Introduction Shell Structures

7/28/2019 Introduction Shell Structures

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By-Milo S. Ketchum&Mark A. Ketchum

INTRODUCTION

The purpose of this book is to provide sketches and descriptions of many types of

shell structures to aid the architect or engineer in the selection of a structure for a

particular use. No claim is made for completeness. Drawings have been used rather

than photographs, first because of the limited number of photographs available, and

second, because there is always a reluctance on the part of the architect to use

something already built because it would seem like copying. Only the structural

features are shown in the sketches and details such as windows, gutters, fascia

members, etcetera, have been omitted. The criterion has always been to picture the

shell after the concrete has been complete but before the brick, stone, windows, or

roofing is placed. Most of the nomenclature is standard in the literature but some of it
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was devised by the writer to fit gaps for which satisfactory terms were not available.

In illustrating shell types, many obvious structures have been omitted because they are

so similar to the basic types. The criterion for showing examples has in all cases been

its usefulness as an architectural or structural unit of construction.

CHAPTER I - FOLDED PLATES

It seems appropriate to start the presentation of examples of shapes and forms for shell

structures with the folded plate because it is the simplest of the shell structures.

The distinguishing feature of the folded plate is the ease in forming plane surfaces.

Therefore, they are more adaptable to smaller areas than curved surfaces which

require multiple use of forms for maximum economy. A folded plate may be formed

for about the same cost as a horizontal slab and has much less steel and concrete for

the same spans. Folded plates are not adapted to as wide bay spacings as barrel vaults.For widths of plate over, say, 12 feet, the thickness of the folded plate must be thicker

than for a barrel vault. Some advantage may be gained by increasing the thickness of

the slab just at the valleys so it will act as a haunched beam and as an I section plate

girder.

BASIC ELEMENTS

The principle components in a folded plate structure are illustrated in the sketch

above. They consist of, 1) the inclined plates, 2) edge plates which must be used to


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stiffen the wide plates, 3) stiffeners to carry the loads to the supports and to hold the

plates in line, and 4) columns to support the structure in the air. A strip across a

folded plate is called a slab element because the plate is designed as a slab in that

direction. The span of the structure is the greater distance between columns and the

bay width is the distance between similar structural units. The structure above is a

two segment folded plate. If several units were placed side by side, the edge plates

sould be omitted except for the first and last plate. If the edge plate is not omitted

on inside edges, the form should be called a two segment folded plate with a

common edge plate.

The structure above may have a simple span, as shown, or multiple spans of varying

length, or the folded plate may cantilever from the supports without a stiffener at the

end.

THREE SEGMENT FOLDED PLATE

This sketch shows a folded plate structure with three segments for each barrel. The

end stiffeners are rigid frames rather than deep girders as in the last example. The

forces from the reactions of the sloping plates on these rigid frames will be quire


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large and at an outside column they will not be balanced by thrusts from the

adjacent plates. The size of the frames may be reduced by using a steel tie between

the tops of the columns which can be concealed in the fenestration.

The dimensions of the plates are dependent on both the width of the barrel and on thespan. The depth of the shell should be about 0.10 times the span and the maximum

slope of a plate should not be greater than 40 degrees. For example, assume for the

above structure that the span is 60 feet and the bay width is 24 feet. The depth of the

shell should be about 6 feet and the horizontal width of each plate with a three

segment plate should be about 8 feet. The slope of the plates is 6/8, which is about 37

degrees and is satisfactory. The thickness of the plates could be about 3 inches.

Z SHELL

Each of the units above has one large sloping plate and two edge plates arranged

with space between the units for windows. This form has been called a Z shell and is

similar to the louver used for window ventilation. The architectural effect is very

dramatic if the structure can be shown by a cantilever projected out beyond the

support. The windows are normally open to the north but most of the light is actually

reflected south light. To increase this effect, the roof surface can be painted with


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aluminum so light from the sun is reflected through the windows to the ceiling and

the windows need not be very large. Adjacent units should be tied together by

structural window mullions. In constructing the Z shell, movable forms need only be

lowered a short vertical distance if construction is started on the right and proceeds

to the left.

The Z shell is not an efficient structural shape since it is discontinuous and its

effective depth is much less than the actual vertical depth. Therefore, the spans are

limited in comparison to the plates having a large number of units side by side.

WALLS CONTINUOUS WITH SHELL

In this structure the walls are of tilt-up concrete construction; concrete is cast flat on

the floor and raised into place by cranes. The walls are designed to be continuous

with the roof plates. Tilt-up walls usually are joined by poured-in-place columns. In

this design, columns are not necessary at the junction of the individual side wall

panels because the walls are braced at the top. Only a simple grouted key slot is

provided.

The tilt-up panels can serve as their own foundation walls so only a continuous

footing pad is used with a notch to receive the tilt-up panel. Dock height interior

floors can be constructed by filling the interior of the building up with dirt to the

required height. The tilt-up walls can be designed for this lateral load because they are

held at the top by the shell and act as vertical beams rather than as cantilever

retaining.


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CANOPIES

A folded plate structure for a small canopy at the entrance of a building is shown.

This folded plate has four segments. A two segment structure is not desirable

because it has very little torsional resistance. This instability can be demonstrated by

a paper model having the ends of the model glued to vertical pieces of cardboard,

acting as stiffening members. If it is absolutely necessary to have a two element

system, a torsion member can be placed in the valley which will carry the unbalanced

loads.

Stiffeners can often be hidden on the top surface so they are not in evidence and the

shell will appear to spring from the vertical column. At the wall of the building there

should also be a stiffener hidden in the wall construction. Provision should be made

for drainage of the center valley.

TAPERED FOLDED PLATES

Folded plate structures may be built with tapered elements and only one of the

many possible combinations is shown here. Another possibility is to place the smaller

depths all at one end so that the entire structure forms a circular ring. The height of

the shells at the center of the span is the critical dimension for bending strength.

Therefore, the structure is not very efficient and not suitable for long spans because

of the excess height required for the large ends. Another weak element in this design


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is the transfer of shear from the small end of the triangular plate to the large end. If

a large number of units are used in each span, the transfer of loads may be difficult.

A folded plate may be used for walls as a thin structural element by casting each plate

flat on the floor and grouting the joints full of concrete. A wall of this type can bemade much thinner than a flat wall.


