Lifetime-Oriented Structural Design Concepts- P23 potx

618 4 Methodological Implementation 5,68 5,68 5,68 5,68 5,68 5,685,685,685,685,68 western arch eastern arch east 13,30 12 3 4 5 78910 6 5,685,685,685,685,685,68 5,68 5,68 5,68 5,685,68 11 64,20 8,57 main- and cross girder west points side view: top view: Fig. 4.157. Location of drilling cores drilled out from various sections of the structure for further laboratory test- ings (Figure 4.157). The total number of drilling cores was limited by logistical reasons within the deconstruction process and in order to ensure the structural stability also during and after the core-drilling. In order to get information on the actual concrete properties, labora- tory tests were performed according to the same methods described in Sec- tion 3.2.1.2. 4.6.5.1.1 Experimental Investigation on Mechanical Concrete Properties According to the available construction drawings the concrete used for all structural members tested now was conform to the formerly strength grade B 450 [3], which correlates at time of construction more or less with a C 30/37 according to the current standards. Due to lack of more detailed information it was assumed that the concrete composition in all parts of the bridge was the same. The maximum size of aggregates could be determined on the drilling cores to 32 mm. 4.6.5.1.1.1 Non-Destructive Tests In order to determine the concrete’s stiffness, resp. the dynamic elastic mod- ulus E dyn non-destructive ultrasonic tests (US-measurements) were performed. Comparing these results, the mean values of E dyn are on a similar level of about 4.6 Application of Lifetime-Oriented Analysis and Design 619 Table 4.17. Dynamic elastic moduli E dyn (mean) and their standard deviations (SD) of the concrete after a service life of 50 years for the different members of the bridge structural member number of specimens E dyn SD [−] [N/mm 2 ] [N/mm 2 ] main girder 12 49,700 4,200 cross girder 8 47,600 2,700 arches 17 50,600 3,300 carriageway slab 5 50,300 2,300 cantilever section 18 52,900 5,300 50, 000 N/mm 2 within all investigated parts of the structure (Table 4.17). The standard deviations (SD) in E dyn vary from 2, 700 N/mm 2 to 5, 300 N/mm 2 . This scatter of in-situ concretes exceeds significantly the experiences on E dyn of laboratory concretes tested at an age of 28 days without former loadings. In the latter typical standard deviations of about 2, 000 N/mm 2 were observed. Thus, it can be assumed, that a main part of the determined standard devia- tions in E dyn from the structural members is affected – besides the deviations due to testing and materials inhomogeneities – by deviations due to concrete’s post-hydratation, environmental impacts and mechanical effects raised by the cyclic loading itself. 4.6.5.1.1.2 Destructive Tests In addition, static compression tests were performed on a couple of speci- mens from the main structural members, mainly to determine the stress-strain relation. The results reveal that the mean values of the Young’s Modulus E stat , the ultimate strain  u as well as the compressive strength f c are approx- imately on the same level for the different structural members (Table 4.18). (This would confirm the assumption, that in all investigated members nearly the same concrete had been used). On the other hand the scatters, e.g. a three- to sixfold standard deviation in f c of the in-situ concrete could be observed in Table 4.18. Relevant mechanical concrete properties E stat , u and f c (mean values) as well as their standard deviations (SD) after a service life of 50 years for the different structural members of the bridge structural member number of specimens E stat SD ε u SD f c SD [-] [N/mm 2 ] [N/mm 2 ] [% 0 ] [% 0 ] [N/mm 2 ] [N/mm 2 ] main girder 7 38,800 6,700 2.17 0.33 72.4 18.5 cross girder 6 39,400 6,300 2.02 0.27 70.6 13.7 arches 5 41,000 8,100 2.28 0.13 86.2 15.4 620 4 Methodological Implementation comparison to results of common static compression 28 days-tests on common separately fabricated concrete specimens. Although f c scattered within a wide range, it became also obvious, that the compressive strength f c has increased due to the post-hardening during 50 years. Assuming, that the concrete has fulfilled the requirements to a B 450, resp. C 30/37, at construction time, a post-hardening of 80 100 % can be