A pipe that carries a hot fluid grows. If it is free to grow, it does so without stress. If it is not, the growth is turned into force, and the force into stress in the pipe and load on everything that holds it. Piping flexibility is the study of that conversion, and the design of lines that keep it within bounds.

This part teaches the discipline itself, written to what a correct calculation should give, whatever program makes it. Where SWBPIPE falls short, a mark headed Today links to the status of this edition. How SWBPIPE builds and solves its model is the subject of Part III.

1.Free growth.

A straight pipe of length L, warmed uniformly by ΔT and free at one end, lengthens by

where α is the coefficient of thermal expansion. For carbon steel α is about 12 × 10⁻⁶ /K near ambient and rising with temperature; for austenitic stainless steel it is about 16 × 10⁻⁶ /K. A carbon steel run 30 m long, warmed by 150 K (270 °F), grows 54 mm (2.1 in). A stainless line of the same length grows about a third more.

A line of any shape, warmed uniformly and free, keeps its shape and grows in size, so its free end moves by αΔT times the straight line from the fixed point to that end, whatever route the pipe takes. The movement a line must absorb between two anchors depends only on where the anchors are. The loops between them decide how much force it takes.

2.Full restraint.

Hold both ends of the same straight run fixed and warm it. It cannot lengthen, so the anchors must push it back by the whole of its free growth. The mechanical strain is equal and opposite to the thermal strain, and the axial stress is

compressive, about 52 ksi, with E the elastic modulus of carbon steel near ambient. That is well above the yield strength of ordinary carbon steel pipe. Notice what is absent from equation (2): the length, the diameter and the wall. A straight run between anchors reaches this stress whatever its size. For a DN200 line (219.1 mm outside diameter, 8.18 mm wall, metal area 5420 mm²) the anchor force is 360 MPa × 5420 mm² ≈ 1.95 MN (about 440 kip). No anchor, nozzle or pipe rack is built for that. A heavier wall does not help: it raises the force in proportion and leaves the stress where it was.

3.Mean and instantaneous coefficients.

The coefficient of expansion is not constant. The instantaneous coefficient is the slope of the length–temperature curve at one temperature; the mean coefficient is the total strain from a reference temperature divided by the whole change. Growth is the mean coefficient times the change. The instantaneous value rises with temperature, so multiplying its hot value by the whole change overstates the growth. The B31 codes avoid the question by tabulating the total expansion from 21 °C (70 °F) to each temperature, in millimetres per metre (inches per 100 feet). With e(T) that total strain, the growth between any two temperatures is

So a flexibility program should ask for temperatures, installed and operating, and look up the total expansion from them. Entering a coefficient invites the error above, and applying a mean coefficient measured from 21 °C to a change that starts elsewhere gives a second one.

4.Relief by bending.

A straight run cannot absorb its growth in direct stress, so the line must take it in bending. Turn the run through a right angle and the growth of each leg becomes a sideways deflection of the other. A pipe is far less stiff in bending than in direct stress, so the same movement is taken with a small fraction of the force. In an L-bend the short leg bends to absorb the growth of the long one; in a Z-bend the offset leg absorbs the growth of the two parallel legs; a U-loop set into a long run absorbs the growth on either side by the bending of its two legs.

5.The guided cantilever.

A leg that absorbs a sideways movement Δ at one end, with both ends prevented from rotating, bends as a guided cantilever. The force and moment at each end are

with I the second moment of area and L the length of the leg. Divide the moment by the section modulus Z = I/(D_o/2), and

The force falls with the cube of the leg length and the stress with its square. Doubling a leg divides its stress by four and its end force by eight. The wall has again dropped out of the stress, and the diameter has come in on top: a larger pipe needs a longer leg for the same movement, in proportion to the square root of its diameter.

Take the DN200 line of section 2, 30 m long, whose 54 mm of growth is taken by a leg at right angles to it. The section has I = 30.19 × 10⁶ mm⁴ and Z = 275.6 × 10³ mm³. With E = 200 000 MPa:

Set the 18 kN of the 6 m leg beside the 1.95 MN of the straight run between anchors. The movement is the same; the shape has cut the force a hundredfold.

The guided cantilever is a sizing tool. It ignores the rotation of the corners and the extra flexibility of the bends, both of which relieve the leg, so it usually overstates the force. It gives the nominal stress in straight pipe and says nothing of the higher stress at the bend, which is the business of section E. Use it to choose a layout; analyse the layout you choose.

6.Flexibility, not strength.