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EDGE SUPPORTED FOLDED PLATES

The usual upturned edge plate can be eliminated and the roof structure can be made

to appear very thin if the edge plate is replaced by a series of columns. The slab

between columns must be designed as a beam and it may be convenient to extend

the main roof slab as a cantilever canopy. The beam element that carries the load of

the roof between columns will then be wider and windows under the slab will have

the same function as in the previous examples of folded plates. Note the vertical

columns in the end walls at the crown of the gable. These take the reactions of the

plates and the horizontal ties may be eliminated. Wind loads are taken by rigid frame

action in the columns and stiffeners.

FOLDED PLATE TRUSS

The term "folded plate truss" is intended to indicate the structural action of this

structure. There are horizontal ties across the width only at the ends of the building

and the structure acts as an edge supported shell as shown in the previous example.

The thrusts from the triangular crossed arches are carried lengthwise to the ends.

The top chord of the inclined truss is formed by the ridge member. The bottom

chords are the ties at the base of the side gables and the diagonals are formed by the

sloping valleys at the intersection of the gables and the triangular plates. The top


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longitudinal compression member may require some additional thickness to form a

compression member of sufficient size to carry the compression force.

This is truly a space structure and its structural action is not as obvious and, therefore,

the architectural appearance is mote subtle that the usual shell structure.

FOLDED PLATE RIGID FRAME

An arch with straight segments is sometimes called a rigid frame. It is not as efficient

as the curved arch because the bending moments are greater. Ties across the platesare required at the knees and at the crown in order to distribute the forces at the

ends of each segment.


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A PHOTO GALLERY OF SHELLS

This set of photographs is to supplement those inMark Ketchum'sphoto gallery and

to point out special problems in shell design or construction.

Click on image to enlarge.

This is a poor photo but it brings up an important problem. The structure is a short

shell for a bakery. The span of the frames that support the shell is 85 feet and the

distance between frames is 35 feet. There are four frame in a series. The shell

thickness is 2 in. at the center and 3 in. near the valleys. Notice the white streaks on

the underside of the shell. They occur where the frames are above the shell. In these

areas the concrete is 3 ft. thick and is much denser than for the 3 in.shell. The concrete

in shells is not compacted in the way thick concrete members are. Design stresses is

shells should be very conservative. Shells get their strength by shape and form, not by

the strength of the material.


This is the previous shell under construction showing the form work. It is movable

and slids on wooden rails on the ground, except for the center which is fixed. This was

done so the movable form could be decentered and moved sooner. Concrete should be

supported as long as possible.
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This is a picture of the famous shell by Felix Candela in Xochimilco, Mexico, and

covers an area of 150 feet round. Candela was a master of geometry. To create a shell

with minimum streses, the shell must have double curvature in all areas. If this is

done, then the calculation of stress is merely incidental.


The photograph of another shell by Candela shows a dome made from three

hyperbolic paraboloidal units separated by skylights. The curvature of the hypars is

not as great as the previous example so he was more conservative. Note the steel

columns under the edges. Deflections are a constant problem with so thin members.


Candela is best known for his many hyperbolic paraboloids, but he was the master of

all types of shells. Here is group picture of a brewery in Mexico. There are both short
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shells and translation shells. The open architecture promotes the appearance of

extreme lightness.


The use of north light shells has not generally been accepted in this country. The

argument is that the light is variable, and therefor you will need artificial illuminationanyway. Any one who has worked under natural light knows that this theory is not

valid. Changes in lighting for various conditions gives relief to the worker. There are

other problems, however, spill of cold air from ovehead windows may be a problem in

cold climates. Also the acoustics of the curved surfaces need to be solved. The next

picture is of north light barrel shells by Felix Candela in Mexico.

One unit consists of a gutter to collect the rain and a barrel, with large slanting

windows between each unit. The windows therefore support the high point of the

barrel. Remember that Candels's shells are all only an inch and a half thick

The construction of airformed shells is a growing industry in this country. The

photograph is of a house and was save from awebsiteentirely devoted to dome

constructors. I selected this picture from one of thereferences
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Here is another dome use for bulk storage. My son, Mark Ketchum, a Principal of

OPAC Engineers, San Francisco, was the engineer of record.


A SECOND GALLERY OF PHOTOGRAPHS

A drawing of the Panetarium in Forest Park, St. Louis. The viewing screen, a spherical

structure, is inside this "hyperboloid of one sheet". After viewing the stars, then it is

possible to climb to the roof and view the stars in nature.

The thickness of the shell is three inches except at the base. This form is used for thehuge power plant cooling towers, which have had a record of several failures, and has

since been subject to considerable research. We were the consultants for the design of

the shell.

This architectural drawing shows the maintaintence facilities for Trans World Arlines

outside of Kansas City. Only two of the latge hangars were built. In adition there are

shops, also shell structures.
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The Engineer on this complex was Dutton Biggs of Kansas City, and my firm was aconsultant on the hangars. All the structures were hyperbolic paraboloids. The shops

are the typical inverted umbrellas and the hangar surfaces are all hypars.

The is the sales and maintenance facilities for the H. W. Moore Company in Denver. It

was one of the first large folded plate structures to be built in this country. A displayarea is to the left and the parts department to the right.

The shop facilities are typical two element folded plates and have spans of 80 ft. Thedisplay areas are a series of Z folded plates with spans of 75 ft.

Candela built hundreds of thousands of inverted umbrella hypars in Mexico. He toldme that he could not charge owners what they cost. They were so inexpensive that it

would undermine the industrial building market. Here is a typical building.

The plan of each unit could be square or rectangular and dimensions varied from 40 ft.
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to 60 ft. In this example, the units were tilted and a clerestory served to add light to the

interior.

The next photos show two different solutions for a similar problem. The first is aconcourse for the St. Louis Airport. In this case it is intersecting cylindrical shells with

triangular slots between elements. The shells are picked up by large ribs at theintersection of the cylinders. This enables the structure to be placed above the platform

on which it rests. The ground plan of the units is 120 ft. and the thickness of the shells

is 4.5 inches, goverend by buckling considerations. The shells were constructed usingmassive movable forms.

The second example is from Candela, and is a manufacturing facility. The shells are

hyperbolic paraboloids, and, presumably, one and one half inches thick. Buckling was

not a consideration. The ground plan is 100 ft. square. Candela always used fixedforms, and erected and dismanteled them so they can be used again. Note how the

shells are picked up by continuation of the end ribs. A much lighter solution than the

previous example.

Here is a picture of the previous shell under construction. Note the long ramp forworkers to carry the concrete in containers up to the top, and put exactly in place. It is a

very efficient system if labor is cheap.
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A THIRD GALLERY OF PHOTOGRAPHS

The next picture shows the construction of precast barrel shells. They were first usedfor the cover of a water collection canal in the mountains of Colorado, but later were

used in a number of industrial and residential buildings. As many as six shells were

made from one form by casting the next shell on the previous one. To acccomplish ths,all of the vertical dimensions must be the same.

