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 engineering discipline and the assumptions behind a flexibility calculation. It does not claim that every method is implemented in SWBPIPE. Statements headed Today identify current routes and limitations, with links to the status of this edition. Part III explains how the program builds and solves a supported model.

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 rises 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 keeps its shape and grows in size when uniformly warmed and free to expand. Its free end moves by αΔT times the straight-line vector from the fixed point to that end. This movement is independent of the pipe route. The movement to absorb between two anchors depends only on their positions. The loops between them determine the required force.

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 temperature change. Growth equals the mean coefficient times that change.

The instantaneous value rises with temperature. Multiplying its hot value by the whole change overstates growth. The B31 codes tabulate total expansion from 21 °C (70 °F), in millimeters per meter (inches per 100 feet). With e(T) as that total strain, growth between two temperatures is

A flexibility program should ask for installed and operating temperatures and obtain the total expansion between them. Entering a coefficient can cause the error described above. A mean coefficient from 21 °C also gives an error if applied to a temperature change that starts elsewhere.

4.Relief by bending.

A straight run cannot absorb its growth through direct stress, so the line must bend. Turn the run through a right angle. The growth of each leg then becomes a sideways deflection of the other. A pipe is much less stiff in bending than in direct stress. Thus, the same movement requires a small fraction of the force.

In an L-bend, the short leg bends to absorb growth of the long leg. In a Z-bend, the offset leg absorbs growth of the two parallel legs. A U-loop absorbs growth on either side through 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 leg length, and stress falls with its square. Doubling a leg divides stress by four and end force by eight. Wall thickness does not appear in the stress expression. Diameter is in the numerator. For the same movement, the required leg length increases in proportion to the square root of pipe 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 corner rotation and extra bend flexibility. Both effects relieve the leg, so the estimate usually overstates force. It gives nominal straight-pipe stress. It does not give the higher bend stress discussed in section E. Use it to choose a layout; analyze the layout you choose.

6.Flexibility, not strength.

Most structural design concerns strength: find the load, then make the member strong enough to carry it. Thermal loading differs. The load is the force needed to impose a movement. That load increases with stiffness. A stronger pipe pushes harder.

The design problem concerns flexibility. The layout must absorb movement with forces and stresses that the pipe, supports and equipment can carry.

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 impose movements. The resulting stress is the stress needed to make the pipe conform to those movements. Where stress reaches yield, local yielding accommodates movement and stops the stress from rising. The load is self-limiting; one application cannot cause collapse.

Repeated application can cause fatigue. Each heating and cooling cycle changes stress at the most highly strained points. Enough cycles of sufficient range crack the pipe, often at a weld, bend or branch. The codes therefore limit secondary stress by its range. The allowable 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. Because the pipe yielded hot, its cold stress has the opposite sign. 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 few cycles, a line within the code range limits cycles elastically between hot and cold stresses. This behavior is shakedown. The separate hot and cold stresses are uncertain. Their difference is the range fixed by 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.

Pressure acts on the closed ends of a run, or on an equivalent area at a bend. The wall carries this force as 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. The applicable code selects the form for a sustained stress calculation. 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, tension in the walls of both legs balances pressure on the outer part of the bend. The elbow is not pushed off its supports. Supports and anchors on a continuous welded line do not see the closed-end pressure force.

Pressure force reaches supports only where the line is not continuous in tension. An untied expansion joint cannot carry that force through its bellows. It transfers pressure times its effective area to the anchors on either side. Other examples include an open end, discharge, slip joint or sleeve coupling. At these points, pressure thrust is a real external load and is often the largest anchor load. It must be applied there; elsewhere, it must not be applied.

6.Pressure elongation.

A pressurized 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 against all length change, as soil restrains a long buried line. Warm and pressurize the run. 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 is measured from the free state. Pressure elongation and thermal growth both lengthen free pipe. Both therefore increase the force on its restraints. A program must represent both effects correctly. It must treat pressure as a wall state with Poisson strain, rather than as an axial force alone.

