Advances in Materials Science and Engineering

Volume 2016 (2016), Article ID 9605450, 8 pages

http://dx.doi.org/10.1155/2016/9605450

## Prediction Intervals for the Failure Time of Prestressed Concrete Beams

TU Dortmund University, 44227 Dortmund, Germany

Received 24 March 2016; Accepted 20 July 2016

Academic Editor: Konstantinos I. Tserpes

Copyright © 2016 Sebastian Szugat et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aim is the prediction of the failure time of prestressed concrete beams under low cyclic load. Since the experiments last long for low load, accelerated failure tests with higher load are conducted. However, the accelerated tests are expensive so that only few tests are available. To obtain a more precise failure time prediction, the additional information of time points of breakage of tension wires is used. These breakage time points are modeled by a nonlinear birth process. This allows not only point prediction of a critical number of broken tension wires but also prediction intervals which express the uncertainty of the prediction.

#### 1. Introduction

Actually, the assessment of existing prestressed concrete bridges by means of recalculation in conjunction with rehabilitation and strengthening is gaining more and more importance compared to the construction of new bridges. The current design codes had been developed over decades always adapting new design approaches that are current at that time. Even for this reason the recalculation of older existing structures often leads to deficiencies concerning load-bearing capacity, durability, and resistance against fatigue. The ongoing increase of traffic concerning heavy trucks underlines the importance of assessment and maintenance of the transport networks and particularly the bridge stock, the latter with regard to structural safety.

Beside corrosion effects, the major influence for time dependent losses of load-bearing capacity is the phenomenon of fatigue failure. Fatigue is caused by frequent cyclic loads due to the crossing of heavy trucks on the bridge deck. Beside steel bridges, prestressed concrete bridges are affected as well; see, for example, [1, 2]. For the design of new bridges against fatigue or the assessment of existing bridges by means of recalculation,* S*-*N* curves are needed. The latter describes the fatigue resistance of the materials. With regard to the prestressed concrete bridges, this refers especially to the embedded reinforcing and prestressing steel in cracked sections. For the design and assessment of bridges,* S*-*N* curves are needed in a range up to load cycles. To obtain values for the whole range and for a better understanding of the fatigue behavior of prestressed concrete bridges, one has to carry out long running tests which are extremely expensive. Hence, there is a great need to optimize these tests procedures.

From historical view, the first documented fatigue tests on prestressed concrete beams will be found in [3]. Larger test series carried out at the University of Texas are described by [4]. Further studies can be found in [5, 6]. A comprehensive survey regarding fatigue tests on prestressing steel in air and embedded in concrete is given in [7]. The latter leads to fatigue strength which is significantly less than studied before.

During the course of the Collaborative Research Center SFB 823 Statistical Modeling of Nonlinear Dynamic Processes, large-scale test series with stress ranges down to 50 MPa and failure times in a range up to load cycles are carried out at TU Dortmund University. The aim of the ongoing experimental studies described subsequently is to investigate fatigue behavior and to provide characteristic* S*-*N* curves for prestressing steel in curved steel ducts embedded in concrete of posttensioned members.* S*-*N* curves belong to the basics which are needed to verify prestressed concrete bridges against fatigue. However, tests under cyclic loading of posttensioned concrete beams may be very time-consuming and expensive. Especially at very low stress ranges with a very high number of cycles, which are of particular interest concerning prestressed concrete bridges, even an optimized test with a realized load frequency of 10 Hz lasts several months. For posttensioned steel, an endurance range in the* S*-*N* curves has not been established by tests up to now. Therefore, the* S*-*N* curve in the range up to cycles can only hypothetically be guessed, until appropriate test results will be available.

For low loads down to 60 MPa, tests in the research project SFB 823 last nearly 100 days so that most experiments are done under higher loads up to 200 MPa. Hence so-called accelerated failure tests (AFT) were conducted. If there are enough AFT experiments, the lifetime at a small load can be estimated from the* S*-*N* curves; see, for example, [8–10]. However, here also these AFT experiments last long and are expensive so that the results of only few experiments are available, in our project, for example, results of ten experiments. Such small numbers of experiments are too small to estimate the lifetime at low load with enough precision. Nevertheless, the main interest lies in the lifetime at low stress at 50 MPa or even lower.

