Bending Stress Fundamentals | Pipeline Engineering

1. Beam Theory Basics

When a pipe spans between supports, it acts as a beam carrying its own weight plus any additional loads. The Euler-Bernoulli beam theory relates bending moment to stress through the flexure formula.

Simply-supported beam with uniform load, reactions, and bending moment diagram — Figure 1: Beam bending diagram showing load, reactions, moment distribution, and neutral axis.

Flexure Formula (Bending Stress): σ = M × c / I = M / Z Where: σ = bending stress (psi) M = bending moment (in-lbf) c = distance from neutral axis to outer fiber = OD/2 I = moment of inertia (in⁴) Z = section modulus = I/c (in³)

The maximum bending stress occurs at the outer fiber of the pipe cross-section. For hollow circular sections (pipes), the stress distribution is linear from zero at the neutral axis to maximum at the outer surface.

Support Configurations

The support type significantly affects both stress and deflection. Common configurations for pipeline spans:

Simply Supported

Pinned-Pinned

Both ends free to rotate. M_max = wL²/8 at midspan.

Fixed-Pinned

Propped Cantilever

One end fixed, one pinned. M_max = wL²/8 at fixed end.

Fixed-Fixed

Both Ends Fixed

Lowest deflection. M_max = wL²/12 at supports.

Cantilever

Fixed-Free

Highest stress. M_max = wL²/2 at fixed end.

Common pipeline support types and configurations — Figure 4: Support configurations - simply supported, fixed-pinned, fixed-fixed, and cantilever.

Key insight: Bending stress increases with the square of span length. Doubling the span quadruples the bending moment and stress (for uniform load).

2. Pipe Section Properties

The geometric properties of the pipe cross-section determine its resistance to bending. For a hollow circular section:

Pipe cross-section showing OD, ID, wall thickness, and stress distribution — Figure 2: Pipe cross-section with geometric properties and bending stress distribution.

Moment of Inertia: I = π(OD⁴ - ID⁴) / 64 Section Modulus: Z = π(OD⁴ - ID⁴) / (32 × OD) Cross-Sectional Area: A = π(OD² - ID²) / 4 Where: OD = outside diameter (in) ID = inside diameter = OD - 2t (in) t = wall thickness (in)

Common Pipe Properties (Schedule 40)

NPS	OD (in)	Wall (in)	Wt (lb/ft)	I (in⁴)	Z (in³)
4	4.500	0.237	10.79	7.23	3.21
6	6.625	0.280	18.97	28.14	8.50
8	8.625	0.322	28.55	72.49	16.81
10	10.750	0.365	40.48	160.8	29.9
12	12.750	0.375	49.56	279.3	43.8
16	16.000	0.375	62.58	560.7	70.1
20	20.000	0.375	78.60	1,113	111.3
24	24.000	0.375	94.62	1,943	162

Pipe Weight Calculation

Steel Pipe Weight (lb/ft): w = 10.68 × (OD - t) × t Alternative using area: w = ρ × A / 144 Where: ρ = material density = 490 lb/ft³ (carbon steel) A = cross-sectional area (in²)

3. Deflection Formulas

Deflection must be checked to ensure the pipe does not sag excessively. The formulas below are for uniform distributed load w over span L.

Deflected beam shapes for different support conditions — Figure 3: Deflection patterns for simply supported, fixed-fixed, fixed-pinned, and cantilever beams.

Support Type	Max Moment	Max Deflection	Location of δ_max
Simply Supported	wL²/8	5wL⁴/(384EI)	Midspan (L/2)
Fixed-Fixed	wL²/12	wL⁴/(384EI)	Midspan (L/2)
Fixed-Pinned	wL²/8	wL⁴/(185EI)	0.4215L from pinned end
Cantilever	wL²/2	wL⁴/(8EI)	Free end

Note on units: For deflection in inches with: - w in lb/in (divide lb/ft by 12) - L in inches - E in psi (ksi × 1000) - I in in⁴ E for steel ≈ 29,000 ksi = 29,000,000 psi

Relative Deflection Comparison

Taking simply supported as baseline (1.0):

Support Type	Relative Moment	Relative Deflection
Simply Supported	1.00	1.00
Fixed-Fixed	0.67	0.20
Fixed-Pinned	1.00	0.42
Cantilever	4.00	9.60

Design tip: Fixed-fixed supports reduce deflection by 80% compared to simply supported, but require moment-resistant connections and create high stress at supports under thermal expansion.

