The introduction in the last decade of new continuous open connectors, the so-called

In 2016, the bridge structure shown in Figs 2–4 was built in Elbląg, Poland [4, 5]. The composite girder was designed with a variable cross section along the beam. In mid-spans, it consisted of a steel T-section with a high steel web connected to the concrete deck slab (see Fig. 1, section no. 2 and Fig. 4), and in regions of hogging bending moments (close to internal supports), the height of T-section's web was reduced, and instead, a concrete web with a variable increasing height toward the internal supports was introduced (see Fig. 1, section no. 4 and Fig. 4). This way, in mid-span regions, the self-weight of the girder was limited, and moreover, the steel part was applied in the tensile zone, while the concrete part was in the compressed zone. In a hogging moments region, the steel part remains to strengthen the narrow, compressed concrete part.

In 2018, another bridge was built [6] (Figs 5 and 6) in Sobieszewo, Poland, where the above solutions were developed. In the zone of hogging moments, the steel part was completely omitted, replacing the composite cross section with a reinforced concrete (RC) cross section. To do so, the so-called transition zone had to be introduced, thanks to which the steel element from the span part could be effectively anchored in the concrete part [7, 8]. This solution allowed for further optimisation of the structure, reducing the use of structural steel only to areas where it is necessary (in places where it would work as compressed, it was replaced with an RC section).

Many bridge structures located along the S3 road in Poland [9] are being designed currently, and a further evolution of hybrid beams is introduced in this design. This is because apart from the transition zone (slightly modified in comparison to the bridge in Sobieszewo), thin concrete webs (20 cm), which were thinner than the ones used so far, were introduced and going outside of ranges provided for composite dowels in the approvals [10,11,12]. To make it possible, a modification of the arrangement of reinforcing bars in the vicinity of the connectors was applied (the new name, strongly reinforced composite dowels [SRCD], was introduced), thus eliminating the destruction mechanism referred to as pry-out cone [1]. Europrojekt Gdańsk, in cooperation with the Wroclaw University of Technology, is responsible for the design of the bridges. Cross section of one of the bridges is presented in Fig. 7. For details of the beam, see also Fig. 12.

At the end of 2019, a railway bridge was built in Dąbrowa Górnicza [4, 13, 14], in which the U-shaped cross section of the structure consisted of two beam girders and an RC deck. Girder's webs are the side edges of the ballast truck (Figs 8 and 9). The main girders consist of steel T-sections placed both in the upper and lower zones of the girder, an RC web and the effective part of the RC deck. This bridge, designed by Fasys Mosty in collaboration with ArcelorMittal [14], appears to be a very economical solution for medium-span railway bridges.

Design rules for objects such as the ones presented in Section 1 go beyond the framework described in the current design standards, for example, [15,16,17]. Apart from the composite dowels, the cross section of the girders has both steel and concrete parts, both of which are responsible for transferring the shear force. Taking into account the significant share of both the steel and concrete parts, it is not possible (or not economically justified) to design the cross sections assuming that the entire shear force is transmitted through the steel part (which is assumed in practice when designing a typical composite cross section according to [17]). A new type of cross section, forcing the application of new design approaches due to the flow of shear stresses, is called hybrid section [4, 9, 18]. The issue of shear force transfer in such a cross section has already been described, and the dimensioning procedure is known at the level of cross section [18,19,20,21]. The cross section described in [20] as a general composite section is now called a hybrid section, and the name has been sanctioned internationally after discussions (Prof. Roger Johnson and Wojciech Lorenc) during work on the European technical approval for composite dowels [11]. While the concept of a hybrid cross section is currently quite precise, the concept of a hybrid beam is not: in particular, it applies to a beam in which both hybrid and RC cross sections are used. In fact, the challenges encountered during the design of beams of the new bridge in Sobieszewo have shown that a new

To sum up the problem of global analysis of hybrid beams, a comparison to both RC and steel–concrete composite beams is made briefly [9, 22]. In RC beams, if linear elastic analysis is assumed, it is allowed in ULS (Ultimate Limit State) to perform an uncracked analysis. Such an approach, presented in

In addition, the application of general rules, for example, those defined in ^{1}p^{2} class grid models, it is currently a standard to use models of higher classes, for example, e^{1+2}p^{2} models with the offset beam technique (deck slab modelled with shell elements, beams with bar elements) or models even more precise in which all elements are modelled with shell elements (e^{2}p^{3}). Such models allow for automatic accounting of shear lag effect (instead of the concept of the effective width of the concrete slab), but their main advantage over the grid models is the automatic transverse distribution of the loads between adjacent girders, as well as obtaining the actual internal forces in the concrete slab. Such models were used as an alternative to the grid models in the design of bridges described in Section 1.

