To calculate the ring bending stress the finite element model of pipe cross-section is used. The vertical and horizontal loads from soil weight are calculated and applied for each point of pipe cross-section at whole perimeter (see picture below).
Soil is modeled as discrete springs around pipe perimeter. The springs are working only for compression and each spring is switched off if tension is detected. The soil peeling usually take place at the top of the pipe cross-section (see picture below).
Also internal pressure and product hydrostatic pressure is applied acting in radial direction on each point of pipe perimeter. The analysis consider geometric nonlinearity, so the effect of additional resistance to the bending caused by internal pressure is also considered.
If open trench type laying is used, then chosen, then soil natural arch of collapse is not considered. The soil pressure on the pipe is calculated using whole depth. The soil pressure increases linearly with increasing depth.
If trenchless laying method is used, and the pipe depth is very high, then a self-supporting arch collapse forms above the pipe (natural arch of collapse). This arch supports all the soil pressure above itself. Only soil inside the natural arch of collapse produce the pressure on the pipe.
The ring bending stress diagram is shown on the picture above. This stress is used in several piping codes to consider additional stresses in the pipe from the soil weight.
The additional vertical load for each point of the pipe insulation casing perimeter from the concentrated force, applied on the ground surface can be obtained from Bousinnesq equation
The additional vertical load for each point of the pipe insulation casing perimeter from the infinite length uniform load across the pipe, applied on the ground surface can be obtained using the equation
START-PROF apply two concentrated loads P at the distance L and one uniform load q, and calculate the sum of soil pressure on each point of the pipe finite element model surface around the upper half of the pipe. Also the pressure from the soil weight and pipe internal pressure are applied to the finite element model. After that the nonlinear finite element analysis is done the same way as mentioned above.
Two options of concentrated forces application are considered. The first option is the application of the two forces at the distance L/2 from the pipe axis. And the second option is when first force is right above the pipe axis. The worst pipe and insulation stresses are calculated and taken into account.
See also Polyurethane (PUR) Insulation Stress Analysis