A flexibility analysis uses several solutions. Each describes a state the line will occupy and is compared with a different limit. Section B explains the distinction: primary loads are assessed as they act, and secondary loads as a range between states.

1.The basic cases.

A line can have several operating conditions: normal, upset, steam-out, startup and cold winter shutdown. Write their temperature states as T₁ … T_n. Write their pressure states as P₁ … P_n. Use W for the weight of pipe, contents, insulation and components. Use D for anchor movements associated 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. For one operating condition, this is normally the operating state minus the sustained state. The difference includes the effect of supports that lift off or slide.

With several operating conditions, each pair of states gives a range. The governing pair is usually the two most distant states. An example is a hot operating state and a cold state below installed temperature.

The range must be formed directly between those states. Do not add their separate stress ranges from 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 lift-off supports, closing gaps or sliding rests is nonlinear. Whether a rest carries load depends on all loads acting together. A thermal case solved alone has no weight on its rests. It can let a rest lift at the first upward movement or hold the pipe down where the real rest cannot.

Each physical state must therefore be solved as a whole, with all its loads together. The operating case is W + T + P in one nonlinear solution. The sustained case is also solved as a whole. The expansion range is the difference of those complete solutions.

The sustained case requires another assessment. Where a rest lifts off when hot, neighboring supports carry its weight share throughout the hot condition. This hot weight distribution is also sustained and must meet the sustained limit. Solve sustained loads in the operating support state, with lifted supports removed, and also in the cold state. Take the worse result.

4.Hydrotest.

A line is tested before service, usually with water above design pressure. For B31.3, test pressure is at least 1.5 times design pressure, adjusted for temperature. A gas or steam line designed for light fluid may lack support for a pipe full of water.

The test case includes water-filled weight, test pressure and test temperature. Spring hangers are locked with travel stops and act as rigid supports. The case assesses support loads, test stresses and the need for temporary supports.

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 behavior described in section B.

1.Why the range governs.

A secondary stress cannot cause collapse; fatigue from cycling is its danger. Self-springing and relaxation shift hot and cold stresses but preserve their difference, which movement fixes. The shakedown limit is roughly the sum of the yield strengths at the two cycle extremes. Within that limit, the line settles into elastic cycling. Stress range and cycle count then determine its life.

Markl then measured the life. The same 1940 work began a program 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 forces and moments, rather than from stresses. Take the two states that bound the cycle. At each point, subtract matching signed moment components. Form the in-plane, out-of-plane and torsional moment ranges, and the axial force range. Then compute 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 several thermal states form real cycles, a range is formed between each applicable pair. The pair with the largest range is compared with the allowable. Smaller cycles are included in the cycle count 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). Line stiffness and the section modulus for the range use nominal wall thickness. Corrosion allowance and mill tolerance do not reduce that wall. Sustained stress uses 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 allowable follows the shakedown principle. It keeps the range within about the sum of cold and hot yield strengths. Sustained stress is present at the hot end of the cycle and consumes part of that margin. The remaining margin is available for 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 uses the reference modulus E_a at the installed temperature, regardless of operating temperature. The allowable range and SIFs come from fatigue tests at room temperature. Those test results are expressed as elastic stresses at the room-temperature modulus. The calculated range must use 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. Its cross-section does not remain round. Longitudinal wall stresses follow the curve and have components toward the section center. These components flatten the section into an oval. The flattened section has a smaller second moment of area.

The bend therefore rotates more under a given moment than beam theory predicts. The wall also 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 flexibility factor k represents the effect on stiffness. It compares component rotation with rotation of straight pipe of the same length under the same moment. A bend with k = 7 bends like seven times its arc length of straight pipe. The bend’s contribution to line flexibility 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 testing rather than stress analysis. It compares the failure moment of straight pipe with a girth butt weld to the failure moment of a component. Both failure moments correspond to the same number of cycles. The factor includes local wall bending, component geometry and welds. The reference butt weld has an SIF of 1.0 by definition. Markl’s fatigue tests gave these classical bend forms:

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 factors differ because ovalization and stress differ between planes. A program must keep the moments separate so that each uses 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. Its fatigue failure moment range is less than half the range that fails the adjacent butt weld.

