Final Project: Pedestrian Bridge AE225 - 2023
December 09, 2023
Skyler Chong, 20886122 Inderpratap Singh Chadha, 20980616
1
Table of Contents 1.0
Introduction ............................................................................................................... 3
2.0
Design Concept ......................................................................................................... 3
2.1
Diagrams & Explanations ..................................................................................... 3
3.0
Methodology ............................................................................................................. 6
3.1
Arch Calculation Package ..................................................................................... 6
3.2
Tributary Calculation Package .............................................................................. 7
3.3
Arched & Vertical Member Sizing Package ......................................................... 9
3.4
Horizontal Beam Sizing Package ........................................................................ 11
4.0
Conclusion .............................................................................................................. 16
References ......................................................................................................................... 17 Appendix A: Drawing Set ................................................................................................. 18
Table of Figures Figure 1: Site Plan Diagram ................................................................................................ 3 Figure 2: Tributary Area Diagram ...................................................................................... 4 Figure 3: Horizontal Member Diagram .............................................................................. 4 Figure 4: Side Elevation Diagram....................................................................................... 5 Figure 5: Section A-A’ Diagram......................................................................................... 5 Figure 6: Funicular shape of the arch. ................................................................................ 6 Figure 7: General Cable Theorem Usage............................................................................ 7 Figure 8: Tributary area and cable specs. ........................................................................... 8 Figure 9: Maximum compression in arch. .......................................................................... 9 Figure 10: Sizing arch members. ........................................................................................ 9 Figure 11: Sizing bridge columns. .................................................................................... 10 Figure 12: Horizontal beam system. ................................................................................. 11 Figure 13: Q-System. ........................................................................................................ 11 Figure 14: P-System and deflection. ................................................................................. 12 Figure 15: Internal forces in beam. ................................................................................... 13 Figure 16: Rotation at node B. .......................................................................................... 13 Figure 17: Planar stress, principal stress and theories of failure....................................... 15 Figure 18: Plan view. ........................................................................................................ 18 Figure 19: Elevation view. ................................................................................................ 19 Figure 20:Section and perspective view. .......................................................................... 20
Table of Tables Table 1: Moments in Arch at given x ................................................................................. 7 Table 2: Tributary Areas and Frequency ............................................................................ 7 Table 3: Tributary Loading on each Category .................................................................... 8
2
1.0
Introduction
The purpose of this report is to demonstrate and apply various structural design techniques by designing a replacement bridge that spans from Carl A. Pollock Hall to a stair tower. This design utilises structural design skills such as finding the load acting on a member based off the tributary area, the general cable theorem to determine the funicular shape of an arch, using equations of buckling to speck member sizes, as well as principle of virtual work to both determine beam dimensions and find deflection/rotation. In addition to determining the shape of the bridge and bridge components, internal shear stresses and bending stresses will be used to determine if the bridge will fail under a set of expected loads. The design of this bridge will demonstrate the engineering toolset developed over the duration of the AE 205 class in one project.
2.0
Design Concept
The idea for this pedestrian bridge was to create an open walkway that feels semi enclosed by having cables and arches surround the user. This effect was achieved by taking a standard arch bridge, and angling the arches together until they meet at the apex of their curves. This creates the effect of an X shaped dome above the pedestrians. In conjunction with this, horizontal bracing members, and cables wrap around the arch system from one side of the bridge, to the other.
2.1
Diagrams & Explanations
The diagrams given below offer explanation and an introspective outlook on the design prepared for the proposed pedestrian bridge spanning Carl A. Pollock Hall and the Stairwell Tower opposite the road. They are not the drawing set however, but a supplemental addendum to it, seeking to aid the calculation packages below this section. The Site Plan diagram displays the overall shape of the arched columns as they lean over the hollow core slab floor. Colour-coded members also assist in differentiating the purpose, use and name of each member in the plan. As such, horizontal bracing is placed in line with cable members to translate the z-axis forces exerted via the orientation of the arches cancels itself. The spacing between each cable is set to be spaced at 5.425 meters consistently while each arch spans 32.55 meters of distance.
Figure 1: Site Plan Diagram
3
The Tributary Area diagram is essential towards understanding the breakdown of the loads undertaken by the cables, arches and vertical columns below. Each arch consists of 5 cables that connect to the edge of the slab while a column in the middle as well as two columns both at the edges of the slabs take all the final loading and transfer them to the ground.
Figure 2: Tributary Area Diagram
The Horizontal Member diagram displays a basic structural plan for how the beams are sized and oriented specifically towards connecting the cables from strung arches to their opposite side across a 6.3-meter gap as well as subsequently supporting the hollow core slab above them.
Figure 3: Horizontal Member Diagram
The Side Elevation diagram’s key-purpose is towards communicating the projected heights of the arches across an x-y plane. It further more shows how the underside of the I-beams would be considered as where the columns would begin, totalling to 5.8 meters in height.