Most structural design is a matter of strength: find the load, then make the member strong enough to carry it. Thermal loading does not work that way. The load is whatever it takes to impose a movement, and it grows with the stiffness of the thing moved. Make the pipe stronger and it pushes harder. The design problem is one of flexibility: to arrange the line so that it takes its movements with forces and stresses that the pipe, its supports and its equipment can bear.

Loads on a pipe are of two kinds, and the whole structure of the piping codes rests on telling them apart. They differ not in size but in what happens when the pipe begins to yield.

1.Primary loads.

Weight and pressure are primary loads. They are forces, and they do not diminish when the pipe deforms. If the stress they cause exceeds what the material can carry, the pipe keeps deforming until it collapses. Primary loads are not self-limiting, and the codes limit them against gross deformation, with the allowable stress at the operating temperature.

2.Secondary loads.

Thermal expansion and anchor movements are secondary loads. They are imposed movements, and the stress they cause is only what it takes to make the pipe conform to them. If it reaches yield somewhere, the pipe yields locally, the movement is accommodated, and the stress stops rising. The load is self-limiting; one application cannot cause collapse. What it can cause is fatigue. Each time the line heats and cools, the stress at its most highly strained points cycles, and enough cycles of a large enough range crack the pipe, most often at a weld, a bend or a branch. So the codes limit secondary stress by its range, with an allowable that depends on the number of cycles.

3.Self-springing and relaxation.

Suppose the calculated hot stress at some point is above yield. On first heating the point yields, and the hot stress settles lower than the elastic calculation said. On cooling, the pipe returns elastically through the full range, and because it was yielded hot it now carries a stress of the opposite sign cold. The line has sprung itself. At high temperature, creep does the same work more slowly: the hot stress relaxes over time and reappears reversed when the line is cold.

After the first cycle or few, a line whose calculated range is within the bounds the codes set cycles elastically between a hot and a cold stress whose difference is the range. This is shakedown. The hot stress and the cold stress are each uncertain, but their difference is not; it is fixed by the movement.

4.Pressure stresses.

Pressure p in a pipe of outside diameter D_o and wall t sets up a hoop stress around the wall,

with D the mean diameter for the average stress through a thin wall, or the outside diameter for the conservative form. Wall thickness for pressure is designed against the hoop stress, by the pressure-design rules of the code, before flexibility is considered.

The pressure acting on the closed ends of a run, or on the equivalent area at a bend, is carried by the wall as a longitudinal stress. The exact mean value, the end force pπD_i²/4 over the metal area, is

The simpler form has long been used in the B31 codes; which form applies in a sustained stress calculation is set by the code you work to. For the DN200 line at 4 MPa, the two give 23.8 and 26.8 MPa. The longitudinal pressure stress is a primary stress and belongs in the sustained stress, added to the bending stress of weight.

5.Where the pressure force goes.

In a continuous line the closed-end force is carried by the wall and is in equilibrium with itself. At an elbow, the pressure on the outer part of the bend is balanced by the tension in the wall of the two legs; the elbow is not pushed off its supports. Supports and anchors on a continuous welded line do not see the closed-end pressure force.

The force reaches the supports only where the line is not continuous in tension: at an untied expansion joint, whose bellows cannot carry it and which transmits the pressure times its effective area to the anchors on either side; at an open end or a discharge; at a slip joint or a sleeve coupling. At such points the pressure thrust is a real external load, often the largest on the anchor, and it must be applied there. Elsewhere it must not be.

6.Pressure elongation.

A pressurised pipe free to move changes length. The hoop stress stretches the circumference and, through Poisson’s ratio ν, shortens the pipe; the longitudinal stress lengthens it. The net axial strain is

taking σ_L = σ_H/2 for a thin wall. Programs often call this the Bourdon effect, after the curved gauge tube that straightens under pressure; bends do the same, opening slightly. For the DN200 line at 4 MPa, σ_H ≈ 52 MPa and the strain is 5.2 × 10⁻⁵: 1.6 mm over 30 m, beside 54 mm of thermal growth. It is small in most plant piping, but not on long high-pressure lines or large thin pipe. Applying the whole closed-end force pA_i as an axial load, without the Poisson contraction, overstates the elongation two and a half times.

7.A fully restrained, heated, pressurised pipe.

