Portal Frame Bridge - Simple Statics-libre

GROUP 3 Traian Carjeu (CID : 00855300) Hammad El Jisr (CID : 00879204) Maria Fligkou (CID : 00869619) Theodoros Litsos (CID : 00881595) Alexandros Nordas (CID :00862602)

PARAMETRIC STUDY ON PORTAL

FRAME BRIDGES

Portal Frame Bridge project 2014 Group 3

Contents

1) Introduction ................................................................................................................................ 2 2) Optimisation of the back-span to main-span ratio ...................................................................... 3 3) Inclination of the piers ................................................................................................................ 6 4) Flexural Pier Stiffness ................................................................................................................ 9 5) Investigation of Soil-Structure Interaction ............................................................................... 12 6) Effect of Uniform Imposed Temperature Variation and Uniform Shrinkage on the Deck ...... 15 7) Conclusion-Case Study ............................................................................................................ 19 REFERENCES ................................................................................................................................. 21


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1) Introduction In this project, an important longitudinal scheme of bridges, portal frames was analysed. Some indicative examples of portal frame bridges are Sfalassa Bridge (steel with 376m main span), Joseph Le Brix Bridge (steel with 147m span). Portal frames are usually made of concrete and have significant advantages with respect to beam bridges, simply supported or continuous. Firstly, as there is a rigid connection between the piers and the deck, the external loads are resisted by both superstructure and substructure, leading to reduced bending moment at decks mid-span. Consequently, a reduced deck depth or a longer span is feasible. Taking advantage of this feature, longer spans are achievable compared to continuous beam bridge longitudinal scheme. However, it is important to point out that the piers are also working under bending and that significantly large horizontal reactions are transferred to the soil rendering necessary a more explicit structural analysis A bridge of this type may be used in cases where constraints to provide a larger clearance under the bridge exist, such as: waterway traffic, avoid disturbances in the water flow and environmental restraints. The advantages provided by the portal frame scheme are dependent on several parameters. This paper consists in a parametric study on a portal frame under uniform distributed load and uniform deck shortening to investigate potential variations in the portal frame behaviour. The parameters that have been taken into consideration for this purpose are given bellow:

Pier height and span lengths; Inclination of the piers; Flexural pier stiffness; Soil-Structure interaction; Uniform shrinkage of the deck. Methodology:

In the parametric study, the initial geometric configuration in which the analysis was based was the following:

Total bridge span L = 150 m Piers Height hpier = 25 m Fixed piers connection to the ground Pinned connections at the abutments High horizontal stiffness of the soil Prestressed concrete box girder section with A = 6 m2, Ix = 12.6 m4 and Iy =21.5m4 was used for both the deck and the pier. This cross section was selected in order to carry out the traffic loads for Load Model 1. Uniform distributed load 100 kN/m

For analysis purposes, a 2D frame complying with the geometrical description given in the introduction was employed and further analysed by specialised computer software under the loading conditions previously described, as well as an imposed uniform temperature variation on the deck. It should be noted that the cross-section of the deck, the cross-section of the piers, as well as the material characteristics, remained unchanged for the purposes of this parametric study, and potential changes are mentioned in the beginning of each chapter.

It should be also noted that for the sake of completeness, over 250 different geometrical configurations were analysed.


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2) Optimisation of the back-span to main-span ratio This chapter focuses on the behaviour of portal frame bridges with varying Objectives:

a) Find the optimal arrangement of the piers such that the sagging moment in the back spans equals the sagging moment in the main spans;

b) Investigate the influence of the structural arrangement by means of ratio upon the hogging moments at the deck-pier connection;

c) Determine the pier loading regime and the base reaction response depending on the pier height and Methodology: The first step consisted in building the geometry for a certain height. The initial height of the pier in this chapter was chosen as After this value was fixed, different portal frame bridges were analysed for different values of the above ratio. Same procedure was employed for different heights of the pier.

The 0 value in the back-span moment for the sagging column represents hogging which was not considered for objective a). Results: For objective a), the results are better illustrated in the following graph in which the vertical axis represents the normalised bending moments at mid span, while the horizontal axis represents the The bending moments were normalised with respect to the continuous beam case, which gives the extreme values.

