# Shell (technical mechanics)

Model of a double-curved bowl

In technical mechanics, a shell is a flat supporting structure that is curved and can absorb loads both vertically (then it is a plate ) and in its plane (then it is a disk ).

Shells owe their load-bearing capacity to their curvature and twist. They can simply be curved, e.g. B. cylindrical, or double curved, e.g. B. spherical .

Shells make optimal use of the load-bearing capacity of their material by transferring loads using membrane forces that are constant across the thickness of the shell. This results in high rigidity with low weight and material use, which makes the bowls important in nature and technology.

In nature, shells occur in:

In technology, bowls can be found in

## definition

In the theory of strength, a shell is understood to be a body formed after a curved surface, the thickness of which is extremely small in relation to its other dimensions. The term “shell” also indicates that it is a body whose thickness is relatively as small as that of the eggshell. "

The shell construction is a form of modern flat structures. Due to the small thickness of its curvature, it combines the advantage of a self-supporting membrane with large spans . In contrast to conventional construction, the loads of such planar structures are absorbed by longitudinal forces and bending . In the case of thin shells, the thickness of which is small compared to the span, the bending can be neglected; in this case the loads are primarily transferred to the supports by longitudinal forces . This state is called the membrane tension state.

Dome-like arches create more usable space than concave , inwardly arched curvatures of membrane and cable network structures. The vaulted room has a protective effect. Primary support structures, such as the pylons of membrane structures, are not necessary for shell structures, so that an undisturbed spatial continuum can be created. Since curved lattice shells can be produced with almost any projection surface, transitions and connections on orthogonal surfaces are seldom a problem. Domes can, for example, be built on a circular floor plan and barrel vaults on a rectangular base.

After 1945, planar structures gained in importance in the field of engineering structures and accommodated new design ideas in architecture. In addition to their function as a building envelope , shell structures are also used in areas such as vehicle and aircraft construction (see: Monocoque ), interior construction and furniture construction . They represent an important design element in contemporary art.

Thin shells are powerful, but also sensitive. In addition to special static-constructive properties, they can also meet special aesthetic requirements. Because of the complicated requirements for statics and material, this type of structure was rarely realized in the decades before the turn of the millennium. It was only with the introduction of new materials that shell construction was given new impulses in recent times.

## history

The analysis of domes and shells shows an interrelationship between construction and form. The desire for wide, column-free room canopies was already a popular topic in antiquity , which at that time encountered great technical difficulties up to further development stages.

In antiquity, cantilever domes prevailed as a design , in which the individual rings were placed on flat horizontal joints . The Mycenaean domed tombs were built in the late Bronze Age . The walls were made of monumental stone blocks . A significant example of this is the Atreus treasury from around 1325 BC. To call Chr. At that time, this structure could already achieve a diameter of 14.50 meters and a height of 13.20 meters. It remained the largest circular dome building for 1400 years until the Pantheon in Rome .

The pantheon, built in 118 AD, consists in the construction of its dome of an inner and an outer shell made of cast concrete. A span of 43.30 meters was achieved for the first time.

The long tradition of brick domes only ended with the introduction of reinforced concrete . The first small reinforced concrete domes were made using the Monier process . Larger domes were mathematically and structurally broken down into beams and ribs . The then largest dome construction of the Centennial Hall in Wroclaw was built in 1911–1913 with a span of 65 meters according to this principle .

The future in domed construction, however, belonged to the thin-walled domes. There followed a transition to thin, light shell structures. The spherical dome-shaped Zeiss planetarium in Jena was certainly decisive for this development . The development of ferro-cement construction in the early 1920s made it possible to create very slim shell structures. An example of this is the second planetarium dome in Jena. With a span of 25 meters, it has a shell thickness of only 6 centimeters. The dome shells built using this process in the late 1920s made it possible to vault ever larger rooms.

There was great progress in the field of shell construction and, in particular, the use of the shell as a roof structure after the Second World War . Before the Second World War, two types were known: the simply curved shells, which mainly resulted in cylindrical shells, and the domes. On the basis of this experience and the knowledge of the pre-war period, the shell construction method was spread and further developed internationally.

The upswing in the post-war period showed the following achievements in this area in the search for new forms and developments:

• Development of double curved surfaces
• Introduction of new forms
• A tendency towards larger spans , which resulted from the attempt to roof over large areas without girders
• Prefabrication of the bowls as a trend towards economic efficiency .

The flexural rigidity of thin-walled or slender structures is significantly less than their tensile rigidity in the tangential direction. In order to optimally utilize the load-bearing behavior of such structures, it is important to avoid bending stress as much as possible.

Curved rods can at Cheap chosen shape and bearing wear bending free loads:

In the case of bowls, the conditions are much more favorable due to the surface effect. In contrast to an arch, a shell can be a support surface for more than one surface load. Conversely, a given load can be carried across the membrane using different shell shapes. There is therefore no such thing as an optimal shell shape.

