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3. Shells 3.1. THE MECHANICS OF SHELLS 3.1.1. Classification Classification based on the stresses The thickness of the shell, the loads, the geometry and the supports determine the stresses that arise in the structure.

- Shells under uniform tension or compression (special membranes). Special geometry, loads and supports can result in uniform tension or compression in a shell without shear stresses. Since a small imperfection would lead to the appearance of shear stresses, this structure is only theoretical. However, designing such structures would help to optimize the stresses and reduce shear stresses.

- Thin shells (membranes). Membrane theory assumes, that only membrane stresses arise in the structure (compressive, tensile and shear stresses). This assumption is valid for thin shells.

- Thick shells. Shell theory assumes that membrane stresses and bending also arise in the structure. This assumption is valid if the bending stiffness of the structure is not negligible.

Classification based on the geometry of the shells We often classify shells based on the Gauss-curvature of its points. The Gauss-curvature is the product of the principal curvatures (𝐾", 𝐾$) and a point is:

• elliptic, if 𝐾"𝐾$ > 0 • hyperbolic, if 𝐾"𝐾$ < 0 • parabolic, if 𝐾"𝐾$ = 0 • planar, if 𝐾" = 𝐾$ = 0

Usually, surfaces have many different types of points (e.g. the torus has elliptic, parabolic and hyperbolic points as well). However, in the case of some special surfaces, all of the points are the same from this point of view. E.g. a surface is elliptic if all of its points are elliptic.

- Elliptic surfaces. In most parts of the structure compressive or tensile stresses arise. Change in the sign of the stresses often occur at the edges. Typical materials: reinforced concrete, concrete, masonry, pneumatic tents and space trusses.

- Parabolic surfaces. Typical materials: reinforced concrete, concrete, masonry, pneumatic tents, cables, space trusses.

Uniform compression Membrane stresses Membrane stresses and bending

Elliptic surface

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- Hyperbolic surfaces. The dominant internal forces are shear forces (the principal stresses have opposite signs). A typical hyperbolic surface is the hyperbolic paraboloid (or saddle roof or hypar). Since the stresses parallel to the rulings are shear stresses, if we rotate the coordinate system with 45° we get the principal stresses (compression and tension). Building a hyperbolic surface from a no-tension or no-compression material is possible because the opposing sign of the principal curvatures of the surface allows the prestress of the surface. Besides hyperbolic paraboloid, the hyperboloid of revolution is another example of hyperbolic surfaces.

Special surfaces - Ruled surfaces. Any point of a ruled surface lies on a line. A ruled surface is either parabolic or hyperbolic (or plane). Some of the ruled surfaces are also developable surfaces.

- Developable surfaces. All developable surfaces are ruled surfaces. The curvature perpendicular to the rulings is constant and the surface has zero Gaussian-curvature. Developable surfaces are cones and cylinders.

- Non-developable ruled surfaces. The curvature perpendicular to the rulings changes. Examples: conoid, hyperbolic paraboloid, hyperboloid of revolution.

If a surface is developable, it is possible to change the external geometry without causing strains (e.g. extension or contraction of the surface). An example of such structure is the Yoshimura origami surface, which is a folded cylinder composed of planar faces. Compared to the cylinder, the Yoshimura surface has larger moment of inertia leading to a larger bending stiffness. As a result, the surface can be built from thin planar elements (which is an advantage if the structure is under compression).

3.1.2. Statics of shells Barlow’s formula The Barlow’s formula gives back the relationship between the internal pressure and the stresses in a pipe. This formula can be used to calculate the stresses in cylindrical or spherical shells. It can be also generalized to a small element of the shell as well. The formula if one of the principal curvatures is zero:

𝑝 =𝑁𝑅,

where 𝑝[𝑁/𝑚𝑚$] is the pressure perpendicular to the surface, 𝑅[𝑚𝑚] is the radius of curvature and 𝑁[𝑁/𝑚𝑚] is the specific force (the product of the stress and the thickness of the shell). In case of doubly curved surfaces, the principal curvatures (principal curvature means, the smallest and the biggest radius of the curvatures) determine the principal 𝜑 and 𝜗 directions and the formula is:

𝑝 =𝑁4𝑅4

+𝑁6𝑅6.

