Crossing Calculation: Two support bridge Case
The calculation addresses the two support bridge crossing design case. It is assumed that each support is of the
same height and that both the pipe and the supports are resting on a rigid foundation. The analysis is built from
the superposition of three cases provided within Roark's formulas (7th Edition) Table 8. These cases are as follows.
Case 1b Concentrated Intermediate Load; Left
End Guided, Right End Fixed
Case 1d Concentrated
Intermediate Load; Left
End Fixed, Right End Fixed
Case 2d Partial Distributed
Load; Left End Fixed,
Right End Fixed
These three cases were selected as their superposition satisfies the boundary conditions for the two support
case. A sketch of the two support case and its associated boundary conditions are shown below.
Additionally
for Case 2d 
and
and for Case 1b
where
L = the distance from the touch down point to the center of the crossing (i.e. the plane of symmetry
D = the bottom of pipe elevation relative to mean seabed level at the center of the crossing
W = the support reaction load
Solution Objective:
The solution objective is to find the combination of W, L and D that satisfies a given submerged weight (w), support offset from center of crossing (a) and support elevation (
).
Unit Definition:
Case Definition:
Input Data
Nominal OD of crossing pipeline [Ref. 7]
Nominal WT of crossing pipeline [Ref. 7]
Yield stress of crossing pipeline [Ref. 7]
Young's modulus [Ref. 7]
Poisson's ratio [Ref. 7]
Average water depth at crossing [Ref. 7]
Steel density [Ref. 7]
Sea water density [Ref. 7]
Contents density [Ref. 7]
Design pressure (MAOP (gas), MOP (oil)) [Ref. 7]
Reference elevation for design pressure [Ref. 7]
as-laid
Internal pressure change relative to as-laid condition
flooded
hydro
op
Support offset from center of crossing (Support offset is taken
to the midpoint of the support group and includes a positional
tolerance of +5-ft.) [Ref. 8]
Support elevation (bottom of pipe to mean seabed level) [Ref. 19]
as-laid
flooded
Temperature differential (degC) between as-laid condition and design phase under consideration.
op
Coefficient of thermal expansion (degC) [Ref. 7]
Residual lay tension
General Calculations
cross sectional area of pipe
internal cross sectional area
external cross sectional area
2nd moment of pipe circular cross section
buoyancy force
contents weight per unit length of pipe
submerged weight per unit length of pipe
flooded weight per unit length of pipe
hydrotest weight per unit length of pipe
operating case weight per unit length of pipe
as-laid
flooded
hydro
op
The following initial guess values are used in the solve block below.
The following equations calculate the beam deflection as a function of distance from the touch down point
for each of the three cases above.
Case 1b:
Case 1d:
Case 2d:
As there are three unknown to be solved for, the solution requires three equations.
The following equalities were selected to define the solution.
at
at
at
where
is the moment at the touch down point for each of the three cases above
is the vertical reaction force at the span center
is the vertical displacement at the support location
The above equalities are defined in terms of the unknowns R, D and L and arranged in
Mathcad solve block as follows.
Case 1b:
Case 1d:
Case 2d:
Sum of moments @ x = 0
Sum of vertical reactions @ x=L
Sum of vertical dispacements @ x=a
. . . double reaction load for single support case
The following functions define the y coordinate as a function of x for the load cases analysed.
n = the case description. Use "as_laid", "flooded" or "op"
Moment distribution
The following function evaluates the moment distribution as a function of x.
Bending moment as a function of x
Bending Stress as a function of x
Maximum Bending Moment as a function of x
Bending Moment at Crossing Center
Maximum Bending Stress as a function of x
True wall and effective axial force calculation
Axial and longitudinal stress calculation
as-laid
flooded
hydro
op
Hoop Stress
as-laid
flooded
hydro
op
Von Mises equivalent stress