### Membrane tension state

Under certain conditions, the loads on a shell are primarily diverted to the supports through stresses that are constantly distributed over the wall thickness and parallel to the central surface . In such cases, one speaks of a state of tensile stress or membrane stress . It is also present in the disk tension state of flat surface structures (disks). When the membrane is in tension, the load-bearing behavior of the material is optimally used.

The prerequisites for the membrane tension state are:

1. The central surface is continuously curved and has no sudden changes in curvature.
2. The shell thickness does not change by leaps and bounds.
3. The loads perpendicular to the middle surface of the shell are applied in a distributed manner .
4. The loads do not change by leaps and bounds.
5. No lateral forces or moments are introduced by the storage , i. H. the storage takes place in the tangential direction of the shell.
6. The transverse deformation and twisting of the shell is not hindered by the bearings.
7. The temperature distribution over the thickness of the shell is constant.
Membrane and bending stress state in a dome shell loaded by a single force

If one of these conditions is violated, the result is a less favorable bending stress condition.

The picture on the right shows a dome, which is loaded with a single force in its center. In the area where the force is introduced, which violates the third condition for the membrane stress state, there is a bending stress state (green). Far away from the introduction of force, however, there is a state of membrane tension (blue), because the dome is continuously curved and the bearing is tangential without preventing the shell from rotating.

### Bending stress state

As a result of the disturbance of the membrane stress state, bending stresses and shear stresses that vary across the thickness develop in the vicinity of the disturbance point perpendicular to the central surface. Such a state of the shell is known as a bending stress state. According to the principle of St. Venant, the disturbances quickly subside as the distance to the disturbance point increases.

The bending stress state can be compared with the plate stress state of planar surface structures. In contrast to these, however, due to the curvature, the membrane and bending stress states do not decouple with the shell problem .

## General properties

### Sensitivity to imperfections

Because the bending stresses can become very large compared to the membrane stresses, the load-bearing behavior of a shell is sensitive to irregularities in shape or load. Insufficient consideration or violations of the prerequisites for the membrane tension state as a result of imperfections can have serious effects. For this reason it is important to map all properties of a shell as precisely as possible in the calculation process.

### Shell shapes

In addition to the largely random and irregularly shaped metal sheets in vehicle construction , simple shell shapes often occur, particularly in construction technology . These arise

• the cylindrical shell (the meridian curve is parallel to the axis of rotation)
• the spherical shell (the meridian curve is a semicircle with the center on the axis of rotation)
• the cone shell (the meridian curve is a straight line that intersects the axis of rotation).

Often different rotary shells are combined with one another, e.g. B. a (finitely long) cylindrical shell with hemispherical shells at the ends. The rotation shells can in many cases be calculated with the membrane theory (see below).

With regard to the main curvature  k, a distinction is made:

• positively curved shells: k> 0
• developable shells: k = 0
• negatively curved shells: k <0.

## calculation

### Assumptions

When calculating shells, the following can often be assumed:

Deviations from these prerequisites usually require a much higher calculation effort.

Special problems, especially in connection with modern materials research ( composite materials, etc.) make the derivation of higher shell theories necessary, e.g. B. of multi-director and multi-layer shell theories, in which mostly all o. G. simplifying assumptions cannot apply.

### Membrane theory

The membrane theory is based on further simplifications. It does not give exact solutions, but is sufficient for many use cases.

• Due to the small wall thickness, the bending stiffness and the internal bending moments are small. Deflections are therefore neglected. The maximum bending moments occur in the edge areas.
• Only normal and shear stresses occur that lie within the plane and are constant over the wall thickness (membrane thickness).
• The influence of the change in shape on the force distribution is neglected. The structure is examined practically in the undeformed state (first order theory).
• The edge of a membrane shell is freely or tangentially supported.
• The forces at the shell edges are directed tangentially to the center plane.

A state of stress in shells that corresponds to the conditions of membrane theory is called a state of membrane stress . It is the basis of the boiler formula .

### Bending theory

If the prerequisites of the membrane theory are not or not nearly given, the bending theory must be applied, i. H. the bending stiffness of the shell has to be taken into account. U. also the finite shear stiffness .

Sometimes it is sufficient to take into account the effects of the bending theory locally after the application of the membrane theory. This is especially true if the edge of the shell is supported in an articulated or clamped manner and thus the conditions of the membrane theory are not met. Other examples are the transitions between different rotary shells.

## materials

Shells are often more complicated to design and manufacture than other supporting structures, but generally require less material.

Shells can be produced monolithically , with reinforced concrete being used as a rule . Alternatively, they are assembled from prefabricated individual parts , for example from curved steel sheets, or as a lattice structure.

The skeleton of the lattice structures is often made of steel, wood (see Zollinger construction method ) or composite materials and can be filled in or spanned with textile building materials , glass or other materials. The infills can assume static functions or just serve as a cladding element or room closure.