Hyperbolic surface Parabolic surface

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Analysis of domes under uniformly distributed load The analysis considers stresses acting in the meridional direction ( 𝜑) and stresses acting in the hoop direction ( 𝜗) (=parallel direction). The stresses in the surface can be derived using Barlow’s formula:

𝑁4 =𝑞𝑅2,

𝑁6 =𝑞𝑅2(2 cos$ 𝜑 − 1).

From the equations, it is clear that in case 𝜑 = 90°, the stresses are compressive and equal in both directions (this is the top of the dome). At 𝜑 = 60°, the sign of the hoop stress changes.

Forces arising at the supports are meridional, the vertical component of 𝑁4 is carried by the vertical supports and the horizontal components are carried by the edge beam (the bottom ring of the dome). The force in the bottom ring can be calculated using Barlow’s formula:

𝑁CDEF = 𝑁4 ⋅ 𝑟 =𝑞𝑅2cos(𝜑) ⋅ 𝑅𝑐𝑜𝑠(𝜑),

where 𝑟 is the radius of the bottom ring. The force in the bottom ring is always tension independent of 𝜑. However, since the hoop stress changes above 𝜑 = 60°, compressive stresses arise in the hoop direction in shallow shells which results in edge disturbance. Edge disturbance means, that the hoop stresses are compressive, but the force in the edge is tension which can lead to cracks. To avoid failure/cracks as a result of edge disturbance, the edges should be reinforced (e.g. with rebars).

Similarly to the calculation above, the internal force in a ring of the structure (in the hoop direction) can be calculated using Barlow’s formula:

𝑁LMMN = 𝑁4O ⋅ 𝑟,

where 𝑁LMMN[𝑁/𝑚𝑚] is the specific force in the ring, 𝑁4P[𝑁/𝑚𝑚] is the meridional stress at the points of the ring and 𝑟[𝑚𝑚] is the radius of the ring.

Analysis of domes under self-weight Using Barlow’s formula, the resulting equations for the meridional and hoop stresses are slightly different:

𝑁4 =−𝑅𝑔R

1 + cos(𝜑),

𝑁6 = 𝑔R𝑅(1

1 + cos(𝜑)− cos(𝜑)),

where 𝑔R[𝑁/𝑚𝑚$] is the self-weight of the structure.

Analysis of a quadrilateral shell under uniformly distributed load Apart from the spherical dome, the quadrilateral shell is the most used shell type. It is a ruled surface with straight rulings, which simplifies the construction of the structure. It is obvious from the structural analysis of the shell, that the normal forces parallel to the rulings are unable to carry the vertical loads. However, the vertical components (𝑁STU ) of the internal shear forces along the edges (𝑁ST) can support the vertical loads. To calculate

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the internal forces, it would be complicated to use the equations of the principal curvatures, as the radii belonging to the principal curvatures are not available automatically. Nevertheless, due to the simple geometry and simple loading, the forces along the semi-rigid edges can be easily calculated. The internal forces along the principal directions are the same at each point of the structure as a result of the geometry. The projection of stresses at the edges (horizontal component):

𝑁STVVVVV =𝑞𝑎𝑏𝑓2

,

where 𝑁STVVVVV[𝑁/𝑚𝑚] is the horizontal component of the stress at the edge, 𝑞[𝑁/𝑚𝑚$] is the uniformly distributed load, 𝑎[𝑚𝑚], 𝑏[𝑚𝑚] are the dimensions of the projection of the surface and 𝑓[𝑚𝑚] is the height. Due to the special geometry and load, it is enough to calculate the stresses at the edges because the stresses in all points are equal.

3.1.3. Edges Free edges Free edges can be constructed only in some special cases. In a shell with free edges, the stresses vanish at the edge. Possible scenarios for free edges:

- if the edge is perpendicular to the rulings of a parabolic surface

- if the edge is circular/elliptical, the shell is elliptic/hyperbolic and the shell has sufficient support (statically determinate or indeterminate).

Free edges are typical for surfaces composed of small segments.

Hyperbolic surface with free edges

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Semi-rigid edges A semi-rigid support means, that the shell is supported only in one direction. In general, this support force is usually a shear force. A semi-rigid edge is typical for quadrilateral shells, where it is enough to support the edge with shear forces. A semi-rigid edge is e.g. an edge beam with large axial but low bending stiffness.

