Pipe Stress and Support Spans: Method

Pressure Thickness, Spans, Guided Cantilever, Stiffness Solution

Scope

The tool makes the three hand checks that come before a formal pipe stress analysis, chosen by the first dropdown: the wall for internal pressure, the spacing of supports for weight, and the flexibility of a plane layout between two anchors for thermal expansion. It is a calculator of the mechanics the piping codes use, with the code quantities (allowable stresses, joint and weld factors, the coefficient Y, occasional-load factors) entered by the user from the code that governs the line. It does not reproduce the allowable stress tables, the hanger-spacing table or any other table of ASME B31.1, ASME B31.3 or MSS SP-58, and it is not a substitute for the analysis those codes require of a line that fails the screening rule, has intermediate restraints, branches, several sizes or three-dimensional routing.

Units and conventions

Lengths, diameters, pressures, stresses, moduli, densities, forces, moments, line loads, the temperature change and the expansion coefficient go through the site's unit dictionary and convert with the unit switch. Factors (E, W, Y, the shape factor, the cold-spring factor), the mill tolerance and the pitch in percent, the earthquake coefficient in g and the number of cycles are plain numbers. The custom leg list is read in metres in SI mode and in feet in USCS mode. In the flexibility analysis x runs along the first leg from anchor A, y is 90° counter-clockwise from it, and the reactions printed are those of the anchors on the pipe.

Internal pressure

The codes write the pressure design thickness of straight pipe as t = P D_o / (2 (S E W + P Y)). A 1965 report of the M. W. Kellogg Company for the Atomic Energy Commission quotes the rule of the B31 code of its day as t_m = P D_o / (2 (S E + P y)) + A, with A the allowance for threading and mechanical strength, for D/t above 6; W is the weld strength reduction factor added by the current editions for welds in the creep range. Y interpolates between the thin-cylinder formula on the outside diameter (Y = 0, the Barlow form that 49 CFR 192.105 and 195.106 use for pipelines with the yield strength and a design factor), on the mean diameter (Y = 0.5) and on the inside diameter (Y = 1). With Y = 0.4 the result stays close to the maximum hoop stress of Lamé's thick cylinder over the range of ordinary pipe. The tool adds the allowance c, divides by one less the mill under-tolerance for the nominal wall to order, and inverts the equation for the pressure a chosen wall carries, P = 2 S E W t_e / (D_o − 2 Y t_e) with t_e the nominal wall less tolerance and allowance. The inside-diameter form of the same equation is printed as a check; the two are algebraically identical.

Stresses are given for the nominal wall by the thin-wall forms and by Lamé: hoop σ_θ = P r_i² (1 + r_o²/r²)/(r_o² − r_i²), radial σ_r = P r_i² (1 − r_o²/r²)/(r_o² − r_i²), and with closed ends the longitudinal stress P r_i²/(r_o² − r_i²), uniform across the wall. The longitudinal pressure stress is half the hoop stress or less, which is why a pipe sized for pressure has longitudinal strength to spare for weight.

External pressure is named only: the tool prints the elastic buckling pressure of a long perfect tube, P_cr = 2 E (t/D_m)³/(1 − ν²). Real collapse depends on out-of-roundness, yield strength and the length between stiffening rings, and the codes use the external-pressure charts of the ASME Boiler and Pressure Vessel Code.

Weight and support spans

The weight per unit length is the pipe metal, the contents, the insulation (from its thickness and bulk density) and any other uniform weight. A span between two supports is one of four beams under that uniform load:

• a single span on simple supports, moment w L²/8 at mid-span, sag 5 w L⁴/(384 E I);
• the end span of a long run of equal spans, solved exactly by the three-moment equation: the moment over the first interior support is (3 − √3)/12 w L² = 0.1057 w L² and the sag 0.00655 w L⁴/(E I);
• the w L²/10 rule, which is the hanger-spacing equation S = 1.2 w l²/Z (w in lb/ft, l in ft, Z in in³) printed in ORNL-TM-3645 for the suggested spans it quotes from the power piping code; the tool takes its sag, slope and frequency from the end span above, which the rule approximates;
• an interior span of a long run, whose ends stay level: moment w L²/12 at the supports, sag w L⁴/(384 E I).

The allowable span is the shorter of L = √(S Z/(c_M w)) for the entered bending stress and L = (δ E I/(c_δ w))^1/4 for the entered sag. ORNL-TM-3645 gives 1,500 psi as the bending stress behind its spans and warns that the weight stress has to share the allowable with the pressure and other sustained stresses. A line drains only if its pitch exceeds the steepest slope of the sagging pipe, c_s w L³/(E I): 3.2 δ/L for the simple span and 16/(3√3) δ/L = 3.08 δ/L for the level-ended span. The lowest natural frequency is f = (λ²/(2π L²)) √(E I g/w), λ² = π² for the simple span and for a continuous run of equal spans (whose spans vibrate alternately up and down), 22.37 for the span with ends held level.

