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A folded plate is an example of a 3-dimensional or space structure. To consider the effectiveness of the folded plate look at the following experiment. When a thin sheet of paper rests between two supports it will bend due to the fact that it has insufficient strength to carry its own weight. If the same piece of paper is folded then it will be capable of supporting one hundred times its own weight. If the load is increased past this point then the structure will fail and the folds will flatten out. This problem can be rectified by using transverse stiffeners at the ends. The folded plate acts as a beam and can support even greater loads. Try this Experiment at home or in school, make sure to use sheets of paper equal area. Folded plates consist of straight pieces joined with sharp edges. It cannot be made as thin as a shell due to the fact that it is subjected to bending.

The result of this bending can be seen in the animation opposite. Folded plates can be seen as a space version of a rigid frame. Folded plates are best formed from reinforced concrete due to the fact that they can be easily cast. Folded plates can take the form of frames as seen above or domes.

Geodesic Domes. Geodesic domes are an example of a curved space frame. They were invented by R.Buckminster Fuller and as was seen in section 1 Structures in Nature examples of Geodesic domes can be found in nature. As was also discussed earlier, domes are not an new development, they have existed for hundreds of years. The geodesic dome is referred to as a braced dome due to the fact that there are bracing members present so that the dome keeps its shape.

The geodesic dome is made up of a network of triangles, this results in the maximum amount of strength and rigidity being obtained while using the minimum amount of materials. The varying number of triangles is referred to as the frequency and the higher the frequency the greater the resistance of the structure to collapse. Click on the icon to see how the different frequencies change the shape of a Geodesic dome. Geodesic domes can be constructed from metals or plastics and they have many advantages. Speed of construction - In 1955 he built domes for the air-force and they only took fourteen hours to erect. Size - the clear span of a dome constructed by Fuller in 1958 was 117m and it had a total weight of 12MN. If you then compare this to St.Peter's Cathedral in Rome which has a diameter of 40m and weighed 100MN. Folded Plate Structure a thin-walled building structure of the shell type. Folded plate structures consist of flat components, or plates, that are interconnected at some dihedral angle. Structures composed of rectangular plates are said to be prismatic. In modern construction practice the most widely used folded plate structures are made of cast-in-situ or precast reinforced concrete (including prestressed and reinforced-cement structures). The structures are used as roofs for industrial and public buildings.

PRELIMINARY DESIGN OF SHELLS The principal purposes for preliminary design of any structure is: (1) To obtain quantities of materials for making estimates of cost. (2) Obtain a clear picture of the structural action, (3) Establish the dimensions of the structure, and, (4) Use the preliminary design as a check on the final design. It is not expected that these preliminary design calculations be precise, but rather they should be within an accepted tolerance. The worst way to start a design is to

immediately set up a finite element analysis. Any new type of structure requires an extended lead time to obtain a thorough understanding of the structural action. The discussion of preliminary analysis here, has been restricted to principals rather than to presentation of calculations. Given these principals, the engineer should be able to set up his own calculations. Do not try to design shells without a thorough study of the relevant sections of the current American Concrete Association regulations. There are differences from the normal structures. Thickness of shells The thickness of the slab elements are normally governed by the number of layers of reinforcing bars. For shells of double curvature, there are usually only two layers so the minimum thickness could be: Two 3/8 in. bars, two 1/2 in. of cover equals 1.75 inches. However a little tolerance should be added. For a barrel shell or a folded plate: Two 1/2 in. bars, one 3/4 in. bar, two 1/2 in. of cover equals 2.75 in. Of course, the concrete stresses should be checked, but they seldom control. Do not think that a shell will be stronger if it is thicker than required. For a description of the structural elements of the shells discussed here, the reader should first study the presentations in Mark Ketchum's Types and Forms of Shell Structures

Preliminary Design for Types of Shells Barrel Shells Folded Plates Umbrella Shells Four Gabled Hypars Domes of Revolution Translation Shells

BARREL SHELLS First find the longitudinal and shear (diagonal tension) reinforcing required for a typical interior element of the structure. 1. A barrel shells acts as a beam in the long direction and as an arch in the curved area.

The arch is supported by internal shears. Approximate values for the bending moments in the arch are summarized in the following sketch.

2. The area of reinforcing is obtained by estimating the effective depth of the beam element, from the center of reinforcing to the center of compression. The force in the reinforcing is equal to the bending moment divided by the effective depth. It may require several approximations to get a fair value. The area of reinforcing is, of course, the force divided by the allowable stress.