Rigid support Rigid support is usually disadvantageous in case of shells. A rigid support means, that the deformation of the shell is constrained perpendicular to the shell. This constraint can result in bending moments at the edges (edge disturbance) which can lead to the failure of the structure. As a result, rigid support is usually avoided in case of shells.

Catenary edge The projection of the edge is a catenary curve and the edge member can bear tension/compression. Vertical support is necessary for these edges but there is no need of horizontal support.

Full support / membrane support The structure is supported not only shear but also normal forces. Support of shear forces is usually needed in case of asymmetric load.

3.1.4. Design Some of the shells, e.g. parabolic shells, carry the loads with membrane forces only in case of special loads. The shape of these shells is usually designed for the typical load. However, if the loads are different from the design load, it can result in large deformations, large bending moments or failure. Nevertheless, most of the shells can carry almost all type of distributed loads with membrane forces, but concentrated forces can lead to singularities (peaks in the stress distribution). The shape of the shell is often optimized for the typical load (which is usually the self-weight).

What is the optimal thickness? Thicker shell means larger self-weight and larger membrane forces (but the membrane stresses are the same). Increasing the thickness of the shell leads a larger bending moment which can lead to failure. Typical thickness is 4-6 cm (which is enough to span 80-100 m) if the buckling of the shell is prevented with a decent construction.

It is important, that bending-free designs are not completely free of bending moments.

• Changes in the curvature leads to bending moments, but if the shell is thin enough, it is negligible. Sometimes, the stresses in the shells and in the edges are incompatible (edge disturbance) which can lead to cracks.

• Similar problem occurs if a membrane support is inappropriate: if the support is able to carry forces perpendicular to the surface it can hinder the deformations of the structure and bending moment arises.

• In other cases, the changes in the curvature as a result of the deformations are large enough to cause non-negligible bending moments.

• Peaks in the stress distribution (caused by concentrated loads or supports) can also lead to bending moments. Such peaks occur in case of translation surfaces supported by shear forces, where the shear forces can be infinitely large at the corners of the structure.

Quadrilateral shell with semi-rigid edges

Catenary curve

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3.2. TYPICAL SHELLS Barrel vaults Barrel vaults are parabolic surfaces, all points lie on a line (a ruling). The curvature of the surface is constant perpendicular to the rulings. They are also translational surfaces, the first generatrix is a line and the second one is a curve.

Barrel vaults are sensitive to the supports.

• If a barrel vault is supported only along the longitudinal edges, it carries the loads as an arch. However, this requires, that the shell follow the shape of the thrust line. Furthermore, if the loads change and differ from the design loads bending moment arises and shear forces are also necessary at the supports.

• If all edges are semi-rigid edges, then the shell is able to carry arbitrary loads with membrane forces. The semi-rigid edges are usually supported at the corners. In case of such support, the second generatrix is not necessary to follow the shape of the thrust line, it can be a circular, elliptical, parabolical segment, a cycloid or a basket-handle curve.

• If the shell is supported along the longitudinal edges and one transversal edge, then the fourth edge can be a free edge. This kind of support also allows the shell to carry arbitrary loads with membrane forces.

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Shells of revolution Shells of revolution can be elliptic or hyperbolic. It is constructed by rotating a curve about one axis. Due to their geometry, it is beneficial to calculate their equilibrium in polar or natural (local) coordinate systems. We usually discuss the internal forces in the meridional direction and along the parallel circles. It is important, that the generatrix should be a smooth curve otherwise bending moment would arise.

The possible supports:

• If we use vertical support only, a flexible bottom ring is also necessary to carry the horizontal forces. This type of support can also be used in case the base curve is an ellipse.

• If the support is rigid, it leads to edge disturbance. However, a technically feasible support is usually rigid, therefore the edge must be reinforced to reduce the edge disturbance.

• The ideal support would be a membrane support, because the shear support is unnecessary and it would avoid the edge disturbance. Such structure is the Sports palace in Rome (Nervi).

• If there is an opening on the top of the shell, the top edge can be free. To avoid bending moment under uniformly distributed load along the edges, it is worth to build an edge beam at the top ring as well.

Sports palace, Rome (Nervi)

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Hyperboloid of revolution It behaves similar to other shells of revolution. It is interesting, that from Barlow’s formula, the two terms regarding the principal direction have opposite sign.

Unless the top ring is stiff enough, horizontal loads (e.g. wind load) could cause large deformations at the top ring which lead to the failure of the structures. This is the problem of all cylindrical shells, but in the engineering practice it occurred for hyperboloids.