This picture shows the precast barrel shells being placed in the mountains of Colorado.They were fabricated in Denver and shipped by truck. The supports for the shells are

cast in place circles. They will carry the end forces of the shells if there is adequate

reinforcing in the shell over the support.

Click onimage to enlarge

Next is a folded plate for a high school gymnasium. This design is the result of the

experience accumulated for a number of previous designs.
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Click to enlarge image

1. The slab elements are thicker at the valleys and ridges, making it possible touse longer spans with the same minimum thickness. In this case over 20 feet.

This, in turn, makes it possible to use less slope on the slabs, resulting in easier

construction. When I wrote my program for folded plate analysis, I wasmotivated to include this feature of variable slab thickness.

2. The normal practice for edge member slab elements was to make them as muchas one third of the width of the other slabs. Even this wide, the stiffness of theedge elements is much less, resulting in excessive deflection. In this case, theedge slabs are small and the edges are supported by steel columns embedded in

the brick walls.

Shell structures need not involve projects with large areas or be of momumental

design. The next picture shows a small shopping unit with a folded plate roof having a

small span. Note, again, that the edge valleys have been supported by steel columnsthat will be embedded in walls. The roof was formed with fixed in place forms, but the

shores were constructed so that they could remain in place while the other forms weremoved quickly and still have the concrete supported agianst deflection.

Clixk on image to enlarge

The Ideal Cement Company needed a hangar to house a company plane, so it built astructure that would use one of it's products, light weight concrete. The building is a

barrel shell, triangular in plan. The rear is supported by a number of columns and thefront, which must have a clear span, by an arch bent in the middle. The thrusts from

this bent arch at the high point, are carried by prestressing cables to a similar rib in the
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back wall. There are, of course, doors to cover the opening.

Click on image to enlarge

A FOURTH GALLERY OF PHOTOGRAPHS

Candela built hundreds of thousands of square feet of this basic design of a short shell,

mostly for open air markets. Note that the ties to take the thrusts form the arches areabove the shell and exposed, thus making a tie free open space. Our architects mightnot like this kind of solution but Candela had complete control of all phases of design

and construction. Some examples had skylights between adjacent arches, so that there

was natural light for the interior.

This project is a series of 40 foot square domes for a discount store. The layout is fourdomes wide by nine domes long. The shape is not a sphere but is a translation shell: see

section onTypes and Forms.There are no internal ties to take the thrusts of the dome

and the end thrusts are carried by external diagonal braces. The walls are precast anderected in place.

The domes are formed by bow string trusses, so no shoring is required amd forms may
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be easily moved. A set of four units were used with nine reuses. The trusses were

salvagable and could be reused. Temporary ties were required until the outside braces

were cast.

One of our architectural clients, Tom Moore, of Denver, Denver was fascinated with

Buckminster Fuller's Geodesic Domes. He commissioned a design for a fraternityhouse using hexagonal units rather than the familiar triangles. Precast units called "dog

bones", a member with a Y shape at each end, were joined to create a sphericalframework over which the shell was cast. The first picture shows the basic geodesic

dome, and the next picture, the external appearance of the fraternity house.

This multiple dome for a church shown under construction, was formed by using a

mond of earth as a form. Then the earth was removed.
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SOME BASIC CONCEPTS FOR SHELL STRUCTURES

By Milo S. Ketchum

If a picture is worth a 1000 words, then a simple model is worth 1000calculations.

There are only a few basic structural systems for post and beam structures, butfor shell structures, there are thousands, each requiring a unique approach to

design.

The supports for a shell are more important than the shell. Shell structures can usually be understood as a set of beams, arches and

catenaries and can be analyzed by that approach.

For any shell structure, there will be a simple method of analysis that can beused to check the more precise analysis.

Stiffest path concepts are useful in understanding shell structures. Support the edges of shells if they are already supported visually by masonry

walls or window walls.

Do not throw away all you structural intuition when you design shell structures. For ordinary structures, an adequate preliminary design should be within 10

percent Shell structures can be estimated to within 5 percent because the only

usual unknown is the amount of reinforcing.

Shell structures can carry relatively large point loads. If you rush into computer calculations without a thorough study of the structure

with several possible arrangements, then you may not get the best structure.

Shell structures get their strength by shape and not by high strength of materialsDo not push stresses to their limit.

In case of doubt, reinforce. Shell structures are very complex and carry forcesby many paths.

Shell structures, because of their complexity and unfamiliarity require a largelead time for developing the design.


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A BRIEF TUTORIAL FOR THE UNINITIATED

This is a shell:

It is called a hyperbolic paraboloid, from a mathematical equation of the same name.

My mother wanted to call it an iambic pentameter. Most now use the name hypar

instead. This one is at the Broadmoor Hotel in Colorado Springs. It is 185 feet square,

(260 feet across the diagonal) and about 50 feet high. The thickness of most of the

structure is a mere 3 inches except for the thick edge members around the edges and

across the top..

This is only one of many types of shells. You could invent a new type every day for

your lifetime and still not invent them all. Some of the common types are hypars,

translations shells, domes, barrel shells, and, folded plates. If you want to see a

presentation of types go toMark Ketchums home page, to the section called Types

and Forms.

Back to the hypar: Why are we able to span 260 feet and be only 3 inches thick? The

reason is double curvature. Note that from the middle of the side of the roof, to the

other middle point,, this shell sags, and from the center point to the lower corner, the

shell humps. The sag is called a catenary, and the hump is called an arch. A catenary

carries load in tension, and an arch carries loads in compression. We have the

structure carrying loads in two ways. Think about this awhile. The other factor is theedge members. They carry the loads from the edges of the shell to the ground. In this

case they are much thicker and are well reinforced. For the Broadmoor shell they are

prestressed with steel cables.

Most shells have this property of carrying load in several ways. That is except for

barrel shells and folded plates. For example, a barrel shell carries load as a beam from
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support to support and crosswise as an arch. If we want to get technical, the barrel

shell is not a true shell.

This is the end of the tutorial but I hope you do not stop here. Look at pictures of

shells, and try to figure how the forces work. Make models. You will learn how they

are built and how they perform. Only then should you turn to the structural theory.

Just for start. Here is a picture of a barrel shell.

Barrel Shells

The elements of a barrel shell are: (1) The the cylinder, (2) The frame or ties at the

ends, including the columns, and (3) The side elements, which may be a cylindrical

element, a folded plate element, columns, or all combined. for the shell shown in the

sketch, the end frame is solid and the side element is a vertical beam.

