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NBS TECHNICAL NOTE 710-1

BUILDING

RESEARCH TRANSLATION

Account of the Principles

of Modular Coordination:

Industrialization in Building

U.S. PARTMENT

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No. 7/0 -

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NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and

technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measure- ment system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Center for Computer Sciences and

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1 Headquarters and Laboratories at Gaithcrsburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

2 Part of the Center for Radiation Research.

I Located at Boulder, Colorado 80302.
\L BUREAU OF STANDARDS MAR 2 2 1972

UNITED STATES DEPARTMENT OF COMMERCE

F^ &cc ' Peter G. Peterson , Secretary

\j.\ .NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

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NBS

-tTECHNICAL NOTE 710-1

ISSUED MARCH 1972

Nat. Bur. Stand. (U.S.), Tech. Note 710-1, 15pages (Mar, 1972) CODEN: NBTNAE

BUILDING RESEARCH TRANSLATION

Account of the Principles of Modular Coordination:

Industrialization in Building

G. Blacfiere, Director

Centre Scientifique et Technique du Batiment

Paris, France

9(/ fleAU 0*

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402. (Order by SD Catalog No. C 13.46:710-1). Price 25 cents
FOREWORD

The United States /French Cooperative Program on Building

Technology entails an exchange of personnel between the

National Bureau of Standards (Building Research Division)

and the Centre Scientifique et Technique du Batiment

(CSTB) of France. The program also involves the exchange

of information between the two research organizations.

It is felt that some of the documented information can

be usefully shared with the U.S. building industry; and,

therefore, certain papers were selected for reproduction

in media on sale to the public by the Government Printing

Office. It should be understood that the CSTB documents

made public througn such media as this TECHNICAL NOTE

do not necessarily represent the views of the National

Bureau of Standards on either policy or technical levels.

At the same time, building researchers at the National

Bureau of Standards consider it a public service to share

with the U.S. building industry certain insights into

French building technology.

O - y '// f

JAMES R. WRIGHT

Chief, Building Research Division

Institute for Applied Technology

National Bureau of Standards
ACCOUNT OF THE PRINCIPLES

OF MODULAR COORDINATION

Industrialization in Building

by

G. Blachere

This paper is translated from the French original

and is published under the Building Research Division/

Centre Scientifique et Technique du Batiment information

exchange program.

Modular co-ordination is indispensable to the

industrialized production of units for assembly in

buildings. It must be subject to the limitations

which are strictly necessary to achieve this end.

A large module must be chosen, the reference for

the co-ordinating dimensions must be fixed, and the

tolerance problem solved. This is the theme of the

report.

Key words: Conventions; dimensional co-ordination;

industrialized production; modular co-ordination;

tolerance.
Necessary Conditions for Industrialization

1. Industrialization of the production of components and elements

used in the erection of building structures presupposes a more extensive

use of machinery, so that skilled labor is eliminated and all labor is

reduced to a minimum. This means that depreciation will form a greater

proportion of the total cost of the building, and that production runs will have to be greater to make the increased investment costs worthwhile and gain the maximum benefit from the reduction in labor.

2. Mass-produced articles may be thought of being incorporated into

structures designed to be composed of preselected components. These

structures may then be either proprietary model systems or "mail-order

catalog" buildings on the "erector set" principle.

In both cases the components are manufactured with a view to being

assembled together. Assembly entails that the dimensions of the

components and the joint details be coordinated; however, these co- ordination conventions are unique to the firm or group of firms producing

the model building or "erector set" parts.

This does not require an agreement between every designer of a building and every manufacturer. The construction of individual dwellings

in the U.S.A. is a case in point: a large number of contractors use one

and the same technique—loadbearing panels with a timber frame, placed

side by side—and thus have an "erector set" system on a national scale.

3. A further possibility may be contemplated, however. Massproduced units may be desired for use in any structure whatsoever. Such

units could be used either for part of the structure (partially-open

system of construction) or they may be used for the whole of the structure,

which could then be an assembly of catalog elements bought from different

manufacturers, (that is, a fully-open construction).

In either case (partially-or fully-opened construction) , some convention is obviously necessary as to the shape, dimensions and quality

of the elements, so that the designer can take into account what products

are available on the market, and conversely so that the manufacturer can

produce units which will fit into the architect's designs.

It is clear, therefore, that these conventions on quality, shape,

and dimensions of open-system units should be accepted by all, whether

designers of buildings or manufacturers who want to establish a market

for the open system. A market of this sort could cover one region of a given country, or the whole country, or a group of countries, or indeed

the whole world.