Hold a straight run so that it cannot change length at all, as a long buried line is held by the soil, and warm and pressurise it. The axial strain is zero, so the mechanical strain must cancel the thermal strain: (σ_L − νσ_H)/E + αΔT = 0, which gives

Pressure puts the wall in longitudinal tension and temperature puts it in compression. In the wall stress they act in opposite senses and do not add. For the DN200 line, 0.3 × 52 − 360 ≈ −344 MPa. The restraint force, measured from the free state, is another matter: both pressure elongation and thermal growth would lengthen the free pipe, so both add to the push on whatever holds it. A program must get both right, and it can only do so by treating pressure as a state of the wall with its Poisson strain, not as an axial force.

A flexibility analysis is not one solution but several, each a state the line will be in, and each checked against a different limit. The load cases follow from the distinction of section B: primary loads are checked as they act; secondary loads are checked as a range between states.

1.The basic cases.

A line may have several operating conditions: a normal state, an upset, a steam-out, a start-up, a cold winter shutdown. Write them as temperature states T₁ … T_n and pressure states P₁ … P_n, with W the weight of pipe, contents, insulation and components, and D any anchor movements that go with a temperature state.

2.Expansion as a difference of states.

The expansion case is not a set of loads. It is the change in forces, moments and displacements between two solved states. With one operating condition it is the operating case less the sustained case: what changed when the line went from cold to hot. With several, each pair of states gives a range, and the one that governs is usually that between the two most distant states, for example a hot operating condition and a cold condition below the installed temperature. It is formed directly between those two states, not by adding the stress ranges each makes with the installed condition.

A difference of two states is not an envelope. An envelope takes, for each quantity at each point, the largest value found in any of several cases. The values it collects may come from different cases and do not describe any one state the line is ever in. That is useful for sizing a support, which must carry the worst load in any case. It is wrong for a stress range, which must be formed between two real states, quantity by quantity with its sign.

3.Superposition and its limits.

For a linear system, the solution for several loads acting together is the sum of the solutions for each alone. Solve W, T and P separately and add, and the result is exact. That is what makes the difference OPE − SUS the same as the thermal case alone.

A line with supports that lift off, gaps that close or rests that slide is not linear. Whether a rest carries load depends on everything acting on it. Solved alone, a thermal case has no weight on the rests, so it lets a rest lift at the first upward movement, or holds the pipe down where no rest can; neither is the truth. So each state the line is actually in is solved as a whole, with all its loads together: the operating case as W + T + P in one nonlinear solution, the sustained case likewise. The expansion range is then the difference of those two complete solutions.

The sustained case needs one more thought. Where a rest lifts off when the line is hot, the weight it carried cold is carried by its neighbours for as long as the line is hot. That hot distribution of weight is also a sustained condition and must also meet the sustained limit. It is found by solving the sustained loads in the operating support state, with the lifted supports removed, as well as in the cold state, and taking the worse.

4.Hydrotest.

A line is tested before it is put in service, usually with water at a pressure above design (for B31.3, at least 1.5 times the design pressure, adjusted for temperature). A gas or steam line designed for a light fluid may never have been supported for a pipe full of water. The test case uses the water-filled weight, the test pressure and the test temperature, with spring hangers locked by their travel stops so that they act as rigid supports. It checks the supports, the stresses under test, and whether temporary supports are needed.

The rule that the range of thermal stress governs, and not its peak, is the foundation of modern piping flexibility. Rossheim and Markl proposed an allowable stress range in 1940, in “The Significance of, and Suggested Limits for, the Stress in Pipe Lines Due to the Combined Effects of Pressure and Expansion” (Trans. ASME, vol. 62), and the rule rests on the behaviour described in section B.

1.Why the range governs.

A secondary stress cannot cause collapse; its danger is fatigue, which is driven by the cycle. Self-springing and relaxation shift the hot and cold stresses but leave the difference between them, which is fixed by the movement. If the range is kept within the shakedown limit, roughly the sum of the yield strengths at the two extremes of the cycle, the line settles into elastic cycling, and the range and the number of cycles decide its life.

Markl then measured the life. The same 1940 work began a programme of fatigue tests on piping components, reported in “Fatigue Tests of Welding Elbows and Comparable Double-Mitre Bends” (Trans. ASME, vol. 69, 1947) and “Fatigue Tests of Piping Components” (vol. 74, 1952). Components were cycled in bending to failure and compared with the fatigue data for straight commercial pipe containing a butt weld. That comparison gave the stress intensification factors of section E, and the shape of the fatigue curve gave the cyclic reduction below. Markl’s “Piping-Flexibility Analysis” (Trans. ASME, vol. 77, 1955) brought the method together, and it entered ASA B31.1-1955.

2.Forming the range.

