Loads and load cases

Load case

A set of loads applied together and solved as one analysis, with the material properties it is solved at. Code stresses come from particular load cases and from combinations of them. See Part II and Part III.

Primary and secondary loads

Primary loads are imposed forces, such as weight and pressure. They do not diminish as the pipe deforms, and excessive primary stress leads to collapse. Secondary loads come from imposed displacement, chiefly thermal expansion. Their stress is self-limiting and relieved by local yielding. The codes limit secondary stress range to guard against fatigue. See Part II.

Sustained case

The load case of the primary loads present through normal operation: weight of pipe, contents, insulation and components, pressure, and other steady forces. Its longitudinal stress, S_L, is checked against the hot allowable.

Operating case

All loads acting together in the operating condition: weight, pressure, thermal expansion and imposed movements, with nonlinear supports and hangers in their operating state. It gives hot displacements, support loads and nozzle loads. B31.1 and B31.3 set no stress limit on it directly.

Expansion case (displacement case)

The case that gives the displacement stress range. It is normally the algebraic difference of operating and sustained cases. This difference includes effects of supports that lift off or slide. Its stress, S_E, is compared with S_A. See Part II and Part III.

Occasional case

A case of loads that act for a small fraction of the operating time: wind, earthquake, relief-valve discharge, fluid hammer, snow. B31.3 (302.3.6) limits the sum of the longitudinal stresses from sustained and occasional loads to 1.33 S_h.

Hydrotest case

The pressure-test case: the line is full of test water at test pressure and ambient temperature. Spring hangers are normally locked (pinned), so they carry the extra weight as rigid supports. The case assesses support loads and test stresses.

Operating condition (T1, P1 …)

One set of temperature and pressure for a line, such as design, normal operating or upset. Programs number them, and build the operating and expansion cases from each.

Algebraic combination

A combination made on the signed components of displacements, forces and moments, before any stress is computed; the expansion range is an algebraic difference. A scalar combination, by contrast, adds stresses already computed, as when sustained and occasional stresses are summed.

Envelope

The largest and smallest value of each result quantity over a set of cases, taken quantity by quantity. The values in an envelope need not occur together, so it serves for design loads and maxima, not as a state to combine further.

Thermal expansion (free expansion)

The change in length of unrestrained pipe with temperature, αLΔT. Flexibility analysis finds the forces, moments and stresses that arise when the system’s supports and geometry resist it. See Part II.

Mean coefficient of thermal expansion

The average expansion coefficient between a reference temperature and the temperature of interest. Expansion equals this coefficient times length times the whole temperature change. It is not the instantaneous coefficient at the higher temperature. Some code tables give total expansion per unit length directly.

Imposed movement (anchor movement)

A displacement or rotation prescribed at a node. Examples include nozzle thermal growth, movement of a vessel or another line, and settlement. Thermal movement enters operating and expansion cases with the line’s own expansion. One-time movements such as settlement follow the applicable code treatment.

Cold spring (cut short, cold pull)

Deliberately making a line short (or long) by a fraction of its expansion and pulling it into place at erection, to lower the hot reactions. The B31 codes give no credit for cold spring in the displacement stress range, and only partial credit in reactions. See Part II.

Pressure thrust

The force p × area that pressure exerts at a closure or discontinuity. Where the wall transfers the closure load, continuous pipe carries it as longitudinal tension. Where the closure is separately supported or compensated, its declared ownership controls the external load. An untied expansion joint or slip joint can transfer thrust to external restraints. A closed end alone does not mean that supports carry the full thrust.

Bourdon effect

The tendency of internal pressure to straighten a bend, as it does in a Bourdon-tube gauge. Programs offer it as an option, which may also include the lengthening of straight pipe under pressure, (σ_L − νσ_H)/E per unit length. SWBPIPE’s exact pressure contracts include qualified pressure strain. They do not model pressure-induced bend opening.

Stresses and code terms

Longitudinal stress

Normal stress along the pipe axis, from axial force, bending moment and pressure. See Part III.

Longitudinal pressure stress

The axial stress caused by pressure in closed-ended pipe. Code work usually takes it as pD_o/4t. Its mean wall value is pD_i²/(D_o² − D_i²). It is part of S_L.