The problem of using such models becomes evident when it is necessary to take into account concrete cracking. A typical example of such a situation is the arch or truss bridge with a deck in the level of tensile elements, but also an ordinary continuous beam, where the concrete slab is in tension in the zone of hogging moments, for example, in the bridge in Elbląg or Sobieszewo (Figs 2–6), or in both sagging and hogging moments, like in the bridge in Dąbrowa (Figs 8, 9). It is necessary to consider the stiffness decrease of shell elements in the main tension direction, which is caused by concrete cracking due to global effects. This problem also occurs when using grid models (e^{1}p^{2}), where it is important to properly reflect the introduction of forces into the tensile elements [24]. It is disputable not only how to determine the proper regions of cracked concrete and how to reduce its stiffness (considering the tension stiffening effect), but also how to apply it to the numerical model with shell elements. The introduction of reduced stiffness in the direction of global tension influences the stiffness in the other directions (the stiffness matrices of shell elements change) [25, 26]. In the grid model (or in an isolated beam), the reduction of the concrete stiffness related to its cracking influences only the bending stiffness EI of the considered beam, which is an elementary case. The opposite is the case when using shell elements. This is particularly noticeable in the e^{2}p^{3} model (3d model with all elements modelled with shells), in which the concrete web is modelled with e^{2} elements in the vertical plane (see, e.g. the bridge in Dąbrowa [13, 14]). If the stiffness in the longitudinal direction is significantly reduced, there is also a significant loss of the shear stiffness in the plane under consideration. The latter influences the shear flow between steel and concrete elements. There are currently no guidelines allowing for efficient (and accepted in engineering practice) modelling of considered types of structures (without considering complicated scientific models of concrete).

In this article, the influence of the following is discussed:

concrete cracking,

concrete rheology and

methods of computational modelling/determination of the range of the cracked zones

In this section, numerical analysis will be carried out to show how the concrete cracked zones develop in a hybrid beam and how this phenomenon affects the bending moments’ distribution along a beam. Influence of creep and shrinkage of concrete is also under investigation.

As an example for numerical investigation, exemplary girder from a newly designed bridge on S3 road was adopted. Geometry of this girder is shown in Fig. 12. For the purpose of analysis, it is assumed that

upper slab of precast element: arranged in one layer, #20/150 and

web of precast element: both sided #10/100.

Reinforcing steel in compressed concrete is omitted in the analysis in approaches A and B. In approach C, reinforcement is considered in every stress state, no matter whether it is compressed or tensioned. The hybrid beam is thus modelled as a structure consisting of five materials: one steel and four types of homogenised RC (

Actions applied to the girder were taken directly from a real bridge analysis as

self-weight of the precast element + ^{3}, of steel 78.5 kN/m^{3});

self-weight of surface layers – as uniformly distributed with a value of 2.65 kPa;

uniformly distributed load (UDL) – 9 kPa, applied to the first, second or both spans and

tandem system (TS) load – 2x 300 kN moving along the entire girder.

These loads allowed to obtain bending moments in a girder that are very close to moments in a real bridge, thus real cracking conditions are expected. The FE model was made in e^{2}p^{3} class with all elements modelized with shell elements, as shown in Fig. 13.

To investigate the influence of concrete cracking on the bending moment distribution along a beam length, three different approaches were adopted. They are as follows:

Approach A: iterative procedure. First, the stress envelope for characteristic combinations is calculated using the uncracked sections’ stiffness (uncracked analysis). In regions where the tensile stress in the concrete exceeds twice the strength f_{ctm} due to the envelope of global effects, the stiffness is reduced to cracked sections’ stiffness and calculations are repeated. Again, the stress layout is checked in all elements, and in regions where the tensile stress in the concrete exceeds twice the strength f_{ctm} due to the envelope of global effects, the stiffness is reduced to cracked sections’ stiffness. This procedure is repeated iteratively until there are no regions with tensile stress more than 2f_{ctm} (where f_{ctm} = 2.90 MPa and 4.07 MPa for concrete C30/37 and C50/60, respectively). This approach is based on a similar approach adopted in

Approach B: simplified method. It is based again on the analogy to

Approach C: automatic non-linear analysis based on pre-defined concrete material laws, considering the changes in concrete stiffness depending on the principal stress in finite elements.