4
Figure 4: Side Elevation Diagram
The Section A-A diagram is designed to showcase how the cable forces run along the plane of the arches themselves which are angled at 60°. This, hence, can give us three different projected heights in the y-axis which correspond to arch heights at certain distances in the y’axis. The z-axis shown utilises the previously mentioned horizontal bracing to nullify any forces in that axis.
Figure 5: Section A-A’ Diagram
5
3.0
Methodology
3.1
Arch Calculation Package
To begin calculating the funicular shape of the arch, a minimum height was established. This was done by assuming an initial height of 2.5m, this height was then projected onto the y’ plane and rounded up, yielding a y’1 value of 3m and an y1 value of 2.598m. Following this, a FBD of the arch was drawn, with 5 equal loads, P, running through the cables spaced 5.425m apart from each other, as seen in Figure 6. An equivalent cable structure was then drawn by applying the general cable theorem, and this was then approximated to a simply supported beam with equivalent loading. The internal moment of the beam was then calculated, and used to solve for H in terms of P. Finally, using y’1 and H, the remaining y’n values were calculated.
Figure 6: Funicular shape of the arch.
6
Figure 7: General Cable Theorem Usage
Following from Figure 7, we can compute the values at M2 at x equals to 10.85 meters to get y’(x) & y(x) values which would equal to slack height in the arch’s plane and projected height in the y-axis, and we can compute M3 similarly at x equals to 16.275 meters for respective y’(x) & y(x) values where the arch would crown. Table 1: Moments in Arch at given x
Moments in Arch H M(x) y'(x) y(x) 4.521 P 13.563 P 3.000 m 2.598 m 4.521 P 21.700 P 4.800 m 4.157 m 4.521 P 24.413 P 5.400 m 4.677 m
3.2
Tributary Calculation Package
Using the previously provided Tributary Area diagram in figure 2 from the Design Concept section, we can find the exact tributary areas of three different load-case scenarios that can be listed into Table 2 below. In this table, the surface area of half of the full floor slab is found via multiplying its width of 6.3 meters with 32.55 meters. Within this surface area, occurrences of tributary areas for a cable (categorised as T1), edge column (T2) and middle column (T3) are counted alongside the values of their tributary areas. Table 2: Tributary Areas and Frequency
Tributary Half Slab Cable T1 Edge Col T2 Mid Col T3
Area
Frequency 205.065 m2 17.089 m2 10.000 2 17.089 m 1.000 2 34.178 m 0.500
7
Further multiplying each of these areas with the loading of 10.2 kPa due to exposure to snow and human traffic will give us individual tributary loads that each unique member of category experiences. We will also ensure that the frequency of the T3 category of middle column gets doubled for an accurate value of the tributary load subjected onto it. Table 3: Tributary Loading on each Category
Tributary Half Slab Cable T1 Edge Col T2 Mid Col T3
Loading 2091.663 kN 174.305 kN 174.305 kN 348.611 kN
Using the tributary load on a cable according to category T1, we can find force P. Since P’s y-component is Py and is equal and opposite to the tributary force in the cable, we can multiply this surmised value of Py by sin (60°) to get the load P equal to 201.27 kilo-newtons. This value of P can be further used to calculate yield stress in the cable via diving P by the cable’s cross-sectional area which is unknown. However, already provided to us is the same stress equal to 600 MPa and a known in P, this lets us shift the equation around to get a cable diameter of 20.7 millimetres.
Figure 8: Tributary area and cable specs.
The maximum compression in the arches can be calculated similarly via the newfound value of P. Inputting this value into our reactions allows us to get a vertical reaction at either A or B and the horizontal force across the arch. Considering that max compressive force occurs at 8
the steepest slope or highest angles, we can tie down onto the ends of the arches since they have the steepest slope. Using Pythagorean theorem to put the squares of both the reaction and the horizontal force and taking their total square root. Doing so, gives us a max compressive force of 1039.77 kilo-newtons present in the arches.
Figure 9: Maximum compression in arch.
3.3
Arched & Vertical Member Sizing Package
In order to size the arch, we need to assume it is bound to fail in compression under a critical load. The previously obtained max compressive load of 1039.77 kilo-newtons alongside a few given knowns such as our E being 200 GPa, K being 1 due to pin-pin connections at both ends of the arch, and a combined length of 6.2 metres. Rearranging the equation would allow us to isolate our unknown on one side, giving us the limit for allowable moment of inertia in the arch members to avoid buckling. This moment of Inertia is an absolute required minimum alongside the similarly obtained HSS 178 X 178 X 6.4 sizing from the CISC Steel Geometric Properties guide (Atkins, 2023).
Figure 10: Sizing arch members.