A barrel shell carries load longitudinally as a beam and transversally as an arch. The

arch, however, is supported by internal shears, and so may be calculated.

Here is a photograph of anorth lignt barrel shell.
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Folded Plates

The elements of a folded plate structure are similar to those of a barrel shell except

that all elements are planar, and the moments in the slab elements are affected by the

differential movement of the joints.

For the structure shown, the end supports and the side supports are both completewalls

To see a picture of a folded plate clickHERE

Short Shells

The elements of a short shell are the barrel, which is relatively short compared to

radius, the element at the base of the cylinder to pick up the arch loads, and the arches

or rigid frame to pick up the entire ensemble. In this case it is a ridgid frame arch. Thesize of the arch could have been reduced by horizontal ties at the springings. There

may be multiple spans.

The short shell carries loads in two ways: (1) As an arch carrying load to the lower

elements. and (2) As as a curved beam to the arches. The thickness of the shell can be

quite thin due to these properties.
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Here is a photograph of ashort shell

Hyperbolic Paraboloid

This is only one of thousands of possibile shapes for hypars. The surfaces are made by

sliding a line over two other lines that are at varying angles. Consequently this surface

can be constructed with straight boards. They have a slight twist depending on their

width. It makes for inexpensive forming.

The hypar carries load in two directions. In this case, the diagonal element that sags is

in tension, and the other element is an arch and is in compression. These forces must

be picked up by the side ribs and delivered to the supports.

Click here to see one of thepossible types of hyperbolic paraboloids.

Domes

Domes are membrane structures, the internal stresses are tension and compression and

are staticaly determinate if the proper edge conditions are fullfilled. In a dome of

uniform thickness, under its own weight, the ring stresses are compression until the

angle to the vertical is about 57 degrees. If the dome is less than a full hemisphere, a

ring is required at the base of the dome to contain the forces.
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Click onDometo see a photo of a dome

Translation Shells

A translation shell is a dome set on four arches. The shape is different from aspherical dome and is generated by a vertical circle moving on another circle. All

vertical slices have the same radius. It is easier to form than a spherical dome.

The stresses in a translation shell are much like a dome at the top, but at the level of

the arches, tension forces are offset by compression in the arch. However there are

high tension forces in the corner.

WHAT HAPPENED TO SHELLS ?

By Milo S. Ketchum

Concrete shell structures were introduced to this country, in the early 1930's, by the

Roberts and Schafer Company of Chicago, who imported the technology from

Germany. Their development and promotion, was very successful, and a large number

of industrial buildings were built during the war. The real impetus after the war, came

from other sources rather than these proprietary methods. Most notable was the

publicity given to the shells of Felix Candela, a Spanish architect and engineer living

in Mexico. For a decade, starting from 1956, the architectural magazines were full of

examples of structures built by many different designers. Then the construction

seemed to stop and little was heard. The movement had run its course.

There is no single reason that accounts for the demise in the construction of shell

structures, rather, it is a result of many factors. I will attempt to list and explain some

of these and show their effect on the construction of shells.

1. I do not accept that the reason was the exorbitant cost. For industrial andcommercial structures, it is true that unfireproofed steel structures of short span
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are much less expensive, but as the span increases, shells become more and

more competitive. If all the cost factors are considered, they may actually cost

less. For monumental structures, the architectural solution is the dominate

factor, not the structural material.

2. The end of the shellbuilding era, it must be noted, coincided with the VietnamWar and all of the social disruption that occurred. Labor also was havingdifferent attitudes, and construction was affected by the removal of men from

the labor market.

3. Most great movements center around strong and charismatic leaders, and thebuilding of shells is no exception. In this case it was Felix Candela, whose

achievements both in number and quality will never be duplicated. If you do

not believe this then go to your library and find, if you can, "Candela, the Shell

Builder" by Colin Faber. Candela was an architect, engineer, contractor,

entrepreneur, and mathematician. As an engineer, he was not dependent on

others for the architectural design, and as a contractor, he was not hedged in by

a engineer or architect. Also, the building climate in Mexico City was favorable

with low cost labor. Furthermore, he took great pains to publicize his work.

4. Architects were quick to get on the band wagon at the time, and many shellswere designed, often to show the extreme forms available. In my opinion,

however, not many architects really understood the possibilities and the proper

function. Shells, to be effective, must use the interior beauty as an asset rather

than depend on the external appearance.

5. Few engineers, of the period, really understood the structural design of shellsand the possibilities, nor did the have the confidence in their knowledge of

structures.6. Coincident with the development of shell structures, was that of precastprestressed concrete. This was a factory product, readily available to the

engineer, without excessive calculations. There was an eager staff of salesmen

to push the product, with a national society to back them up. No such support

was available to the designers of shell structures.

7. The Portland Cement Association put on a vigorous promotional campaignwith sales engineers of a high caliber available to any engineer. When the

financial situation in the country got tough in the cement industry during the

1960's, this support was withdrawn.

At the present time, shells have a great many advantages going for them that they did

not have in the past. Pumped concrete makes it easier to place at the heights required

for roofs. With the computer, analysis has become much more precise, and

construction estimating is less laborious. Preliminary estimates for the costs of shell

structure are easy to estimate.


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Some time in the future, the cost of structural steel will rise beyond reason, and some

one will discover the utility and beauty of shell structures, will design them, find they

are salable, publicize them, and will start the cycle of popularity again. Things are not

built or done because they are economical, beautiful, or utilitarian. The are built or

done because someone wants to build or do them, and in the process then become

economical or beautiful or utilitarian.

CONSTRUCTION OF CONCRETE SHELLS

By Milo S. Ketchum

The construction of a reinforced concrete shell involves many problems, the designand construction of forms, reinforcement selection and placing, concrete materials and

placing, and curing and decentering. All of these problems must be understood in

order to make the structure safe and economically feasible. More than almost anyother structural system, shells depend upon the ability of the architect and engineer to

foresee the design problems and upon the ingenuity of the contractor to solve the

mechanical problems of construction. To build a satisfactory shell requires a detailedstudy of the methods of construction, well prepared plans, and good supervision.

The normal standards for the construction of concrete structures are, of course,absolutely necessary for shells. Particularly, the production, placing, and curing of

concrete must be under firm control. Only the highest standards should be acceptable

for shell structures. These are outlined in the codes, standards, and publications of theAmerican Concrete Institute.

ECONOMY

By the term economy, we mean the design and construction of the best building at the

least cost. This criterion is not always useful, because it is difficult to define the bestbuilding, especially if there are intangibles that cannot be evaluated in terms of money.