In approaches A and B for uncracked sections, material stiffness was calculated automatically on the basis of Young modulus (_{cm} = 32.8 and 37.3 GPa for C30/37 and C50/60, respectively) and Poisson's coefficient _{s}_{s} stands for modulus of elasticity and area of reinforcement, _{0} for _{a}/_{cm} ratio and _{s} for the area of rebar to concrete ratio (_{s}/_{c}). The Poisson coefficient was adopted to be 0. For transversal and diagonal directions, the stiffness was defined like for uncracked sections, not to influence the shearing stiffness in a plane of shell elements.

For approach C, the following law for concrete elements was adopted (with distinction to elements with different reinforcement layouts). RC parts were modelled using a homogenised material that enabled considering tension stiffening effect for reinforcement and tension softening of the concrete by applying one stress–strain curve for each part of the RC members. This model is based on the stress–strain and force–strain relationships presented in [27,28,29]. These relationships were originally defined for embedded reinforcing steel, but the authors adopted them to concrete parts.

This approach of modelling of cracked reinforced parts is not strict and has some apparent inconsistencies. Firstly, longitudinal reinforcement may not be oriented in the same direction as principial stresses in the concrete. Tension stiffening effect is calculated on the basis of longitudinal reinfocement, but appears in finite element model in the direction of principal stresses. Moreover, when crack appears, Poisson's coefficient in the concrete could be reduced to 0 value, which also would affect the stiffness matrix in a two-dimensional shell finite element. Despite the abovementioned doubts presented in Fig. 14, the model law for concrete materials was adopted in approach C analysis as a sufficient and relatively easy method in implementation of engineering approximation.

The models in approaches A–C are different also due to fact that they induce concrete cracking at different tensile stress levels: 2_{ctm} in approach A, 0 but at _{ctm} in approach C.

Similar approaches were considered for estimation of concrete creeping and shrinkage in a hybrid beam. Results presented hereafter were obtained with the following assumptions:

For approaches A and B, creep coefficients 2.13 and 1.52 were taken for C30/37 and C50/60 elements, respectively, and shrinkage strain was assumed to be 0.34‰ and 0.36‰, respectively. Creep coefficients were taken to reduce the E_{cm} of uncracked concrete elements (as a simplification using the analogy to commonly used regulation of EC4 [17]).

For approach C, a rheology of concrete was considered by the so-called strain approach, default for CSM (Construction Stage Manager) in SOFiSTiK software [30].

Shrinkage strains were applied only to uncracked concrete, which is an analogy to [17] regulation which states that ‘

First, the results of analysis with approach A are presented. In the following figures, it is presented how the tensile zones in concrete develop. Maps of tensile stress are presented in an upper _{ctm}, green-coloured regions indicate tensile stress in the range of _{ctm}–2_{ctm}, and in yellow-marked regions, the tension does not exceed _{ctm}. For the uncracked analysis (Fig. 15), it is observed that stresses exceeding 2_{ctm} are present only in the

After reducing the stiffness in red highlighted regions, calculations were redone and new stress layouts were analysed (Fig. 16). As expected, a new analysis caused reduction of tensile stress in the _{ctm} in this region, so they will undergo cracking (reduction of their stiffness).

Repeated calculations with decreased stiffness of newly cracked elements showed decrease of tensile stress in the prefab upper slab, noticeable increase of tension in the

Multiple updates of cracked zone range and recalculations cause a minor increase of the cracked zone in the web close to the internal support, and finally, the stable state can be found (let it be named step 4; despite that, in fact, seven iterations were needed to get the final state of cracking in case of considered beam). Changes in stress layout are hardly visible compared to this in Fig. 17, so no further maps of tensile stress are presented. Approach A makes it possible to observe a continuous development of the cracked zones until their final state. In approach B, the cracked zones are

Changes in bending moment envelope depend on the assumed approach and the cracking zone development in approach A, which are presented in Fig. 19 and Table 1. In Table 1, relative changes in extreme sagging and hogging bending moments in relation to results from uncracked analysis are presented.

Bending moment values (kN m) along the girder's length (m), depending on the assumed approach (A, B, C). Numerical interpretation of Fig. 19.

No. | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | M+ / M0+ | M− / M0− | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Approach | x [m] | 0 | 2,03 | 4,06 | 6,09 | 8,12 | 10,15 | 12,18 | 14,21 | 16,24 | 18,27 | 20,3 | ||

M+ uncracked (M0+) | 0 | 1782 | 2940 | 3652 | 3749 | 3401 | 2626 | 1505 | 1 | −1902 | −4129 | 100,0% | ||

M− uncracked (M0−) | 0 | 529 | 787 | 789 | 520 | −16 | −820 | −1892 | −3252 | −4922 | −6865 | 100,0% | ||

M+ cracked A (Step 2) | 0 | 1804 | 2984 | 3727 | 3853 | 3535 | 2786 | 1684 | 183 | −1730 | −3947 | 102,8% | ||

M− cracked A (Step 2) | 0 | 561 | 850 | 885 | 648 | 143 | −628 | −1668 | −2996 | −4634 | −6547 | 95,4% | ||

M+ cracked A (Step 3) | 0 | 1816 | 3009 | 3768 | 3909 | 3607 | 2870 | 1779 | 279 | −1637 | −3845 | 104,3% | ||

M− cracked A (Step 3) | 0 | 579 | 884 | 936 | 716 | 229 | −526 | −1548 | −2860 | −4481 | −6369 | 92,8% | ||

M+ cracked A (Step 4) | 0 | 1820 | 3017 | 3782 | 3928 | 3630 | 2898 | 1811 | 312 | −1607 | −3819 | 104,8% | ||

M− cracked A (Step 4) | 0 | 584 | 896 | 953 | 739 | 257 | −492 | −1508 | −2814 | −4432 | −6321 | 92,1% | ||

M+ cracked B (15%) | 0 | 1829 | 3035 | 3811 | 3968 | 3682 | 2954 | 1873 | 372 | −1544 | −3748 | 105,8% | ||

M− cracked B (15%) | 0 | 597 | 920 | 990 | 787 | 318 | −419 | −1423 | −2717 | −4321 | −6194 | 90,2% | ||

M+ C (TS) | 0 | 1781 | 3012 | 3760 | 3951 | 3590 | 2828 | 1616 | 125 | −1902 | −4068 | 105,4% | ||

M− C (TS) | 0 | 602 | 947 | 1019 | 824 | 343 | −366 | −1415 | −2719 | −4346 | −6274 | 91,4% |

Similar simulation is made to show how the concrete creeping influences bending moment distribution along the beam (Fig. 20 and Table 2).

Bending moment values (kN m) along the girder's length (m0 due to creep in dependence of the assumed approach (A, B, C). Numerical interpretation of Fig. 20.

No. | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | M+ / M0+ | M− / M0− | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Approach | x [m] | 0 | 2,03 | 4,06 | 6,09 | 8,12 | 10,15 | 12,18 | 14,21 | 16,24 | 18,27 | 20,3 | ||

M uncracked t = 0 | 0 | 391 | 631 | 733 | 685 | 489 | 147 | −345 | −1006 | −1853 | −2868 | 100,0% | 100,0% | |

M uncracked t = 100 y | 0 | 442 | 730 | 885 | 887 | 741 | 449 | 8 | −603 | −1401 | −2367 | 120,7% | 82,5% | |

M cracked A (Step 4) t = 100 y | 0 | 442 | 731 | 886 | 888 | 844 | 452 | 12 | −599 | −1397 | −2364 | 109,3% | 92,2% | |

M cracked A (Step 4) t = 100 y | 0 | 444 | 735 | 892 | 896 | 753 | 464 | 25 | −584 | −1379 | −2342 | 121,7% | 81,7% | |

M cracked B (15%) t = 0 | 0 | 419 | 685 | 815 | 794 | 626 | 311 | −154 | −788 | −1607 | −2595 | 111,2% | 90,5% | |

M cracked B (15%) t = 100 y | 0 | 442 | 731 | 887 | 889 | 744 | 453 | 13 | −598 | −1393 | −2360 | 121,0% | 82,3% | |

M cracked C (TS) t = 0 | 0 | 0 | 427 | 715 | 851 | 845 | 672 | 370 | −105 | −746 | −1584 | 116,1% | 89,9% | |

M cracked C (TS) t = 100 y | 0 | 0 | 459 | 782 | 951 | 982 | 838 | 569 | 128 | −480 | −1286 | 129,7% | 78,4% |

Fig. 21 and Tab. 3 shows secondary bending moments due to shrinkage.