3. The tension in the diagonal direction is determined first by equating the longitudinal force to the shear forces.

4. The sum of the shearing forces equals the longitudinal forces. Let S equal the unit shear at the end of the beam. Then: S times the width of the shell times the length divided by 4 equals the longitudinal force. If there are no other forces on an element at the neutral axis of the beam, then the diagonal tension equals the shear. From this information, a pattern of diagonal tension bars can be constructed. 5. The horizontal reaction of the arch elements of the shell must be contained by an rigid frame and a horizontal tie. Assume that this is simply a wide arch equal to half of the span. An approximation for the horizontal force would be equal to the load per foot on this arch times the arch span, squared divided by 8 and the rise. The thrust in the arch can be determined from this and the vertical reaction. 6. The edge spans of the shell should be supported by intermediate columns. The stiffness of a barrel shell at the outside edges is simply not stiff or strong enough to carry the required loads. The shell reinforcing at the edge members acts more like a

typical arch and should be reinforced with two layers of bars. Return to start of program Return to types of shells

FOLDED PLATES The design of folded plate roof structures follows the design of barrel shells, but is much simpler because the elements are all essentially beams. 1. Support the folded plate at its longitudinal edges by frequent columns as was suggested for barrel shells. 2. Analyse and design the slab element as a continuous beam on fixed supports, including the first spans, normally a simple support. If it is haunched, then as a continuous haunched beam.

3. Design a typical longitudinal interior element as a beam by the usual methods. 4. Support the ends of the folded plates by rigid frames. In this case the frames are loaded by the shear forces from the slab element and are in the plane of the frame members. Return to start of program Return to types of shells

UMBRELLA SHELLS Following is a sketch of a typical inverted umbrella hypar. The principal elements are:

• • • •

The shell element with stresses predicted by the membrane equation. The interior rib created by the intersection of the shell elements. The exterior rib supporting the shell, particularly in the exterior corners The cental column and the connection to the shell. The membrane equation for a hypar gives the direct stresses in the shell: Shear = Tension = wab/2f, where w = unit load, a and b = the dimensions of the individual panel, and f is the vertical height of the panel. These loads are transfered directly to the supporting ribs through shear, and are used to design the ribs. The internal ribs are in compression and the external ribs are in tension. In both cases, the direct stress varies from zero at the edges to maximum at the center. If the external ribs are placed above the shell then the edge member will be prestressed in positive moment and the edge of the shell will tend to deflect upward which is most desirable. It is also desirable to design this member for the additional weight of the edge member. The deflection at the end of the rib is critical. The central column should be designed for some unbalanced load. The connection to the shell defies analysis, but tests by the Portland Cement Association have proved the strength of these types of joints. Be sure to include adequate reinforcing for any contingency. Return to start of program Return to types of shells

FOUR GABLED HYPARS The design of this structure follows, with exceptions, the design of the umbrella hypar. Please refer to the previous example. The sketch shows the essential

elements:

The shell acts as an arch in one direction and as a catenary in the other. The membrane theory would predict that the stresses would be the same but of different sign. Studies by the finite element method have demonstrated that if the abutments are fixed, the compression stresses are greater, but if the abutments move because of, for example, a steel tie stretching, then the catenary stresses are larger. Which brings us to the conclusion that for the first case it would be advisable to increase the thickness of the shell near the supports to take the load off the rib elements. The top ridge member is in compression and may require additional area above that of the shell. This is a long compression member and is free to deflect downward with the possibility of ultimate buckling, (Which has happened.) It is, therefore, advisable to camber this member upward to offset this tendency. The slanting side ribs are also in compression and to some extent in bending, and sould be designed for some of the weight of the rib, say one quarter for a start. Return to start of program Return to types of shells

DOMES OF REVOLUTION

The rules described are suitable for domes of revolution of any configuration or variable thickness, not just cylindrical domes. The steps are as follows:

o o o

Determine the total weight, P, above a series of horizontal sections The total vertical stress,V, at any section will be equal to the vertical force, P. The radial force at any section can be obtained for the freebody diagram for an element as shown in the sketch. The symbol, Z, is perpendicular to the element. For a cylindrical dome the radial force can be obtained form the equation: (T(vertical) + T(horizontal))/R = Z

o

o

If the shell is not vertical, or nearly vertical, at the base, then a ring beam will be required. The force in the ring beam is obtained from the horizontal component, H, of the force at the base as shown in the sketch, and the cylinder formula: P = HR, where R is the horizontal radius of the shell. There will be some bending moment at the junction of the shell and the ring beam, so it is usual to gradually increase the thickness at this point and add moment reinforcing.

Return to start of program

Return to types of shells

TRANSLATION SHELLS The translation shell is simply a square dome as shown by the sketch. The shape is generated by a curve moving along another curve. If the curves are circles, then every vertical section is the same. The dome is usually supported by arches. There are three principal design areas:

o o o

The central dome area which is designed like a spherical dome. The corners where there is considerable tension from the ring beam affect. The arches which take their share of the total load. They are loaded in shear including the weight of the arches themselves. Return to start of program Return to types of shells

The main advantage of folded plate structures over other shells (such as cylindrical) is the simplicity of manufacture. Approximate static calculations of folded plate structures are based on the membrane theory of shells; more exact static calculations are based on limit equilibrium and on P. L. Pasternak’s and V. Z. Vlasov’s general theory of shells.

