Reduced wall to section stress — SWBPIPE verification

Part VI · VerificationVI–C14

Part VI

Verification

C. Stress recovery · Reduced wall to section stress

A pipe of outside diameter 0.2 m has a nominal wall of 0.01 m. The user takes off 0.002 m for corrosion and a further 0.00125 m for mill tolerance, both entered as thicknesses. The pipe carries an axial force, two bending moments, a torque and internal pressure. What section properties does the thinner wall give, what stresses follow, and does the extra thinning lower the section modulus?

The mill tolerance is an absolute thickness entered by the user, not a fraction, and the solver supplies no value of its own. If none is entered, no reduction is taken. The numbers are invented.

Read with

Reducing the wall is right for stress. The program also reduces it for stiffness and weight, which flexibility analysis does not do (Status).

Fig. VI–C14.—Reduced wall to section stress

1.Inputs.

Illustrative values, taken from no standard and chosen so the arithmetic can be followed by hand. Most do not describe a real pipe; read them in any consistent set of units.

Quantity	Symbol	Value
Outside diameter	D_o	0.2 m
Nominal wall thickness	t_nom	0.01 m
Corrosion allowance	c	0.002 m
Mill tolerance	m	0.00125 m
Axial force	F	5000.0 N
Bending moment about local y	M_y	1000.0 N·m
Bending moment about local z	M_z	−400.0 N·m
Torsional moment	T	250.0 N·m
Internal pressure	p	2000.0 Pa

2.Method.

The effective wall is the nominal wall less the corrosion allowance and the mill tolerance. The outside diameter is held and the bore grows. The annulus formulas give the area and second moment; the section modulus is the second moment over the outer radius, and the torsion constant is twice the second moment.

The stresses are force over area, moment over section modulus, and torque times outer radius over torsion constant. The hoop stress uses the mean radius and the effective wall; the longitudinal pressure stress is half of it. For comparison, the section with the corrosion allowance alone is also worked out.

In this case the section properties are computed from the effective wall in the benchmark itself and passed to the solver’s stress recovery. The note records that the reference values were also worked independently in decimal arithmetic.

t_eff = 0.01 − 0.002 − 0.00125 = 0.00675 m(1)

D_i = D_o − 2t_eff = 0.2 − 0.0135 = 0.1865 m(2)

A = (π/4)(D_o² − D_i²) = (π/4)(0.04 − 0.03478225) = 4.098011267067042 × 10⁻³ m²(3)

I = (π/64)(D_o⁴ − D_i⁴) = 1.9153656442289003 × 10⁻⁵ m⁴(4)

Z = I / (D_o/2) = 1.9153656442289 × 10⁻⁴ m³, J = 2I = 3.8307312884578006 × 10⁻⁵ m⁴(5)

r_torsion = D_o/2 = 0.1 m, r_m = (D_o − t_eff)/2 = 0.096625 m(6)

σ_hoop = p r_m / t_eff = 2000.0 × 0.096625 / 0.00675 = 2.862962962962963 × 10⁴ Pa(7)

Corrosion only: t_eff,c = 0.01 − 0.002 = 0.008 m, D_i,c = 0.184 m, Z_c = 2.2274444834258686 × 10⁻⁴ m³(8)

3.Results.

Expected section properties and stresses
Quantity	Expected
Effective wall, t_eff	0.00675 m
Area, A	4.098011 × 10⁻³ m²
Section modulus, Z	1.915366 × 10⁻⁴ m³
Section modulus, corrosion only, Z_c	2.227444 × 10⁻⁴ m³
Axial normal stress	1.220104 × 10⁶ Pa
Bending stress about y	5.220935 × 10⁶ Pa
Bending stress about z	−2.088374 × 10⁶ Pa
Torsional shear stress	6.526169 × 10⁵ Pa
Hoop pressure stress	2.862963 × 10⁴ Pa
Longitudinal pressure stress	1.431481 × 10⁴ Pa

The tests check that:

The stress recovery on the effective-wall section completes without being blocked.
The axial, both bending, torsional shear, hoop and longitudinal stresses each match the table.
The section modulus with the mill tolerance taken off is smaller than with the corrosion allowance alone.
The section area with the mill tolerance taken off is also smaller.
The two section comparisons are made on the benchmark’s own section arithmetic, not on a solver section-property routine.

What it shows. Compared with an independent hand calculation.

Path exercised. The benchmark calls the solver’s components directly: elements, loads, frame solver and stress recovery. It does not go through the program’s own model-to-solve path.

Agreement. Each stress must match the hand calculation to within an absolute difference of 1.0 × 10⁻⁹ Pa. The two section comparisons are strict inequalities, checked exactly. Long values are shown here to seven significant figures; the tests compare the full values in the record.

For the student

The wall is thinned from the bore while the outside diameter is held, so the distance to the outer fibre does not change. The bending stress rises only because the second moment, and with it the section modulus, falls. The hoop stress rises for two reasons: the wall is thinner and the mean radius is slightly larger.

4.Run it yourself.

cd projects/chirality-piping
cargo test --manifest-path validation/benchmarks/stress/Cargo.toml recovers_milltol_effective_wall_fixture
cargo test --manifest-path validation/benchmarks/stress/Cargo.toml milltol_reduction_strictly_reduces_section_modulus

Hand calculation: validation/hand_calcs/stress/tp_pmm_p3_milltol_effective_wall_stress.md. Test record, with the recorded run of 2026-07-10: stress-tp-pmm-p3-milltol-effective-wall-stress.md.