Bridge deck design - Proposed solutions and initial analyses

3.2 Proposed solutions and initial analyses

3.2.1 Bridge deck design

The bridge deck used by all the different bridge designs consist of a thin 10 𝑚 × 4 𝑚 metal plate to serve as a surface for the live load, three longitudinal beams of 10 𝑚 length that will carry the main weight of the thin plate and two or three 4 𝑚 long beams to act as girders on each end and placed below the beams depending on the situation.

This will create a one-way slab system [1, p. 58] that will distribute the load as shown in Figure 7 being so that the dark grey area will be the load supported by the central beam, represented as CD in the figure, and the rest of it distributed evenly to the side beams AB and EF. It is then observable that the central beam will be the critical component from the bridge deck struc-ture as it will have to support 8 𝑘𝑁/𝑚 while the side beams will support half the amount.

Figure 7. Weight distribution on a one-way slab system. Adapted from [1, fig.2-11(b)]

36 Thus, the central beam can be represented as:

The maximum design shear force 𝑉_𝑒𝑑 and moment 𝑀_𝑒𝑑 can then be calculated as 𝑉_𝑒𝑑 =

8𝑘𝑁 𝑚⁄ ·10 𝑚

2 = 40 𝑘𝑁 and 𝑀_𝑒𝑑 = ^{8𝑘𝑁 𝑚}^{⁄ ·(10 𝑚)}²

8 = 100 𝑘𝑁 · 𝑚.

According to the aforementioned one-way slab system, it is manifest that the shear and moment forces acting on the side beams will be half of those acting on the central beam.

Like so, the girder beams could be represented as:

Which will produce a maximum shear of 𝑉_𝑒𝑑 = 40 𝑘𝑁 and moment of 𝑀_𝑒𝑑 = 40 𝑘𝑁 · 𝑚 on the beam. Despite this direct analysis not being accurate for every bridge model as it will depend on the column placement, it is the worst-case scenario when assuming evenly spaced supports.

Figure 10. Bridge deck 3D design. Dimensions in meters. (Jordi Mata Garcia, 2021)

Figure 8. Deck’s central beam ideal representation. (Jordi Mata Garcia, 2021)

Figure 9. Deck girder-beam ideal representation. (Jordi Mata Garcia, 2021)

37 3.2.2 Simply supported bridge

On an initial investigation and due to the deck spans being 10 𝑚 long, the most logical column placement is at the union of said spans thus resulting in two-column joints as depicted in Figure 11.

At each column joint, there are then different possibilities concerning the number of supporting members. Being the deck light material-wise due to it being a pedestrian bridge it is safe to assume that both the environmental forces and the live loads travers-ing it will create a considerable rocktravers-ing motion effect on the platform. In consequence, a single column system is not recommended as it will be exposed to sudden unexpected bending forces that may cause the beam to suddenly fail, hence a double-column system is the next best option cost-effective wise since it will provide enough stability to com-pensate the bending effort without compromising the stability of the structure. Its placement, nevertheless, will be on the edges of the light and dark grey areas depicted in Figure 7 (at 1 𝑚 from the side edge of the deck) for a more even distribution of the act-ing forces.

The compressive force acting on each of the columns will then be each of the reactions on the girder beam, which due to symmetry are both of 40 𝑘𝑁.

Figure 11. Simply supported bridge representation. (Jordi Mata Garcia, 2021)

38 3.2.3 Simple cantilever bridge

Despite most of the cantilever bridges being built with the help of trusses to adequately distribute and balance the loads, on a span of 30 𝑚 that is very ineffective since the space required by both the support structures and the suspended middle part are very large. Therefore, a simple cantilever structure will be used:

As can be seen from Figure 12, the structure is in fact a double cantilever, each of the side loads balancing each other.

This structure offers the possibility of a single column base at the cost of the column beams taking not only pure com-pressive force as in the simply supported bridge but also bending stress created by the cantilever beam situation pro-duced once the force is divided. Assuming again a double-column system, the 40 𝑘𝑁 received from the girder beam will be divided as shown in Figure 13, rendering that 𝐹_𝑐 = 35.78 𝑘𝑁 and 𝐹_𝑏 = 17.89 𝑘𝑁, the latter creating a bending moment of 200 𝑘𝑁 · 𝑚 on the beam.