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The result of this bending can be seen in the animation opposite. Folded plates can be seen as a space version of a rigid frame. Folded plates are best formed from reinforced concrete due to the fact that they can be easily cast. Folded plates can take the form of frames as seen above or domes.

Geodesic Domes. Geodesic domes are an example of a curved space frame. They were invented by R.Buckminster Fuller and as was seen in section 1 Structures in Nature examples of Geodesic domes can be found in nature. As was also discussed earlier, domes are not an new development, they have existed for hundreds of years. The geodesic dome is referred to as a braced dome due to the fact that there are bracing members present so that the dome keeps its shape.

The geodesic dome is made up of a network of triangles, this results in the maximum amount of strength and rigidity being obtained while using the minimum amount of materials. The varying number of triangles is referred to as the frequency and the higher the frequency the greater the resistance of the structure to collapse. Click on the icon to see how the different frequencies change the shape of a Geodesic dome. Geodesic domes can be constructed from metals or plastics and they have many advantages. Speed of construction - In 1955 he built domes for the air-force and they only took fourteen hours to erect. Size - the clear span of a dome constructed by Fuller in 1958 was 117m and it had a total weight of 12MN. If you then compare this to St.Peter's Cathedral in Rome which has a diameter of 40m and weighed 100MN. Folded Plate Structure a thin-walled building structure of the shell type. Folded plate structures consist of flat components, or plates, that are interconnected at some dihedral angle. Structures composed of rectangular plates are said to be prismatic. In modern construction practice the most widely used folded plate structures are made of cast-in-situ or precast reinforced concrete (including prestressed and reinforced-cement structures). The structures are used as roofs for industrial and public buildings.

PRELIMINARY DESIGN OF SHELLS The principal purposes for preliminary design of any structure is: (1) To obtain quantities of materials for making estimates of cost. (2) Obtain a clear picture of the structural action, (3) Establish the dimensions of the structure, and, (4) Use the preliminary design as a check on the final design. It is not expected that these preliminary design calculations be precise, but rather they should be within an accepted tolerance. The worst way to start a design is to

immediately set up a finite element analysis. Any new type of structure requires an extended lead time to obtain a thorough understanding of the structural action. The discussion of preliminary analysis here, has been restricted to principals rather than to presentation of calculations. Given these principals, the engineer should be able to set up his own calculations. Do not try to design shells without a thorough study of the relevant sections of the current American Concrete Association regulations. There are differences from the normal structures. Thickness of shells The thickness of the slab elements are normally governed by the number of layers of reinforcing bars. For shells of double curvature, there are usually only two layers so the minimum thickness could be: Two 3/8 in. bars, two 1/2 in. of cover equals 1.75 inches. However a little tolerance should be added. For a barrel shell or a folded plate: Two 1/2 in. bars, one 3/4 in. bar, two 1/2 in. of cover equals 2.75 in. Of course, the concrete stresses should be checked, but they seldom control. Do not think that a shell will be stronger if it is thicker than required. For a description of the structural elements of the shells discussed here, the reader should first study the presentations in Mark Ketchum's Types and Forms of Shell Structures

Preliminary Design for Types of Shells Barrel Shells Folded Plates Umbrella Shells Four Gabled Hypars Domes of Revolution Translation Shells

BARREL SHELLS First find the longitudinal and shear (diagonal tension) reinforcing required for a typical interior element of the structure. 1. A barrel shells acts as a beam in the long direction and as an arch in the curved area.

The arch is supported by internal shears. Approximate values for the bending moments in the arch are summarized in the following sketch.

2. The area of reinforcing is obtained by estimating the effective depth of the beam element, from the center of reinforcing to the center of compression. The force in the reinforcing is equal to the bending moment divided by the effective depth. It may require several approximations to get a fair value. The area of reinforcing is, of course, the force divided by the allowable stress.