Hoop stress (circumferential stress)

The stress around the circumference from internal pressure, about pD/2t, roughly twice the longitudinal pressure stress. It governs wall thickness, which is set by the code’s pressure design rules, not by the flexibility analysis.

Section modulus, Z

Second moment of area divided by the outer radius: Z = π(D_o⁴ − D_i⁴)/32D_o. Bending stress is M/Z. Codes say whether the nominal wall, or the wall less allowances, is used for each stress.

Effective section modulus, Z_e

A replacement for the branch pipe’s own Z in a reduced-outlet branch connection, where the applicable code edition requires it. Its classical form is πr₂²T_s, from branch mean radius and an effective wall. Check the edition in use.

S_L, sustained longitudinal stress

The longitudinal stress from sustained loads: pressure, weight and other sustained forces. In B31.3 it is limited to S_h. The terms it includes, and the stress indices applied, differ by code and edition.

S_h, hot allowable

The basic allowable stress of the material at the maximum metal temperature of the condition, or of the displacement cycle, under analysis.

S_c, cold allowable

The basic allowable stress at the minimum metal temperature of the displacement cycle under analysis, often ambient.

S_A, allowable displacement stress range

The limit on S_E. B31.3 (302.3.5) gives S_A = f(1.25S_c + 0.25S_h), and allows the unused part of S_h, that is S_h − S_L, to be added to it.

Cyclic reduction factor, f

The factor on S_A for equivalent full displacement cycles over line life. It is 1.0 for 7000 cycles or fewer and falls as cycle count rises. Recent B31.3 editions permit values above 1.0 for some materials at low cycle counts.

S_E, computed displacement stress range

The code stress of the expansion case (B31.3, 319.4.4). It uses bending and torsional moment ranges, with the SIFs applied. Recent B31.3 editions also include the axial force range. It is computed with reference modulus E_a.

Displacement stress range

The change in stress between two states of the displacement cycle, usually cold and hot, computed from the algebraic difference of the moments and forces. Because the stress is self-limiting, it is the range, not the stress in either state, that the code limits. See Part II.

Self-springing (relaxation)

A stress shift caused by local yielding or creep in the first cycles. Hot stress falls, and cold stress of the opposite sign appears. The range remains unchanged, which is why the code limits it.

Reference modulus E_a and hot modulus E_m

E_a is the modulus of elasticity at a reference temperature, 21 °C (70 °F) in B31.3, used to compute S_E. E_m is the modulus at the maximum or minimum metal temperature, used to scale reactions to that condition.

Code check

The comparison, at every node and component, of each code stress with its allowable, with the SIFs of the component applied. See Part II.

Components and flexibility

Bend (elbow)

A curved length of pipe defined by its bend radius and angle; a long-radius elbow has a radius of 1.5 nominal diameters. A bend is more flexible than straight pipe and carries higher local stress, both through ovalization. See Part II.

Mitre bend

A change of direction made from straight segments cut and welded. Codes give its flexibility factor and SIF according to whether the cuts are closely or widely spaced.

Ovalization (ovalisation)

The flattening of a bend’s cross-section under bending moment. It makes the bend more flexible and raises the local stress. Internal pressure resists it, and codes give corrections that reduce k and the SIF of large, thin-walled bends under pressure.

Flexibility characteristic, h (pipe factor)

The classical bend parameter for flexibility factor and SIF: h = tR/r². Its inputs are wall t, bend radius R and mean pipe radius r. Branch connections and mitres have their own forms.

Flexibility factor, k

The ratio of a component’s rotation under a moment to that of straight pipe of the same length and section. For a bend the codes give k = 1.65/h, not less than 1, applied over the arc. See Part II.

Stress intensification factor (SIF), i

A ratio of fatigue-significant component stress to nominal straight-pipe bending stress M/Z under the same moment. Markl’s fatigue tests originally established it. It multiplies the moment in code stress equations.

In-plane and out-of-plane SIF, i_i and i_o

Separate SIFs for bending in the plane of a bend or tee and bending across it. For a bend, in-plane bending opens or closes it; for a tee, the plane is that of the header and branch. B31.1 uses a single SIF where B31.3 uses the pair.