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. At high pressure in large, thin-walled bends, it appreciably reduces k and i. The codes have long permitted a correction for this effect. Flanges or stiff components welded near bend ends also restrain ovalization. They reduce k and i for the same reason.

For many years, the B31 codes gave flexibility and stress intensification factors in an appendix table. They now obtain those factors from ASME B31J. Its 2017 edition gives in-plane, out-of-plane and torsional factors for bends, tees and branch connections. It also gives test methods for components outside its coverage. The classical forms above teach the subject. Design values come from the standard required by the applicable code.

5.Branch connections.

At a branch, three pipes meet. The simplest model joins them rigidly where their centerlines cross. That model is stiffer than the real connection. The run wall deforms locally under branch moments, like a shallow shell. The branch rotates slightly relative to the run.

In a thin run wall, this local flexibility can substantially reduce branch moments. A rotational spring at the run surface represents the effect, using branch-connection flexibility factors.

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 uses the branch section modulus. It does not use the run section modulus. B31.3 has long used an effective section modulus Z_e for this purpose. 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 statically determinate cantilever from A. Under a uniform temperature rise, it expands freely. By section A, B moves by αΔT times the straight-line vector 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 center 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 classical: the thermal thrust of a two-anchor line is one force through the elastic center. The bending moment equals that force times the point’s distance from its line of action. Points farthest from the force line therefore have the largest moments.

Take force axes along the principal axes of the elastic weight, where I_xy = 0. The two force equations also separate. Each redundant can then be found alone.

In space, the construction uses six redundants and three-dimensional moments of inertia of the elastic weight. Torsion contributes 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 this method. It releases anchor B and integrates the flexibility coefficients in closed form. The arc contribution is multiplied by k. The three equations are solved together for k = 1, 5, 10 and 20. Moving the redundants to the elastic center separates the moment from the forces and gives the same reactions.

The recorded hand comparison exercises the bend element and frame kernel directly. It does not exercise the desktop or product-physics load-generation path. The case records the comparison tolerances and limitations; it is not a new run for this edition.

SWBPIPE uses the stiffness method (Part III). Its unknowns are nodal displacements rather than forces at a released anchor. The two methods give the same answer for the same model and assumptions. The curved bend element uses flexibility integrals evaluated around the arc and inverted to form stiffness.

Cold spring deliberately pre-strains a line during erection. It takes up part of the thermal movement before heating. A line that will grow is cut short and pulled together at the last joint. A line that will shrink below its 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 anchor reaction is about CR, opposite in sense to the hot reaction. Heating changes the reaction 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. The codes therefore give no cold-spring credit in the displacement stress range (B31.3 clause 319.2.4). The range is computed as if cold spring were absent.

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 represent the closing-weld location. Their gap equals the cut length, and closing that gap is a load. After closure, the cut is present in both cold and hot states. It therefore cancels from their range, as the codes require.

Cold spring is a displacement load rather than a primary load. The sustained case excludes it. The cold-spring load alone gives the force needed to close the joint during erection.

Supports hold a line in some directions and permit movement in others. The model must describe each support’s actual behavior. That description includes restraint directions, possible release, stiffness and sliding behavior.

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 can be on one side only, as at a rest. It can also be on both sides, as at 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 in any direction and restraints defined relative to the pipe. A guide then remains a guide for any pipe direction.

2.Stiffness.

A support is usually modeled as rigid. A rigid support attracts more load than a soft support. This assumption is conservative for that support, but can understate loads on neighboring supports. A cantilever bracket, long rack beam or vessel shell can move appreciably under load. Where that movement matters, the model uses the support’s actual 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. Friction acts in the support plane against sliding. Under Coulomb friction, the pipe sticks until the resultant tangential force reaches μN. It then slides against a force of that magnitude, opposite 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 the result depends on the load path. A static solution assumes one-step loading from a stated initial state. Friction reverses on cooling, and the line does not return exactly to its initial position. Treat friction results as estimates, rather than 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 neighbors. A rod hanger behaves similarly. It can hold a pipe up but cannot hold it down. Upward line growth makes the rod go slack.

