FEXLA

Membrane and bending in a shell

Why a curved wall carries pressure by stretching, where it has to bend, and how the two are told apart.

Principle 12 of 20 · 5 min read ·

A stress through the wall split into membrane, bending and peakFour profiles of stress through a wall, inside face at the top and outside face at the bottom. The total is about 189 MPa at the inside face and 65 at the outside. It equals a uniform membrane block of 100 MPa, plus a straight bending line of plus and minus 40 MPa, plus a peak that is a sharp rise of about 49 MPa at the inside face with a small self-balancing remainder elsewhere, about 5 MPa at the outside face.Total≈ 189 to 65Membraneσm = 100Bendingσb = ± 40Peak≈ 49 at the face=++Inside face at the top, outside face at the bottom; MPa
Figure 1. A stress through the wall split into membrane, bending and peak.
Contents

A shell carries load in two ways: by membrane action, a stress that is even through the wall and stretches the mid-surface, and by bending, a stress that changes through the wall from tension on one face to compression on the other. Curved shells carry pressure mostly by membrane action, which is why they can be thin. Bending appears where the shape or the load forces the wall to change its curvature, such as at joints, flat parts and attachments.

This article shows why curvature favours stretching, where a shell has no choice but to bend, and how a computed stress is split into its membrane and bending parts before it is checked.

Solid detail of a nozzle junction in brick elements with the weld, inside the translucent shell model
Figure 2. A solid detail of the junction in bricks, with the weld, inside the see-through shell model: the through-wall stress is read here.

Stretching and bending

Cut a strip of wall one unit wide. Whatever loads it carries can be summed into a force along the mid-surface, N per unit length, and a moment, M per unit length. The force spreads evenly over the thickness; the moment needs a stress that changes sign through it. For a wall of thickness t:

σm = N / t, σb = 6·M / t²σb is the bending stress at each face, + on one and − on the other

Membrane action uses all of the metal at the same stress. Bending uses the faces fully and the middle hardly at all, so for the same peak stress a wall carries far less by bending than by stretching. A thin wall is good at one and poor at the other: N grows with t, but the moment it can take grows with t².

Why curvature favours stretching

Put pressure on a curved wall and the pressure can be balanced by tension along the curve. Each small piece of a cylinder is pulled outward by p and held by the hoop tension at its two edges, which meet at a slight angle. Nothing needs to bend. The classic result is the hoop membrane force and stress:

N = p·R, σ = p·R / tcylinder of mean radius R, away from its ends

A flat plate has no curve for tension to work along. Pressure on it can only be carried by shear and bending, with the plate sagging between its supports. The difference is large, as the numbers below show.

The same pressure on a flat strip and on a curved wallTwo sketches side by side. Left, a flat strip held at both ends with pressure arrows pressing down on it; a dashed line shows it sagging between its supports, so it carries the pressure by bending. Right, a curved wall in section with pressure arrows pushing outward from inside; arrows along the wall show hoop tension N = p·R carrying the pressure without bending.Flat: it bendspsags between supportsbending at edges and middleCurved: it stretchesNNphoop tension N = p·R
Figure 3. The same pressure on a flat strip and on a curved wall. The strip sags and bends; the curved wall carries it as hoop tension.

Where a shell has to bend

A curved shell carries pressure by stretching only where nothing disturbs it. Bending appears wherever the shell is forced to change its curvature:

  • Joints between different shapes, such as a cylinder and a head or a cylinder and a cone, where the two parts would grow apart if they were free.
  • Flat parts: flat heads, flat cover plates and the flat sides of a box, which have no curvature to work with.
  • Attachments: a load at a nozzle or clip bends the wall round the patch it acts on.
  • Thickness changes, out-of-roundness and peaking at weld seams, where the mid-surface is not where an ideal shell would put it.
From a local load to stresses: The shell stretches and bendsThe same section with the deflection exaggerated: the nozzle is pulled out, the shell lifts next to it and settles back to its original line a short way off.Shell wallNozzlePush or pullShearMomentIt all enters through this patch of wallStretches: membraneBends: one face stretched,the other squeezedDistance from the nozzle edgeStress√(R·t)2√(R·t)3√(R·t)most has fadednearly allHighest at the jointouter faceinner faceTotalMembrane=Bending+Peak+
Figure 4. At an attachment the shell both stretches and bends. The bending is largest at the joint and dies out along the shell.

In each case the bending is local. It fades within a few times √(R·t) of the cause, and beyond that the wall goes back to membrane action.

Splitting a computed stress into the two

A model of a shell made of shell elements reports membrane and bending directly, because those elements are built from N and M. A solid model gives the full stress at every point through the wall instead, and that distribution is rarely a straight line. To check it, the stress is taken along a line through the wall and split into three parts. With x measured from one face:

σm = (1/t)·∫ σ dxmembrane: the average over the wall
σb = (6/t²)·∫ σ·(t/2 − x) dxbending: the straight line with the same moment
peak = σ − σm − σb·(1 − 2x/t)what is left; it adds no force and no moment

This is called linearisation. The membrane and bending parts carry the whole force and moment on the section; the peak is a self-balancing remainder, usually a sharp rise near one face at a notch or weld toe.

Why the split matters

Membrane and bending get different limits because they fail differently. A wall that yields in membrane is yielded right through, and further load simply stretches it. A wall that yields in bending yields first at the faces only. The middle is still elastic, and the section can carry about half as much moment again before it is yielded through.

Bending reserve of a rectangular wall sectionTwo stress profiles through a wall of thickness t in bending. Left, at first yield: a straight line from plus yield on one face to minus yield on the other, moment σy·t²/6 per unit width. Right, fully plastic: a uniform block of plus yield over half the wall and minus yield over the other half, moment σy·t²/4. The ratio is 1.5.tFirst yield+σy−σyM = σy·t²/6Fully plastic+σy−σyM = σy·t²/4Ratio 1.5, per unit width of a rectangular section
Figure 5. A rectangular wall section in bending: the moment at first yield and the fully plastic moment, 1.5 times larger.

That reserve is why design-by-analysis rules typically allow primary membrane plus bending to reach about 1.5 times the allowable stress used for general membrane. Local membrane stress, which is confined to a small region, gets a similar typical increase. The values are code-specific, so check your governing code.

Key points

  • Membrane stress is even through the wall; bending stress changes sign from face to face.
  • Curved shells carry pressure mainly by stretching, which is why they can be thin.
  • Flat parts, joints, attachments and thickness changes force the wall to bend locally.
  • Linearisation splits a through-wall stress into membrane, bending and a self-balancing peak.
  • Bending gets a higher typical limit than membrane because a bent section has a plastic reserve.

Where this matters in practice

  • Shell-to-head and cone junctions, where bending adds to the membrane stress close to the joint.
  • Flat heads, cover plates and box headers, which carry pressure mainly by bending.
  • The shell round a nozzle or clip, where the local load produces both membrane and bending stress.
  • Reading results from a solid model, where each check needs the stress linearised through the wall.

FEXLA reports the local stress at a nozzle or clip as membrane and membrane-plus-bending values and compares each with its limit.

See the nozzle check

Related principles

Next principle: Stress concentration and the weld toe

Educational article. It explains principles and is not design guidance; follow your governing code and edition.