Torsional SIF, i_t

A SIF on the torsional moment, given in recent code editions and in ASME B31J where applicable; older practice takes it as 1.0.

Tee, branch connection

The junction of a branch with a header: a welding tee, a reinforced or unreinforced fabricated tee, or an integrally reinforced branch fitting. Each type has its own SIFs. It is modeled as three elements meeting at the node where the centerlines cross, with the SIFs applied there; ASME B31J also gives local flexibility.

Reducer

A concentric or eccentric fitting joining two pipe sizes, modeled as a short element of changing section, or as a step. Codes give an SIF for concentric reducers within stated limits.

Rigid element

An element far stiffer than the pipe, used for valves, flange pairs and other stiff components. It carries its own weight, entered separately, and programs normally let it expand with the line temperature.

Expansion joint (bellows)

A flexible element that absorbs movement in its convolutions, with axial, lateral, angular and torsional behavior. SWBPIPE’s current connector uses an authored symmetric 6 × 6 work matrix on a v3 document. It replaces one straight span, with small movements, untied hardware and empty pressure-region lists in every case. Legacy four-rate joints require re-authoring. See Part III A 4.

Untied expansion joint

A joint without tie rods. It passes pressure thrust, equal to the pressure times its effective area, to the main anchors on either side, which must be designed for it.

Tied expansion joint

A joint whose tie rods carry the pressure thrust, so the anchors do not. The rods prevent axial movement across the joint, which then takes lateral and, if hinged or gimballed, angular movement.

Effective area

The area on which pressure acts to give a bellows’ thrust, based on its mean convolution diameter and stated by the maker.

Nozzle

The connection of a pipe to a vessel, tank, pump, turbine or compressor. Nozzle loads are compared with equipment-standard or manufacturer allowables. Examples are API 610 for centrifugal pumps and NEMA SM 23 for steam turbines. Alternatively, loads enter a local shell stress assessment such as WRC 107 or WRC 297. See Part II.

Nozzle flexibility

The local flexibility of a vessel or tank shell at a nozzle, modeled as springs at the connection in place of a rigid anchor. Stiffnesses come from methods such as WRC 297, API 650 Annex P for tanks, or finite elements. A rigid anchor usually overstates nozzle loads from displacement.

Supports and restraints

Anchor

A restraint on all six degrees of freedom at a node: rigid unless given a stiffness. See Part II and Part III.

Rest (+Y support, resting support)

A one-way vertical support: it holds the pipe up but cannot hold it down, so the pipe may lift off. CAESAR II’s name, +Y, follows its default of Y vertical.

Guide

A restraint across the pipe axis that permits axial movement. On horizontal pipe, it usually acts horizontally; a rest at the same point carries vertical load. On vertical pipe, the guide acts in both horizontal directions. It often has a gap.

Line stop (axial stop)

A restraint along the pipe axis, in one or both directions, that leaves lateral movement free.

Limit stop

A restraint, usually axial, that leaves the pipe free until it has moved a stated gap in one or both directions, and then stops it.

Gap (clearance)

The clearance, in a stated direction, between the pipe or its shoe and a restraint. The restraint exerts no force until the gap closes, which makes it nonlinear.

Skewed restraint

A restraint acting along a direction that is not a global axis, given by direction cosines or a vector. Guides and line stops on skewed runs need it.

Friction coefficient, μ

The ratio of friction force to normal force at a sliding support. The support holds while the sliding force is no more than μN, and otherwise slides against a force μN. Values of about 0.3 for steel on steel and 0.1 for PTFE slide plates are common; project practice governs. See Part III.

Lift-off

The loss of contact at a rest when thermal movement raises the pipe. The rest then carries nothing, and its share of the weight passes to neighboring supports or nozzles.

Snubber

A device that resists rapid movement, as in an earthquake or fluid hammer, but lets slow thermal movement pass. It acts in occasional and dynamic cases only.

Hot and cold reactions

The forces and moments at a support or anchor in the operating condition (hot) and in the installed condition after operation (cold). With cold spring or self-springing the cold reaction need not be zero. For simple two-anchor systems the B31 codes estimate both from the range reaction, the cold-spring factor and, for the hot reaction, the ratio E_m/E_a.

