SWBPIPE solves the problem of Part II by the direct stiffness method. The centerline 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. The stiffness matrix relates twelve end movements to twelve end forces and moments. Assembly combines the element matrices into one system for the line. The solution gives the movements of the nodes.

The classical force method of Part II F solves the same problem by another method. The bend element below uses the same flexibility integrals.

This part separates beam theory from the supported software routes at revision 462f66975d. The ordinary route publishes preview-physics-1 semantics. Models 0.3.0 and 0.4.0 can select exact_straight_pressure_v2; model 0.4.0 requires that route for load/reference states. 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 the element’s reference direction, with its x component removed. Local z completes a right-handed set. For a round straight pipe, the reference direction selects the reporting axes for forces and moments. It does not change the stiffness.

The direction cosines transform the element stiffness to global 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, movement-rotation terms change sign. A positive θ_y lowers the z coordinate ahead of it. The axial and torsional terms are EA/L and GJ/L.

3.Bends.

The curved bend element uses a flexibility formulation, as in the classical methods. With one end fixed, unit-load moments and forces define the flexibility at the other end. The formulation integrates these terms around the arc of radius R. The flexibility factor k multiplies the bending terms (Part II E):

Here, M is the in-plane bending moment. The out-of-plane bending moment is M′. The product mapping applies the same user-entered k in both planes. The element interface can accept separate factors.

The formulation evaluates the integrals in closed form and inverts the 6 × 6 flexibility matrix. Equilibrium transfers the result to a 12 × 12 stiffness matrix between the tangent points. Weight is applied along the arc. The expansion loop records a bending-only reference at k = 1, 5, 10 and 20.

4.Rigid elements, expansion joints and tees.

Rigid elements. In the engineering model, valves, flanges and other stiff components use elements much stiffer than the adjoining pipe. Their weight acts at their center 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 transmits pressure thrust to the adjoining pipe and anchors. The thrust equals pressure times effective area. Tie rods carry the thrust across a tied joint.

Tees join the header and branch at a common node; the branch SIF is applied with the branch section modulus.

5.Assembly and solution.

Each element adds its stiffness to the model system, Ku = F. Elimination removes rigidly restrained degrees of freedom. Prescribed movements contribute to the reduced load. For a properly restrained model the reduced K is symmetric and positive definite. The sparse direct method reorders equations by reverse Cuthill–McKee and factors the matrix as LDL^T.

A zero, negative or relatively small pivot can indicate an unconstrained mechanism. Numerical admission also requires the applicable integrity checks. 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.

Solved movements give restraint reactions, R = Ku − F. Recovery also gives element end forces and moments, plus section forces at the quarter points and middle. Reaction attribution and output limits are stated below. 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 distributed load becomes consistent (equivalent) nodal loads. 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 center of gravity.

2.Thermal expansion.

For a uniform straight member, thermal load uses the mean expansion coefficient over its temperature change, or thermal strain directly (Part II A). The equivalent axial nodal load is

pushing outward at the two ends of the straight member. The solution gives the element force. With both ends fixed, the compressive force is EAαΔT. With free thermal growth, the force is zero.

A realized curved macro-element uses the free uniform thermal expansion field, u_free. Its equivalent load is K_macro*u_free. Its recovered elastic action uses K_macro*(d-u_free). It does not apply the straight member’s axial end-pair as the complete arc load.

3.Pressure.

Pressure is treated in the usual way for beam models of piping. In a continuous line, the pipe wall carries the closed-end pressure force. Unbalanced points can transmit thrust to the system: 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 a heated, pressurized pipe fixed at both ends, temperature causes wall compression. Pressure causes wall tension of about νσ_H. The effects 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.

Equipment growth at a nozzle, anchor movement and settlement can be represented by prescribed movements. Each load case assigns these movements to restrained degrees of freedom. 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. It uses wind pressure q, drag coefficient C_d and exposed diameter, including insulation. The engineering model uses the wind component normal to the pipe.

Static seismic load is w = m′·a·g. It uses mass per length and seismic coefficient a in g. The engineering model applies each direction and sense, with the selected combination rule.

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 this engineering description, each support acts on one node, rigidly or through a stiffness. Its type defines directions relative to the pipe, or the user specifies a direction.

1.Nonlinear supports.

Whether a rest, a gap, a rod hanger or a friction support acts depends on the answer. The program assumes a state for each nonlinear support and solves the corresponding linear system. It compares each support state with the result. A rest releases if its reaction would pull the pipe. A released rest re-engages if the pipe would pass through it. A gap closes when movement reaches its clearance.

A friction support slides when tangential force would exceed μN. The sliding force is μN against the motion. Supports that disagree change state, and the program solves the system again. Iteration seeks unchanged states and balanced forces.

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 uses differences of section forces and moments between two states. 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 supply code stress rules, allowables or code acceptance. User-authored formula rules can evaluate declared inputs, including intervals. Nonfinite intermediate arithmetic blocks the formula evaluation. A formula gate does not establish standards approval or professional acceptance. That check is made in the program you validate in (Part V).

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