Buried Pipe Loads: Method

Buried Pipe Loads

What the tool does

The tool finds the loads on a buried pipe and checks the two kinds of pipe wall against them. A rigid pipe (concrete, clay, cast iron) carries the load by the strength of its own ring and is bought by its three-edge-bearing strength; a flexible pipe (steel, ductile iron, plastic) deflects until the soil at its sides carries the load with it, and is checked for deflection, ring compression and buckling. The earth load follows Marston's theory as written in the Soil Conservation Service's Technical Release 5 (1958) and the Corps of Engineers manual EM 1110-2-2902 (1997); the live load is Boussinesq's stress under a wheel footprint with the footprints and impact factors of the Bureau of Reclamation's M-25 (2019); the flexible-pipe deflection is the modified Iowa formula and Reclamation's own equation with its table of the modulus of soil reaction. Every factor that comes from a table (Kμ′, the bedding load factor, E′, the lag and impact factors) is an input with a sourced default.

Conventions

H is the cover from the top of the pipe to the surface, B_c the outside width of the pipe (inside diameter plus twice the wall), B_d the width of the trench at the top of the pipe, γ the unit weight of the fill. Loads W are per unit length of pipe; a pressure P on the pipe is the load divided by B_c. The live load is placed with its footprint centred over the pipe axis.

Earth load in a trench

The backfill in a trench settles against the trench walls and hangs part of its weight on them. Marston's balance of a horizontal slice gives the load on a rigid pipe (TR-5 eq. 1-1; EM 1110-2-2902 eqs 2-5 and 2-7)

W = C_d γ B_d²,   C_d = [1 − e^{−2Kμ′H/B_d}] / (2Kμ′),

where K = (√(μ² + 1) − μ)/(√(μ² + 1) + μ) is the ratio of lateral to vertical pressure and μ′ the friction coefficient between the backfill and the trench wall. Marston's chart, reproduced as EM 1110-2-2902 Figure 2-5, carries five values of the product: 0.1924 for granular materials without cohesion (the largest value the product can take, hence the smallest load), 0.165 for sand and gravel, 0.150 for saturated top soil, 0.130 for ordinary clay and 0.110 for saturated clay (the largest load). A flexible pipe whose side fills are compacted to the stiffness of the pipe shares the load with them and carries C_d γ B_c B_d (TR-5 eq. 1-2). The prism load γ H B_c is the weight of the soil directly over the pipe.

Earth load under an embankment

A pipe laid on the natural ground and covered by an embankment (a positive projecting conduit) is loaded by the prism over it plus or minus the shear on the sides of that prism. If the fill beside the pipe settles more than the top of the pipe (a rigid pipe on a firm foundation) the shear drags down and the load exceeds the prism: the projection condition. If the top of the pipe settles more, the load is less: the ditch condition. The settlement ratio r_sd measures which, and the projection ratio p is the height of the pipe above the natural ground over its width. The shear acts up to the plane of equal settlement, H_e above the pipe, which TR-5 eq. 1-4 (Marston's assumption) gives as the root of

e^{±2KμH_e/B_c} ∓ 2Kμ H_e/B_c = ±2Kμ r_sd p + 1,

and the load is W = C_p γ B_c² with (TR-5 eqs 1-5a, 1-6a)

C_p = [e^±2KμH/B_c − 1] / (±2Kμ) when H ≤ H_e (complete condition),

C_p = [e^{±2KμH_e/B_c} − 1] / (±2Kμ) + (H/B_c − H_e/B_c) e^{±2KμH_e/B_c} when H > H_e (incomplete condition),

upper signs for the projection condition, lower for the ditch condition; r_sd p = 0 returns the prism load. TR-5 notes that Marston's assumption gives slightly greater loads than Spangler's later expression for H_e. The settlement ratio is an input: TR-5 computes it from the consolidation of the foundation and the fill (its sheets ES-115), and design values are usually taken from tabulated recommendations.

Transition width. Widening a trench raises the trench load until it reaches the embankment load; beyond that width the pipe is loaded as if under an embankment (TR-5 page 1-10). The tool solves C_d γ B_d′² = C_p γ B_c² for the transition width B_d′ and, for a trench installation, designs the rigid pipe for the smaller of the two loads. EM 1110-2-2902, written for conduits through dams and levees, instead takes the larger of the trench load and the prism load; that value is printed beside it.

