Why can a vessel that easily withstands 10 bar of internal pressure collapse under full vacuum?

Under internal pressure the wall is in tension and fails only when stress reaches the material strength. Under external pressure the wall is in compression and a thin shell becomes elastically unstable: at a critical pressure, set by E and (t/D)³ for a long cylinder, the round shape buckles into lobes. For a thin shell this collapse pressure can be well below one atmosphere even though its internal-pressure rating is high.

What is the role of stiffening rings in vacuum vessels?

Rings hold the shell circular at intervals, which reduces the unsupported length L. A shell shorter than its critical length must buckle into more lobes, so its collapse pressure rises roughly as 1/L. This lets the designer use a much thinner shell; the rings themselves must have enough moment of inertia (ring plus effective shell strip) not to collapse first.

How do codes such as ASME VIII Div. 1 handle external pressure design?

They use charts instead of the closed-form buckling equations. From L/Do and Do/t you read a geometric strain factor A; from the material chart at design temperature you read a stress factor B; the allowable pressure is then P_a = 4B/(3·Do/t) for Do/t ≥ 10. The charts include the effect of plasticity and fabrication imperfections, and give a factor of safety of about 3 on collapse.

What is the critical length of a cylinder under external pressure?

It is the unsupported length beyond which the collapse pressure no longer depends on length, because the shell buckles into two lobes like an infinitely long tube. For steel it is about L_c = 1.11·Do·(Do/t)^0.5. Below L_c the end supports raise the collapse pressure, so placing rings closer than L_c is what makes them effective.

What design external pressure would you use for the inner shell of a jacketed reactor that may also run under vacuum?

The inner shell sees the jacket pressure acting on its outside plus the loss of internal pressure, so the design external pressure is the jacket design pressure plus full vacuum, 0.1 MPa. It must be checked for buckling at that pressure with the effective length between jacket closures or rings, and separately for internal pressure when the jacket is empty.

Why are fabrication tolerances on roundness important for vacuum vessels?

Buckling is very sensitive to initial imperfections: a slightly oval shell starts bending as soon as load is applied and collapses well below the theoretical pressure. Codes therefore limit out-of-roundness and local flat spots and base their charts on imperfect shells. A shell that fails the roundness check must be corrected or re-rated before service.

Should you use the inside or outside diameter in external pressure calculations, and why does a higher-strength steel not help much?

External-pressure formulas and code charts are written in terms of the outside diameter Do, and the corroded thickness. Elastic collapse depends on Young's modulus, which is about 200 GPa for all carbon and low-alloy steels, so a higher-yield grade barely changes P_c; strength matters only for short, thick shells where collapse occurs by yielding.

Vessels under external pressure and stiffening rings

Why thin shells under vacuum or jacket pressure fail by elastic buckling, collapse pressure of long and short cylinders and spheres, critical length, code chart design and how stiffening rings are sized.

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Why it matters

Vacuum distillation columns, evaporators, jacketed reactors and any tank that can be sucked in when it is pumped out or steamed and cooled all see external pressure. A shell that is many times strong enough for internal pressure can collapse like a crushed can under only one atmosphere of external pressure, because the failure mode is buckling, not yielding. Stiffening rings let the designer keep a thin shell instead of buying a much thicker one.

Key ideas

Buckling, not strength. Under internal pressure a shell fails when stress reaches the material's strength, so thickness grows linearly with pressure. Under external pressure a thin shell becomes elastically unstable at a critical (collapse) pressure P_c at which the circular shape snaps into two, three or more lobes. P_c depends on Young's modulus E (stiffness), not on yield strength, and on the geometry ratios t/D and L/D. That is why high-strength steel gives almost no advantage for vacuum service.

Long and short cylinders. A cylinder with ends far apart buckles into two lobes (an oval) and its collapse pressure is independent of length. If the ends — heads or stiffening rings — are close enough, they hold the shell round and force it to buckle into more lobes, which needs a much higher pressure. The dividing length is the critical length L_c. For L < L_c the shell is "short" and P_c rises roughly in proportion to 1/L.

Effective length. L is the distance between lines of support: between stiffening rings, or between a ring and a head. For a dished head the support line is usually taken at one-third of the head depth inside the head-to-shell joint; check the definition in your code book.

Imperfections. Real shells are slightly out of round, and buckling is very sensitive to this. Codes therefore limit out-of-roundness during fabrication and apply a factor of safety (about 3 for cylinders in ASME VIII Division 1) to theoretical collapse pressures. Spheres are even more imperfection-sensitive: their real collapse pressure is far below the classical value.