4. Conventions on quality may be expressed by means of an agrement

or by a similar procedure.
5. The convention on shape appears principally at the moment as an agreement about jointing. In these partially open systems, in which the

manufactured units are not directly assembled with one another but only

with structures erected in situ and according to specification, no convention on shape is necessary; every manufacturer can specify how

the "made in situ" parts are to be designed to fit the components he

produces.

If, on the other hand, prefabricated components of different origins

or playing different roles in the structure are to be fitted together, a general convention on jointing will be necessary.

Research towards establishing such a general convention is still in

its early stages. Some remarks on this subject are appended to this

note.

Dimensional Convention

6. It could be supposed that the dimensions to be given to the

various components and structures are fixed arbitrarily, attention

being paid only to the following: that there be no ambiguity as to the

values of the dimensions given, the points between which they are to be

measured, and the degree of accuracy they must meet; and the dimensions

of the component and of the structure to be assembled be equal. For

instance, it could be conventionally agreed that the distances between

the axes of the trusses in a curtain-wall structure must be one of the

following dimensions: 93, 147, 168, 252 cm, etc... and that the dimensions

of the curtain-wall panels must be fixed accordingly.

Such a solution is possible in principle. But it clearly has fewer

advantages than a choice of dimensions such that the sum of two accepted

dimensions is itself an accepted dimension. Indeed it is current in

building that a given structure must have one dimension equal to the sum

of two others. For example, a floor-span equal to the sum of several

widths of wall panels or vice-versa, one wall panel equal to the sum of

several partitions, panels, etc.

In other words, the selection of accepted dimensions should have the

property of additivity, that is the sum of two accepted dimensions is an accepted dimension.

7. If integers are being considered, only the product of integers

possesses this property, together with the sizes deduced therefrom by

multiplying every integer by the same factor. Any selection from among

these loses this property.

If the common factor is termed a "module," it will be seen that in

order to have the benefit of additivity, all dimensions which are the

product of the module and any whole number must be accepted.

3
We shall see later how this module is to be selected. First, however,

it should be emphasized that if additivity is to be preserved, there can

be only one module in a given field of production and distribution of

open-system components.

Choice of the Module

8. Since the coordinating dimensions are multiples of the module,

the larger the module, the smaller the number of permitted dimensions

within any given range.

A reduction in variety is often of great advantage to the manufacturer,

but is always a limitation for the architect. The choice of the module

results from a compromise between these two opposing standpoints.

The precise value of the module should, on the other hand, be chosen

with due regard for industrial considerations. If it should happen that

certain sizes are particularly common among semi-finished products,

account should be taken of this fact. This is indeed the case: a width

of one metre twenty (four feet) is very common for plates, fabrics, etc.

It would, therefore, be advantageous to tie the module to this dimension

of 1.20 m.

Furthermore, aesthetic considerations come into the question.

Horizontal coordinating dimensions will often have to agree with dimensions

for height of story and height of room, for which official minimum values

exist in most countries. It is important that the module or its multiples

should harmonize pleasantly with these heights. This is one of the

arguments which led to rejection of lm for the horizontal module, since

the dimension of lm forms less harmonious rectangles with the heights of

2.50 and 2.70 m than those formed with the dimension of 1.2m.

9. The compromise between the architect's wishes and those of the

manufacturers first led to the choice of the dimension of 10 cm for the

module (this is the value now accepted by the I.S.O.).

The choice of such a small dimension for the module results also

from the desire of some people to modulate every dimension, even the

"technical dimensions" such as wall, column or floor thicknesses.

10. The idea of modulating technical dimensions has now been

abandoned. The 10 cm dimension consequently seems too small, or rather

the number of accepted dimensions which would result is too large to enable the fabrication of large one-piece components by means of a dimensional equipment assembly. This is the case put forth for injection

molds and stamping dies. It would be too expensive to have sets of molds

or dies dimensioned at 10 cm by 10.

sThe most common dimensions seem to be 2.50 m room-height and 2.70 m

story-height.
Larger modules have, therefore, been chosen, at least for horizontal

dimensions: 30 cm for dwellings, 60 cm for industrial buildings. In the IMG glossary these are termed multimodules. In fact, they are just

modules, large ones.

In some countries, the question has been raised of having larger

modules, for example 3 or 6m. In practice, however, there are few

manufacturers who would consider, independently of one another, supplying

construction units for industrial buildings with dimensions which are a multiple of 3 or 6m. The truth of the matter is that in these cases, given a modular system by 60 cm, a construction grid of, say, 6 m would

be chosen. On this basis, taken from all the possibilities offered by

modular coordination, a small number of catalog structures or an "erector

set" system has been developed.