Figure 12. Simple cantilever bridge representation. (Jordi Mata Garcia, 2021)

Figure 13. Cantilever column forces. (Jordi Mata Garcia, 2021)

Figure 15. Pratt truss bridge design with forces. Red is compression, blue is tension. (Jordi Mata Garcia, 2020)

3.2.4 Truss bridge

Truss bridges are likely one of the preferred options when it comes to short gaps due to their ease of assembly and their weight-carrying capabilities without the need for sup-ports.

Despite the many different truss bridge designs, the Pratt truss was the one chosen for the initial design due to the critical components in the bridge being purely in compres-sion and the possibility of below-deck clearance to keep things in the simplest of ways.

The bridge will then be divided into 6 sections, two of them being triangular sections at each end of the bridge and 5 𝑚 × 5 𝑚 square sections for the middle four sections with diagonal supports as described in Figure 14.

The reason for choosing squared sections is that having 45° angles on the diagonal beams help minimise the forces in both tension and compression members. Since modi-fying that angle decreases the forces in either the tension or compression members while increasing the opposite, the most optimal design to have the structure based on members under compression yet keeping it minimal is the chosen 45° angle.

Breaking down the forces inside the design following the traditional analysis results in:

Figure 14. Pratt truss bridge design. (Jordi Mata Garcia, 2020)

As observed in Figure 15, most of the longest members of the structure are kept under tensile load. This is beneficial as beams are prone to buckle under compressive forces, an effect that gets increased dramatically fast with the length of the member. Thus, upon initial inspection, the critical components to be studied will either be the end diagonal members under 169.7 𝑘𝑁 of compressive force or the top members enduring 216 𝑘𝑁 of compressive force.

It is important to note that this calculation is made assuming that the total load of 4 𝑘𝑁/𝑚² is distributed evenly between each side of the truss and then acting solely on the joints. With the help of COMSOL software, it is possible to evaluate the effects of an evenly distributed load of 8 𝑘𝑁/𝑚 on the half truss structure analysed in Figure 15, observing that all the forces detailed in the figure get reduced by about 15%. For the sake of assuming the worst-case scenario, the forces detailed in Figure 13 will be taken as the ones acting in the structure.

3.2.5 Failure analysis applied to the critical components

According to the traditional calculations, the maximum bending stress 𝜎_𝑚𝑎𝑥 [7, Eq. (6-12)] can be calculated according to:

𝜎_𝑚𝑎𝑥 =𝑀_𝐸𝑑 · 𝑐

𝐼 (28) where c is the distance from the neutral axis to the outermost side on the bending axis on the cross-section.

Similarly, the maximum shear stress 𝜏_𝑚𝑎𝑥 caused by the shear force 𝑉_𝐸𝑑 [7, Eq. (7-3)] is obtained as follows:

𝜏_𝑚𝑎𝑥 =𝑉 · 𝑄_𝑚𝑎𝑥

𝐼 · 𝑡 (29) where 𝑄_𝑚𝑎𝑥 is the maximum first moment of inertia of the cross-section.

Note: all the calculations will be made with the use of an Excel sheet where all the re-quired calculations have been automated depending on loads, boundary conditions, cross-section profiles and steel category. Refer to Appendix E for more information.

41 3.2.5.1 Bridge deck – Central Beam

According to the values obtained in section 3.2.1 and using Equations 28 and 29, there is a possibility of evaluating the impact that 𝑀_𝐸𝑑 and 𝑉_𝐸𝑑 may have on the I-beam cross-section according to the IPE standards in terms of bending and shear stress based on a traditional stress analysis as shown in Table 9. It is easily observable that the limiting stress is the bending stress, which is an expected outcome when having a lengthy simply supported beam.

Table 9. Profile - stresses based on the traditional analysis. (Jordi Mata Garcia, 2021)

Profile IPE220 IPE240 IPE270 IPE300 IPE330

𝝈_𝒎𝒂𝒙 [𝑴𝑷𝒂] 396.83 308.33 233.16 179.51 140.19

𝝉_𝒎𝒂𝒙 [𝑴𝑷𝒂] 42.90 37.18 31.15 26.18 22.53

There are many different steel categories depending mostly on its yielding stress, rang-ing from 235 𝑀𝑃𝑎 up to 450 𝑀𝑃𝑎. For the sake of simplicity, the lesser yieldrang-ing stress steel (S 235) will be assumed thus establishing the material’s yielding stress at 235 𝑀𝑃𝑎.