The shape of the structure is ideal for cooling towers because it facilitates an ideal airflow: the air flows intensively from the bottom to the top.

Quadrilateral shells They are non-developable ruled surfaces. If the loads are vertical and the projection of the surface is a rectangle, then it is enough to support the surface with shear forces. If the projection is a parallelogram or the load has components that are not vertical, then normal forces are also needed at the supports.

A usual support of quadrilateral shells is composed of edge beams along the edges carrying shear forces. Shear forces cause axial forces in the edge beams and they are supported at the corners. Sometimes they are also supported by walls or additional columns. They can help to carry asymmetric loads but they also cause edge disturbance!

Saddle surfaces They are similar to quadrilateral shells because both of them are hyperbolic paraboloids. However, the edges of saddle surfaces are curved.

If the shell is supported along two opposing edges, then it behaves like a parabolic arch and it can carry uniformly distributed loads with membrane forces (because it follows the shape of the thrust line).

To provide full membrane support to the structure, two opposing edges should be fully supported and the other two edges should be supported with semi-rigid edges.

The stress distribution of the surface depends on the geometry! We can classify the saddle surfaces based on the aspect ratio of their projection:

Projection of saddle surfaces with abnormal, semi-normal and normal aspect ratios

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• Normal aspect ratio: the length of the long sides are multiples of the short sides (1/2, 1/3…). There are no shear stresses on the sides with full membrane support. The shear stresses change sign along the semi-rigid edges.

• Semi-normal aspect ratio: the length of the long sides are multiples of the shorts sides plus one half (2/3, 2/5…). Shear stresses arise on the sides with full membrane support.

• Abnormal aspect ratio: any other aspect ratio. The shear stresses change sign in multiple points along the semi-rigid edges.

Saddle surfaces are also translation surfaces.

Translation surfaces Translation surfaces are generated by translating one curve (first generatrix) along another curve (second generatrix). The generatrices can be any smooth curves, but in practice they are usually parabolas.

Possible supports:

• Full membrane support would be beneficial but in practice, it is almost impossible to support the edges without hindering the deformations perpendicular to the surface which would lead to bending moments.

• The typical is to provide semi-rigid edges (with edge beams). In this case there is no need to provide horizontal support.

Sail vault The sail vault is a spherical shell with square plan.

Possible supports:

• Full membrane support results in a similar behavior to the dome. • The most typical type of support is to use semi-rigid edges on all sides. • It is also possible to support only the corners leading to an arch-like behavior.

Conoid Conoids are non-developable ruled surfaces, but they can deform without strains (i.e. without extension or contraction). It is constructed by connecting a curve to an axis with lines lying on the directrix plane. If the structure has full membrane support along the curved edge, then the axis can be a free edge. There are other possible support types, some of them are illustrated in the figures.

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Segment shells The segments can be elliptic, parabolic and hyperbolic. However, at the intersection of the neighboring segments the tangent of the surface is not smooth, therefore edges beams are necessary. Depending on the shape of the intersection curve and the segments these edges can be semi-rigid or full membrane supports. Practically, they are ribs under or above the shell.

The external edges can be free for some special geometries, full membrane support is also possible, and semi-rigid edges are typical if the projection of the surface is polygonal.

Free-form surfaces The shape of the free-form surfaces is determined with a form-finding process. It is an inverse problem, where some of the parameters are fixed and the shape of the shell is unknown. The fixed parameters are usually: the edges, the load and we assume a stress distribution. The load is usually the self-weight of the structure or the uniformly distributed load.

In case of tents the self-weight is negligible and the loads are uncertain: the live loads (snow and wind loads) can change. However, to provide a shape for the structure, it must be prestressed. As a result, the stress distribution is a fixed parameter.

Nevertheless, the stress distribution depends on the designer. It is preferred to have uniform stress distribution, which is the hydrostatic stress distribution and it leads to a minimal surface. Such surface is the soap film. Unfortunately, it is not possible to create a minimal surface above all geometries. However, almost any stress distribution can be prescribed in advance which is smooth.

The edges also depend on the designer: rigid edges can be constructed with almost any shape. Semi-rigid edges and free edges are not typical because they need extra conditions regarding the stress distribution. Apart from rigid edges, catenary edges are also typical.