Figure 2-1: Normalised Sagging Bending Moments

In which is: main span moment at mid span if i = 1, and back span moment if i = 2. The bending moments were normalised w.r.t. a value of 48.24 MN-m which is the maximum value from the recorded ones, and corresponds to the continuous beam case.

Similarly, the hogging moments were collected and plotted in the following figure, but this time the normalisation was done with respect to 10m pier height. For this case the maximum bending moment according to which all other values were normalised has a value of 72.6 MN-m.

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Figure 2-2: Normalised Hogging Bending Moments

This was done to investigate the response in terms of bending at the pier crown, and easily evaluate the horizontal reaction at the foundation level. The results collected for objective c), are plotted in the following two diagrams. It is to be noted the high sensitivity of the horizontal reaction for different ratios and the height of the pier.

Figure 2-3: Normalised Pier Bending Moments

Figure 2-4: Normalised Horizontal Reaction

The maximum horizontal reaction value recorded has a value of 2481 kN and corresponds to a value of 1 on the vertical axis on the previous figure.

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Conclusion: For objective a) the study shows that the maximum sagging moments in the deck are very sensitive to variations in in the position of the piers. The height of the piers varies the bending moments, but is not as important as the main span to back span ratio. Nevertheless, the pier height will be contrasted with main to back-span ratio further in this section of this part of the study.

An important result given by the parametric study is that at a value of equal to 1.25, the portal frame effect vanishes, and all the curves collapse at the value predicted by the continuous beam model. At this point, the sagging moments are equal in main and back span. As we shift to the right from 1.25 on the horizontal axis, an increase in the bending moments in the main span of the deck is observed. It can be also noticed that the bending moment curves tend to diverge, hence the height of the pier becomes important as the portal frame effect is mobilised. As it can be seen from Figure 2-1 , the continuous beam model represent a sort of envelope for the bending moments at mid-span of the deck, therefore it can be concluded that the portal frame effect is desirable in design of bridges over the continuous beam case. This can be explained by the increased redundancy of the structure. Moreover, as the mainspan is increased, smaller and smaller pier height show less variation on the graph, leading to smaller bending moments; therefore the pier height/stiffness is important in portal frame effect which is allowed for if is different than 1.25. Moreover, for a given height of the pier, there is a point on the horizontal axis for which the sagging moment in the back-span is zero; hence, any potential prestress is eliminated. The only part of the deck requiring these kinds of techniques is the main span, with possible debonding in the back-span.

The analysis conducted for objective b) of this paragraph shows the exact same point, 1.25 on horizontal axis, balances the hogging moments in the main and back-span, hence the portal frame effect is inexistent. As the main to back-span ratio is increased to achieve reasonable portal frame geometries, the hogging moments in the main span are observed to be larger than in the case of continuous beam deck. The bending moments at the back-span increase with slower rate, and for small values of the pier height, the variation is almost constant. This indicates that the piers are bended to keep the node in equilibrium. The portal frame effect can be therefore quantified directly from Figure 2-2 by the difference between the beam case and the lowest curve at 3.5 on horizontal axis.

This makes the point of objective c), and as can be observed from Figure 2-3, the pier bending moments are equal to zero at 1.25 It is now very clear the importance of the piers stiffness which increases by decreasing the height of the pier. If the portal frame effect is mobilised by moving away from the point equal to 1.25 on the horizontal axis of the graph in Figure 2-3 the stiffer piers are loaded more as the structure is statically indeterminate. It is noticed that for large pier stiffness (i.e. small heights) the bending moments at the piers crown increase drastically as the portal frame effect becomes pervasive. It can be therefore concluded that the behaviour of the portal frame bridge can be controlled by variations in the height or cross section of the pier.

Finally, the study conducted for objective c), shows that the curves describing the variation of the horizontal reaction at the base of the pier have a high sensitivity to peak values. Therefore, controlling the variation of the bending moments in the deck by both pier stiffness and ratio, heavily affects the reaction of the foundation. This aspect governs the design of portal frame bridges in some cases as the bearing capacity of the soil may be exceeded even for small variations in the parameters of interest in this chapter. All the parameters should be carefully considered when seeking for the best portal frame geometry as the relationship between various components of the bridges response is not linear. Moreover, for 1.25 main to back-span ratio, the horizontal reactions are changing their direction. This has a major effect on the axial stresses transmitted in the deck, as further discussed in the following chapters.

