SWBPIPE solves the problem of Part II by the direct stiffness method. The centreline is divided into elements that meet at nodes. Every node has six degrees of freedom, three movements and three rotations, in the order u_x, u_y, u_z, θ_x, θ_y, θ_z. Each element relates the twelve movements at its two ends to the twelve forces and moments there through its stiffness matrix; the matrices are assembled into one system for the whole line and solved for the movements of the nodes. The classical force method of Part II F solves the same problem by other means, and the bend element below is built from the same flexibility integrals.

This part describes the method as it should be computed. Where the program does not yet compute it that way, a mark headed Today says so and links to the status of this edition.

1.Axes.

Each element has its own axes. Local x runs from the first node to the second. Local y comes from a reference direction given for the element, with its x component removed, and local z completes a right-handed set. For a round pipe the reference direction affects only the axes in which forces and moments are reported, not the stiffness. The element stiffness is turned into global axes by the direction cosines of the local axes,

where T repeats the 3 × 3 direction-cosine matrix four times along its diagonal. The inclined member case checks this transformation.

2.Straight pipe.

A straight element carries axial force, torsion and bending about both local axes. Its section comes from the nominal outside diameter D_o and nominal wall t, with D_i = D_o − 2t:

J is the torsion constant, equal to the polar moment for a circular tube. Stiffness and weight use the nominal wall. Corrosion and other allowances reduce the section used for stress only, as the codes require (Part II K).

For bending in the local x–y plane, with the degrees of freedom ordered v_i, θ_z,i, v_j, θ_z,j, an element of length L has the stiffness

In the x–z plane the terms coupling a movement to a rotation change sign, because a positive θ_y lowers the z coordinate ahead of it. The axial and torsional terms are EA/L and GJ/L.

3.Bends.

A bend is built the way the classical methods built it, from its flexibility. With one end held, the flexibility at the other end is found by integrating the unit-load moments and forces around the arc of radius R. The bending terms are multiplied by the flexibility factor k (Part II E):

where M is the in-plane and M′ the out-of-plane bending moment. For an elbow the classical theory gives the same k in both planes. The integrals are evaluated in closed form, the 6 × 6 flexibility is inverted, and the result is carried by equilibrium to a 12 × 12 stiffness joining the two tangent points. Weight is applied along the arc. The expansion loop checks the element at k = 1, 5, 10 and 20.

4.Rigid elements, expansion joints and tees.

Rigid elements. Valves, flanges and other stiff components are modelled as elements many times stiffer than the pipe they join, with their weight lumped at their centre of gravity. Expansion joints replace the pipe over their length with the joint’s own axial, lateral, angular and torsional stiffness. A joint without tie rods also transmits its pressure thrust, the pressure times its effective area, to the adjoining pipe and anchors; tie rods carry that thrust across the joint instead. Tees join the header and branch at a common node; the branch SIF is applied with the branch section modulus.

5.Assembly and solution.

Every element adds its stiffness into one system for the whole model, Ku = F. Restrained degrees of freedom are removed exactly, not represented by stiff springs, and prescribed movements carry their effect into the load. For a properly restrained model the reduced K is symmetric and positive definite. It is solved by a sparse direct method: the equations are reordered to keep the matrix narrow (reverse Cuthill–McKee), then factored as LDL^T. A pivot that is zero or negative, or tiny against the largest diagonal term, means the model can still move as a mechanism; the solve stops there and names the free node and direction. A dense solve is available as a check.

No support is ever assumed. A model with no supports, or one missing a material, an orientation or a unit, is not solved, and the message names what is missing.

6.What comes back.

From the solved movements the program recovers the reaction at every restraint, R = Ku − F, with its three force and three moment components; the forces and moments at the ends of each element; and the section forces at intermediate points along each element: its quarter points and middle. Element forces are recovered as the element stiffness times its end movements, less the equivalent loads that were applied to its nodes.

Checked by: cantilever, branch assembly, loads to resultants, intermediate points, one linear static path, torsion.

Loads are applied at the nodes. A load spread along an element is replaced by its consistent (equivalent) nodal loads, so that a uniform load w over a straight element of length L gives wL/2 and wL²/12 at each end. The forces the element actually carries then come from the solution, as in A 6.

1.Weight.

Weight is the mass per length of pipe, contents and insulation times a gravity you enter, with its sign and axis. There is no default gravity and no default density. On a bend it is applied along the arc. Each rigid component adds its weight at its centre of gravity.

2.Thermal expansion.