Shells require a minimum of structural materials. The volume of concrete in the roof,

is usually less than the concrete in the floor slab. It is fairly easy to estimate quantities.

For example, for hyperbolic paraboloid (HP) umbrella shells, the average thickness

per square foot of projected area is about 3.5 inches and for a gabled HP shells or

saddle shells, 4.5 inches including the edge members. The weight of steel for aninverted umbrella is about 3 pounds per square foot of horizontal area. For square

gabled hypars or saddle shell it is about 17 pounds per square foot of horizontal area.

These quantities vary little with the span. Other types of shells have similar quantities.

The factor next in importance is the cost of formwork; the contractor must use hisingenuity to devise form systems that can be erected, moved forward and re-erected at


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minimum cost. The next factor is the time to erect forms, set reinforcing, place andcure concrete, dismantle forms, and be ready to the next cycle. The contractor must

have a well-planned organization with good supervision, to achieve acceptable results

and costs.

There are intangible advantages to shell structures that are difficult to estimate, but arenevertheless real. The undersurface is uncluttered, clean, light, and dust free. Other

structural systems may require large additional costs for hung ceilings. There are

certain industries, particularly food handling and processing, where such dust free

surfaces are most desirable. Light interior surfaces may reduce the cost of lighting andbright interiors may be better for the morale of the workers.

It is difficult to put a price on intangibles, especially if they are not absolutelynecessary. An increase in the efficiency of the workers should have a monetary value.

The entire cost of the building, must be considered, and not just the cost of the

structure. For example, the use of a shell structure may raise or lower the cost ofheating, lighting, and finish materials, so the cost of the structure alone is no measure

of the economy. These factors are not always properly evaluated by owners and

architects, and the engineer should point them out.

Comparison with other materials.

For short spans, costs may be greater than other materials; the cost of shells of 50 to

100 foot span for roof systems for industrial or commercial use may be less than the

cost of other structural systems such as steel or timber. There are many factors thatmake this possible. The quantity of materials is almost independent of the span, so the

cost of longer spans may be only a little higher than for short spans. The weight andcost of steel structures. however, increases with the span. For this type of shell

structure, it is important that the forming system be movable and that the constructionbe planned so that steel setting, concrete placing, concrete curing, and moving ahead is

rapid.

Long span shells.

The economy of long span shells depends on many factors. The aesthetics and

functional solution of the architectural problems often have a more important effect on

the selection of the structure than have the costs. The overriding element in the cost, is

the forming system. A large shell will require a single use form unit unless there aremany repeated uses. Without the savings of multiple use forms, the economy may be

lost. On the other hand, alternate materials are also expensive. One cannot make an

out-of- hand judgment that shells are too expensive for a particular design. The only

accurate way to determine the least cost of alternates, is to make separate, complete,designs, and to obtain accurate formal bids from contractors. The costs obtained will

apply only to a particular bidding climate and do not necessarily hold for some other


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bidding situation.

FORMS

The design and construction of forms is a major consideration of costs, and involves asignificant proportion of the total cost. It is important to understand the various typesof forming and their advantages and disadvantages.

Single use forms.

The entire roof is formed at one time, and the forms are not reused. This method is

satisfactory, either for small or large shells, where there is a single structural element.It requires a large amount of forming materials in comparison to the final area of the

shell. An advantage is that it is not necessary to have an elaborate schedule of

sequential forming, reinforcement pacing, concrete placing, curing, and decentering.

All of the forming can be done at one time, then the reinforcing, and so forth. On largeshells, patented steel scaffolding is often used and may be rented. Single use forms

should be considered if there are fewer than, say, 4 units.

Demountable panels.

The form is constructed with panels supported by shores arranged so that the shores

are built into and support the shell directly. Then some of the panels can be taken

down and moved ahead in one or two days without disturbing the shores. A crew canbe kept continuously busy removing and reerecting panels and shores. It is important

to have available more panels and shores than required for each concrete placing

operation, otherwise this system will have no particular advantage in terms of laborefficiency. Some of the built-in shores should remain in place until adequate concretestrength precludes excessive deflection from overloaded young concrete. The

contractor and the engineer should fully agree upon the schedule for removal of panels

and shores. This method is most useful when, say, three or four structural units are tobe built which require some mutual support.

Movable forms - small decentering

For inverted HP umbrella shells, for example, forms for one unit can be constructed in

quadrants so it is necessary to decenter only a few inches. Then the quadrants may be

separated and moved to the next unit to be constructed. Fortunately this type of shell isstructurally self-supporting so it is not necessary to leave shores in place until the next

element is joined to the structure. However, for umbrella shells, the corners may tendto sag, so re-shores should be used at these places. If large numbers of elements are to

be built, as for an industrial building, then it will pay to design these form units so they

can be raised and lowered by hydraulic or mechanical jacks and have wheels to move


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them around.

Movable forms - large decentering.

On shells other than umbrellas, it may be necessary to decenter the forms for a majorportion of the full height of the shell. This method requires a considerable investmentin mechanical equipment such as long hydraulic jacks or the repeated use of short

jacks. The mechanical ingenuity of the contractor is very important for proper design

of this type of forming system.

Precast shells.

Precasting has the advantage that material and construction conditions are under the

best control, forms may be constructed for repeated use, and concrete materials may

be better controlled. The disadvantage is that it is usually necessary to transport the

shell units over a considerable distance if the are built in a precasting yard. If they areprecast on the construction site, then the transportation is easier, but it is necessary to

have large cranes to move and to lift them into place. A structural problem is theconnection of these shells to the supporting columns. Precasting should be considered

only for small units.

Earth.

A number of shells have been built by using earth as a forming material. The surfaceof the earth mound is covered with a suitable contact material such as plywood to

make the under surface of the concrete acceptable. After casting the concrete, the earth

is excavated. Most of the structures built in this manner have been domes. Shells havealso been built without forms by using a close grid of reinforcing bars with theconcrete placed by shotcrete

The form surface.

The most convenient and least expensive material for the surface of the forms isplywood. Loose boards may look better to the architect, but they are considerably

more expensive, and must be replaced more often. Usually the curvature of the surface

is such that plywood can be twisted to the required shape, if not, then four foot by

eight-foot sheets must be cut into two-foot widths. It will be necessary either to trim

the large sheets slightly to obtain double curvature, or provide some method of closingthe gaps between sheets. It is expensive to trim all the sheets, and afterward the are

difficult to use again. Strips of plywood, say four inches wide, placed on the top of theplywood, have been used to cover these gaps. The underside of the shell will show as

rectangular panels if additional strips are placed at the middle of the eight-foot length.