That assumption establishes, then, that IPE270 is the smallest profile that can be used since the maximum stress that it endures is barely below the yielding stress. If a certain safety margin is to be supposed, IPE330 is the next safest option since it is below 75%

that of the yielding stress of the material.

However, when the maximum permitted deflection 𝛿_𝑚𝑎𝑥 obtained from Figure 3 is compared to the limit condition deflection caused by a uniform distributed load on the beam 𝛿_𝐿𝑖𝑚 as in Table 6: The safety condition is failed; thus a different profile is needed. The smallest profile to

satisfy that condition is the IPE400, hence the analysis will proceed with it.

Having it classified as a Class 1 cross-section the plastic analysis can be carried out on its own without any reduction affecting its resistance.

Through Equation 9 and Table 5 it is obtained that the design plastic shear resistance is 𝑉_{𝑐,𝑅𝑑}_{𝐼𝑃𝐸400} = 579.21 𝑘𝑁 and the moment capacity 𝑀_{𝑐,𝑅𝑑}_{𝐼𝑃𝐸400} = 307.15 𝑘𝑁 · 𝑚. Both values being bigger than the design values, it is confirmed that the beam will retain its full plastic capacities with ease.

As per the elastic critical moment, knowing that it is a simply supported beam with free ends and with a uniform distributed load applied on the top flange of the beam, it is ob-tained through Equation 20 that 𝑀_𝑐𝑟_{𝐼𝑃𝐸400} = 107.85 𝑘𝑁 · 𝑚. That value is greater than the design moment value 𝑀_𝐸𝑑 proving that the beam is indeed resistant to elastic defor-mation.

Finally, for the buckling resistance verification of the beam, there are two alternatives as specified in section “2.3.3.3 – Buckling resistance”: the method for I-shaped profiles or the general approach. If the method for I-shaped profiles is used, Equation 13 renders that 𝑀_{𝑏,𝑅𝑑}_{𝐼𝑃𝐸400} = 100 𝑘𝑁 · 𝑚. Despite that being deemed as safe per the method, when using the general approach it is obtained that 𝑀_{𝑏,𝑅𝑑}_{𝐼𝑃𝐸400} = 86.5 𝑘𝑁 · 𝑚, making it fail.

This does not explicitly mean that the beam is unsafe as it fulfils the analysis dedicated exclusively for its profile shape. Yet, to guarantee maximum safety in the most critical component, the IPE450 profile will be chosen as the definitive profile shape for all the beams present in the bridge deck component.

3.2.5.2 Simply supported bridge – Column

The main threat to a member that endures compressive forces, such as columns and pil-lars, is buckling failure. As per the traditional analysis, the elastic critical buckling stress 𝑁_𝑐𝑟 can be obtained through Equation 21 while applying a certain effective length factor to the length of the member depending on the boundary conditions.

To serve as an example, a cantilevered beam would have a factor of 𝐾 = 2. Using said values and knowing the column would sustain an axial load of 𝑁_𝐸𝑑 = 40 𝑘𝑁, an IPE160 profile can endure a buckling force of 𝑁_𝑐𝑟_{𝐼𝑃𝐸160} = 45 𝑘𝑁 in its strong axis mak-ing it a good candidate to a reasonable extent.

Nonetheless, if proceeded with the Eurocode 3 safety conditions it is revealed through Equation 23 that this profile can only offer a total column strength of 𝑁_{𝑏,𝑅𝑑}_{𝐼𝑃𝐸160} = 13.38 𝑘𝑁, making it inadequate for safety.

The smallest profile that can offer an available column strength so that 𝑁_{𝑏,𝑅𝑑} ≥ 𝑁_𝐸𝑑 is an IPE240, with a total amount of 𝑁_{𝑏,𝑅𝑑}_{𝐼𝑃𝐸240} = 54.09 𝑘𝑁.

3.2.5.3 Simple cantilever bridge – Column

As in the previous situation, the column will withstand a vertical force coming from the girder of 𝐹 = 40 𝑘𝑁, yet as described in section “3.2.3 – Simple cantilever bridge” that force is divided into two components: an axial load of 𝑁_𝐸𝑑 = 35.78 𝑘𝑁 and a bending point load of 𝑃 = 17.89 𝑘𝑁, the latter creating a shear force of 𝑉_𝐸𝑑 = 17.89 𝑘𝑁 and a bending moment of 𝑀_𝐸𝑑 = 200 𝑘𝑁 · 𝑚 at the base of the beam.