For preliminary design, one can rely on the conclusions drawn in this chapter and on the curves plotted in this paper, but for detailed analysis, more sophisticated methods have to be employed to analyse the behaviour of the portal frame bridge with the geometry predicted by the results in this article.

Design around the 1.25 main to back-span ratio is desirable; however due to various reasons this cannot be always achieved. In this cases the portal frame is preferred over continuous beam scheme, as the bending


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moments in the deck are reduced. Inclined piers or curved ones may reduce the main to back-span ratio, and is a value of 1.25 is not achievable, the portal frame effect may be enhanced by means discussed further in this paper.

The main to back-span ratio has a limitation though, and this can be seen from Figure 2-2. The points corresponding to zero hogging in the back span represent the points where uplift in the abutment is induced. In the case where abutments are not anchored, the ratio may go that far and no further.

3) Inclination of the piers The inclination of the piers has also significant influence in the design of portal frame bridges and therefore, this chapter focuses on this parameter.

Objectives: Define the effect of piers inclinations on the maximum sagging and hogging moment, the horizontal reaction at the foundation of piers, the bending moment at the pier crown and the axial force on the deck.

Methodology: The geometric configuration, material properties and the cross-sections of the deck and the piers remained unchanged in this analysis. The only variable is the inclination of the piers.

Results: The results of the analysis are illustrated in the following graphs. The bending moments and the internal axial forces in the deck as well as the reactions at the foundation of the pier were normalised based on the respective value in 90 inclination.

Inclination (kN-m) (kN-m) (kN-m) H (kN) Compression in mainspan (kN)

Tensile in backspan

(kN) 90 26380 43930 17570 1053 526.4 526.4

Figure 3-1: Normalised sagging moment at the main span, hogging moment at the pier-deck connection and moment reaction at the pier foundation

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Figure 3-2: Normalised horizontal reaction at the pier foundation

Figure 3-3: Normalised compressive/tensile axial force at the main/back span

Conclusions: It is evident by Figure 3-1 that the inclination of the piers has a negligible effect on the bending moments in the deck and a small one in the moment at the crown of pier. However, it affects significantly the axial forces on the deck and the horizontal reaction at the pier. Due to ratio the compressive and the tensile force on the main and back span respectively are equal but it must be underlined that this is not a general case.

Comments: It is important to point out that in a long span bridge, the axial force that is introduced in the deck is of very small magnitude comparing to the bending moments. Therefore, the axial stress is almost negligible with respect to the flexural stresses. The figure below presents the axial and bending stresses for different inclinations.

Figure 3-4: Maximum Axial and Bending stresses at the deck

Consequently, in long span portal frame bridges, the inclination of the piers mainly affects the foundations and not the structure.

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On the contrary, in shorter bridges where flexure is not that dominant, the effect of axial stresses is more significant. By analysing a bridge with 30m span, =1.25, 40 inclination of the piers and a rectangular long term concrete section A = 4 m2, I = 1.333m4 for the deck and A = 6m2, Ix = m4, Iy = 4.5m4 for the piers, the influence of the axial stress is obvious:

Figure 3-5: Stress Distribution at the deck. In the right diagram, the combination of axial and bending stresses is presented.

Based on the previous chapter, a further parametric analysis of the inclination of the piers was made with . As expected, only the horizontal reaction at the piers and the axial force in the deck were affected. It is interesting to underline that in this specific case, where the piers do not have bending moment, the compressive axial force in the deck is negligible when the inclination is 90. However, as the inclination increases, the values of the compressive axial force in the deck are converging to those found by the first parametric analysis where . As far as the tensile force is concerned, it is obviously that it decreases as is decreased.