stated. On the other hand, the ultimate strain  u remained more or less on the value of a C 30/37 at an age of 28 days (Figure 4.158). At first, these results seem to be in contrast to the typical stress-strain relations of laboratory concrete at an age of 28 days (Figure 4.158). In the latter, the ultimate strain  u also increases with increasing compressive strength f c . Although the post- hardening effect on the one hand leads to a significant increase in strength in the bridge’s in-situ concrete, on the other hand the cyclic loadings reduced – analogue to investigations on laboratory test concretes – the ultimate strain  u of the in-situ concrete significantly (Section 3.2.1.2). (The strength f c is impaired by cyclic loadings only barely). Additionally also the shape of the stress-strain relation diverges significantly between laboratory concretes (at 28 days) and the in-situ concrete of the 50 year old bridge (Figure 4.158). The typical concave shape towards the strain axis was not observable at the in- situ concrete. Thus, it could be proved also by these tests, that cyclic loadings 0 Strain [‰]e Cylinder compressive strength [N/mm²] Ultimate strains e u 80 60 40 20 0-4-3-2-1 C 80/95 C 50/60 C 35/45 C 20/25 Bridge concrete Fig. 4.158. Comparison of stress-strain curves between bridge concrete (dashed line) and laboratory concretes with different strengths at the age of 28 days (solid lines) [193] 4.6 Application of Lifetime-Oriented Analysis and Design 621 300 µm Fig. 4.159. LM-micrograph of in-situ concrete change the stress-strain curve from a concave form towards the strain axis to a straight line, as it was also observed in cyclic tests on laboratory concretes (Section 3.2.1.2). 4.6.5.1.1.3 Microscopic Analysis Furthermore, microscopic analyses partly proved the existence of micro- cracks within the concrete microstructure caused by cyclic loading (Fig- ure 4.159). The path of these microcracks are similar to those of laboratory concretes (Section 3.2.1.2), which were subjected to about 600,000 load cy- cles at a stress regime of S max /S min =0.675/0.10. In both cases microcrack- ing starts in the transition zone between cement paste and coarse aggregate grains. By the continuous cyclic loadings a prolongation of these microcracks through the cement paste was raised. However, it must be emphasised that the existence of microcracks significantly was depending on the extraction point within the respective drilling core, i.e. samples without any microcracks were observed as well. 4.6.5.1.1.4 Cyclic Tests The further changes in the concrete properties were also investigated by applying further cyclic loads on specimens taken from the bridge. Initially, the upper and lower stress levels S max (= σ max · f c )andS min (= σ min · f c ) for the cyclic test regime were determined on the basis of actual determined concrete strength. For all cyclic tests the upper and lower stress levels were ad- justed to 0.675 f c and 0.10 f c , respectively. Altogether, sixteen specimens were subjected to cyclic loading using the test setup described in Section 3.2.1.2. 622 4 Methodological Implementation arch main and cross girder S max /S min = 0.675/0.10 S max /S min = 0.675/0.10 QT3-1 HT11 O3 Fatigue strain [‰] Total strain [‰] 0.0 -0.1 -0.2 -0.3 0 10.0 20.0 30.0 0 200,000 400,000 600,000 0.0 -2.0 -2.5 -1.5 -1.0 -0.5 Number of cycles N [in million] Number of cyclesN[-] O1 O2 O3 O5 W3 QT3-1 HT3 HT8 HT11 Fig. 4.160. Total longitudinal (left) and fatigue strain (right) at S max Seven of these in-situ specimens failed already within a comparatively low number of load cycles between 652 and 307,000. Other ones, however, resisted millions of load cycles without any occurrence of failure. In comparison to these in-situ specimens, none of the laboratory concrete specimens of grade C 30/37 failed before applying about 800,000 load cycles at the same test regime. Hence, it can be revealed that the bridge’s in-situ concrete has a more sensitive behaviour to further cyclic loading in comparison to laboratory con- crete of C 30/37 at an age of 28 days. The development in total longitudinal strain at S max of all nine (unfailed) specimens during the cyclic tests is illustrated (Figure 4.160, left). On three of these specimens the cyclic tests were carried out as long-term tests for 12.0 millions to 27.8 millions load