This trick makes an interesting under surface of the shell.


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Top Forms.

Except for very steep slopes and thick walls, top forms are unnecessary. Candela hasbuilt these shells with practically a vertical slope. The thickness was only 1.5 inches,

and there was a grid of reinforcing bars to support the concrete. The shell was virtuallyplastered. It top forms become necessary, the most convenient method is to use wiremesh or metal lath panels that can be removed as soon as the concrete has been placed,

so that the surface can be finished with the rest of the roof.

Form uses.

For movable forms, the cost of forms per square foot is reduced if the forms are used

repeatedly during construction. The optimum number of uses appears to be from 6 to10, depending on the total length for the time of construction, and the curing time

between uses. By this number of uses, the original total cost will be divided by the

number of uses, and the form surface may not have to be rebuilt at an additional cost.

Construction joints.

Small to medium size shells may be placed at one time and construction joints may not

be a problem. On the other hand, a shell may be so large, that is not possible to placeall of the concrete at one operation, and construction joints become necessary. These

joints should be planned and specified by the engineer, and indicated on the

construction drawings, together with the details of the method of stress transfer such as

the inclusion of keyways, special reinforcing, and possible thickening of members. Ifthe shell membrane is thin, usually the stresses are fairly small, and no special

reinforcing is required at construction joints. The most important detail is that thescreed at the edge of the placing area is carefully fitted to the reinforcing, and thatcareful preparation of the surface for the next concrete placing is provided. In edge

members, the usual good construction methods for beams, girders, and columns

should be followed.

REINFORCING

In many cases there is a grid of reinforcing bars that must be held in place. On a

sloping surface, steel setters find it more convenient to place the lower bars vertically.

If this is done, then the slab bolsters will run horizontally. Fresh concrete on slopes

tends to slide downward, and if the bolsters are horizontal, a gap will form in theconcrete below the bolster. On steep slopes, this may be unsightly and be an

unsatisfactory structural condition. The reinforcing should be detailed so the bolsters

are vertical. It is important to provide adequate lapping of the bars, or if thereinforcing is heavily stressed, the bars may be welded. It must be specified and

detailed by the engineer, and may be expensive. With curved surfaces, it is easy to


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underestimate the length of bars.

CONCRETE MATERIALS

The compression stresses in shells are usually quite low, so the concrete strength is notthe most important element in designing the concrete mixture. However, there aresome cases where high early strength concrete becomes necessary in order to move

forms rapidly for maximum production. Placeability and low shrinkage are important.

The use of admixtures that make the concrete more fluid for pumping may cause theconcrete on steep slopes to move downward, so that the engineer in writing the

specifications must consider this problem. For slanting surfaces, the water cement

ratio tends to become adjusted to the optimum value. It there is too much water, the

concrete will run down the slope, and if there is too little, the concrete becomesunplacable. The use of lightweight concrete for shells may save some weight, but this

saving will result in little reduction in stresses, so there will be little reduction in the

quantity of reinforcing. If soil conditions are marginal, the reduced weight on thefootings may save some concrete.

CONCRETE PLACING

The objective of concrete placing is the production of a smooth dense solid texture on

the under surface of the shell with no pockets or honeycombing. Shells are so thin thatthe under surface will not have the advantage of the weight above the surface as in

beams or columns, so extra care must be taken. Placing concrete in shells is hard work

for the placing crew who must work on a sloping surface and often shovel heavyconcrete uphill, so every effort should be made to make the operation simple and

convenient, with an adequate number of workers and finishers. It is important that theconcrete be placed on the form at the place where it is required. Otherwise the placing

will be greatly slowed.

There is a general agreement among engineers that concrete should be placed from thebottom of the shell upward. Then the concrete will not sag downward and cause

pockets. There is considerably more work for the crew to place the concrete from the

bottom up than from the top down, because gravity is a very convenient concrete

mover. Both the contractor and the crew may resist placing from the top down. Thereshould be a thorough understanding, both in the specifications and in the supervision

on this point.

Screeds.

There are several methods to establish the thickness of the shell as the concrete isplaced. One method is to use long screed rails set on blocks on the form, so the

reinforcing can be placed underneath. They may run either horizontally or vertically,

and should be close enough together so that the screed board will rest on the rails. The


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screed rails are removed as soon as the surface is established, and the depressions leftby the boards and blocks are filled up so there are no marks on the under surface.

Another method is to use concrete blocks or short posts nailed to the form and placed

close enough together so the finisher can establish the proper thickness by eye. On a

large production job, mechanical screeds are desirable.

Vibration.

In order to produce a smooth dense texture, the vibration of the concrete must be undercareful control. Shells are thin and vibrations are transmitted only short distance, the

vibrator must cover so every square foot of the surface. If this is not done, an air

pocket or rough texture will result. One method that has been used, is to construct a

rectangular grid of 3/4 inch square wood strips fastened with s single nail or bolt at theintersections. The vibrator should be placed at the center of each square for only a

short time. One touch of the vibrator is required to achieve the desired result. This grid

will fold into a compact unit so it can be moved from spot to spot. The vibratoroperator must be given precise directions.

Placing method

The selection of the type of placing equipment, whether by pumping, by bucket and

mobile crane, by movable wood runways and carts, should be the decision of thecontractor, based on factors such as the equipment available, the steepness of the

slopes, the form and shape of the shell, and the distance above the ground.

CURING

The accepted standards for the curing of concrete apply to the construction of shells.They are thin and do not generate or retain heat, so cold weather protection, both

above and below the surface is essential. It is often possible to build enclosures under

the forms that can be insulated and heated. On relatively small inverted umbrellas,where movable forms are used, the enclosure may be part of the forming system. In

hot weather, the thin surface is susceptible to plastic cracking, so precautions must be

taken.

Time of decentering.

In general, for any concrete structure, the longer the forms remain in place, the betterthe results. Too early removal of forms may not affect the strength, but may have

serious consequences for deflections, especially for thin cantilevers at the corners ofan umbrella shell. However, rapid decentering may be directly involved with the

economy of the project, especially for industrial buildings where large areas are to be

covered. Special precautions must be made, if rapid movement is involved. A decisionmust be made on the minimum cylinder strength allowable in the concrete before


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decentering, so it can be written into the specifications. Then the contractor can makehis decision on the type of concrete and the rapidity of movement of the forms. It may

be possible to move forms on a 24 hour curing schedule if high early strength concrete

is used, and the critical thin shell element are reshored. On large projects, the time of

decentering is sometimes controlled by deflection tests to establish an acceptablemodulus of elasticity at which forms may be moved.

