elements show, that the alternative load combinations may lead to considerably different reliability levels. Probabilistic methods of structural reliability theory are used to identify characteristic features of each combination and to formulate recommendations. However, further calibration studies are urgently needed in order to prepare National annexes to EN 1990 on time

Credibility of Design Procedures

Theory of structural reliability enables comprehensive analysis of structural elements with respect to various limit states, and provides valuable insights into the methodology of applied standards. In addition to reliability analysis of the structural element, a new concept of the credibility of theoretical models used to calculate the design value of basic variables is introduced. The presented example of structural verification for limit states of cracking shows that the credibility of commonly applied formulas and reliability of a reinforced concrete slab have a great scatter and are in some cases inadequate

Target reliability levels in Eurocodes and ISO standards

TARGET RELIABILITY LEVELS IN EUROCODES AND ISO STANDARDS. M. HOLICKÝ, J. MARKOVÁ, M. SÝKORA (Czech Technical University, Prague, Klokner Institute)The target reliability levels recommended in national and international documents vary within a broad range, while the reference to relevant costs and failure consequences is vague only. In some documents the target reliability index is indicated for one or two reference periods (1 year, 50 years or life-time) without providing appropriate links to the design working life. This contribution attempts to clarify the relationship between the target reliability levels, costs of safety measures, failure consequences, reference periods and the design working life. For ultimate limit states of common buildings and bridges (RC2), it is recommended to consider reliability index of 3.8 for a reference period equal to the design working life (50 years for buildings, 100 for bridges).= Рассматриваются условные уровни надежности, рекомендованные в национальных и международных документах. Показано, что эти уровни варьируются в широком диапазоне, в то время как ссылки на соответствующие затраты и последовательность отказов недостаточно изучена. В некоторых документах индекс целевой надежности определяется для одного или двух базовых периодов (1 год, 50 лет или в течение всего жизненного цикла) без предоставления соответствующих ссылок на проекты. В данной работе делается попытка прояснить отношения между уровнями целевой надежности, затратами на обеспечение безопасности, последовательностью отказов, базовыми периодами и сроком эксплуатации. Для предельных уровней состояния общественных зданий и мостов (RC2) рекомендовано использовать целевой индекс 3.8 для периода, равного проектному сроку службы: 50 лет – для зданий,100 лет – для мостов

Target reliability levels in Eurocodes and ISO standards

TARGET RELIABILITY LEVELS IN EUROCODES AND ISO STANDARDS. M. HOLICKÝ, J. MARKOVÁ, M. SÝKORA (Czech Technical University, Prague, Klokner Institute)The target reliability levels recommended in national and international documents vary within a broad range, while the reference to relevant costs and failure consequences is vague only. In some documents the target reliability index is indicated for one or two reference periods (1 year, 50 years or life-time) without providing appropriate links to the design working life. This contribution attempts to clarify the relationship between the target reliability levels, costs of safety measures, failure consequences, reference periods and the design working life. For ultimate limit states of common buildings and bridges (RC2), it is recommended to consider reliability index of 3.8 for a reference period equal to the design working life (50 years for buildings, 100 for bridges).= Рассматриваются условные уровни надежности, рекомендованные в национальных и международных документах. Показано, что эти уровни варьируются в широком диапазоне, в то время как ссылки на соответствующие затраты и последовательность отказов недостаточно изучена. В некоторых документах индекс целевой надежности определяется для одного или двух базовых периодов (1 год, 50 лет или в течение всего жизненного цикла) без предоставления соответствующих ссылок на проекты. В данной работе делается попытка прояснить отношения между уровнями целевой надежности, затратами на обеспечение безопасности, последовательностью отказов, базовыми периодами и сроком эксплуатации. Для предельных уровней состояния общественных зданий и мостов (RC2) рекомендовано использовать целевой индекс 3.8 для периода, равного проектному сроку службы: 50 лет – для зданий,100 лет – для мостов

A lattice model for the line tension of a sessile drop

Within a semi--infinite thre--dimensional lattice gas model describing the coexistence of two phases on a substrate, we study, by cluster expansion techniques, the free energy (line tension) associated with the contact line between the two phases and the substrate. We show that this line tension, is given at low temperature by a convergent series whose leading term is negative, and equals 0 at zero temperature

A compact null set containing a differentiability point of every Lipschitz function

We prove that in a Euclidean space of dimension at least two, there exists a compact set of Lebesgue measure zero such that any real-valued Lipschitz function defined on the space is differentiable at some point in the set. Such a set is constructed explicitly.Comment: 28 pages; minor modifications throughout; Lemma 4.2 is proved for general Banach space rather than for Hilbert spac

SURFACE INDUCED FINITE-SIZE EFFECTS FOR FIRST ORDER PHASE TRANSITIONS

We consider classical lattice models describing first-order phase transitions, and study the finite-size scaling of the magnetization and susceptibility. In order to model the effects of an actual surface in systems like small magnetic clusters, we consider models with free boundary conditions. For a field driven transition with two coexisting phases at the infinite volume transition point $h=h_t$, we prove that the low temperature finite volume magnetization m_{\free}(L,h) per site in a cubic volume of size $L^d$ behaves like m_\free(L,h)=\frac{m_++m_-}2 + \frac{m_+-m_-}2 \tanh \bigl(\frac{m_+-m_-}2\,L^d\, (h-h_\chi(L))\bigr)+O(1/L), where $h_\chi(L)$ is the position of the maximum of the (finite volume) susceptibility and $m_\pm$ are the infinite volume magnetizations at $h=h_t+0$ and $h=h_t-0$, respectively. We show that $h_\chi(L)$ is shifted by an amount proportional to $1/L$ with respect to the infinite volume transitions point $h_t$ provided the surface free energies of the two phases at the transition point are different. This should be compared with the shift for periodic boun\- dary conditons, which for an asymmetric transition with two coexisting phases is proportional only to $1/L^{2d}$. One also consider the position $h_U(L)$ of the maximum of the so called Binder cummulant U_\free(L,h). While it is again shifted by an amount proportional to $1/L$ with respect to the infinite volume transition point $h_t$, its shift with respect to $h_\chi(L)$ is of the much smaller order $1/L^{2d}$. We give explicit formulas for the proportionality factors, and show that, in the leading $1/L^{2d}$ term, the relative shift is the same as that for periodic boundary conditions.Comment: 65 pages, amstex, 1 PostScript figur

Probabilistic Analysis of Crack Width

Probabilistic analysis of crack width of a reinforced concrete element is based on the formulas accepted in Eurocode 2 and European Model Code 90. Obtained values of reliability index b seem to be satisfactory for the reinforced concrete slab that fulfils requirements for the crack width specified in Eurocode 2. However, the reliability of the slab seems to be insufficient when the European Model Code 90 is considered; reliability index is less than recommended value 1.5 for serviceability limit states indicated in Eurocode 1. Analysis of sensitivity factors of basic variables enables to find out variables significantly affecting the total crack width

Probabilistic Analysis of Crack Width

Probabilistic analysis of crack width of a reinforced concrete element is based on the formulas accepted in Eurocode 2 and European Model Code 90. Obtained values of reliability index b seem to be satisfactory for the reinforced concrete slab that fulfils requirements for the crack width specified in Eurocode 2. However, the reliability of the slab seems to be insufficient when the European Model Code 90 is considered; reliability index is less than recommended value 1.5 for serviceability limit states indicated in Eurocode 1. Analysis of sensitivity factors of basic variables enables to find out variables significantly affecting the total crack width