Figure 3-6: Compressive axial force in the main span

Figure 3-7: Tensile axial force in the back span


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4) Flexural Pier Stiffness Objective The purpose of this parametric study is to find the sensitivity of the portal frame effect to the variation of the flexural stiffness of the piers. The flexural stiffness of the piers was varied in 2 ways: Changing the second moment of area, of the piers while maintaining the same axial stiffness Changing the height, of the piers

The effect of the piers flexural stiffness on the sagging moment at midspan, the moments in the pier and the reactions at the support, was considered.

For the control portal frame the results were as follows: . Influence of on the Portal Frame Effect The flexural stiffness of the pier was varied by changing the ratio from and the following results were obtained:

Figure 4-1: Sensitivity of the moment at midspan, Mmidspan, to the variation of Ipier/Ideck

From the above curves, it is clear that the moment at midspan tends to decrease although not significantly (up to 11%) with increasing that is, the portal frame effect will be enhanced. This is expected as a higher flexural stiffness means that a higher moment will be taken by the pier which leads to a reduction in . Consequently, since the total static moment is constant ( the hogging moment at the ends of the main span will increase.

It is also important to note that % variation in is high for and low for . Therefore, the sensitivity of the portal frame effect to the variation of increases as decreases, and decreases as the pier becomes stiffer in flexural relative to the deck (see Figure 4-1) o Convergence of as

In the parametric study conducted, the axial stiffness was kept the same and the flexural stiffness was varied. In reality, increasing the pier flexural stiffness by increasing the second moment of area results in a higher pier cross-sectional area, and thus a higher axial stiffness and vice versa. In the limit case, when both the axial and the flexural stiffness are very high ( ), the piers will act as full fixity supports Figure 4-3. Accordingly, as shown in Figure 4-1 while the hogging moments will be . Theoretically, if the pier axial stiffness is very low (~0) while the flexural stiffness is very high ( ), the mainspan will still be fixed against rotation but can freely move in the vertical direction.

In all cases, for high pier flexural stiffness (high ), will converge to regardless of the axial stiffness of the piers.

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o Convergence of as On the other hand, for piers with very low flexural stiffness (~0), the axial stiffness should also be very low and the whole deck will act as 1 simply supported beam (Figure 4-3). In this extreme case, which is very high (Figure 4-1). In addition, theoretically speaking, if the axial stiffness is very high ( ) but the flexural stiffness is negligible (~0) the deck will behave as a continuous beam and . In reality, for low pier flexural stiffness (low ) will be between the 2 values but much closer to where the axial stiffness is also very low:

Figure 4-2: Sensitivity of the moments in the pier and the horizontal reaction at the support to the variation of Ipier/Ideck

Figure 4-3: The 2 limit cases for Ipier/Ideck Left: Simple supported beam with a span of 2L (Ipier/Ideck 0), Right: Beam with fully fixed supports at the main span (Ipier/Ideck )

As stated previously, the higher the flexural stiffness of the pier the more moment taken by the pier. This can be seen in the Figure 4-2 above where the moment in the pier crown, , the moment at the support

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and increase with increasing . It is important to note that the variation in , and is much more significant than that of the midspan moment (up to 67%). The figures above also confirms what was stated before regarding the sensitivity of the portal frame effect to the variation of .

o Convergence of , and as Practically speaking, increasing the pier flexural stiffness by varying results in very high pier axial stiffness. In the limit case, the piers will act as fixed supports as stated before. Therefore, the moment at the crown of the pier will be: . (see Figure 4-2) If the axial stiffness of the piers was very low (~0), (which is not realistic in this case because increasing will result in high axial stiffness), then the supports will be free to move vertically as stated previously and theoretically: In reality, for high pier flexural stiffness (high ), will be between the 2 values but much closer to respectively, where the axial stiffness is also very high: For very high , is half the value of (Figure 4-2) The horizontal reaction is . Therefore, o Convergence of , and as In the case where the flexural stiffness is very low, the portal frame effect will be eliminated and the moment in the pier will be negligible (Figure 4-2). Influence of the Pier Height on the Portal Frame Effect

The height of the piers was varied from to 3 and the results were plotted on the same graph as shown below.