9
Similarly, to how we sized the members in the arch by ensuring they don’t buckle under a certain critical load, the middle and edge vertical columns can be sized by assuming buckling under critical loads determined by their tributary loads. One method to get these loads would be to calculate the arch loads, divide them by 2 to get the loads at each end and then multiply by 4 or either 2 depending on whether you’re calculating for a middle or edge vertical column. Another easier method would be to observe that the tributary area of the middle vertical column is double the tributary areas of edge vertical columns and equal them to the load subjected on the entire slab. This gives us a F1 (critical load for middle column) equal to 2091.663 kilo-newtons, while also F2 (critical load for edge column) equal to 1045.832 kilonewtons. Sharing the same other knowns such as E being 17 GPa, height equal to 5.8 meters and K is equal to 0.7 as there a fix-pin joint condition. These give us the moments of inertia listed below, alongside column size specifications. However, we will ultimately oversize our columns to account for any dead loads, worse-case loading scenarios, etc according to the data from the CISC Steel Geometric Properties guide (Atkins, 2023).
Figure 11: Sizing bridge columns.
10
3.4
Horizontal Beam Sizing Package
To determine the shape of the horizontal beam, the principle of virtual work was used. As seen in Figure 13, a Q-System was established applying a 1kN load at the midspan of the beam and the moment at distance x from node A was calculated. The same process was them applied to the P-System, to obtain the moment equations required for the PVW formula.
Figure 12: Horizontal beam system.
Figure 13: Q-System.
11
Figure 14: P-System and deflection.
Using the PVW equation, the moment of inertia was obtained with respect to the deflection at C. Then the max deflection was calculated and used to solve for the moment of inertia, Figure 14. As seen in Figure 15, the bending moment diagram of the horizontal beam, the maximum moment was obtained. The beam shape and a new moment of inertia value was identified to be W310x97 using the maximum moment and calculated moment of inertia as reference. With the new beam shape and I obtained from the beam selection table (Atkins, 2023), the deflection at C was found to be 25.56mm, Figure 15.
12
Figure 15: Internal forces in beam.
Next, a Q-System was established with a 1kNm load at node B. A moment equation was derived for the Q-System, and using Mp, that was found while solving for deflection, the rotation at point B was calculated using PVW, Figure 16.
Figure 16: Rotation at node B.
13
Due to the location of the cut, and the point of analysis, a, two planar stresses are known to be zero. Firstly, as the beam is being analysed at its midspan, there is no shear force, thus shear stress must also be zero. Secondly, since point “a” lies on a neutral axis, the distance from the neutral axis to point a in the x direction is zero, making sx zero as well. Thus, making sy the only planar stress. This also indicates that s2 is zero, and s1 is equivalent to sy. After determining the principal stresses of the beam, the maximum shear was calculated and fount to be half the value of s1. Finally, using the yield stress of the steel beam (350MPa) and the principal stresses (s1), Von Mises’ criterion was used to determine of the beam would fail. As sy and s1 are the only nonzero components of the equation, yield stress can be compared directly to the principal stress. Since sy is greater than s1, the beam will not fail under the loads it is subjected to in this analysis.
14
Figure 17: Planar stress, principal stress and theories of failure.
15
4.0
Conclusion
The final design of the pedestrian bridge is a culmination of the learnings made towards honing the structural design capabilities and engineering toolset developed through the duration of AE205. The opportunity to size individual members based on their moments of inertia, applied forces, and other factors led members needing to be oversized, particularly the vertical columns, to ensure structural stability and project longevity. There has been a process of an intense and constant review of each aspect of the design to ensure that no calculations led to failure either via deflection, buckling or through the applications of different theories of failure. The proposed pedestrian bridge has been calculated to be able to be nominal under all considered load cases learnt through the duration of AE205, and as such, can be recommended for consideration towards replacing the previous pedestrian bridge spanning the road between Carl A. Pollock Hall and the Stairwell across A lot Parking. However, it is to be acknowledged that various external and internal factors such as dead loads, seismic loading, wind, and rain loads as well as possible impact loading cases weren’t able to be considered in the design process due to a lack of skill sets towards navigating such scenarios.
16
References Atkins, A. (2023a, December 8) Final Project Fall 2023, University of Waterloo Atkins, A. (2023b, December 8) Appendix A, University of Waterloo. Atkins, A. (2023c, December 8) Hollowcore Precast Concrete Deck Span Tables, University of Waterloo. Atkins, A. (2023d, December 8) Approximate Sizing Guidelines, University of Waterloo. Atkins, A. (2023e, December 8) Arch Studio Companion Package, University of Waterloo. Atkins, A. (2023f, December 8) CISC Steel Geometric Properties, University of Waterloo. Atkins, A. (2023g, December 8) Steel Sizing W beams and W and HSS Columns, University of Waterloo. American wide flange beams. Engineering ToolBox. (n.d.). https://www.engineeringtoolbox.com/american-wide-flange-steel-beams-d_1318.html Leet, K., Uang, C.-M., Lanning, J. T., & Gilbert, A. M. (2018). Fundamentals of structural analysis. McGraw-Hill Education. Hibbeler, R. C., & Yap, K. B. (2019). Statics and mechanics of materials. Pearson Education Limited.
17
Appendix A: Drawing Set
Figure 18: Plan view.
18
Figure 19: Elevation view.
19
Figure 20:Section and perspective view.
20