SUMMARY AND CONCLUSIONS

Shells have a great economic potential for the construction of low-cost industrial or

commercial buildings to cover large areas. Long spans are more expensive to build

because they must usually be constructed with a single use form. However, they have

other advantages that may outweigh the initial cost of the structure. The only answerto the economic question is to take bids from contractors on several competitive

systems.

The construction of shells is not difficult, but it requires teamwork and the cooperation

of the contractor and engineer. The latter must design the structure so it is easy to

build, and must show sufficient details so the contractor can construct it economically.The engineer must be aware of the economic factors in shell construction. The

specifications must be carefully written to reflect the problems in thin shell

construction. The contractor should study the plans in order to solve the constructionproblems before they become difficult in the field. He should use a competent

superintendent, one with mechanical ingenuity and perseverance. The engineer should,

during construction, be free to make decisions on plans and specifications that will

expedite the construction without any reduction in quality. The joint efforts of the

contractor and engineer will create a structure bringing pride and a sense ofaccomplishment to all parties.

COSTS OF SHELL STRUCTURES

Much of the material in this discussion was taken from my paper Economic Factors in

Shell Roof Construction in the proceedings of the World Conference on Shell

Structures, San Francisco, California, 1962, the heyday of shell construction. I have notincluded any of the prices quoted and have omitted many of the arguments for the use of

shell structures. I hope that you will find it of value.

There are three factors to consider in discussing costs of shells:

Designing shells to reduce costs and to use them effectively. Comparison with costs of other materials. Accurate estimation of quantities of materials, and subsequent costs.

With respect to designing shells effectively, the word to both architects and engineers is


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Keep it Simple. Stupid. From the point of view of aesthetics, engineering, and

construction, there is little to be gained by elaboration. Shells do not require elaboration

because they are strong forms in themselves. There is a way to promote shells so thatthey will be used more often. The first building in the area should be simple in design

with relatively simple finishes. When the bids come in low, the architect can prove that

shell structures are inexpensive and their economic future is assured.

There are many advantages to shells, but it is difficult to put a price on intangibles.

Shells offer uncluttered, clean, light, dust free surfaces, resulting from the smooth

undersurface of shells that have no girders, beams or trusses. Other structures mayrequire hung ceilings, that can be avoided with a shell roof. Certainly the increase in

efficiency of the workers in a building should have a monetary value.

Comparison of costs with other systems

It should be emphasized that the only accurate method of cost comparison is to make

alternate designs and take bids. All other data is useful but is always in question. Thedifficulty in comparison of shells with other systems of construction, is that the quantity

of materials will vary little with the span while others, increase with the span. Here,however, are some of the results of a study made in the late 1950s, sponsored by the

Ideal Cement Company.

Ratio of cost of shell to steel frame, steel deck, and steel purlins

Barrel Shells, 20 ft. spacing, 60 ft. span 1.25

Barrel Shells, 40 ft. spacing. 60 ft. span 1.10

The obvious conclusion is shells are increasing competitive with longer spans. Another

study was made for umbrella hyperbolic paraboloids. In all cases, the area covered(1600 square feet) was the same. Here are some more statistics developed.

Ratio of Cost of 40 ft. square Hypars Compared to Other Systems

Hypars, 40 x40 ft. 1.00

Timber frame 0.91

Steel frame wood joists 0.78

Steel frame wood purlins 0.77

Steel frame, steel purlins, steel deck 1.04

Steel frame, open web joists 0.90

Prestressed Concrete 1.01

Estimating costs

Surprisingly it is less difficult estimate costs of shell roof buildings than it is toestimate costs of other types. The biggest factor is the unit cost of forms, and here the

ingenuity and experience of the contractor plays an important part. The area of forms


36/46

and the volume of concrete are easy to estimate. The quantity of reinforcing does notvary greatly with the spans and is a relatively small percentage of the total cost.

Following are some of the results of studies on the quantities of reinforcing steel in thefollowing shell structures. These shells were design for 30 pounds per square foot.

Folded plates, spans to 70 ft., span half the width, slope of plate: 4/12 Barrel shells, spans: 50 to 100 ft, radius of shell 25 ft., width: 30 ft. Square inverted umbrella hypars, 30 ft. to 60ft., slope: 4/12 Square hypar dome shells, 40ft. to 100 ft., slope: 4/12

Quantities of Reinforcing Steel for Typical Interior Bays of Shell Roof Structures

Type Thickness, inches Steel, psf

Folded Plate 3.25 1.71-0.007L

Barrel Shells 3.50 1.10-0.004L

Umbrella Hypars 2.25 1.38-0.003L

Dome Hypars 3.50 1.80-0.002L

Guidance forselecting the forming systemis given in the section on construction of

shells, In general, unless the forms can be used at least five or six times, it is better to

stick to single use forms.

Determination of the cost of forms is the most difficult part of the estimation of thecost of shells. Published costs are virtually worthless. Therefore, it is necessary to go

back to fundamentals, to design and price forms for the particular project, and to

collect local costs based on systems you know to be satisfactory for the particular job,

Shells of minimum cost

Shells can compete with other structural systems, but a system for construction at a

minimum cost must be devised. Four conditions must be fulfilled:

There must be minimum quantities of materials, both steel and concrete. The formwork must be inexpensive.

The building must be extremely simple, with no extras for light, heat aesthetics,or expensive finish materials.

There must be a real desire on the part of the contractor to reduce costs.Summary

Shell structures may cost only slightly more than competitive materials, especially ifadditional costs, for example for hung ceilings, are considered. They are not difficult to

estimate. The thickness is based on minimum values, and the reinforcing is only a fractionof the total cost. The most difficult element is the cost of forms. There are many solutionswhich must be carefully studied to arrive at the most desireable solution.
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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 Associationregulations. 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'sTypes and Forms of Shell

Structures

Preliminary Design for Types of Shells

Barrel Shells

Folded Plates
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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.
http://www.ketchum.org/-milo/design.html#umbrellahttp://www.ketchum.org/-milo/design.html#umbrellahttp://www.ketchum.org/-milo/design.html#fghyparshttp://www.ketchum.org/-milo/design.html#fghyparshttp://www.ketchum.org/-milo/design.html#Domeshttp://www.ketchum.org/-milo/design.html#Domeshttp://www.ketchum.org/-milo/design.html#Transhttp://www.ketchum.org/-milo/design.html#Transhttp://www.ketchum.org/-milo/design.html#Transhttp://www.ketchum.org/-milo/design.html#Domeshttp://www.ketchum.org/-milo/design.html#fghyparshttp://www.ketchum.org/-milo/design.html#umbrella


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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 ofthe 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.