Figure 4-4: Moments at midspan pier crown and support reactions

For the moments in the portal frame, the effect of varying the height is inverse to varying the second moment of area. That is the sensitivity of the moments to changing the pier second moment of area by a factor of is

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the same as changing the pier height by a factor of . On the other hand, the sensitivity of the horizontal reaction to changing the pier height is much higher (2nd order); changing the pier second moment of area by a factor of is the same as changing the pier height by a factor of . Therefore, while both changing the second moment of area and the height of the pier have an inverse effect in terms of the frame moments (as the flexural stiffness is ), the sensitivity of the horizontal support reactions is much higher to the variation of the pier height. For an extreme variation in the pier height, the convergence of the moments will be the same as the variation of the second moment of area. However, for a very short pier (0), the flexural stiffness will be very high and the portal frame will act as a beam with 2 fixities (Figure 4-3). The horizontal reaction will collapse towards 0. This is shown in Figure 5-2. Finally, it is important to note that enhancing the portal frame effect by increasing the second moment of area is more recommended than shortening the piers. This is because it results in smaller horizontal reactions at the fixed supports to be taken by the foundation. If however, the foundation is stiff enough (e.g. rock), and it is more feasible to use shorter piers, then increasing the portal frame effect can be done by shortening the piers.

5) Investigation of Soil-Structure Interaction This chapter deals with the behaviour of portal frame bridges with varying the stiffness of the ground conditions. In order to investigate in detail the soil-structure interaction, more than 200 combinations of different soil conditions and pier heights were analysed.

Objectives: a) Investigate the influence of the horizontal soil stiffness on the portal frame effect b) Investigate the influence of the rotational soil stiffness on the portal frame effect

Methodology: In order to simulate the stiffness of the ground, springs were introduced in the base of the piers. For the purpose of this study only the horizontal and rotational stiffness of the soil are taken into consideration by horizontal and rotational springs. The vertical soil settlement in both piers is considered uniform and does not affect the analysis.

Figure 5-1: Spring Modelling of soil stiffness.

Results: The results of the analysis which are presented are focused on the horizontal reaction in the base of the piers, the moments in the crown of the piers and the sagging moment in the main mid-span.

For case study a), the horizontal soil stiffness was varied from Kx = 100MN/m for soft soils to very high Kx for hard rocks. Concerning the pier height, it was varied to h = 25m, while considering values for short piers.

The horizontal reaction in the base of the pier (H), for the above soil and height combinations, is presented in the next graph. It must be mentioned that H is normalised with respect to the Hmax=16.4 MN, which is the maximum value for the recorded ones, representing the horizontal reaction of short piers ( h = 7m) and hard rock supports (ux fixed) .


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Figure 5-2: Normalised Horizontal Reaction at the base of the piers

It is clear that for a given pier height, the higher the horizontal stiffness of the soil the higher the horizontal reaction. However, the horizontal reaction is more sensitive to the soil stiffness variation for short piers, rather than for high piers. In our case, if h>15m the horizontal reaction converges to the same value for both soft soils and hard rocks, tending to zero for extremely high piers.

The moment in the crown of the pier, presented below, is normalised with respect to the Mmax = 69.9 MN-m, which is the maximum value for the recorded ones, corresponding to infinite horizontal springs stiffness and piers with zero length ( fixed beam). As the horizontal soil stiffness decreases the moment in the crown of the pier decreases as well. What is more, the moment in the crown of the short piers is significant sensitive in soil stiffness variation.

Figure 5-3: Normalised Moment in the crown of the pier.

The moment at mid-span shown in Figure 5-4 is normalised with respect to the minimum recorded sagging moment Mmin = 56.7 MN-m, which is the moment in the case of fixed beam (zero pier height). As the stiffness of the soil decreases the sagging moment at mid-span also decreases, but there is no significant variation related to soil stiffness degradation.

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Figure 5-4: Normalised Moment in the middle of the mid-span.

For the case study b), the rotational soil stiffness is varying from Krot = 1000MN-m for soft soils to extremely high Krot for hard rocks. As for the height of the piers, it was varied up to h = 25m, while considering values for short piers, as well.

In this case, the horizontal reaction in the base of the pier (H), presented in the graph below, is normalised with respect to the maximum recorded value as in the first case.