Bending moment values (kN m) along the girder's length (m) due to shrinkage in dependence of the assumed approach (A, B, C). Numerical interpretation of Fig. 21.

No. | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | M− / M0− | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Approach | x [m] | 0 | 2,03 | 4,06 | 6,09 | 8,12 | 10,15 | 12,18 | 14,21 | 16,24 | 18,27 | 20,3 | |

M uncracked Shrinkage | 0 | −35 | −70 | −106 | −141 | −177 | −212 | −247 | −282 | −319 | −365 | 100,0% | |

M cracked A (Step 4) Shrinkage | 0 | −34 | −67 | −101 | −134 | −168 | −202 | −235 | −268 | −304 | −340 | 93,2% | |

M cracked B (15%) Shrinkage | 0 | −26 | −51 | −77 | −103 | −129 | −154 | −180 | −204 | −233 | −256 | 70,1% | |

M cracked C (TS) Shrinkage | 0 | −29 | −58 | −88 | −116 | −147 | −175 | −204 | −234 | −263 | −294 | 80,5% |

Cracking of concrete influences the bending moment distribution along a beam. In fact, the range of cracked zones due to external loads is much bigger in support zone sections that are entirely made out of concrete than in mid-span sections that are predominantly made out of structural steel in a tensile zone. This means that the reduction of beam's inertia at the internal support is bigger than that in the mid-span and, due to cracking, the hogging moments decrease (about 8%–10%) and the sagging ones increase (about 5%–6%) in comparison to uncracked analysis.

Irrespective of the assumed approach (A, B or C), the obtained results are in a good convergence. Differences in final bending moments envelopes do not exceed 2%. This means that, if cracking is considered, the cracked range itself has a minor impact on the results of analysis.

Creep of concrete causes further redistribution of bending moments. Approaches A and B give nearly the same results, while in approach C, the obtained results differ slightly (bigger shift of the entire bending moment diagram towards the sagging moments). This can be explained by the fact that in a combination of self-weight loads, a model including automatic reduction of stiffness due to cracking (approach C) has a greater crack range (cracking occurs at _{ctm} stress) that the models in approaches A and B (cracking occurs at 2_{ctm}). This can be clearly observed in Table 2 approach C: even without consideration of creep (t = 0), the bending moment diagram is shifted towards sagging moments (compared to approaches A and B). The change due to creep itself is similar for all approaches (A, B, C). For uncracked analysis creep influence is slightly bigger (comparing to approaches A, B, C), what can be explained by the fact that creep strains increase also in tensile uncracked concrete sections’ parts (mainly close to the internal support).

Final bending moments after creep of concrete are very similar for both cracked and uncracked analyses. The convergence of these results obviously depends on the reinforcement ratio and creep coefficient, but it can be generally concluded that for some limits of the latter, the cracking itself will not influence significantly moments’ redistribution due to creeping.

For the considered beam, the actual decrease in hogging bending moment due to creep varies (depending on the approach A–C for cracked analysis) from 230 to 330 kN m, while hogging moment from external loads is of about 6300 kN m. For uncracked analysis the decrease of bending moments due to creep is about 500 kN m comparing to 6870 kN m of the total hogging moment. Thus, the change is up to approx. 4%–5% for cracked analysis and about 7% for uncracked one. Similarly, increase in sagging moment of 100–140 kN m in relation to moment from external loads of about 3900 kN m gives a change of about 3.5% for cracked analysis and, by analogy, about 200 / 3750 = 5.5% for uncracked one.

Secondary moments caused by concrete shrinkage depend on the cracked zone ranges along the beam length. Because shrinkage strains were applied to the structure's parts that are not cracked, direct comparison of secondary moments cannot be made (different assumptions for every model). For approach C, the direction of concrete softening due to cracking is different, as it depends on the principal stress direction. But it can be noticed that despite the fact that there are significant percentage differences between particular approaches, the actual results are quite similar. Relative differences in bending moments up to 30% (between uncracked analysis and approach B) turn out to be of minor importance, considering the real differences of up to 110 kN m (for the considered beam), which is about 2% of the hogging moments induced by the external loads (about 6300 kN m for cracked analysis). The overall share of secondary shrinkage moments in entire bending moments is about 4%–5%.

