Tall vessels: wind and seismic loads

Why it matters

Distillation columns, absorbers and stacks can be 30–80 m tall but only a few metres wide. For them, the shell thickness at the bottom is often set not by pressure but by wind or earthquake bending combined with weight. A column designed for pressure alone can buckle on its leeward side, sway excessively, or oscillate dangerously in a steady breeze through vortex shedding.

Key ideas

A tall vessel is a vertical cantilever fixed at its skirt base. Wind or seismic forces act along its height, producing a shear force and a bending moment that grow towards the base. The bending moment creates axial (longitudinal) stress in the shell, tensile on the windward side and compressive on the leeward side, which adds to the longitudinal stresses from pressure and weight.

Wind load (IS 875 Part 3 approach). The design wind speed at height z is the basic wind speed for the site multiplied by factors for risk (life), terrain and height, and topography: V_z = V_b·k1·k2·k3. The design wind pressure is p_z = 0.6·V_z² (N/m² with V_z in m/s), which is just ½·ρ_air·V² with ρ_air ≈ 1.2 kg/m³. The force on a section is F = C_f·A_e·p_z, where C_f is the force coefficient for a cylinder (about 0.7 for a smooth tall cylinder in most conditions; take it from the code for your Reynolds number and surface) and A_e is the projected area using an effective diameter that includes insulation, platforms, ladders and attached piping. Because V_z increases with height, the column is divided into zones and the moment at any section is the sum of each zone force times its lever arm.

Seismic load (IS 1893 approach). The design horizontal acceleration coefficient is A_h = (Z/2)·(I/R)·(S_a/g), where Z is the zone factor, I the importance factor, R the response-reduction factor and S_a/g the spectral acceleration for the vessel's natural period and soil type. The base shear is V_B = A_h·W, distributed up the height (more at the top for a cantilever), giving the seismic moment. All factors come from the code tables. Wind and earthquake are not assumed to act together; the worse governs.

Combining stresses. At any section the longitudinal stress is pressure (+P·D/(4t)) ± bending (4M/(π·D²·t)) − weight (W/(π·D·t)). The cases to check are: maximum tension on the windward side (operating pressure, minimum weight) and maximum compression on the leeward side (no internal pressure or under vacuum, full weight). The tensile combination is compared with the allowable stress times joint efficiency of the circumferential seam; the compressive combination with the code's axial buckling limit. The hoop stress P·D/(2t) is checked separately. Thickness is often increased in steps down the column.

Deflection and vibration. The top deflection under wind is limited (a common criterion is about 150 mm per 30 m of height) to protect trays, piping and platforms. Wind can also make a column oscillate across the wind when vortices shed at a frequency matching its natural frequency. The vortex-shedding frequency is f_v = S·V/D with Strouhal number S ≈ 0.2. If the critical wind speed V_c = f_n·D/S falls within the site's normal wind range, the column is stiffened, made heavier, damped, or fitted with helical strakes. Columns in a row can also excite each other.

Other loads. Weight during erection (empty, no insulation), hydrotest (full of water, possibly horizontal), and piping thrust. Loads also flow down into the skirt, base ring and anchor bolts.

Formulas

V_z = V_b·k1·k2·k3 (design wind speed, m/s; factors from IS 875 Part 3) p_z = 0.6·V_z² (design wind pressure, N/m²) F = C_f·A_e·p_z (wind force on a section, N); w = C_f·D_eff·p_z (force per metre height, N/m) M = w·H² / 2 (base moment for uniform w over height H, N·m) A_h = (Z/2)·(I/R)·(S_a/g), V_B = A_h·W (seismic coefficient and base shear; factors from IS 1893) σ_p = P·D / (4·t), σ_b = 4·M / (π·D²·t), σ_w = W / (π·D·t) (longitudinal stresses) σ_z = σ_p ± σ_b − σ_w (+ windward, − leeward) f_n = (3.52 / (2π))·√(E·I / (m·H⁴)) (first natural frequency of a uniform cantilever) V_c = f_n·D / S, S ≈ 0.2 (critical wind speed for vortex shedding)

• V in m/s; p in N/m²; C_f force coefficient (–); D_eff effective diameter (m); H height (m); W weight above the section (N); M bending moment at the section (N·mm when used with mm); P design pressure (MPa); D shell diameter (mm); t corroded thickness (mm); E Young's modulus (Pa); I second moment of area (m⁴); m mass per unit height (kg/m).

Worked examples

Example 1 (standard): wind force and base moment Given: column height H = 30 m, effective diameter 2.0 m (with insulation), design wind speed taken uniform at V_z = 39 m/s (given), C_f = 0.7.

1. p_z = 0.6 × 39² = 912.6 N/m².
2. Load per metre: w = 0.7 × 2.0 × 912.6 = 1278 N/m.
3. Total wind force: F = w·H = 1278 × 30 = 38.3 kN.
4. Base moment: M = w·H²/2 = 1278 × 30² / 2 = 574.9 kN·m. Answer: F = 38.3 kN, M = 575 kN·m (in a real design V_z varies with height and the column is split into zones).

Example 2 (GATE level): combined longitudinal stresses at the bottom of the shell Given: same column, D = 1800 mm (use throughout), corroded t = 10 mm, P = 1.0 MPa, weight above section W = 250 kN, M = 574.9 kN·m = 5.749 × 10⁸ N·mm.

1. Pressure: σ_p = 1.0 × 1800 / (4 × 10) = 45.0 MPa.
2. Weight: σ_w = 250 000 / (π × 1800 × 10) = 4.42 MPa.
3. Bending: σ_b = 4 × 5.749 × 10⁸ / (π × 1800² × 10) = 22.6 MPa.
4. Windward, operating: σ_z = 45.0 + 22.6 − 4.42 = 63.2 MPa (tension) — compare with allowable × circumferential-seam efficiency.
5. Leeward, operating: 45.0 − 22.6 − 4.42 = 18.0 MPa (still tension).
6. Leeward, not pressurised: −22.6 − 4.42 = −27.0 MPa (compression) — check against the axial buckling allowable.
7. Hoop: P·D/(2t) = 90 MPa, so here pressure still governs the hoop design, but the compressive case of −27.0 MPa must be checked separately.

Common mistakes

• Writing p_z = 0.6·V²·C_f·A — that is a force, not a pressure; keep pressure (N/m²) and force (N) separate.
• Using the bare shell diameter and ignoring insulation, platforms and ladders in the projected area.
• Forgetting that the pressure stress disappears during shutdown, so the leeward compressive case must be checked unpressurised.
• Adding wind and seismic moments together.
• Using operating weight for the windward tension case; the minimum weight is more severe.
• Ignoring vortex-shedding resonance for slender, lightly damped columns.

For GATE CH

Expect numericals on wind pressure and force from a given speed and coefficient, base moment for a uniform load, seismic base shear from a given coefficient, and combined longitudinal stress (pressure ± bending − weight) on windward and leeward sides. Conceptual questions cover why the leeward side is critical for buckling and how vortex shedding is avoided.

Quick check

1. Design wind speed 45 m/s: what is p_z?
2. A 20 m column carries a uniform wind load of 1.5 kN/m. What is the base moment?
3. Which side of a column under wind is checked for buckling?
4. A vessel weighs 300 kN and A_h = 0.12. What is the seismic base shear?

Answers: 1. 0.6 × 45² = 1215 N/m². 2. 1.5 × 20²/2 = 300 kN·m. 3. The leeward (compression) side. 4. 36 kN.