Figure 5-5: Normalised Horizontal Reaction at the base of the piers

Figure 5-6: Normalised Moment in the crown of the pier

The moment in the crown of the pier, presented above, is normalised again with respect to the Mmax = 69.9 MN, corresponded to infinite rotational springs stiffness and piers with zero length (fixed beam). As the rotational soil stiffness increases the moment in the crown of the pier increases as well. It is worth to mention that the moment in the crown of only short piers is sensitive in rotational soil stiffness degradation. The moment at midspan is normalized as in the first case. As the stiffness of the soil increases the sagging moment at mid-span also increases, but there is small variation related to soil stiffness degradation, as it is noticed in the next graph.

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Figure 5-7: Normalised Moment in the middle of the mid-span.

Finally, it must be mentioned that the previous graphs and results are also representative of the behaviour of the structure taking into account the stiffness of the link between the piers and the foundation. In this case the springs represent the stiffness of the connection, varying from pinned to fixed connections and the stiffness of the foundation is considered infinite.

Conclusion The soil structure interaction is a very important parameter with a huge impact on the portal frame effect. The designers must take into account the stiffness of the soil in order to estimate the real internal forces in the structure and check the capacity of the foundation. The horizontal reaction is a control parameter which is directly related to the soil stiffness, as presented in Figure 5-2 and Figure 5-5. For a given height, when the horizontal stiffness increases and/or the rotational stiffness degrades the horizontal reaction increases, as well. The effect of soil degradation is more significant in short piers rather than in high ones. Moreover, in bridges with short piers the moments in the crown of the piers and the moments in the main mid-span are more sensitive to soil stiffness variation. However as it is shown in Figure 5-3 and Figure 5-4, the soil stiffness affects the pier crown moment more than the mid-span moment. As a result, the improvement of the foundation does not always guarantee significant reduction in the above moments. The soil structure interaction impact in portal frames is a complicated combination of the soil stiffness and the pier flexibility.

6) Effect of Uniform Imposed Temperature Variation and Uniform Shrinkage on the Deck

This chapter focuses on the main behavioural aspects of portal frame bridges subjected to a uniform shrinkage of the deck. Essentially, shrinkage induces a uniform compressive strain in the deck, which results in its shortening. Hence, shrinkage can be considered to have an equivalent effect to an imposed uniform temperature variation of a specific amount, imposed on the deck.

According to BS 5400, a typical value of Shrinkage which is usually taken into account for the design of concrete bridge decks is . For a long term thermal coefficient of the concrete approximately equal to , an equivalent temperature variation resulting in shortening of the deck can be determined, such that it creates the same strain effect as the above typical value of shrinkage. The equivalent temperature variation is T = -13.5 C. Hence, our parametric study will mainly focus on imposing a temperature variation on the deck of the portal frame, and observing the variation of the Horizontal Reactions at the foundation, the bending moment at the pier crown and the compressive force on the deck, as well as their sensitivity to various parameters. Due to space limitations, only the results regarding the variation of the horizontal reactions at the foundation will be demonstrated. The latter is highly representative of the variation of the portal frame effect, while it is also an index of the effect on the foundation in each case, which is a matter of utter significance in bridge design. The main objectives of this parametric study are presented below:

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Krot= 1000 MNm

Krot= 2000 MNm

Krot= 4000 MNm

Krot= 6000 MNm

Krot= 8000 MNm

Krot= 10000 MNm

Krot= 20000 MNm

Krot= 40000 MNm

Krot= 60000 MNm

Krot= 80000 MNm

Krot= 10000 MNm

Krot= fixed

Krot=0 (pinned)


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HORIZONTAL REACTION VARIATION AT THE FOUNDATION FOR DIFFERENT SUPPORT CONDITIONS AT THE ABUTMENTS

Horizontal Reaction at the foundation closer to the hinged abutment support-1 Hinged/1 RollerSupportHorizontal Reaction at the foundation closer to the roller abutment support-1 Hinged/1 RollerSupportHorizontal Reaction at Supports-2 Roller Supports