cycles. (The others were tested only up to 600,000 load cycles). At first, it became evident that the initial strains scatter within a wide range. In order to reveal these differences it is more suitable to take into account only the increase in strains during the cyclic loadings (”fatigue strains”, see also Section 3.2.1.2). For this purpose the fatigue strains of the nine in-situ specimens up to 600,000 cycles are separately illustrated in Figure 4.160, right. The development of the fatigue strains within the first 600,000 cycles are quite differently shaped for each specimen, which indicates also a wide scattering in the maximal bearable number of cycles up to failure N f . Before the cyclic tests were started the mean value of the compressive strength f c had been determined. This averaged strength was taken as the reference value to adjust the stress levels S max and S min . Since the compres- sive strength f c of the bridge’s concrete varied significant (Table 4.18), the parameters S max and S min of the cyclic stress regime could not precisely be adjusted to 0.675 and 0.10, respectively. 4.6 Application of Lifetime-Oriented Analysis and Design 623 60 65 70 75 80 Residual Young’s modulus [%] 60 65 70 75 80 85 90 95 100 Residual dynamic elastic modulus [%] O3 QT3-1 HT11 O3 QT3-1 HT11 S */S * = 0.55 - 0.64/0.10 max min S */S * = 0.55 - 0.64/0.10 max min 85 90 95 100 -0.8 -0.6 -0.4 -0.2 0 Fatigue strain e fat,max [‰] -0.8 -0.6 -0.4 -0.2 0 Fatigue strain e fat,max [‰] t=50a t=50a Fig. 4.161. Correlation between fatigue strain and the residual stiffness for S max /S min =0.675/0.10 Since the compressive strength f c remained nearly constant during the cyclic tests (Section 3.2.1.2), it was possible to calculate almost the actual specific values for both stress levels S ∗ max (S ∗ max =0.675 f c /f ∗ c )andS ∗ min (S ∗ min =0.10 f c /f ∗ c ) by determining the specific compressive strength f ∗ c after the cyclic test. The specific strengths f ∗ c of the three long-term cyclic loaded specimens amount to 86.6 N/mm 2 for QT 3-1, 88.9 N/mm 2 for HT 11 and 90.9 N/mm 2 for O 3. Thus, the real values of the upper stress level S ∗ max amount to 0.55 (QT 3-1, HT 11) as well as 0.64 (O 3). Cyclic loadings lead to degradation processes combined with changes in the mechanical concrete properties. An adequate description of the changes in the Young’s modulus, referred to the fatigue strain, is given in Section 3.2.1.2. Following this approach here, the Young’s modulus as well as the dynamic elastic modulus versus the fatigue strain are illustrated in Figure 4.161 for the three long-term cyclic tests. Thereby, it has to be considered, that the concrete specimens in this case are already about 50 years old and had imprinted already a certain amount of fatigue strain within this period. However, this amount of the accumulated fatigue strains remains unknown in value and is surely different for each specimen. Nevertheless, at first roughly an almost linear relationship between the residual Young’s modulus/dynamic elastic modulus resp. and the fatigue strain has been observed. Although, the actual stress levels of the applied loads on each specimen are not equal, the changes in the stiffness can be ap- proximated adequately by a common trendline. This underlines again that the linear relationship between the residual stiffness and the fatigue strain at S ∗ max is also valid for lower load levels as observed in other tests (Sec- tion 3.2.1.2). Furthermore, it could be proved that quite different accumulated fatigue strains – as it can be assumed within the 50 years of service lifetime 624 4 Methodological Implementation Fig. 4.162. Three dimensional Finite Element model of the road bridge at H¨unxe Table 4.19. Number of elements of structural members structural member deck slab main girders cantilevers arches cross girders hangers sum number of elements 1632 816 1428 204 736 224 5040 of the bridge – have no significant influence on further development of the ratios between residual stiffness and fatigue strain. Additionally, it could be observed that the dynamic elastic modulus is quite more influenced by the cyclic loading than the Young’s one. In comparison to investigated normal and high strength laboratory concretes without any pre-loadings (Section 3.2.1.2), the development of the residual Young’s modulus versus fatigue strain of the bridge’s concrete follows nearly the same trendline. 