Analysis of the obtained results confirms that both cracking of concrete and rheology of concrete influence bending moment distribution along the beam. Cracking of concrete induces the most significant changes, while changes due to creep and shrinkage affect the bending moment less severely. Moreover, creep shifts the bending moment diagram towards sagging moments, while shrinkage shifts it towards hogging ones, reducing the influence of the previous one. Considering that creep affects the bending moment values of 4%–7% and shrinkage of 4%–5% (in opposite directions), one can state that the rheology of concrete is of minor importance in a design of hybrid beams. In classic composite beams, in which concrete slab is placed on the top of steel I-beam, after cracking of concrete in the internal support zones, the rheology of concrete is induced only in mid-span sections (where concrete is under compression). Support sections (I-beam + rebar in the slab) restrain deformation due to concrete rheology, and significant secondary moments appear. On the contrary, in hybrid beams, uncracked concrete parts are both in mid-span and support zones, and due to concrete rheology, the entire beam undergoes deformations with limited internal restraint. Secondary moments due to creep and shrinkage are thus limited to less significant values.

Regions where concrete should be assumed to be cracked cannot be directly established according to the procedure presented in [17]. It means it is not enough to perform at first uncracked analysis, then to reduce axial stiffness for places where the tensile stress exceeds 2f_{ctm} and then to recalculate a static system. It has been proven in Section 3.4 that such an approach should be iterative, as changes in stiffness of one part influence the tensile force redistribution in the entire element, including the web of a beam. Such an approach seems to be impractical for engineering application. Similarly, approaches based on non-linear analysis (like the presented approach C) seem to be too complicated for a design based on many combination of actions, simply too much of computational power would be required. Another problem is proper determination of stiffness matrix for shell elements undergoing cracking process in a certain direction. This is problematic to ensure a proper axial stiffness and, at the same time, proper stiffness for in-plane shearing of planar finite elements.

Taking the abovementioned into account, it seems to be reasonable to accept the design concept in which for global elastic analysis, the entire structure would be

for dimensioning of internal support sections: to limit a redistribution of hogging moments to, for example, 5% (which is on the safe side because real decrease of hogging moment comparing to the one obtained from uncracked analysis is about 8%–10% due to cracking, 4%–7% due to creeping and −4%–5% due to shrinkage, and overall, it is up to 5%–10%, so more than the assumed 5%);

for dimensioning of mid-span sections: to limit a redistribution of hogging moments to, for example, 15% (which is on the safe side because real decrease of hogging moment comparing to the one obtained from uncracked analysis is about 8%–10% due to cracking, 4%–7% due to creeping and −4%–5% due to shrinkage, and overall, it is up to 5%–10%, so less than the assumed 15%).

Determination of exact limits of redistribution should be based on the analysis of many different hybrid beams and, therefore, parametric studies. Such studies are currently the subject of the author's work.

Thus, in this article, the authors actually propose a complete reversal of the concept used as standard in EC4 for the needs of the hybrid beam: uncracked analysis. The presented concept of global elastic analysis enables easy modelling of hybrid beams with safe-sided assumptions for both hogging and sagging moment regions. The above is the result of a broader look at the experience in the design and construction of structures presented in [4], and the beam analysis presented in this paper is an example that enables a quantitative assessment of the problem.

#### Bending moment values (kN m) along the girder's length (m), depending on the assumed approach (A, B, C). Numerical interpretation of Fig. 19.

No. | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | M+ / M0+ | M− / M0− | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Approach | x [m] | 0 | 2,03 | 4,06 | 6,09 | 8,12 | 10,15 | 12,18 | 14,21 | 16,24 | 18,27 | 20,3 | ||

M+ uncracked (M0+) | 0 | 1782 | 2940 | 3652 | 3749 | 3401 | 2626 | 1505 | 1 | −1902 | −4129 | 100,0% | ||

M− uncracked (M0−) | 0 | 529 | 787 | 789 | 520 | −16 | −820 | −1892 | −3252 | −4922 | −6865 | 100,0% | ||

M+ cracked A (Step 2) | 0 | 1804 | 2984 | 3727 | 3853 | 3535 | 2786 | 1684 | 183 | −1730 | −3947 | 102,8% | ||

M− cracked A (Step 2) | 0 | 561 | 850 | 885 | 648 | 143 | −628 | −1668 | −2996 | −4634 | −6547 | 95,4% | ||