The range is formed from the forces and moments, not from stresses. Take the two states that bound the cycle. At each point, subtract the moments of one state from those of the other, component by component, with their signs: the in-plane moment range, the out-of-plane moment range, the torsion range, and the axial force range. Then compute the stress from those ranges.

Never difference a summary or combined stress. A combined stress is built from squares and absolute values; the signs and directions of its parts are gone. Suppose the in-plane moment at an elbow is +10 kN·m hot and −10 kN·m cold. The moment range is 20 kN·m. The combined stress is the same in both states, and differencing it gives a range of zero. A moment that turns from one plane to another does the same thing less obviously.

Where there are several thermal states, the range is formed between each pair that makes a real cycle, and the pair with the largest range is checked against the allowable. Smaller cycles of other sizes are counted into the number of cycles, as described below.

3.The computed range in B31.3.

The current form of B31.3 (clause 319.4.4) computes the displacement stress range as

where S_a is the axial stress range from the axial force range, intensified by an axial SIF where one applies, and the bending and torsional ranges are

with M_i, M_o and M_t the in-plane, out-of-plane and torsional moment ranges, i_i, i_o and i_t the corresponding SIFs, and Z the section modulus. Earlier editions omitted the axial term and wrote S_E = √(S_b² + 4S_t²). B31.1 forms the range differently in detail: a single SIF applied to the resultant of all three moment ranges, torsion included.

Flexibility calculations use nominal dimensions (B31.3 clause 319.3.5): the stiffness of the line, and the section modulus for the range, come from the nominal wall, not a wall reduced by corrosion allowance or mill tolerance. The sustained stress, by contrast, is computed on a section reduced by the allowances.

4.The allowable range.

The allowable displacement stress range (B31.3 clause 302.3.5) is

with S_c and S_h the basic allowable stresses at the minimum and maximum metal temperatures of the cycle. Where the sustained stress S_L is less than S_h, the unused part may be added, which gives the liberal form

The logic is that of shakedown. The allowable is built to keep the range within about the sum of the cold and hot yield strengths; the sustained stress, which is present at the hot end of the cycle, uses up part of that margin, and what it leaves is available to the range.

The factor f is the cyclic stress range reduction factor. It comes from Markl’s fatigue curve and falls as the number of equivalent full cycles N rises:

so f is 1.0 up to about 7000 cycles, roughly one a day for twenty years, and falls below it for lines that cycle more often. Cycles of smaller range are counted into N by weighting each with the fifth power of its ratio to the largest range. Recent editions of B31.3 let f rise a little above 1.0 for some lower-strength steels at moderate temperature. B31.1 sets out the same framework, with its stress checks under clause 104.8.

5.The modulus for the range, and hot reactions.

The range is computed with the reference modulus E_a, the modulus at the installed temperature, whatever the operating temperature. This is not an oversight. The allowable range and the SIFs come from fatigue tests at room temperature, their results expressed as elastic stresses at the room-temperature modulus, and the calculated range must be expressed on the same basis.

The forces and moments the line exerts hot are smaller, because the hot metal is less stiff. B31.3 clause 319.5.1 estimates the reactions of a two-anchor line without intermediate restraints from the range R computed at E_a:

with E_m the modulus at the extreme (usually hot) temperature and C the cold-spring factor of section G, zero when there is no cold spring. The cold reaction is estimated from the cold spring, or from the self-springing the line will undergo if its range is large, whichever is greater. A general analysis computes the reactions directly, but the principle is the same: the stress range at E_a, the hot loads at the hot modulus.

A pipe bend is more flexible than a straight pipe of the same length, and more highly stressed. Both facts come from the same cause, and both have to be in any flexibility calculation that is to be believed.

1.Ovalization.

Bend a curved tube and its cross-section does not stay round. The longitudinal stresses in the wall, following the curve, have components toward the middle of the section, and they flatten the section into an oval. A flattened section has a smaller second moment of area, so the bend rotates more under a given moment than beam theory predicts, and the wall carries high local bending stresses around its circumference.

Bantlin observed the ovalization in 1910, and found bends much more flexible than the theory of bars predicts. Von Kármán gave the first theory in 1911, for in-plane bending, and his analysis is the beginning of the theory of curved pipe. Vigness extended it to out-of-plane bending in 1943.

2.The flexibility factor.

The effect on stiffness is carried by the flexibility factor k: the ratio of the rotation of the component to that of a straight pipe of the same length under the same moment. A bend with k = 7 bends as seven times its own arc length of straight pipe would. In a calculation, the bend’s contribution to the flexibility of the line is multiplied by k.