Negative projecting and induced-trench installations (a pipe in a shallow trench under an embankment, or a compressible layer placed over the pipe to throw the load onto the side fills) follow the same theory with a different geometry and are not included.

Live load

Boussinesq's solution for a point load P on an elastic half-space gives the vertical stress 3Pz³/(2πR⁵) at depth z and distance R (EM 1110-2-2902 eq. 2-3). Integrated over a uniformly loaded rectangle it gives, under a corner of a rectangle M by N at depth H, the pressure I p_s with Newmark's coefficient

I = (1/4π) { [2mn√(m² + n² + 1)/(m² + n² + 1 + m²n²)] (m² + n² + 2)/(m² + n² + 1) + arctan[2mn√(m² + n² + 1)/(m² + n² + 1 − m²n²)] },   m = M/H, n = N/H,

which is the coefficient tabulated in Reclamation M-25 Table E-3 (and, read the other way round, Holl's integration of a point load over a rectangular area of pipe). Under the centre of a footprint the pressure is four corner values, 4 I p_s (M-25 eq. E.2). The tool evaluates the stress under the footprint at the pipe crown, at the centre and averaged across the pipe width (sixteen-point Gauss quadrature of the closed form), and takes the live line load as W_L = I_f × mean pressure × B_c at the most loaded section, without spreading it along an effective length of pipe.

Default loads (Reclamation M-25 Appendix E): an HS-20 wheel of 16,000 lb on a 10 in by 20 in tire print (the H-20 truck has the same rear wheel), and the Cooper E-80 locomotive as four 80,000 lb axles uniformly spread over 8 ft by 20 ft. Impact factors: highway 1.5 for 2 to 3 ft of cover, 1.4, 1.3, 1.2, 1.1 for each further foot and 1.0 from 7 ft; railroad 40 percent at the ties falling linearly to zero at 10 ft. Reclamation requires 5 ft of cover under roads and railroads. NAVFAC DM-7.01 places a second 16,000 lb wheel 6 ft from the first; the optional second load does this. For an aircraft, enter the wheel load and tire print; wheel groups are not assembled by the tool.

Rigid pipe

A rigid pipe is rated by the load per unit length per unit of inside diameter that it carries in the three-edge-bearing test at a 0.01 in crack. In the ground the load is spread by the bedding, so the field strength is the test strength times a bedding (load) factor L_f. The required D-load is (NAVFAC DM-7.01 eq. 4-2; EM 1110-2-2902 eq. 3-2)

D_0.01 = (W_E + W_L) FS / (L_f D_i).

Trench load factors of NAVFAC DM-7.01 Figure 18(e): Class A concrete cradle 4.8, 3.4 or 2.8 (1.0 percent, 0.4 percent or no reinforcing steel), Class B compacted granular bedding 1.9, Class C 1.5, Class D flat subgrade 1.1; EM 1110-2-2902 Table 3-1 lists 1.5 ordinary, 1.9 first class and 2.5 concrete cradle. The factor of safety is usually 1.25 (DM-7.01); the EM uses 1.3. Under an embankment the lateral pressure on the pipe helps, and EM 1110-2-2902 eq. 3-1 gives B_f = 1.431/(X_p − X_a/3) with the constants of its Table 3-2; the page prints it for information and uses the entered factor.

Flexible pipe: deflection

Spangler's Iowa formula, as modified by Watkins and Spangler (Reclamation M-25 eq. C.1), gives the horizontal deflection

Δx = D_L K W r³ / (E I + 0.061 E′ r³),

with the deflection lag factor D_L, the bedding constant K, the load W per unit length, the mean radius r, the wall stiffness E I per unit length and the modulus of soil reaction E′. Dividing by the diameter and writing the load as a pressure on the pipe width, with the lag factor on the earth load only (M-25 Appendix C; EM 1110-2-2902 eq. 6-3 with 0.149 PS = E I/r³),

Δx / D = K (D_L P_E + P_L) / (E I/r³ + 0.061 E′).