4.6.5.1.2 Finite Element Model A three dimensional finite element model of the bridge has been developed for numerical analysis of the structural state after 50 years of service (Fig- ure 4.162). To match the geometrical shape of the bridge as well as possible and to model the connections of all structural members correctly, a quite large number of elements according to Table 4.19 has been required. The size of the resulting stiffness matrix is about one billion entries, which only could be handled using bandwidth optimization and sparse storage schemes offered by the finite element program [788]. A three dimensional shell element suit- able for geometrically and physically nonlinear analyses has been implemented for calculation purposes (see e.g. [421, 443]). This element employs a layered approach to combine the both composites of reinforced concrete. The for- mulation of the finite element allows for up to four uniaxial steel layers to model reinforcement bars as well as the prestressing tendons in an accordant position. 4.6 Application of Lifetime-Oriented Analysis and Design 625 4.6.5.1.3 Material Model To mirror the complex material behaviour of concrete correctly, a three dimensional material model developed by Kr ¨ atzig and P ¨ olling (see e.g. [444]) was used for the nonlinear finite element simulations of the structure. To avoid mesh-dependencies, the crack band and fracture energy approach has been incorporated into the material model [95]. Both, reinforcement steel bars as well as the prestressing tendons are predominantly subjected to tension. Therefore, they are modeled, according to the layered element concept, as dimensionless steel layers, using an uniaxial elasto-plastic material law with a damage component d. Hence, the resulting stress-strain relations for reinforcement bars and tendons read: σ s = E s (1 −d) s σ s = E s (1 −d)( s +  ps ) (4.423) In the upper eq. (4.423) σ s , s denote stress and strain of the steel due to loading. The prestrains of the tendons are termed  ps ,whereasE s stands for the Young’s modulus of steel. 4.6.5.1.4 Damage Mechanisms According to the expertise two damage mechanisms are considered as relevant for the time-dependent degradation of the structure, namely fatigue of the prestressing tendons and corrosion of the reinforcement steel bars. For both appropriate numerical models are incorporated into the basic material model of reinforced/prestressed concrete for structural simulations. 4.6.5.1.4.1 Corrosion of the Reinforcement Steel Bars A first impact of corrosion on structural response is the reduction of the reinforcement bars’ cross-section during time. Assuming a constant corrosion rate k s for the entire perimeter of the reinforcement bars according to Fig- ure 4.163, the cross-section area A s of each steel bar at time t reads: A s = π(D 0 − 2k s (t −t i )) 2 4 (4.424) In eq. (4.424) D 0 denotes the initial undamaged diameter of the steel bars cross-section and t i the initiation time. For structural elements without con- crete cracks corrosion is assumed to start after initiation time t i is passed. Within that time the corrosion attack front is presumed to permeate through the concrete cover to the steel bars. If concrete cracks appear in structural elements due to mechanical loading, corrosion initiates immediately thereafter (t i =0). The second effect of corrosion concerns the damage of the bond between concrete and steel bars due to expanding rust products, which can reach up to the ninefold of the original steel volume. The impact of bond damage on 626 4 Methodological Implementation i n i t i a t i o n t 0 d 0 , d t , d n o c r a c k s a t t a c k f r o n t c o n c r e t e c o v e r w i t h c r a c k s @ d a m a g e , d = 2 k s t g l o b a l c o r r o s i o n k s  0 0 , 1 0 , 2 0 , 3 0 , 4 0 , 5 0 , 6 0 , 7 0 , 8 0 , 9 1 , 0 0 , 1 0 , 2 0 , 3 0 , 4 0 , 5 0 , 6 0 , 7 0 , 8 0 , 9 1 , 0 b o n d d a m a g e  , A S / A S 0 0 l o c a l c o r r o s i o n d b Fig. 4.163. Applied corrosion model the structural response is a reduction of the part of the concrete strains  c which is transferred to the steel strains  s . Within our approach it is modeled as follows:  s =(1−d b ) c (4.425) Further, the evolution of bond damage is guided by a variable d b taken from