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.


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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 thevertical height of the panel.

These loads are transfered directly to the supporting ribs through shear, and are used todesign 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 edgemember. 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 thestrength of these types of joints. Be sure to include adequate reinforcing for any

contingency.


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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, thecompression 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 thepossibility 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.

DOMES OF REVOLUTION


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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 Determine the total weight, P, above a series of horizontal sectionso The total vertical stress,V, at any section will be equal to the vertical force, P.o 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 If the shell is not vertical, or nearly vertical, at the base, then a ring beam will berequired. 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.

o There will be some bending moment at the junction of the shell and the ringbeam, so it is usual to gradually increase the thickness at this point and add

moment reinforcing.

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 everyvertical section is the same. The dome is usually supported by arches. There are three

principal design areas:


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o The central dome area which is designed like a spherical dome.o The corners where there is considerable tension from the ring beam affect.o The arches which take their share of the total load. They are loaded in shear

including the weight of the arches themselves. INTERESTING POSSIBILITIES FOR SHELL STRUCTURES

There are several types of shell structures that offer advantages in terms ofutility and cost and perhaps in aesthetics.

LONG SPAN DOMES Constructions of spherical domes by using inflated membranes as forms has

proven to be a viable solution to many problems. The largest shells constructed

are now less than 300 feet in diameter. Some uses, such as sports facilities

demand greater spans, perhaps as much as 1000 feet. The design and

construction of such a shell is a challenge, and using inflatable forms and a

uniform depth may not be possible. For very large spans, a grid constructedwith pans is necessary so that the dome will be stiff enough and still weigh less

than a uniform depth.

The construction of this structure will require the greatest ingenuity. Onemethod will be to construct the dome by a form on a track that moves around

the inside of the dome, placing concrete at a short horizontal and vertical

section, and then moving around to place the next section. After each circle is

completed, then the rig would be raised, and the ring above placed. The grid

could be formed by using semirigid insulation thus ensuring acoustical

treatment

VAULTS A vault may be defined as a single barrel shell, supported on its side by walls or

columns. The virtue of the vault is that half of the load on the shell is carried by

the walls, and the other half is carried to the ends and at that point the usual

arch and tie are required. The thickness of a shell can be much less that for a

normal arch of the same span because the shell carries loads as a space

structure.

This structural system can be used, for example, for sports facilities with widthsup to 300 feet and lengths to 500 feet. These spans will require a ribbed

structure created by pans or insulating blocks of foam set on the forms. Again

ingenuity will be required for maximum economy. FOLDED PLATE The usual folded plate structure, has been constructed with two or three

element folded plates, with slopes of slightly less than 45 degrees, and tied at

the ends by frames and horizontal ties. A much more interesting system is to

use plates with much less slope and use vertical columns for end support rather

than the usual ties. The spans of the slab elements can be of relatively long, (25


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to 30 feet), if the slabs are haunched. The optimum distance valley to valley

would then be, say, 50 ft., and the span of the folded plates, 50 ft. with a height

of 5 ft. The slope of the slabs is the a ratio of 1 to 5. The concrete would be

much easier to place than The usual steep slope, thus leading to better

economy. In my opinion the shape would be more esthetically interesting,

leading to a de-emphasis of the repetitious nature of some folded platestructures.

FUNICULAR SHELLS A dome of any size or shape can be designed for any plan by using the

differential equation of a bubble. Finite differences are used to solve the

equations. The photo shows a stick model of a tear drop shaped dome. Theconstruction of this model was possible because the dimensions were generated

by the program.

I have written a paper on "Funicular Frame Works" which shows how togenerate and solve the equations. It is available bysending me an Email.

SHELL ARCHES A shell arch has a longitudinal cross section of a barrel shell or a folded plate,

but is a circular arch or other shape in profile. This arch is suitable for

extremely long spans, say to 1000 feet, and is one of the most efficient

structural systems possible. To put it another way, there will be less concrete in

the roof than in the floor system, and the reinforcing will be minimum

The most interesting aspect of the design is the selection of the forming system.The usual method would be to use a curved form moving on a track that wouldform two or three units, and then decenter, move to the next set of shells,

recenter and cast the next unit. The decentering of such a large structure for

would be a real chore and would require expensive manual or hydraulic jacks.

My idea is to form the final curves with air inflated bags, then the bags could

be deflated to move to the next placement, and the massive forms would not

need to be decentered. This type of structure deserves further study.
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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A SIGNIFICANT FAILURE On September 14, 1970, the gymnasium ot The Tucker High School in

Hennison Virginia, a four element hyperbolic paraboloidal roof structure

collapsed. The structure was 155 ft. by 162 ft in plan. The following

photographs were borrowed from Engineering News Record, Sept. 24. 1970.

The collapse was complete, as you see, and there were injuries to the school

children. I was called in by the investigating firm, Wiss, Janney, and Elsner togive my opinion. After tests on materials and further analysis, the conclusion

was that the failure was due to progressive deflection caused by the eccentricity

of the compressive forces in the central ridge members. Three other similar

structures were subsequently torn down. I looked at one of them that had a

deflection of 18 inches at the center.

It was a good ten years before it finally occurred to me what the real designfault was. It shows that when you design shells, all your normal design instincts

flee. The center point should have been cambered upward . We designed a

similar structure in Colorado Springs in 1962 except that it was 185 ft. in plan.

The Owner proposed to hang a heavy curtain from the ridge member, so wecarefully prestressed it as well as all edge members. This structure has had no

problems and is in excellent condition. This comment was received from a reader in December 2007: Correction to Milo Ketchum's web archive... http://www.ketchum.org/-

milo/failure.html


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Tucker High School is in Henrico County, a suburb of Richmond, Virginia, notHennison (which I've never heard of). I was a student at Henrico High School

(from 1962-1968) which was built at the same time and same architecture as

Tucker HS. In fact, it is the Henrico HS gym pictured in the "before" photo.

Construction of the Henrico HS gym was still in progress when I started the 7th

grade, as the contractor was behind schedule. I watched the construction rightoutside my classroom window. Whenever I think back to all the gym classes I

attended there, and all the basketball games with thunderous cheering and

bleacher stomping, I shudder to think how close to death we all were. There

was a third high school, Varina HS, in eastern Henrico County, also built at the

same time and same architecture. After the Tucker collapse, all three gyms

were demolished and rebuilt in a boxy, traditional style.

-Jim Garner (Contact information provided)

Introduction Shell Structures

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