For a bend of radius R, with pipe of mean radius r and wall t, the governing parameter is the flexibility characteristic or pipe factor

Thin walls and tight bends make h small, and the bend flexible. The classical result for the flexibility factor is the asymptotic closed form of Clark and Reissner (“Bending of Curved Tubes”, Advances in Applied Mechanics, vol. 2, 1951):

These analyses all give the same flexibility factor for in-plane and out-of-plane bending, and one k for both planes is the classical practice.

3.The stress intensification factor.

The stress intensification factor, or SIF, is defined by fatigue test, not by stress analysis. It is the ratio of the moment that fails straight pipe with a girth butt weld to the moment that fails the component in the same number of cycles. It therefore takes in the local bending of the wall, the geometry and the welds of the component, and it is measured relative to a butt weld, whose SIF is 1.0 by definition. Markl’s fatigue tests gave the classical forms for a bend:

for in-plane and out-of-plane bending. In-plane bending acts in the plane of the bend, opening or closing it; out-of-plane bending acts across that plane. The two are different because the ovalization, and so the stress, is different, and a program must keep the moments apart so that each can be multiplied by its own factor.

A long-radius elbow in the DN200 line makes the numbers concrete. With R = 304.8 mm, t = 8.18 mm and r = 105.5 mm, h = 0.224, so k ≈ 7.4, i_i ≈ 2.44 and i_o ≈ 2.03. The elbow bends like 7.4 times its arc length of pipe, and fails in fatigue under a moment range less than half of what fails the butt weld beside it.

SIFs multiply moments, and only moments (with, in recent practice, axial force where a factor for it is given). They are never applied to the pressure stress, which has its own rules.

4.Pressure, flanges and the modern source.

Internal pressure resists ovalization. In large, thin-walled bends at high pressure it stiffens the bend and lowers both k and i appreciably, and the codes have long allowed a correction for it. Flanges or other stiff components welded close to the ends of a bend also restrain the ovalization, and lower k and i for the same reason.

For many years the B31 codes carried their flexibility and stress intensification factors in an appendix table. They now take them from ASME B31J, whose 2017 edition gives in-plane, out-of-plane and torsional factors for bends, tees and branch connections, with test methods for components it does not cover. The classical forms above are the right place to learn the subject; the values for design come from the standard your code calls up.

5.Branch connections.

At a branch, three pipes meet. The simplest model joins them rigidly at the point where their centrelines cross. That is stiffer than the truth. The wall of the run pipe deforms locally under the branch’s moments, like a shallow shell, and the branch rotates a little relative to the run. In a thin-walled run this local flexibility can reduce the branch moments considerably; it is modelled as a rotational spring at the surface of the run, with flexibility factors for branch connections.

The branch SIF and the header SIF are applied to the moments in the branch and in the run respectively. Stress on the branch side of a reduced branch is computed with a section modulus belonging to the branch, not the run; B31.3 has long used an effective section modulus Z_e for it. The SIF and its section modulus come as a pair from the same source, and must be used together.

Before computers, flexibility calculations were made by hand by the force method, and the classical manuals set it out for lines of every shape. The 1941 Kellogg manual, Design of Piping Systems, written for The M. W. Kellogg Company by D. B. Rossheim, A. R. C. Markl, H. V. Wallstrom and E. Slezak, is the classic of that tradition and the model for this one. The method remains the clearest way to see how a line absorbs its growth, and the best check on a program.

1.Release and restore.

Take a line between two anchors, A and B. Release anchor B. The line is now a cantilever from A, statically determinate, and under a uniform temperature rise it simply grows: by section A, B moves by αΔT times the straight line from A to B. Anchor B must put it back. The forces and moments it applies to do so are the redundants: three in a plane (two forces and a moment), six in space.

Let X_j be the redundants and m_j(s) the bending moment along the line due to a unit value of each. The movement at B in the direction of redundant i caused by a unit value of redundant j is the flexibility coefficient

the integral taken along the whole line, with each bend’s arc length multiplied by its flexibility factor k. Axial and shear deformations are small in ordinary lines and are usually left out; in space, torsion adds a term in ds/GJ. Compatibility at B then reads

with Δ_i the free thermal movement of B relative to A, less any movement imposed on B relative to A. Solve these three or six equations, and every force and moment in the line follows by statics.

2.The elastic centre.