Hence, we propose here two methods which use additional information beside the failure times of former tests to predict the failure time at low stress. The additional information is given by a degradation measure. Usually the sizes of cracks are used as degradation measures; see, for example, [11–13]. But here we have the advantage that the time points of the breaking of the tension wires in the prestressed concrete beams are available since acoustic signals obtained by a microphone indicate clearly the breakage of a wire. We model the time points of the breaking of the tension wires with a point process where the waiting times for the next breaking of a tension wire follow an exponential distribution depending only on the number of wires which are broken before. Such point processes are also called birth processes (see, e.g., [14]).

Point processes as Poisson processes and renewal processes are often considered in reliability and lifetime analysis; see, for example, [15, 16]. Reference [15] treats also a linear birth process for fatigue accumulation in Chapter 18 and uses birth processes with time-varying intensity for crack growth in Chapter 26. However, our birth process is nonlinear in the number of broken tension wires. The nonlinearity is due to the redistribution of the load on the tension wires. There are several approaches for load sharing systems as those of [17, 18] or [19]. But they assume several systems exposed to the same stress so that accelerated failure tests cannot be treated.

Linking the nonlinear birth process of each experiment with its underling stress, we provide two types of prediction intervals for the time of a critical number of broken tension wires. The critical number of broken tension wires has a direct relation to the failure time of the concrete beam so that its lifetime can be derived from the time of the critical number of broken tension wires. We use the times between successive breaks of the accelerated experiments and some optional first breaking times of the concrete beam for which we want to obtain the prediction interval.

Although prediction intervals provide not only a prediction but also its precision, they are often not derived. Most prediction intervals are only derived for the simple situation that all experiments are conducted under the same conditions; see, for example, [8, 9, 20–25]. Only few prediction intervals for accelerated experiments are available as those of [8] for normal distributed lifetimes and [26] for exponential distributed lifetimes while the prediction intervals of [26] are based on simulations. Our prediction intervals are simulation-free and thus faster to calculate.

In Section 2 the description of the experiments with the concrete beams is given. Section 3 provides the statistical model with the birth process and its link to the stress while Section 4 treats the two proposed prediction intervals. The results for our experiments with the concrete beams and some simulations are given in Section 5. At last, Section 6 provides a conclusion.

#### 2. Test Setup and Procedure

The tests on prestressed concrete girders (hereinafter SB01–SB05) within the Collaborative Research Center SFB 823 have been carried out at TU Dortmund University. The experimental setup is based on the setup of already conducted experiments, also carried out at TU Dortmund University (see [27]).

The series described in [27] consisted of five concrete girders (TR01–TR05) with tendons for posttensioning. They had been tested with different stress ranges from 455 MPa to 98 MPa for the prestressing steel in curved steel ducts. The prestressing steel of the tendons used for these test girders has been taken from an existing bridge which was built in 1957 and demolished in 2007. Each of the taken 3/8′′ strands consists of seven single wires. Each strand had been consisted of a steel grade St1570/1770 with a diameter of 9.3 mm and a cross-sectional area of 52 mm^{2}. The prestressing steel had been strained at a length of 2 m for the curved tendon with a minimum radius of m in a region of the test girder with pure bending without shear. Hence, the influence of fretting corrosion between the tensioned strand and steel duct is included.

The experimental setup consists of steel frames, the concrete girder, and a hydraulic press in a four-column testing machine, which can apply a cyclic load at maximum ±2500 kN (see Figures 1 and 2). The overall dimensions of the concrete girder are 4.00 m × 1.00 m × 0.30 m. A recess in midspan of the girder in conjunction with a steel contact element ensures the unambiguous definition of the center of the compression zone in the upper cross-sectional part and from this the exact inner lever arm and tension force in the tendon.