Code design. ASME VIII Division 1 (UG-28) and IS 2825 do not ask you to use the closed-form equations. They provide charts: from L/Do and Do/t read a strain factor A; from the material chart at design temperature read a stress factor B; then the allowable external pressure is P_a = 4B / (3·Do/t) for Do/t ≥ 10. Use the theory below to understand trends and for preliminary sizing, and take the final answer from the code charts.

Stiffening rings. Rings (flat bars, angles or tees welded around the shell, inside or outside) shorten the effective length. Each ring must itself be stiff enough not to buckle together with its share of shell, so the code specifies a minimum moment of inertia for the ring-plus-shell section. Rings are welded continuously or with intermittent welds within code limits; internal rings must not block trays or packing support.

Design external pressure. For a vessel that can be evacuated the design external pressure is full vacuum, 0.1 MPa (1 bar). For a jacketed vessel the inner shell sees the jacket pressure plus any internal vacuum. Long thin vessels must also be checked for axial compression from wind and weight, a separate buckling mode.

Heads. Heads under external pressure are on their convex side. Codes check them as an equivalent sphere and often require them to be thicker than for internal pressure.

Formulas

P_c = (2·E / (1 − ν²))·(t / Do)³ (long cylinder, L > L_c)

For steel with ν = 0.3: P_c ≈ 2.2·E·(t / Do)³.

P_c = 2.6·E·(t / Do)^2.5 / (L / Do − 0.45·(t / Do)^0.5) (short cylinder, L < L_c, ν = 0.3)

L_c = 1.11·Do·(Do / t)^0.5 (critical length, ν = 0.3)

P_c = 2·E·(t / R)² / √(3·(1 − ν²)) ≈ 1.21·E·(t / R)² (sphere, classical; real shells collapse far lower)

P_a = 4·B / (3·Do / t) (ASME VIII Div. 1 allowable external pressure, Do/t ≥ 10; B from code chart)

I_req = P_c·Ls·Do³ / (24·E) (ring-plus-shell section needed so the ring does not collapse first)

P_c = collapse pressure (MPa); P_a = allowable external pressure (MPa); E = Young's modulus at design temperature (MPa); ν = Poisson's ratio; t = shell thickness (mm, corroded); Do = outside diameter (mm); R = radius (mm); L = unsupported length (mm); Ls = ring spacing (mm); I = second moment of area (mm⁴). All valid only while the stress at collapse stays below the yield stress (elastic buckling).

Worked examples

Example 1 (standard): unstiffened vacuum shell Given: Do = 2000 mm, t = 10 mm (corroded), steel E = 2 × 10⁵ MPa, ν = 0.3, long shell (no rings), full vacuum design 0.1 MPa, factor of safety 3 on collapse.

P_c = 2.2·E·(t/Do)³ = 2.2 × 2 × 10⁵ × (0.005)³ = 0.055 MPa.
Allowable = 0.055 / 3 = 0.018 MPa ≪ 0.1 MPa — the shell would collapse.
Thickness needed without rings: (t/Do)³ = 3 × 0.1 / (2.2 × 2 × 10⁵) = 6.82 × 10⁻⁷, so t/Do = 0.00880 and t ≈ 17.6 mm (order 18 mm plus corrosion allowance).

Example 2 (GATE level): adding stiffening rings Same shell, t = 10 mm, with rings at Ls = 1500 mm.

Critical length: L_c = 1.11 × 2000 × (2000/10)^0.5 = 1.11 × 2000 × 14.14 = 31 400 mm. Ls ≪ L_c, so the shell is short.
P_c = 2.6 × 2 × 10⁵ × (0.005)^2.5 / (1500/2000 − 0.45 × (0.005)^0.5) (0.005)^2.5 = 1.768 × 10⁻⁶; numerator = 0.919 MPa; denominator = 0.75 − 0.0318 = 0.718. P_c = 0.919 / 0.718 = 1.28 MPa.
Allowable = 1.28 / 3 = 0.43 MPa > 0.1 MPa ✓. Elastic check: stress at collapse = P_c·Do/(2t) = 1.28 × 2000/20 = 128 MPa, below the yield of ordinary steel ✓.
Ring stiffness for a collapse pressure of 3 × 0.1 = 0.3 MPa: I_req = 0.3 × 1500 × 2000³ / (24 × 2 × 10⁵) = 7.5 × 10⁵ mm⁴ (75 cm⁴) for the ring with its effective strip of shell.
Result: 10 mm shell with rings every 1.5 m is adequate, saving about 7.6 mm of plate over the whole shell. Confirm with the code charts before final design.