The equations are coupled: each redundant appears in every equation. The coupling can be removed by moving the point at which the redundants act. Think of each element of the line as carrying an elastic weight dw = ds/EI, with a bend’s weight multiplied by k. The elastic centre is the centroid of that weight:

Tingey introduced the idea into piping analysis, as the “virtual centre of gravity” of the line.

Attach a rigid arm from B to the elastic centre and let the redundants act there, as a force (H_x, H_y) and a moment M₀. Measure x′, y′ from the elastic centre and write I_xx = ∫y′²dw, I_yy = ∫x′²dw and I_xy = ∫x′y′dw, the moments of inertia of the elastic weight. Because the first moments about the centroid vanish, the moment equation separates from the force equations, and for a plane line in uniform expansion, with e = αΔT(r_B − r_A) the free movement of B,

The first result is the classical one: the thermal thrust of a two-anchor line is a single force whose line of action passes through the elastic centre. The bending moment at any point is that force times the point’s distance from its line of action, so the largest moments are at the points farthest from it. Take the force axes along the principal axes of the elastic weight, where I_xy = 0, and the two force equations separate as well; each redundant is then found alone.

In space the same construction holds with six redundants and the moments of inertia of the elastic weight in three dimensions, with torsion entering through ds/GJ. It is lengthy by hand.

3.A worked case, and the stiffness method.

The verification case Expansion loop with a curved bend solves a plane L-bend between two anchors by exactly this method: anchor B released, the flexibility coefficients integrated in closed form with the arc’s share multiplied by k, and the three equations solved together for k = 1, 5, 10 and 20. Moving the redundants to the elastic centre would separate the moment from the forces and give the same reactions. The hand result then checks the solver.

SWBPIPE, like every modern program, solves the same problem by the stiffness method (Part III): unknown displacements at the nodes, rather than unknown forces at a released anchor. The two methods give the same answer for the same model. Its curved bend element is built from the same flexibility integrals, evaluated around the arc and then inverted to a stiffness.

Cold spring is the deliberate pre-straining of a line when it is erected, so that part of its thermal movement is taken up before it is ever heated. A line that will grow is cut short and pulled together at the last joint; a line that will shrink, one operating below the installed temperature, is cut long and pushed together.

1.What it does.

Let the expansion range produce a reaction R at an anchor. Cut the line short by a fraction C of the movement it must absorb, and close the gap at erection. The cold line now carries a reaction of about CR at its anchors, opposite in sense to the hot reaction, and when it is heated the reaction swings through the full range to about (1 − C)R hot. With C = ½, the hot and cold reactions are about equal and half the uncorrected hot value. The usual purpose is to bring down the hot load on a nozzle or anchor that cannot take the full value.

2.No credit in the range.

Cold spring moves both ends of the cycle by the same amount. It changes the hot and cold stresses but not the range between them, and the range is what governs fatigue. So the codes give no credit for cold spring in the displacement stress range (B31.3 clause 319.2.4); the range is computed as if there were none.

3.Two-thirds credit in the reactions.

Cold spring is hard to achieve exactly and hard to verify once the line is closed. The codes therefore credit only part of it in the hot reactions: equation (15) of section D takes two-thirds of C, giving R_m = R(1 − 2C/3)(E_m/E_a), while the cold reaction is taken at the full CR. Both are estimates for two-anchor lines; a general analysis applies the cold spring explicitly and applies the same caution to the result.

4.Modelling it.

In a model, cold spring is an imposed relative displacement across the cut: two nodes at the location of the closing weld, a gap between them equal to the cut, and the gap closed as a load. Once closed, the cut is present in every state, cold and hot, so it cancels from the range between them, as the codes require. It is a displacement load, not a primary one, and the sustained case leaves it out. The cold-spring load alone shows the force needed to close the joint at erection.

A line is held by supports, and each support acts in some directions and not in others. A model is only as good as its description of what each support actually does: which way it holds, whether it can let go, how much it gives, and whether the pipe slides on it.

1.The kinds of restraint.

Any of these may have a gap: a clearance through which the pipe moves freely before the restraint takes hold. A gap may be on one side only, as with a rest, or on both sides, as with a guide with clearance on each side of the shoe. The two behave quite differently and must be described separately.

Supports do not always act along global axes. A guide on a run that is not parallel to the plant axes acts perpendicular to the pipe, not along north or east. A program must allow restraints along any direction, and restraints defined relative to the pipe, so that a guide stays a guide whatever way the pipe runs.

2.Stiffness.