Horizontal Reaction at the Foundation-2 Hinged Supports

Objectives Demonstrate the effect of the support conditions at the abutments on the portal frames behaviour, when the deck is subjected to a uniform temperature variation/Shrinkage. Investigate the sensitivity of the Portal Frame effect to varying pier height, for the worst case scenario, i.e. different support conditions at both abutments, when the deck is subjected to a uniform temperature variation/Shrinkage. Investigate the sensitivity of the portal frame effect to different geometric arrangements of the piers, i.e. different , when the deck is subjected to a given temperature variation/shrinkage

Methodology, Results and Interpretation It should be noted that the cross-section of the deck, the cross-section of the piers, as well as the material characteristics, remained unchanged. Three different parametric studies were conducted.

The first parametric study considers the case of a uniform temperature variation of the bridges deck, resulting in shortening, for a typical range of temperatures commonly observed in bridge structures. In general, the support conditions at the abutments of portal frame depend on multiple factors, such as the geometry of the bridge, the topography, the design considerations, etc. Therefore, it is essential to demonstrate how various support conditions affect the portal frames behaviour under imposed deck deformation due to temperature variation or shrinkage.

The results shown in Figure 6-1 are obtained from analyses performed on portal frames of the same geometrical and loading arrangement, with 3 different support conditions at the abutments:

2 Hinged Supports 1 Hinged Support and 1 Roller Support 2 Roller Supports

Figure 6-1: Relative Variation of Horizontal Reaction in the Foundation, for 3 different support conditions at the abutments


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By observing Figure 6-1, we can define 3 different behavioural bounds of the Portal Frame, depending on the support conditions at the abutments:

1. 2 Hinged Supports at the Abutments: In this case, a uniform imposed temperature variation/shrinkage, induces an axial force in the deck, which is directly taken by the abutments. Hence, the piers and the foundation remain unaffected, and the portal frame effect remains essentially unchanged.

2. Different Support Conditions at the Abutments1 Roller Support/1 Hinged Support: In this case, the induced deck deformation is resisted by both the hinged abutment and the foundation-base of the piers. The behaviour of the portal frame under uniform imposed load/ shrinkage is asymmetric, and both piers and the foundation are affected. As demonstrated in Figure 1, the foundation closer to the abutment which is supported on roller support is more sensitive to temperature variations and deck shrinkage, than the support closer to the abutment which is hinged.

3. 2 Roller Supports at the Abutments: In this case, there is no means by which the induced axial force in the deck can be resisted by the abutments; hence it is evenly resisted by the foundation. The response of the portal frame is symmetric, and the change in the portal frame effect is expressed through the equal variation of the horizontal reactions at both piers bases. However, in this case, due to the fact that the load is shared among the two piers, the sensitivity of the latter to imposed temperature variation and deck shrinkage is smaller than the second case described above

It should also be mentioned that the response of the portal frame varies linearly with increasing temperature in the case of roller supports at the abutments, while there is a deviation from the linear variation model in the case of different support conditions.

For the second parametric study, the worst case scenario, in terms of the temperatures effect on the portal frame and the foundation (i.e. different support conditions at the abutments), is considered. For the same range of temperatures as in the first study, analyses for reducing pier height were performed, in order to demonstrate the temperature variation/shrinkage effects sensitivity to different pier heights. The pier heights considered, where 25 m, 20 m, 15 m, 10 m, 5 m, and 1 m. The last 2 values, especially the last one, although not realistic in practice, were considered for completeness, as well as to demonstrate a behavioural limit boundary for the portal frame subjected to temperature variation/ shrinkage of the deck.

Figure 6-2: Relative Variation of Horizontal Reaction in the Foundation under Uniform Temperature Variation imposed on the Deck, for variable pier Height

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VARIATION OF THE HORIZONTAL REACTION AT THE FOUNDATION FOR VARYING PIER HEIGHT


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As shown in Figure 6-2, the pier height has a dominant effect on the response of the portal frame. In general, the variation of the pier height, and the corresponding variation of the pier stiffness, leads to an enhancement of the portal frame effect (i.e. moment resisted at the crown of the piers and horizontal reaction at the foundation). Hence, by observing the diagram shown above, we can infer that this enhancement of the portal frame effect through the increase of the pier stiffness, leads to a greater sensitivity of the structural system to imposed temperature variation/ shrinkage. It is clear that for the same amount of height reduction, i.e. 5m, the relative variation of the horizontal reaction becomes greater as the pier height decreases, for a given imposed temperature variation. Moreover, the pier closer to the abutment where the portal frame is supported on roller support, is affected most, as it was expected.