For the compressive part, the analysis proceeds in the same fashion as in the previous case yet considering that the length of the beam is now 𝐿 = 11.18 𝑚. This agrees with the previous analysis, indicating that IPE240 is the smallest profile able to produce an available column strength greater than the axial load applied.

As per the buckling resistance, it is uttermost important to establish the alignment or orientation of the beam. If the bending moment happens on the strong axis y-y, the re-sistance of the beam will be several times higher than if the bending moment takes plac-es along the weak axis z-z.

Just on the grounds of showing an example, the biggest IPE profile IPE600 will have a moment capacity of 𝑀_{𝑐,𝑅𝑑}_{𝐼𝑃𝐸600} = 825.32 𝑘𝑁 · 𝑚 on its strong axis but only of 𝑀_{𝑐,𝑅𝑑}_{𝐼𝑃𝐸600} = 114.12 𝑘𝑁 · 𝑚 on its weak axis, making it unsuitable to take the design moment of the beam. Similarly, the maximum deflection in the beam in its strong axis will be 𝛿_𝑀𝑎𝑥 = 0.0431 𝑚 just under the limit deflection value 𝛿_𝐿𝑖𝑚 = 0.0447, mean-while on its weak axis could reach 𝛿_𝑀𝑎𝑥 = 1.18 𝑚.

Based on the maximum deflection of the beam, the only suitable profile is the IPE600, which will prove to be a suitable candidate by having a buckling resistance of 𝑀_{𝑏,𝑅𝑑}_{𝐼𝑃𝐸600} = 589.72 𝑘𝑁 · 𝑚. As observable, the limiting factor in this critical compo-nent is the deflection endured due to the bending moment more than the moment itself.

3.2.5.4 Truss Bridge – Members under compression

In the case of the truss bridge, there are two possible critical components:

1. Horizontal member: 𝑁_𝐸𝑑₁ = 216 𝑘𝑁, 𝐿_𝑜₁ = 5 𝑚.

2. Diagonal member: 𝑁_𝐸𝑑₂ = 169.7 𝑘𝑁, 𝐿_𝑜₂ = 7.071 𝑚.

For case number 1, the smallest profile that can endure 𝑁_𝐸𝑑₁ is IPE270 with an available column strength of 𝑁_{𝑏,𝑅𝑑}₁ = 282.61 𝑘𝑁.

For case number 2, the smallest profile that can endure 𝑁_𝐸𝑑₂ is IPE300 with an available column strength of 𝑁_{𝑏,𝑅𝑑}₁ = 214.67 𝑘𝑁.

As predicted, the diagonal members are the critical components in truss bridges. Thus, the IPE300 profile will be the one used for the beams that make up the bridge.

3.2.6 Summary of beam profile selection according to Eurocode 3

Bridge deck: IPE450.

Simply supported bridge column: IPE240.

Simple cantilever bridge column: IPE600.

Truss bridge members: IPE300.

3.3 COMSOL Methodology

The COMSOL analyses will be performed using the Structural Mechanics module ad-don.

A 3D stationary study will be performed on a multi-physics interface composed of a beam interface to act as the beam skeleton of the structure and a shell interface to act as the thin plate that carries the load applied to the structure. This will enable the “Mul-tiphysics” interaction analysis, where “Shell-Beam” connections can be established over the shared edges of the beams and the thin plate with an offset established at half the height of the beam to indicate that the load of the thin plate is applied on the top flange of the beam.

In the study, it is very important to establish the correct orientation of the cross sections as the structures are designed according to the specifications of the I-beam profiles and using their strong axis. A wrong orientation may result in weaker structures.

As per the simulation of the deck beam interactions, a symmetry physic is applied to the joint between the longitudinal beams using axis 3 as the symmetry plane normal. Since a free end physic behaviour cannot be applied as the software does not accept an easy implantation of two beams resting on the same girder section, the symmetry physics al-low the beams not to be continuous and thus creating wrong bending moments.