A support is usually modelled as rigid. A rigid support draws more load than a soft one, so this is conservative for the support itself, but it passes less load to its neighbours than they will see in the plant. A cantilevered bracket, a long rack beam or a vessel shell can move appreciably under load; where it matters, the support is given its real stiffness.

3.Friction.

A pipe that slides on a rest is resisted by friction. The friction belongs to the rest itself and depends on that rest’s own normal reaction N, which changes with the load case. It acts in the plane of the support, against the sliding, and it is Coulomb friction: the pipe sticks until the resultant tangential force reaches μN, then slides against a force of that size opposing the direction of slip. With T₁ and T₂ the two tangential components,

The limit is a circle, not a square: a pipe sliding diagonally is resisted by μN, not by μN in each direction. Typical coefficients are about 0.3 for steel on steel and about 0.1 for PTFE slide plates.

Friction usually raises the operating loads on anchors and nozzles, because it resists the growth that the line was laid out to accommodate. It can also lower them, by holding a leg that would otherwise push on a nozzle. So the operating case is commonly run both with and without friction, and the worse result taken for each quantity.

Friction dissipates energy, so its result depends on the path. A static solution assumes the loads were applied in one step from a stated starting state; on cooling the friction reverses, and the line does not return exactly to where it began. Treat friction results as estimates, not as one exact answer.

4.Lift-off.

A rest carries weight only while the pipe presses on it. As a line heats, parts of it rise, and a rest there stops carrying load. Its share of the weight goes to its neighbours. The same happens with a rod hanger: a rod can hold a pipe up but not hold it down, and when the line grows upward the rod goes slack.

This is the classic way nozzle loads go wrong. A pump discharge line rises from the nozzle, turns and runs toward the rack, with a rod hanger or a rest near the pump carrying the weight of the valves and the riser. When the line heats, the riser grows upward from the nozzle and lifts the line off that support. The weight it carried now has nowhere to go but the next support along and the pump nozzle. An analysis that treats the support as two-way has it holding the pipe down, which no rest or rod can do; the plant shows a misaligned coupling. The remedy is a spring, which section I takes up.

Lift-off makes the analysis nonlinear. A program finds the support states by iteration: assume a state for each support, solve, check each support against the result, change those that disagree, and solve again until none changes.

Where a line must be supported at a point that moves vertically between cold and hot, a rigid support will either lift off or hold the pipe against its growth. A spring carries the weight and lets the point move. The price is that a spring’s load changes with its travel, and the change goes somewhere.

1.The procedure.

Spring hangers are designed, not guessed, by a standard procedure in five steps.

Step 1. Restrained weight. Solve the weight case, W, with a rigid vertical support at each hanger location. The load found at each is the hot load: the share of weight the spring is to carry when the line is hot. A spring that carries exactly this in the hot position leaves the rest of the line, and the equipment, with the weight distribution of a rigidly supported line.

Step 2. Travel. Solve the operating case with each hanger removed and its hot load applied as an upward force in its place. Under weight alone this force holds the pipe exactly where the rigid support did, so the movement found at the hanger is the thermal movement: the travel. Removing the hanger, or giving it zero stiffness, is what lets the travel appear.

Step 3. Rate and variability. A spring of rate k changes its load by k times its travel δ. The variability is that change as a fraction of the hot load,

and it is commonly held to 25 %, following MSS SP-58 practice, or lower near sensitive equipment. Choose a rate low enough to meet the limit, and a spring whose working range contains both the hot and the cold load. If no variable spring will do, use a constant-effort support.

Step 4. Cold load. The load the spring is set to at installation, the cold or installed load, follows from the hot load and the travel. A spring’s load falls as the pipe rises, because the spring extends, and rises as the pipe falls. So for travel measured from cold to hot, positive upward,

a point that moves up is set cold at more than its hot load; a point that moves down is set cold at less. It is worth stating in words on every hanger schedule: a sign error here doubles the load change the spring was chosen to limit, and puts it on the equipment.

Step 5. Final analysis. Put each spring into the model as its rate with its cold load applied as a preload, and run every load case again. In the operating case each spring should now carry close to its hot load. In the sustained case it is common practice to carry each spring as a force equal to its hot load, the weight share it was designed for; the difference between cold and hot load is caused by thermal movement, and belongs with the displacement stresses.

2.A worked variability check.

A hanger location has a hot load of 12.0 kN and moves 20 mm upward from cold to hot. For 25 % variability the rate may not exceed 0.25 × 12 000 N / 20 mm = 150 N/mm. Choose a spring of 120 N/mm (about 685 lbf/in). Its variability is 120 × 20 / 12 000 = 20 %, and its cold load is 12.0 + 0.120 × 20 = 14.4 kN. The 2.4 kN it carries cold above its hot load is taken off the neighbouring supports and nozzles when the line is cold, and handed back to them as it heats. The variability limit is a limit on that exchange.