experiments of [181] and depicted here on the right hand side of Figure 4.163: d b =  0forΔA s /A s < 0.03 1 − 1 33·ΔA s for ΔA s /A s ≥ 0.03 (4.426) 4.6.5.1.4.2 Fatigue of the Prestressing Tendons The second long-term damage mechanism, namely fatigue of the steel rein- forcement bars, is modeled within the W ¨ ohler-approach anchored in struc- tural design codes [182]. The failure criterion is defined by the bilinear S-N curve (Figure 4.164), which relates the stress amplitudes Δσ Rsk resulting from each truck crossing, to the number of load cycles to fatigue failure N f : Δσ Rsk =  Δσ ∗ Rsk1 [ 10 6 N f ] 1 k 1 for N f ≤ 10 6 Δσ ∗ Rsk2 [ 10 8 N f ] 1 k 2 for N f > 10 6 (4.427) Herein, Δσ ∗ Rsk1 ,Δσ ∗ Rsk2 denote the limit values of the stress amplitude for 10 6 and 10 8 load cycles, respectively, depending on the diameter of steel bars; (k 1 =5)and(k 2 = 9) are parameters defining the logarithmic slope of the S-N curves. This relationship has been modified to account for uncertainties of the fa- tigue life N f by a parameter κ s affecting the slope of both sections of the fatigue curve [627]. The evolution of the fatigue damage variable d fat s is de- scribed by a nonlinear function: d fat s = − 1 ϑ s · ⎡ ⎣ 1 −(1 −e −ϑ s ) m  j=1 N j (Δσ s,j ) N fj (Δσ s,j ) ⎤ ⎦ (4.428) 4.6 Application of Lifetime-Oriented Analysis and Design 627 4 5 6 7 8 9 1 0 1 0 2 0 3 0 5 0 1 0 0 5 0 0 2 0 0 3 0 0 n u m b e r o f c y c l e s l o g N f D s R s k [ M N / m ² ] s t r e s s m a g n i t u d e 1 0 0 0 = 9 . 0 c s 8 . 0 7 . 0 6 . 0 5 . 0 4 . 0 3 . 0 2 . 0 = 1 . 0 c s M o d i f i c a t i o n , c s ¹ 5 . 0 M o d e l C o d e , c s = 5 . 0 5 1 0 2 0 5 0 d a m a g e 0 0 . 2 0 . 4 0 . 6 0 . 8 0 . 2 0 . 4 0 . 6 0 . 8 0 d s f a t J N j ( d s , j ) j = 1 m N f j ( d s , j ) Fig. 4.164. Modified S-N curves for steel and fatigue damage evolution function where ϑ s defines the degree of nonlinearity (Figure 4.164). The sum argument in the brackets reflects a normalized fatigue life accumulated at different stress amplitudes Δσ s,j . Furthermore, the impact of fatigue damage is taken into account by a reduction of the material stiffness as follows: E fat s =(1−d fat s )E s (4.429) 4.6.5.1.5 Modelling of Uncertainties During an ordinary design process, all input parameters are usually treated in a deterministic way using just mean values as input. Such an approach denies the stochastic character of material properties and damage driving forces ab initio. In the context of generating input data for numerical simulations two important questions arise. The first one is, how many data sets have to be generated to ensure a good representation of the population characteristics, namely mean value, standard deviation and type of distribution. Thereby, it should be considered, that the higher the number of sets is chosen, the more expensive – in terms of computation time – the presented approach will be. The second question concerns the method to be used for this purpose. Therefore the statistical moments mean, standard deviation, skewness and kurtosis have been regarded (Figure 4.165). Obviously, an impressive small number of simulations seems to be sufficient for generation of input data with the postulated characteristics taking Latin Hypercube sampling instead of pure Monte Carlo method. This even holds for the higher order statistical moments skewness and kurtosis. Further, the generated data sets have been compared to the expected values in Figure 4.166. The Gaussian shape of the distributions of expected and generated values are in great accordance. Just little differences in mean and standard deviation of the material parameters compressive strength and elastic modulus were found. It is assumed that all material parameters obtained by testing reflect the bridge’s structural state after 50 years of service, shortly before its deconstruc- tion. Therefore, the properties have to be transformed back to the structural virgin state, to serve as realistic input data for the lifetime simulations. [...]