M+ cracked A (Step 3) | 0 | 1816 | 3009 | 3768 | 3909 | 3607 | 2870 | 1779 | 279 | −1637 | −3845 | 104,3% | ||

M− cracked A (Step 3) | 0 | 579 | 884 | 936 | 716 | 229 | −526 | −1548 | −2860 | −4481 | −6369 | 92,8% | ||

M+ cracked A (Step 4) | 0 | 1820 | 3017 | 3782 | 3928 | 3630 | 2898 | 1811 | 312 | −1607 | −3819 | 104,8% | ||

M− cracked A (Step 4) | 0 | 584 | 896 | 953 | 739 | 257 | −492 | −1508 | −2814 | −4432 | −6321 | 92,1% | ||

M+ cracked B (15%) | 0 | 1829 | 3035 | 3811 | 3968 | 3682 | 2954 | 1873 | 372 | −1544 | −3748 | 105,8% | ||

M− cracked B (15%) | 0 | 597 | 920 | 990 | 787 | 318 | −419 | −1423 | −2717 | −4321 | −6194 | 90,2% | ||

M+ C (TS) | 0 | 1781 | 3012 | 3760 | 3951 | 3590 | 2828 | 1616 | 125 | −1902 | −4068 | 105,4% | ||

M− C (TS) | 0 | 602 | 947 | 1019 | 824 | 343 | −366 | −1415 | −2719 | −4346 | −6274 | 91,4% |

#### Bending moment values (kN m) along the girder's length (m0 due to creep in dependence of the assumed approach (A, B, C). Numerical interpretation of Fig. 20.

No. | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | M+ / M0+ | M− / M0− | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Approach | x [m] | 0 | 2,03 | 4,06 | 6,09 | 8,12 | 10,15 | 12,18 | 14,21 | 16,24 | 18,27 | 20,3 | ||

M uncracked t = 0 | 0 | 391 | 631 | 733 | 685 | 489 | 147 | −345 | −1006 | −1853 | −2868 | 100,0% | 100,0% | |

M uncracked t = 100 y | 0 | 442 | 730 | 885 | 887 | 741 | 449 | 8 | −603 | −1401 | −2367 | 120,7% | 82,5% | |

M cracked A (Step 4) t = 100 y | 0 | 442 | 731 | 886 | 888 | 844 | 452 | 12 | −599 | −1397 | −2364 | 109,3% | 92,2% | |

M cracked A (Step 4) t = 100 y | 0 | 444 | 735 | 892 | 896 | 753 | 464 | 25 | −584 | −1379 | −2342 | 121,7% | 81,7% | |

M cracked B (15%) t = 0 | 0 | 419 | 685 | 815 | 794 | 626 | 311 | −154 | −788 | −1607 | −2595 | 111,2% | 90,5% | |

M cracked B (15%) t = 100 y | 0 | 442 | 731 | 887 | 889 | 744 | 453 | 13 | −598 | −1393 | −2360 | 121,0% | 82,3% | |

M cracked C (TS) t = 0 | 0 | 0 | 427 | 715 | 851 | 845 | 672 | 370 | −105 | −746 | −1584 | 116,1% | 89,9% | |

M cracked C (TS) t = 100 y | 0 | 0 | 459 | 782 | 951 | 982 | 838 | 569 | 128 | −480 | −1286 | 129,7% | 78,4% |

#### Bending moment values (kN m) along the girder's length (m) due to shrinkage in dependence of the assumed approach (A, B, C). Numerical interpretation of Fig. 21.

No. | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | M− / M0− | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Approach | x [m] | 0 | 2,03 | 4,06 | 6,09 | 8,12 | 10,15 | 12,18 | 14,21 | 16,24 | 18,27 | 20,3 | |

M uncracked Shrinkage | 0 | −35 | −70 | −106 | −141 | −177 | −212 | −247 | −282 | −319 | −365 | 100,0% | |

M cracked A (Step 4) Shrinkage | 0 | −34 | −67 | −101 | −134 | −168 | −202 | −235 | −268 | −304 | −340 | 93,2% | |

M cracked B (15%) Shrinkage | 0 | −26 | −51 | −77 | −103 | −129 | −154 | −180 | −204 | −233 | −256 | 70,1% | |

M cracked C (TS) Shrinkage | 0 | −29 | −58 | −88 | −116 | −147 | −175 | −204 | −234 | −263 | −294 | 80,5% |

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