Common mistakes

Using the internal-pressure formula for vacuum service; buckling governs at a far lower pressure.
Thinking a higher-strength steel cures buckling: P_c depends on E, which is nearly the same for all steels.
Using inside diameter in the buckling formulas, which are written in Do.
Taking L as the tangent-to-tangent length when rings are present, or ignoring the head support line.
Forgetting that a jacketed inner shell sees jacket pressure plus vacuum.
Applying elastic formulas when the stress at collapse exceeds yield (thick, short shells fail by yielding).

For GATE CH

Expect conceptual questions on why external pressure is a stability problem, which parameters P_c depends on (E, t/D cubed for long cylinders, L/D), the effect of stiffening rings, and simple numericals with the long-cylinder formula or ratios (doubling t raises the long-cylinder P_c eightfold). Practise identifying long versus short cylinders from L_c.

Quick check

By what factor does doubling t change P_c of a long cylinder?
Does replacing carbon steel by a higher-yield steel help a vacuum shell? Why?
What do stiffening rings change in the collapse formula?
What design external pressure is used for a vessel that may be evacuated?

Answers: 1. Eight times (t cubed). 2. Hardly; elastic buckling depends on E, not yield strength. 3. They reduce the unsupported length L, so the shell behaves as a short cylinder. 4. Full vacuum, 0.1 MPa.

Interview questions

All Process Equipment Design interview questions

Try answering each one aloud before you open it.

1.Why can a vessel that easily withstands 10 bar of internal pressure collapse under full vacuum?Concept
Under internal pressure the wall is in tension and fails only when stress reaches the material strength. Under external pressure the wall is in compression and a thin shell becomes elastically unstable: at a critical pressure, set by E and (t/D)³ for a long cylinder, the round shape buckles into lobes. For a thin shell this collapse pressure can be well below one atmosphere even though its internal-pressure rating is high.
2.What is the role of stiffening rings in vacuum vessels?Concept
Rings hold the shell circular at intervals, which reduces the unsupported length L. A shell shorter than its critical length must buckle into more lobes, so its collapse pressure rises roughly as 1/L. This lets the designer use a much thinner shell; the rings themselves must have enough moment of inertia (ring plus effective shell strip) not to collapse first.
3.How do codes such as ASME VIII Div. 1 handle external pressure design?Concept
They use charts instead of the closed-form buckling equations. From L/Do and Do/t you read a geometric strain factor A; from the material chart at design temperature you read a stress factor B; the allowable pressure is then P_a = 4B/(3·Do/t) for Do/t ≥ 10. The charts include the effect of plasticity and fabrication imperfections, and give a factor of safety of about 3 on collapse.
4.What is the critical length of a cylinder under external pressure?Concept
It is the unsupported length beyond which the collapse pressure no longer depends on length, because the shell buckles into two lobes like an infinitely long tube. For steel it is about L_c = 1.11·Do·(Do/t)^0.5. Below L_c the end supports raise the collapse pressure, so placing rings closer than L_c is what makes them effective.
5.What design external pressure would you use for the inner shell of a jacketed reactor that may also run under vacuum?Concept
The inner shell sees the jacket pressure acting on its outside plus the loss of internal pressure, so the design external pressure is the jacket design pressure plus full vacuum, 0.1 MPa. It must be checked for buckling at that pressure with the effective length between jacket closures or rings, and separately for internal pressure when the jacket is empty.
6.Why are fabrication tolerances on roundness important for vacuum vessels?Concept
Buckling is very sensitive to initial imperfections: a slightly oval shell starts bending as soon as load is applied and collapses well below the theoretical pressure. Codes therefore limit out-of-roundness and local flat spots and base their charts on imperfect shells. A shell that fails the roundness check must be corrected or re-rated before service.
7.Should you use the inside or outside diameter in external pressure calculations, and why does a higher-strength steel not help much?Concept
External-pressure formulas and code charts are written in terms of the outside diameter Do, and the corroded thickness. Elastic collapse depends on Young's modulus, which is about 200 GPa for all carbon and low-alloy steels, so a higher-yield grade barely changes P_c; strength matters only for short, thick shells where collapse occurs by yielding.

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