Hangers

Variable spring hanger (spring support)

A spring that carries weight and lets the pipe move vertically. Its load changes with travel, by the spring rate times the movement. See Part II.

Constant support hanger (constant-effort hanger)

A spring and lever mechanism that carries nearly the same load over its whole travel. It is used where vertical movement is large, or where a variable spring would change the load too much.

Rod hanger

A rigid rod from the structure above. It carries load in tension only, goes slack if the pipe rises, and swings with horizontal movement.

Restrained-weight case

A weight case solved with a rigid vertical support at each proposed hanger, to find the load each hanger must carry.

Hot load (operating load)

The load a hanger carries in the operating condition. It is normally set to the load found in the restrained-weight case. This supports the pipe as designed when hot.

Cold load (installed load)

The hanger load at installation. It equals hot load plus spring rate times travel, with upward travel from cold to hot positive. This load is set on the hanger’s scale during erection.

Travel

Vertical pipe movement at a hanger between cold and hot conditions. An operating analysis finds it with each hanger replaced by a force equal to its hot load.

Variability

The change of a variable spring’s load between cold and hot, as a fraction of the hot load: spring rate times travel over hot load. It is commonly limited to 25 %.

Hanger design

The sequence used to size springs. A restrained-weight case gives hot loads, and an operating case gives travels. Spring size and rate are then selected, followed by cold load and assessment of variability and travel range.

Model and results

Centreline model

The piping representation along the pipe centerline. Bends are arcs; components have their own stiffness and weight. Branches meet at centerline intersections. See Part III.

Node (point)

A point on the centerline where elements join, supports and loads act, and results are reported. Programs number or name them.

Element

The piece of the model between two nodes, described by a stiffness matrix: straight pipe, bend, rigid element or expansion joint.

Degree of freedom

One of the six independent movements of a node: three translations and three rotations. A restraint removes or constrains degrees of freedom. See Part III.

Elastic centre

In the classical analysis of a line between two anchors, the centroid of the line with each part weighted by its flexibility. The thermal force acts through it, and the moment at any point is that force times the distance from its line of action. See Part II.

Consistent (equivalent) nodal loads

The end forces and moments that do the same work as a load spread along an element. A uniform load w on a span L gives wL/2 and wL²/12 at each end. See Part III.

Shear deformation

Deflection from transverse shear, added to bending deflection in the Timoshenko beam. It matters for short, thick-walled lengths. The ordinary SWBPIPE frame uses Euler–Bernoulli bending without this contribution; a receiving program may include it.

Active set

The set of nonlinear supports acting in a solution: rests in contact, gaps closed, and friction holding or sliding. A program assumes a set, solves, compares support states with the result, and updates inconsistent states. It repeats until the state converges or an iteration limit stops it. SWBPIPE’s product limit remains four iterations. Nonconvergence is reported. A separate confirmation checks the converged support state and member bound. Cases that pass can publish checked rows; an unavailable confirmation keeps Sensitive with a named reason. See Part III.

Member-relative recovery bound

A conditional bound on recovered member actions against the intended re-formed linear system. Qualified recovery uses corrected movements; displayed nodal movements keep the original binary64 solution. The bound depends on an inverse-norm assumption and retains an unproved arc distance premise. See Part III A 6.

Mechanism (unstable model)

A model with too few restraints to prevent rigid-body movement in some direction. Its stiffness matrix is singular and it has no unique solution.

Program equivalents

A blank (–) means the program’s own term is not certain here, not that it lacks the feature.

Older names in the program

SWBPIPE’s own screens may still use a few names this manual has replaced.

One-way support, lift-off support

A rest.

Range envelope

An envelope.

Result-state subtraction

An algebraic difference of matching signed result-state components. It is a labeled difference of solved states, not a solved equilibrium state.

Mechanics combination

An algebraic sum of cases, each with a factor. Ordinary mechanics combinations are withheld for nonlinear supports, mixed moduli or a constant-effort factor sum other than 1. The current exact pressure contract refuses combinations.

Constant-effort hanger

A constant support hanger.

Station

A point along an element, such as a quarter point, at which section forces are reported. It is not a node: it has no degrees of freedom, and nothing can be attached to it.