This is the classic way nozzle loads go wrong. A pump discharge line rises from the nozzle, turns, and runs toward the rack. A nearby rod hanger or rest carries the valves and riser weight. 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. A two-way support model holds the pipe down, which a rest or rod cannot 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 iterates to find support states. It assumes a state for each support, solves, and compares each state with the result. It changes inconsistent states and solves again until no support changes.

A rigid support at a point that moves vertically between cold and hot can lift off or restrain thermal growth. A spring carries weight while permitting movement. Its load changes with travel. That load change transfers to other parts of the line.

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. Each support reaction gives the hot load: the weight share the spring must carry when hot. A spring carrying that load in the hot position preserves the weight distribution of the rigidly supported line. This includes the weight loads on equipment.

Step 2. Travel. Remove each hanger and apply its hot load as an upward force in its place. Solve the operating case. Under weight alone, this force holds the pipe where the rigid support did. The movement at the hanger is therefore the thermal movement, or travel. Removing the hanger or setting its stiffness to zero permits this travel.

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. This sign relationship should be stated in words on every hanger schedule. A sign error doubles the load change that the spring was selected to limit. That change loads the equipment.

Step 5. Final analysis. Put each spring into the model with its rate and cold-load preload. Run every load case again. In the operating case, each spring should carry close to its hot load.

Common sustained-case practice represents each spring as a force equal to its hot load, the weight share for which it was designed. The cold-to-hot load difference comes from thermal movement. That difference belongs with 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 extra 2.4 kN cold load reduces loads on neighboring supports and nozzles when cold. Heating transfers this load back to them. The variability limit is a limit on that exchange.

3.Constant-effort supports.

A constant-effort support uses a spring and lever or cam to keep its load nearly constant throughout travel. It is selected when a variable spring cannot meet the variability limit over the required travel. It is also selected where nearby equipment load must not change. It is set to the hot load. It carries a fixed force in every case that includes weight. It contributes nothing to a thermal range because its load does not change with movement.

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 a nozzle anchor. This usually suits the stiff casings of pumps, turbines and compressors. A vessel or tank shell deforms under nozzle load. A rigid anchor can overstate the load and place stress incorrectly.

Rotational and axial springs at the shell represent nozzle flexibility. Their data can come from WRC 297 for cylindrical vessels, API 650 Annex P for tank nozzles, or finite elements.

3.The equipment grows too.

The equipment also changes temperature, so its nozzles move. A pump nozzle moves as the casing grows from its restraint points to the nozzle face. Those restraints include feet or centerline supports and the locating key. A vertical vessel grows upward from its skirt base. A tank shell bulges under liquid head and rotates at the nozzle.

Each motion is an imposed displacement at the pipe end and belongs to the operating state that causes it. A vessel at T₁ has one nozzle movement; at T₂, it has another. Like thermal expansion, anchor movement is secondary and contributes to displacement stress range.

4.Which cases to check.

Nozzle loads are assessed in the operating case, which is usually largest, and in other applicable cases. The sustained case gives equipment load when the line is cold. With friction, both frictional and frictionless cases count. Parallel pumps require each hot/cold arrangement: one operating and one spare. The cold spare remains on a line pushed by its hot neighbor. The vendor and standard specify applicable cases; the analysis must supply all of them.

Occasional loads are the last of the loads a static flexibility analysis carries. Each code check needs particular quantities from particular cases. The analysis is organized to supply those quantities.

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 governing standard specifies how to combine directional results. Examples are the square root of the sum of squares and percentage rules. 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.

The table distinguishes engineering inputs from currently supported mechanics quantities. SWBPIPE does not supply a built-in code stress, allowable or compliance judgment. User formula criteria can evaluate intervals. Nonfinite intermediate arithmetic blocks evaluation; a passing formula gate is not standards approval. The code check is made in the program used for validation, as Part V explains.