3. The tension in the diagonal direction is determined first by equating the longitudinal force to the shear forces.

4. The sum of the shearing forces equals the longitudinal forces. Let S equal the unit shear at the end of the beam. Then: S times the width of the shell times the length divided by 4 equals the longitudinal force. If there are no other forces on an element at the neutral axis of the beam, then the diagonal tension equals the shear. From this information, a pattern of diagonal tension bars can be constructed. 5. The horizontal reaction of the arch elements of the shell must be contained by an rigid frame and a horizontal tie. Assume that this is simply a wide arch equal to half of the span. An approximation for the horizontal force would be equal to the load per foot on this arch times the arch span, squared divided by 8 and the rise. The thrust in the arch can be determined from this and the vertical reaction. 6. The edge spans of the shell should be supported by intermediate columns. The stiffness of a barrel shell at the outside edges is simply not stiff or strong enough to carry the required loads. The shell reinforcing at the edge members acts more like a

typical arch and should be reinforced with two layers of bars. Return to start of program Return to types of shells

FOLDED PLATES The design of folded plate roof structures follows the design of barrel shells, but is much simpler because the elements are all essentially beams. 1. Support the folded plate at its longitudinal edges by frequent columns as was suggested for barrel shells. 2. Analyse and design the slab element as a continuous beam on fixed supports, including the first spans, normally a simple support. If it is haunched, then as a continuous haunched beam.

3. Design a typical longitudinal interior element as a beam by the usual methods. 4. Support the ends of the folded plates by rigid frames. In this case the frames are loaded by the shear forces from the slab element and are in the plane of the frame members. Return to start of program Return to types of shells

UMBRELLA SHELLS Following is a sketch of a typical inverted umbrella hypar. The principal elements are:

• • • •

The shell element with stresses predicted by the membrane equation. The interior rib created by the intersection of the shell elements. The exterior rib supporting the shell, particularly in the exterior corners The cental column and the connection to the shell. The membrane equation for a hypar gives the direct stresses in the shell: Shear = Tension = wab/2f, where w = unit load, a and b = the dimensions of the individual panel, and f is the vertical height of the panel. These loads are transfered directly to the supporting ribs through shear, and are used to design the ribs. The internal ribs are in compression and the external ribs are in tension. In both cases, the direct stress varies from zero at the edges to maximum at the center. If the external ribs are placed above the shell then the edge member will be prestressed in positive moment and the edge of the shell will tend to deflect upward which is most desirable. It is also desirable to design this member for the additional weight of the edge member. The deflection at the end of the rib is critical. The central column should be designed for some unbalanced load. The connection to the shell defies analysis, but tests by the Portland Cement Association have proved the strength of these types of joints. Be sure to include adequate reinforcing for any contingency. Return to start of program Return to types of shells

FOUR GABLED HYPARS The design of this structure follows, with exceptions, the design of the umbrella hypar. Please refer to the previous example. The sketch shows the essential

elements:

The shell acts as an arch in one direction and as a catenary in the other. The membrane theory would predict that the stresses would be the same but of different sign. Studies by the finite element method have demonstrated that if the abutments are fixed, the compression stresses are greater, but if the abutments move because of, for example, a steel tie stretching, then the catenary stresses are larger. Which brings us to the conclusion that for the first case it would be advisable to increase the thickness of the shell near the supports to take the load off the rib elements. The top ridge member is in compression and may require additional area above that of the shell. This is a long compression member and is free to deflect downward with the possibility of ultimate buckling, (Which has happened.) It is, therefore, advisable to camber this member upward to offset this tendency. The slanting side ribs are also in compression and to some extent in bending, and sould be designed for some of the weight of the rib, say one quarter for a start. Return to start of program Return to types of shells

DOMES OF REVOLUTION

The rules described are suitable for domes of revolution of any configuration or variable thickness, not just cylindrical domes. The steps are as follows:

o o o

Determine the total weight, P, above a series of horizontal sections The total vertical stress,V, at any section will be equal to the vertical force, P. The radial force at any section can be obtained for the freebody diagram for an element as shown in the sketch. The symbol, Z, is perpendicular to the element. For a cylindrical dome the radial force can be obtained form the equation: (T(vertical) + T(horizontal))/R = Z

o

o

If the shell is not vertical, or nearly vertical, at the base, then a ring beam will be required. The force in the ring beam is obtained from the horizontal component, H, of the force at the base as shown in the sketch, and the cylinder formula: P = HR, where R is the horizontal radius of the shell. There will be some bending moment at the junction of the shell and the ring beam, so it is usual to gradually increase the thickness at this point and add moment reinforcing.

Return to start of program

Return to types of shells

TRANSLATION SHELLS The translation shell is simply a square dome as shown by the sketch. The shape is generated by a curve moving along another curve. If the curves are circles, then every vertical section is the same. The dome is usually supported by arches. There are three principal design areas:

o o o

The central dome area which is designed like a spherical dome. The corners where there is considerable tension from the ring beam affect. The arches which take their share of the total load. They are loaded in shear including the weight of the arches themselves. Return to start of program Return to types of shells

The main advantage of folded plate structures over other shells (such as cylindrical) is the simplicity of manufacture. Approximate static calculations of folded plate structures are based on the membrane theory of shells; more exact static calculations are based on limit equilibrium and on P. L. Pasternak’s and V. Z. Vlasov’s general theory of shells.