The span table applies the same limits to every nominal size of the chosen schedule and sets the result beside the Support Spacing Schedule of UFC 3-430-09. That schedule and the similar tables of ASME B31.1 and MSS SP-58 are for straight horizontal runs without concentrated loads; valves, flanges and risers need their own supports and changes of direction a shorter span.

The occasional-load check is deliberately simple: wind as the velocity pressure times a shape factor on the insulated diameter, an earthquake as a static horizontal load of a_h times the weight, each combined with the weight as a resultant on the span entered. The codes add the result to the longitudinal pressure stress and allow a stated increase of the basic allowable for occasional loads. Wind-induced vibration, response spectra and support stiffness are outside the check.

Expansion: bends, the allowable range

A leg heated by ΔT grows α L ΔT; the growth of the line between two fixed anchors, y = α ΔT U, has to be taken up by bending. A curved pipe ovalizes when bent and is therefore more flexible and more highly stressed than a straight beam of the same section. With the flexibility characteristic h = t R/r², the codes use the flexibility factor k = 1.65/h, the asymptotic result of the Kármán and Clark–Reissner analyses, and Markl's stress intensification factor i = 0.9/h^2/3, a fatigue-based factor referred to straight pipe with a girth butt weld. Welding tees take h = 4.4 t/r, reinforced fabricated tees h = (t + t_e/2)^5/2/(r t^3/2) and unreinforced ones the same with no pad, all with i = 0.9/h^2/3. Internal pressure reduces k and i of large thin bends; the tool does not apply that reduction.

Expansion stress is displacement-controlled. The codes therefore limit its range, S_E = √(S_b² + 4 S_t²) with S_b = i M/Z, to S_A = f (1.25 S_c + 0.25 S_h), f = 1 up to 7,000 full cycles, and compute it with the cold modulus over the whole temperature change. The total 1.25 (S_c + S_h) is available when no sustained stress uses part of S_h.

Guided cantilever

The classical estimate treats each leg as a cantilever whose free end moves sideways without rotating. A movement δ then needs the force 12 E I δ/L³ and end moments 6 E I δ/L², a stress 3 E D_o δ/L². The growth along one axis is assigned to the legs at right angles to it; legs that follow one another carry the same force and so divide the movement in proportion to L³. Solving for the length gives the familiar L = √(3 E D_o δ/S_A), with i under the root when the leg ends on a bend. The method is exact only if the corners cannot rotate. Real corners do rotate, which relieves the moments, so the estimate is normally high, often by a factor of two or more for a short leg on a long run.

The screening rule D y/(L − U)² ≤ 30 S_A/E_a of ASME B31.3 (inches for D and y, feet for L and U; 0.2083 S_A/E in consistent units) is evaluated for the layout. It is empirical and the code states that it is not always conservative.

Stiffness solution

The tool then solves the same layout as a plane frame fixed at both anchors. Straight legs are beam elements with axial and bending stiffness (shear deformation neglected); every bend is a circular arc tangent to its two legs, cut into 24 elements per 90° with the bending stiffness E I/k. The load is the free thermal strain α ΔT of every element. The solution gives the three reactions at each anchor, the moment everywhere, the expansion stress i M/Z (i over the bends, 1 on straight pipe and at the anchors) and the displaced shape, drawn magnified over the layout. A second run with square corners and rigid joints shows what the bend flexibility is worth. The independent check of the solver is a flexibility-method (elastic-centre) solution integrated along the exact arcs; the two agree to six figures or better with square corners and to about one part in ten thousand with bends of ordinary radius.

Cold spring does not change the stress range. With the cold-spring factor C the tool applies the code rule to the reactions: hot (1 − 2C/3)(E_h/E_c) R, cold the larger of C R and (1 − (S_h/S_E)(E_c/E_h)) R.

Limits

One pipe size and material per run; two anchors with no intermediate guides, stops or hangers in the flexibility analysis; plane layouts loaded in their own plane (no torsion, no out-of-plane bending); up to five legs; anchors that do not move; no weight, pressure thrust, friction or spring supports in the frame; no branch connections in the layout (their factors are tabulated for reference only); no elbow pressure stiffening; no creep relaxation. Expansion joints, buried pipe and plastic pipe, whose modulus depends on time and temperature, are outside the tool.