Thermal load is a temperature change on each element with the mean expansion coefficient over that change, or the thermal strain directly (Part II A). It is applied as the equivalent nodal load

pushing outwards at the two ends of each element. The force the element actually carries comes from the solution: EAαΔT in compression if both ends are held, nothing if the element is free to grow.

3.Pressure.

Pressure is treated in the usual way for beam models of piping. In a continuous line the closed-end pressure force is carried by the pipe wall and balances itself, so it moves the system only at unbalanced points: untied expansion joints, open ends and slip joints. There it acts as a thrust equal to the pressure times the effective area. The longitudinal pressure stress is always added in the stress recovery (E 1).

Pressure also lengthens a closed pipe slightly (the Bourdon effect). With the hoop stress σ_H and the longitudinal pressure stress σ_L = σ_H/2, the axial strain is

about a fifth of σ_H/E for steel. It is small, and programs usually make it optional. In the wall of a heated, pressurised pipe held at both ends, temperature causes compression and pressure causes tension of about νσ_H: the two act in opposite senses. On the anchors both push, and the pressure’s share is (1 − 2ν)pA_i, about 40 % of the closed-end force.

4.Imposed movements and cold spring.

The thermal growth of equipment at a nozzle, the movement of an anchor, and settlement are applied as prescribed movements of the restrained degrees of freedom, per load case. Cold spring is a prescribed closing of a gap left at the cut (Part II G).

5.Wind and seismic.

Both are equivalent static loads. Wind on a span is w = q·C_d·D, from the wind pressure q, a drag coefficient C_d and the exposed diameter including insulation, applied with the component of the wind normal to the pipe. Static seismic load is w = m′·a·g, from the mass per length and a seismic coefficient a in g, applied in each direction and each sense and combined by the rule you choose.

6.Material at temperature.

Elastic modulus, shear modulus and expansion coefficient can be entered at several temperatures. A load case takes its properties at one of them, or at a temperature between two, interpolated linearly:

It never extrapolates. For code work the displacement stress range is computed with the reference (installed) modulus, and hot reactions are scaled from it (Part II D).

Checked by: restrained thermal growth, partial-span load, weight, seismic and wind loads, properties at temperature.

Supports are described in Part II H and I as the practitioner uses them. In the model each acts on the degrees of freedom of one node, rigidly or through a stiffness, in the directions its type implies relative to the pipe, or along a direction you give.

1.Nonlinear supports.

Whether a rest, a gap, a rod hanger or a friction support acts depends on the answer. The program finds the active set by iteration: it assumes a state for each such support, solves the linear system for that state, and checks every support against the result. A rest whose reaction would pull is released; a released rest the pipe would pass through is re-engaged; a gap closes when the movement reaches its clearance; a friction support slides when the tangential force would exceed μN, and is then loaded by μN against the motion. The supports that disagree change state and the system is solved again, until no support changes and the forces balance.

With rests and gaps alone this settles in a few passes for most lines. Friction makes the answer depend on the order of loading, and the iteration is not guaranteed to converge. When it does not, the result is reported as not converged, never as solved.

Checked by: mixed supports, spring boundary, rest re-engaging, gap, friction, and the other cases of Part VI D.

A load case is a set of loads with the material properties it is solved at. Each is solved on its own, as a linear solve or, with nonlinear supports, as an iteration on their states. Combinations are then formed from the results:

A displacement stress range is formed from the difference of section forces and moments between two states, and the stress is then computed from those differences (E 2). The usual set of cases for a code check is set out in Part II C.

Stresses are recovered from the section forces at each end and intermediate point, by open mechanics:

The pressure stresses use the mean radius of a thin wall. The codes use their own forms, such as pD_o/4t for the longitudinal stress, so the validating program will differ slightly.

1.Peak longitudinal stress.

For a quick reading at each point, the peak longitudinal stress on the circumference is

with torsion reported beside it. It is a mechanics value, not a code stress. With in-plane and out-of-plane SIFs entered for a bend or branch, the intensified bending term is √((i_iM_i)² + (i_oM_o)²)/Z. SIFs apply to the moments, never to the pressure stress.

2.Ranges.

The range between two states is recovered from the difference of their forces and moments: the moment range first, then the stress. Differencing two summary values, or two intensified values, gives the wrong answer whenever the moments turn between the states.

3.What it does not compute.

SWBPIPE computes movements, loads, forces, moments and the stress components above. It does not compute code stresses, allowables or stress ratios, and it does not judge compliance with any code. That check is made in the program you validate in (Part V).

Checked by: axial, bending, torsion, pressure, range, end forces to stress.