4. ASME B31.8 Stress Criteria

ASME B31.8 distinguishes between restrained (buried) and unrestrained (above-ground) piping. Above-ground spans are typically unrestrained, governed by §833.6.

Unrestrained Pipe (§833.6)

Longitudinal Stress Limit: S_L ≤ 0.75 × SMYS × T Combined Longitudinal Stress: S_L = S_P + S_b Where: S_P = P × D / (4t) — longitudinal stress from pressure S_b = M / Z — bending stress SMYS = Specified Minimum Yield Strength (psi) T = temperature derating factor (Table 841.1.8-1)

Design Factors (Table 841.1.6-1)

Location Class	Design Factor (F)	Description
Class 1, Div 1	0.72	Sparsely populated, no buildings
Class 1, Div 2	0.60	Offshore, platform gathering
Class 2	0.60	Fringe areas, 11-45 buildings
Class 3	0.50	Suburban, 46+ buildings
Class 4	0.40	4+ story buildings common

Temperature Derating (Table 841.1.8-1)

Temperature (°F)	Derating Factor (T)
≤ 250	1.000
300	0.967
350	0.933
400	0.900
450	0.867

Joint Factors (Table 841.1.7-1)

Pipe Type	Joint Factor (E)
Seamless	1.00
ERW (Electric Resistance Welded)	1.00
Submerged Arc Welded (SAW)	1.00
Furnace Butt Welded	0.60

Important: For bending stress alone, many engineers use F×E×T×SMYS as the allowable. For combined stress (pressure + bending), the limit is 0.75×SMYS×T per §833.6.

5. Design Practice

Maximum Span Calculation

Rearranging the bending stress equation to solve for allowable span:

Simply Supported Span (stress-limited): L_max = √(8 × σ_allow × Z / w) Where: σ_allow = allowable bending stress (psi) Z = section modulus (in³) w = total load (lb/ft) L_max = maximum span (ft)

Typical Support Spacing Guidelines

NPS	Gas (Empty)	Liquid (Full)	Hydrotest
4–6	18–22 ft	14–18 ft	12–15 ft
8–10	22–28 ft	18–22 ft	15–18 ft
12–16	28–35 ft	22–28 ft	18–22 ft
20–24	35–42 ft	28–35 ft	22–28 ft

Design Considerations

Thermal Expansion: Above-ground pipelines expand/contract with temperature changes. Provide at least one sliding support per span to accommodate axial movement. Fixed supports at both ends create high thermal stress.

Hydrotest Conditions: Water-filled lines weigh significantly more than gas-filled. Always check span adequacy for hydrotest. Water adds approximately:

4" pipe: +5.5 lb/ft
8" pipe: +20 lb/ft
12" pipe: +45 lb/ft
24" pipe: +180 lb/ft

Support Types:

Pipe shoes: Welded saddles, high capacity, allows sliding
U-bolts: Simple, adjustable, lower capacity
Roller supports: For thermal expansion accommodation
Guides: Restrain lateral movement, allow axial

Rule of thumb: For initial estimates, simply-supported spans of 2.5× to 3× the pipe diameter (in feet) are often acceptable for gas service. Always verify with calculations.

Common Mistakes to Avoid

Ignoring fluid weight during hydrotest
Assuming fixed supports without designing moment connections
Not checking deflection (sag can cause drainage issues)
Over-restraining thermal expansion
Using wrong section modulus (Z, not plastic modulus Z_p)

Bending Stress in Pipelines

0.75 × SMYS

20–40 ft

L/240

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