Finally, for the third parametric study, the same conditions and temperature variation as in the above parametric study were considered, and the piers position varied, in order to demonstrate the temperature variation/shrinkage effects sensitivity to different main span to back span ratio . The values considered for , correspond to the increase of the length of the 2 back spans by 5m, with a decrease in the main spans length, accordingly. Figure 6-3 shows that the relative variation of the horizontal reaction at the Foundation varies highly nonlinearly with varying main span to back span ratio .

Figure 6-3: Relative Variation of the Horizontal Reaction in the Foundation under uniform Temperature variation imposed on the Deck, for variable Main Span to Back Span Ratio

For a given uniform temperature variation/ shrinkage imposed on the bridge deck, we can see that up to a ratio of , the horizontal reactions sensitivity to this ratio increases, and right after this ratio, a great discontinuity occurs, leading to a degradation branch, where the sensitivity of the horizontal reaction variation is less. Hence, the behaviour of the portal frame under varying is more sensitive for values below 1.25, than for values greater than this. Furthermore, the greatest sensitivity of the horizontal reaction variation is observed at this ratio, and the main reason behind this lies in section 2 of the project. At a ratio equal to 1.25, no flexure of the piers occurs, while for values of greater than this, the direction of the Horizontal Reaction at the Foundation is reversed, hence leading to a completely different behaviour of the portal frame.

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Relative Variation of Horizontal Reaction at the Foundation Level - Back Span to Main Span Ratio


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7) Conclusion-Case Study

To conclude, the parametric study presented before was employed to propose a portal frame geometry for a bridge crossing river Thames. The total span was considered 250 m, which corresponds to the average width of Thames in London. The minimum pier height is 20 m from which half is above the water level.

Based on Figure 2-1 for rationalising the bending moment distribution on the deck, the following geometric arrangement would be desired. Nevertheless, the minimum distance between the piers should be 164m due to fluvial traffic.

Based on Figure 6-1, in order to avoid the effect of uniform shrinkage of the deck on the foundation, hinged supports were considered in both abutments. In that manner, the induced axial force in the deck due to shrinkage is directly resisted by the abutments.

Based on Figure 2-1 and Figure 2-2, with the current main to back-span ratio (3.81), a potential uplift in the abutments was considered. Moreover, the sagging moment in the main-span was to be reduced.

It is presented on chapter 2 that the most uniform bending moment distribution over the deck is achieved when the main to back-span ratio has a value of 1.25. For this reason and the purposes presented in the previous bullet, the piers were inclined to 500, decreasing the main to back-span ratio to 2.167. Hence, the uplift peril was prevented as well. Besides the reduction of main to back-span ratio, the inclination of the piers introduced a higher compressive/tensile force on the main/back span


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(Figure 3-3). However, based on Figure 3-4, the axial stresses are negligible comparing to the bending stresses and therefore, it did not influence our design.

Based on Figure 5-2, considering the current pier height, an analysis with fixed piers gives the same value of the horizontal reaction as the one from an analysis with any horizontal stiffness of the soil. For an increase portal frame effect, high stiffness in the piers is desirable. According to Figure 4-1, the bending moment at midspan was decreased even further by increasing the stiffness of the pier 10 times.

Based only on results from this paper, the following geometry was predicted and as it can be seen, it shows an elegant portal frame behaviour:

- Decreased bending moment at midspan;

- Decreased flexure effects at back-span;

- Piers loaded in both flexure and compression (bending may be reduced by curved piers);

- Considerable horizontal reactions.


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REFERENCES

Design of Bridges Lecture Notes, Dr. Alfredo Camara, Dr. Ana M. Ruiz-Teran, Imperial College London 2014

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