As a final note, since the software does not allow a roller support as a standard support system, a “prescribed displacement” fixture is applied to one end of the bridge allowing the displacement only in the longitudinal direction of the beam to create the roller sup-port effect.

4 RESULTS

All analyses were performed with an Extra-fine meshing to guarantee accurate results.

4.1 Bridge deck

An initial analysis using COMSOL standard of the critical component with an IPE450 cross-section profile under an edge load of 8𝑘𝑁/𝑚 with a pinned near end and roller-supported on the far end rendering 𝜎_𝑀𝑎𝑥 = 70𝑀𝑃𝑎, 𝜏_𝑀𝑎𝑥 = 10.54 𝑀𝑃𝑎 and a deflec-tion of 𝛿_𝑀𝑎𝑥 = 15.63 𝑚𝑚, agreeing greatly with the results obtained in section 3.2.1 – Bridge deck design taking into consideration the cross-sectional properties of the beam.

When analysing the whole bridge deck, a thin plate of 10 𝑚𝑚 of the same steel S235 material will be used on top of the beams to simulate the beam flooring. This approach is not real since, in a real-life application, the deck would be a corrugated plate of very light metal, concrete or even a sandwich layered metal to serve as the base for the pavement, yet the weight carried by the plate to the beams is enough to produce a realis-tic dead load.

Nevertheless, when the whole deck is simulated taking into consideration the weight of the component members the values in the same critical member are modified, rendering a maximum deflection of 𝛿_𝑀𝑎𝑥 = 9.90 𝑚𝑚, a maximum bending of 𝑀_𝑀𝑎𝑥 = 65.7 𝑘𝑁 · 𝑚 and a maximum shear force of 𝑉_𝑀𝑎𝑥 = 19.73 𝑘𝑁. Upon inspecting the stresses in the same element, a slight increase is appreciated in the normal stress appreciated where 𝜎_𝑀𝑎𝑥 = 76.13𝑀𝑃𝑎 meanwhile, on the other hand, the shear stress halved its value for 𝜏_𝑀𝑎𝑥 = 5.30 𝑀𝑃𝑎. This is due to not only the effect of gravity on the members but also the interaction between the members creating an axial load in the members.

Also, as seen in the same Figure as the natural deflection of the plate also creates torsion in the side members as depicted in Figure 16, generating a lateral displacement of 10 𝑚𝑚. This will create some warping stress on the side members, represented through the first principal stress in Figure 17. It can be observable in Figures 18 (Normal Stress) and 19 (von Mises stress), though, that the central beam is still the critical component.

Figure 16. Bridge deck - Torsional moment analysis in COMSOL. (Jordi Mata Garcia, 2021)

Figure 17. Bridge deck – first principal stress analysis in COMSOL (Jordi Mata Garcia, 2021).

Figure 19. Bridge deck - von Mises stress analysis in COMSOL (Jordi Mata Garcia, 2021).

Figure 18. Bridge deck - Normal stress analysis in COMSOL (Jordi Mata Garcia, 2021).

4.2 Simple bridge

The simulation of an individual support beam shows an evenly spread axial force load equal to the input thus adding no information to the situation.

In an ideal situation, due to the even load from both sides as the spans are both equal in shape and load the only force the column will endure is a vertical load as seen in Figure 20. In a more realistic approach, small lateral bending can occur due to the live load crossing the bridge. To simulate such a situation, the bridge structure can be simulated having the central span with a slightly higher load so each column section has to bend and warp, as seen in Figure 21.

The structure shows no big change, except on an increase of the von Mises stress espe-cially on the joints of the side beams, indicating the shear stress to be the main cause of possible failure. The central beam seems to remain stable, most likely due to the whole deck structure balancing its stresses out. The appearance of such a small torsion moment in the column section indicates that the structure is very stable despite the live load loca-tion.

Figure 20. Simply supported bridge - whole structure von Mises stress analysis with equal loads in COMSOL. De-formation scaled by 150x. (Jordi Mata Garcia, 2021)

There is an increase, though, in the axial load sustained by the columns due to the con-sideration of the dead load of the structure itself. The increase is, though, not substan-tial, and the now 86 𝑘𝑁 are easily sustained by the strong axis of the column, which has an actual strength capacity of around 760 𝑘𝑁. The weak axis of the column would only

In document Beam Structure Design Analysis (sivua 35-0)