3.Constant-effort supports.

A constant-effort support uses a spring and a lever or cam so that its load stays nearly the same through its whole travel. It is chosen where the travel is too large for a variable spring to stay within the variability limit, or where the load on nearby equipment must not change at all. It is set to the hot load. It carries a fixed force in every case that includes weight, and contributes nothing to a thermal range: its load does not change when the line moves.

The piping is often the least of what a flexibility analysis protects. A pump casing distorted by its piping loses alignment and wears its seals; a compressor misaligns; a vessel nozzle cracks at its weld. The loads the line puts on equipment are therefore limited, and often they govern the layout more than any pipe stress.

1.Where the limits come from.

Allowable nozzle loads are not set by the piping codes. They come from the equipment standards and from the equipment vendor:

The limits are given in the axes the standard defines, usually the nozzle axis and two directions across it, or the equipment’s own axes. Loads must be reported in those axes, as forces and moments with their signs, not as resultants in global axes.

2.Anchor or flexible nozzle.

The simplest model ends the line at the nozzle with an anchor. For pumps, turbines and compressors, whose casings are stiff, that is usually right. A vessel or tank shell is not stiff: it gives under the nozzle load, and a rigid anchor there overstates the load and puts the stress in the wrong place. Nozzle flexibility is modelled with rotational and axial springs at the shell, from WRC 297 for cylindrical vessels or from the stiffness coefficients of API 650 Annex P for tank nozzles, or from finite elements.

3.The equipment grows too.

The equipment is at temperature as well, and its nozzles move. A pump nozzle moves by the growth of the casing from the point where the pump is held, its feet or its centreline supports and the key that locates it, to the nozzle face. A vertical vessel grows upward from the base of its skirt. A tank shell bulges outward under its liquid head and rotates at the nozzle. Each is an imposed displacement at the end of the line, belonging to the operating state that causes it: a vessel at T₁ moves its nozzle by one amount, at T₂ by another. Like thermal expansion, an anchor movement is a secondary load and part of the displacement stress range.

4.Which cases to check.

Nozzle loads are checked in the operating case, which is usually the largest, but not only there. The sustained case gives the load the equipment carries whenever the line is cold. Where friction is present, the cases with and without it both count. Two pumps in parallel, one running and one spare, are checked with each hot and the other cold, because the cold spare still sits on a line pushed by its hot neighbour. The vendor and the standard decide which cases apply; the analysis must supply them all.

Occasional loads are the last of the loads a static flexibility analysis carries. After them comes the use of all the results: each code check needs particular quantities from particular cases, and the analysis is organised to supply them.

1.Wind.

Wind on a pipe is a drag load per unit length,

where q is the velocity pressure at the height of the pipe, C_d the drag (force) coefficient of a cylinder, which depends on the Reynolds number and the surface, and D_e the exposed diameter, including insulation. Wind speed rises with height, and the wind standard in use gives the profile. The load acts on the component of the wind perpendicular to each pipe; a run parallel to the wind takes little. Wind is applied from each direction that matters, and in both senses.

2.Static seismic.

The simplest seismic analysis applies a static load equal to a seismic coefficient, in g, times the weight of pipe, contents and insulation. It is applied in each horizontal direction and, where required, vertically, each in both senses, as separate cases. The directional results are then combined by the rule the governing standard gives, such as the square root of the sum of squares or a percentage rule. B31.3 does not require wind and earthquake to be taken as acting together.

3.Relief thrust.

A relief valve discharging to atmosphere pushes back on the line with a force equal to the momentum of the jet plus the excess pressure at the outlet times its area,

with ṁ the mass flow, and V_e, p_e and A_e the velocity, pressure and area at the exit. It arrives suddenly, so as a static load it is multiplied by a dynamic load factor, up to 2. B31.1 sets out a method in a nonmandatory appendix.

Response spectrum and time-history analysis, fluid hammer, slug flow and vibration are dynamic problems, and outside the scope of this part.

4.From results to a code check.

Each category of code check draws on particular cases. The table sets them out, with what SWBPIPE supplies for each today.

What SWBPIPE supplies in every row is mechanics: forces, moments and stress components. It does not compute a code stress, apply a code allowable, or judge compliance with any code. The code check is made in the program you validate in, as Part V describes.