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Parametric structures, box culvert

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Results.

ExampleReferenceRecordsLargest deviation
End-to-end flow, section forces checked against staticsClosed form30%
Vertical equilibrium balance for a box culvert below the water tableClosed form20%
Partly submerged culvert, uplift from the submerged volumeClosed form10%
Box culvert, shell model against a plane frameIndependent calculation20.366%
Section at mid-height, total axial force in the wallsClosed form10%

Problem. The box culvert is tested four ways. The shell model is compared with a plane frame model, the section forces with statics, the total axial force in the walls at half height with free-body equilibrium and the flotation balance with Archimedes' principle.

Reference document. The section forces come from free-body equilibrium and flotation from Archimedes' principle. The shell is a mid-surface model, so the reference volume is the mid-surface box as well.

In the roof slab the sum of the mid-span and corner moments and the corner shear force take these values from statics.

∣Morta∣+∣Mkose∣=wL28,Vkose=wL2\lvert M_{orta}\rvert + \lvert M_{kose}\rvert = \dfrac{wL^2}{8},\qquad V_{kose} = \dfrac{wL}{2}

The axial force at the wall base and the buoyancy force of the water are computed from these relations.

N=wL2+γbeton t H,U=γsu VN = \dfrac{wL}{2} + \gamma_{beton}\,t\,H,\qquad U = \gamma_{su}\,V

Numerical model. The culvert is modelled with quadrilateral shell elements at two mesh densities. An odd number of divisions puts an element centre at mid-span.

End-to-end flow, section forces checked against statics

Parameters. span = 3 · height = 2.5 · t = 0.3 · w = 66

Closed form. Free body equilibrium, closed form

QuantityReferenceComputedDeviation
For the top slab, ∣Mmid∣+∣Mcorner∣=wL2/8\lvert M_{mid}\rvert + \lvert M_{corner}\rvert = wL^2/874.2574.250%
Shear force at the corner, wL/2wL/299990%
Axial force at the wall base, N=wL/2+γtHN = wL/2 + \gamma t H117.75117.750%

Vertical equilibrium balance for a box culvert below the water table

Parameters. span = 3 · height = 2 · length = 6 · t = 0.3 · water level = 3

Closed form. Archimedes' principle

QuantityReferenceComputedDeviation
Resultant uplift force, γwaterV\gamma_{water} V353.16353.160%
Weight of the structure, γconcreteAt\gamma_{concrete} A t4504500%

Note. The drawing shows the model with its supports and loads, not a solution.

Partly submerged culvert, uplift from the submerged volume

Parameters. water level = 1.4 · height = 2

Closed form. Archimedes' principle, over the submerged volume

QuantityReferenceComputedDeviation
Resultant uplift force247.212247.2120%

Note. With the water table below the roof the roof load is zero. The drawing shows the model with its supports and loads, not a solution.

Box culvert, shell model against a plane frame

Independent calculation. An independent plane frame model with the same section and the same load

QuantityConditionReferenceComputedDeviation
Bending moment at the middle of the top slab, per unit width, h = L/15span = 3 · height = 2 · t = 0.25 · ν = 0 · mesh = 15×1023.666723.580.366%
Bending moment at the middle of the top slab, per unit width, h = L/31span = 3 · height = 2 · t = 0.25 · ν = 0 · mesh = 31×2123.666723.64610.0868%

Note. The remaining deviation falls with the square of the thickness.

Section at mid-height, total axial force in the walls

Parameters. span = 3 · length = 1 · q = -40

Closed form. Static equilibrium

QuantityReferenceComputedDeviation
Resultant vertical force at the section1201200%

Note. The section nodes are the nodes of the two walls at half height.

Shell and plane frame are compared on the same culvert section. The element value of the shell is the average over the element. Once both are read at the same station, the remaining deviation decays with the square of the element size, so it comes from the discretisation.Shell and plane frame are compared on the same culvert section. The element value of the shell is the average over the element. Once both are read at the same station, the remaining deviation decays with the square of the element size, so it comes from the discretisation.0123-20020Roof moment, shell element against plane framex [m]Mₓₓ [per unit width]0.050.10.22e-45e-40.0010.0020.0050.010.02Convergence of the shell solution to the frame solutionElement size h [m]|M(shell)/M(frame) − 1|Plane frameShell, element valuesStations matchedSecond order, reference slopeStations not matched

Shell and plane frame are compared on the same culvert section. The element value of the shell is the average over the element. Once both are read at the same station, the remaining deviation decays with the square of the element size, so it comes from the discretisation.