Reclamation's equation (M-25 eq. 3.1) is the same balance with the prism load, K = 0.1, a time-lag factor T_f in place of the deflection lag factor and a design factor F_d on E′; it predicts the long-term vertical deflection, and its constant 0.07 is 0.1 × 100/144 rounded. The tool evaluates it in consistent units, ΔY/D = 0.1 (T_f γ H + P_L)/(E I/r³ + 0.061 F_d E′). The Embedment list fills E′, F_d and T_f from M-25 Table 5 (soil classes I to IV by compaction; Howard's field-derived values); with a native trench-wall modulus entered, the composite E′ = S_c E_b of M-25 eq. 3.6 and Table 7 is used. M-25 Table 10 allows 5 percent for flexibly lined steel, PVC and fiberglass pipe and 3 percent for mortar-lined steel and ductile iron. EM 1110-2-2902 notes that the Iowa formula overpredicts the deflection of stiff pipes and underpredicts that of very flexible ones by a factor of about two.

Flexible pipe: ring compression and buckling

The pressure P on the pipe puts the wall in compression with the thrust T = P B_c/2 per unit length (NAVFAC DM-7.01 eq. 4-5; EM 1110-2-2902 eq. 6-4); the wall stress T/A is compared with the wall strength over a factor of safety (1.5 to 2 in DM-7.01, 2 in the EM). A plain wall has A = t and I = t³/12 per unit length.

The soil-supported ring buckles at the external pressure

q_cr = √(32 R_w B′ E′ E I / D³),   R_w = 1 − 0.33 h_w/H,

where h_w is the height of the ground water above the pipe. EM 1110-2-2902 eq. 6-7 writes this as a wall stress, f = 0.77 (R/A) √(B M_s E I/(0.149 R³)), which is the same expression with B′ = 1 (0.77/√0.149 = 1.995 against √(32/8) = 2), and applies a factor of safety of 2. AWWA Manual M11 uses the same form with the empirical support coefficient B′ = 1/(1 + 4e^−0.065H), H in feet; that expression is offered as an option and is cited, not reproduced from a public document. The external pressure is the water head γ_w h_w plus the buoyancy-reduced soil pressure R_w γ H plus either the internal vacuum or the live load, whichever is greater. A ring with no soil support collapses at 3 E I/[(1 − ν²) r³]; EM 1110-2-2902 eq. 6-8 prints this with a constant that is half as large (K = 216 in its units, which corresponds to 1.5 rather than 3).

Flotation and thrust

An empty pipe below the water table is lifted by γ_w π B_c²/4 per unit length and held down by its own weight and the soil over it. The tool takes the water table at the ground surface, the prism of buoyant soil (γ − γ_w) H B_c over the pipe width only, and reports the factor of safety and the cover needed for the entered factor. At a bend of deflection angle θ the internal pressure P on the bore area A leaves the unbalanced force T = 2 P A sin(θ/2) along the bisector of the bend (the resultant of the two forces P A along the legs); a thrust block needs the bearing area FS T/q against undisturbed soil of allowable bearing q.

Checks made on the tool

The library is checked against the worked example of TR-5 (36 in pipe, 30 ft of fill: C_d 2.92, H_e/b_c 1.62, C_p 15.13, 22,240 lb/ft, transition width 2.95 b_c), the D-load, plastic-pipe deflection, wall-area and ring-buckling examples of EM 1110-2-2902 Appendix B, the deflection examples of Reclamation M-25 Appendix A and its live-load tables E-2, E-3 and E-4, and an independent numpy / scipy evaluation (the Boussinesq integral by numerical quadrature, the embankment equations by a root finder, twenty seeded installations).

Limits

Circular pipe with a plain wall; one pipe in the trench; vertical trench walls; a level surface. Not included: negative projecting and induced-trench installations, jacked pipe, the lateral pressure and reinforcement design of concrete pipe, corrugated profiles, the wall strain and long-term moduli of plastics, longitudinal bending, frost and seismic loads, and restrained joints. The default case (24 in pipe, 8 ft of cover, 4.5 ft trench in ordinary clay, H-20 wheel; a 3 in concrete wall on Class B bedding and a 3/16 in steel wall in moderately compacted clean sand) is illustrative.