... general, structural degradation modelling in our concept follows a two step procedure, displayed in Figure 4.169 First, the structure is stepwise subjected to a design load combination consisting of dead (G) and traffic (Q) load as well as the prestressing (V ) of the tendons At that load level, the external forces are kept constant A further augmentation of the external load would lead to structural. .. (n + 1) A nonlinear structural simulation over the lifetime T , under the fixed load combination G + V + Q and the long-term degradation mechanisms described above, follows in the second step The corresponding time-dependent system governing eq (4.439) reads [622]: KT (u, d)Δu(Tm+1 ) = G + V + Q − (FI (Tm ) + ΔFI (Tm+1 )) (4.439) 4.6 Application of Lifetime-Oriented Analysis and Design 633 Fig 4.169... Developement of Concrete Strength The long-term evolution of the compressive strength fc depends on the cement type, the curing conditions and the ambient temperature For a 4.6 Application of Lifetime-Oriented Analysis and Design 629 fc5% 50 compression strenght f fcm fc95% 100 80 ~1.60 60 c [N/mm²] 120 40 60 30 Washa, Seamann, Cramer fc,50a ~ 1.60 x f c 20 CEB-FIP Modelcode 90 10 fc,50a ~ 1.45 x f c fc,50a... (4.432) and eq (4.433) to estimate those quantities with respect to fck This is reasonable to keep the number of uncertain independent parameters as small as possible 4.6 Application of Lifetime-Oriented Analysis and Design fck + 8 E(fck ) = 21500 · 10 fct (fck ) = 1.40 · fck 10 631 1 3 (4.432) 2 3 (4.433) 4.6.5.1.5.3 Modelling of Spatial Scatter by Random Fields The spatial variability of relevant... forces ΔFI (Tm+1 ) due to damage The structural degradation in time is reflected in the left part of Figure 4.169 as an increase of the deflections with time, solely caused by action of damage mechanisms introduced above All long-term simulations are performed until a computational limit state is reached Each corresponding time instant provides a discrete estimation of structural lifetime under given conditions... to obtain an analytical function of structural lifetime [623] This function enables for reliability analysis of the structure not requiring any further time consuming nonlinear finite element calculations [622] 634 4 Methodological Implementation 4.6.5.1.7 Conclusions The presented approach allows a combination of stochastic modelling and nonlinear damage-oriented structural analyses by the means of... structural analyses by the means of the finite element method considering relevant scattering properties A method to determine from samples, reflecting the structural state after 50 years of service, to the corresponding material properties at the structural virgin state is presented The relevant damage mechanisms and the corresponding implementation into numerical models have been introduced Thereby,... vibration measurements eight one-dimensional acceleration sensors were attached equidistant The mechanical structure was excited by impulse loads Furthermore experiments on a 4.6 Application of Lifetime-Oriented Analysis and Design 635 prestressed concrete tied-arch bridge in H¨ nxe (Germany) will be presented u The bridge (built in 1952) had a span of 62.5 meters and was deconstructed in 2005 Main- and cross-girder,... and B can be transformed by arbitrarily selectable transformation matrices T to a new ˜ ˜ ˜ equivalent state space system characterized trough the matrices C, A and B 4.6 Application of Lifetime-Oriented Analysis and Design ˜ ˜n˜ CA B = CT T−1 An T T−1 B = CAn B 637 (4.449) Because of the possibility of transformation identified state space matrices do not contain the mechanical interpretable structure... for identification of measured structures excited by impulse loads The presented algorithm is based on the ideal theoretic impulse function with infinitesimal duration 4.6 Application of Lifetime-Oriented Analysis and Design 639 20 acceleration in [m/s2] 0 −20 −40 −60 −80 0 (a) experimental setup 100 200 300 400 discrete time steps − fs=10kHz 500 600 (b) measured impulse and system reaction (8 channels) . results, the mean values of E dyn are on a similar level of about 4.6 Application of Lifetime-Oriented Analysis and Design 619 Table 4.17. Dynamic elastic moduli E dyn (mean) and their standard deviations (SD). their standard deviations (SD) after a service life of 50 years for the different structural members of the bridge structural member number of specimens E stat SD ε u SD f c SD [-] [N/mm 2 ] [N/mm 2 ]. different strengths at the age of 28 days (solid lines) [193] 4.6 Application of Lifetime-Oriented Analysis and Design 621 300 µm Fig. 4.159. LM-micrograph of in-situ concrete change the stress-strain