Since Frei Otto , wide-spanning, light surface structures have been a household name. Research groups dealt not only with the further development of the types of construction, but also with the combination of different building materials. In "membrane-concrete composite construction ", the special material properties of concrete and membrane are combined with one another.

In the past, vaults were made from materials such as stone , clay and masonry .

### concrete

Concrete is particularly ideal for free, non-geometric shapes. It is easy to work with and enables the best possible implementation of the design. Concrete is economical and can be used in any climate. But it also has disadvantages: problems can arise from drying out too quickly, especially with shell structures . With a polystyrene cover or other measures, the evaporation of the water can be delayed, so that the concrete hardens more slowly and a higher strength is achieved.

### Reinforced concrete

In cases where bending and tensile stresses occur in concrete parts, steel and concrete work together as a composite construction material, reinforced concrete . Because both building materials have almost the same thermal expansion , this composite effect is retained even with temperature fluctuations. Up to a certain limit, the load-bearing behavior of reinforced concrete components corresponds to that of steel or unreinforced concrete. The bond between steel and concrete ensures that the necessary crack formation is limited to a harmless level. Reinforced concrete is particularly suitable for the production of monolithic planar structures, which are characterized by high rigidity and low deformations.

### Glass

Since glass as a building material has a very brittle material behavior, it requires the reduction of bending and tensile stresses, especially for structural panes .

Glass is mainly used in lattice shell structures as a covering for large curved roof structures.

### plastic

In the construction sector, plastics will certainly be used more and more to make load-bearing components in the future. Thermoplastics (acrylates) and thermosets in particular have gained great importance. These plastics have a number of advantageous properties: weatherproof, usually very resistant to chemicals, high strength, low weight and partly translucent.

### Wood

Large barrels, domes and other free shell shapes with widths of 20 meters and lengths of 50 meters can in future be made from wood-based materials . This aspect means that in contemporary architecture, free surface structures can be implemented using simple systems. The wood-based composite is best suited for this project. The starting material for this is a material composite made of three-layer, plywood or sandwich panels. These woods are glued to a textile carrier fabric made of polyester and glass fibers with a two-component adhesive . As a result, the structure can be produced in terms of area, the individual tracks can be transported to the construction site and installed there.

## Construction methods

The shell construction is a form of modern space structure .

To this day, it is common to design the bowls to choose mathematically defined geometric shapes. However, the time of simply curved shapes is over; Now the architect has doubly curved surfaces at his disposal, from domes to hyperbolic paraboloids . Almost all of these shapes are square, which also means that related problems like tension, shape changes, and more have been better explored than before.

One principle could be: “ If you choose the right shape (in the architectural design), half the work is already done. This means assigning the geometry its right place. "

The manufacturing process can also influence the shape of bowls. Often they are prefabricated , wooden bowls have been for decades. The individual elements are manufactured in a size that is easy to transport and lift. With the prefabrication, the saving in material expenditure is a priority in the development. The relationship between shape and load-bearing capacity is hardly anywhere closer in construction than in shell constructions. Thanks to the further developments, shell constructions require minimal material expenditure compared to other building constructions.

Prefabricated bowls have been found in industrial construction since the late 1940s . Since prefabrication was also to be made possible for general buildings, they were divided into the following types:

• “Prefabricated shell constructions with special tasks for construction, function, design and representation. In this case it is often justified to use the prefabricated parts produced as individual production on the construction site or in the factory "
• “Prefabricated shell constructions that can be used very frequently in industrial buildings. In addition to the roofs of hall and low-rise buildings, where large spacing between columns is required, this also includes roof structures ... "

A further classification could be made according to factory or construction site production as well as industrial or individual production .

A big step has also been taken in favor of shell construction for the work equipment : decades ago it was a great deal of effort to clad mostly geometrically complex forms with elaborate wooden formwork , but now relocatable, reusable and inexpensive metal scaffolding is available. In this way, static-constructive forms can be produced monolithically.

### Grid shells

In addition to closed shells, there are lattice shells , rigid open planar structures. They are particularly suitable for roofs such as large, curved glass roofs. They are all thin-walled and curved shells. Their own weight and external loads are largely carried by normal force loads in the middle plane of the shell (membrane tension state). Tensile and bending loads can be absorbed by pulling glass surfaces into place as stiffeners.

Lattice bowls can also be described as a bowl with many large openings. However, the load-bearing behavior must not be impaired. For example, if the grid shell is made from individual rods , the rod connections in the nodes and the mesh type must ensure the load-bearing effect.

Stability problems such as buckling can occur with very wide lattice shells . Such shells have to be provided with complex additional measures such as arches, frames , cable spans or the like. Frame-like nodes must be implemented if diagonals are not desired.

A prominent example of lattice shells is the Multihalle in Mannheim.