11. Vertical dimensioning of components is independent of their

horizontal dimensioning*

Different large modules may, therefore, be chosen for these two directions. No agreement has yet been reached on the vertical direction

where 20 or 30 cm are proposed, at least not for housing structures and

depending on the country. For industrial structures, 60 cm has been

agreed upon for facade components. But, while large building components

offer variable width and length, most of them are either of room height,

i.e. the height from floor to ceiling, or of story height, i.e. the height

from floor to floor. Hence the important thing is to limit the variety

of room and story heights. This can be done without choosing between

the vertical modules of 20, 30 or 60 cm: room and/or story heights are

the vertical module indeed.

No significant limitation of room and story heights has so far been

achieved. Heights of 230, 240, 250, 260, 270, 280, 290 and 300 are all

currently accepted, 230 and 250 are accepted only for room heights and

270 only for story heights.

Restricting the variety of these dimensions is made difficult by the

fact that these heights are laid down in official regulations which are difficult to get changed.

However, these heights do not correspond to either health or safety

requirements. They result merely from a compromise between habit and

economy.

§Except in the case of the dimensions of semi-finished products. A

plate, for instance, may be used with its largest dimension made vertical

or horizontal in two different components.

5
Expressing Modular Dimensions

12. Modular dimensions may be expressed in metric units. The units

currently used in building are, depending on the country, the millimeter, the centimeter or the meter. Many users of modular coordination

use the symbol M to designate the decimeter. Thus, a length of 230 cm would be written 23 M.

The problem remains as to what symbol should be used for large modules.

Symbols such as M , 3, etc., have been proposed. We would recommend the

M

form 3 M. The dimension 120 cm would then be written 4.3M. This mode of

presentation is readily comprehensible even for those who are unfamiliar

with the convention and no confusion is possible. This is not true of

all the other formulas using indices or exponents, since the indices

generally do not indicate a multiplicator but merely an order number.

Thus M would tend to be understood as a third-order module rather than

three times the module, and exponents show the power MJ would be read as M cubed.

Which Dimensions Should Be Modulated?

13. It is necessary to decide which dimensions should be determined

according to the formula n x M.

It is obvious that not every dimension could be used to measure

distance between any two points in a structure. It is quite possible to limit the use of modular dimensions to those dimensions which are essential for the possibility of assembling components to one another.

These are known as "coordinating dimensions."

This coordinating dimension is measured on each of the components to be assembled or rather on the structure to be formed by the components

already erected or the one to be erected. If the same manufactured

product can be assembled or erected following different procedures, it

is likely to have different coordinating dimensions for each mode of

assembly. For instance, a concrete block which can either be laid dry

or with a mortar joint will have two coordinating dimensions, a panel

which could just as well be used either with built up rabbetes, or installed flush, has two coordinating dimensions (see diagram 21).

14. It is necessary that those dimensions which are coordinating ones

in the structure under consideration should be modular; that is, those

dimensions which determine the possibility of assembling components which

are actually going to be connected in the structure being considered and

not in all the possible assemblies in every structure. Thus in a facade

which is inserted between the floors, the floor thickness is not a co- ordinating dimension for the facade since it does not affect the

connections in the facade. On the contrary, it is a coordinating

dimension for the height of the staircase.

6
However, it is possible to vary the incline of a prefabricated

flight of stairs during its erection so that with a given flight of

stairs, not one given height, but any height within a given range, may be

met. Consequently, the thickness of floor is no longer a coordinating

dimension and it becomes unnecessary to modulate it.

We have, therefore, two different reasons for not modulating a dimension. The first is that this dimension plays no part in assembly.

The second is that a possibility of taking up may enable the use of non- modulated components.

15. These possible ways of avoiding modulating certain dimensions

should be carefully looked for and used. They help to avoid having to give a dimension greater than absolutely essential merely for reasons

of modulation, where the increase would entail higher consumption of

materials and hence higher price without any concomitant improvement in

value.

When the length of a slab is increased for modulation reasons, the

dimensions of the room over this slab are increased simultaneously. The

expense is greater, but so is the value for a greater useful area

provided. But if such a slab only needs to be 14 cm thick, increasing

this for modular reasons to 30 cm entails an increase in the volume of

concrete required in the proportion of 7 to 10, and hence an increase in

price without any added advantage to the future user. Increases in these

"technical dimensions," are really inadmissible, and for this reason it

is essential that there be a means of avoiding having to modulate them.

In order to avoid modulating technical dimensions, building systems

should be chosen in which the width and height of a panel are coor- dinating dimensions but not its thickness; the length and breadth of a slab, but not its thickness; the height and length of an internal wall

but not its thickness, the height of a column but not its thickness, etc.

If the plan of a construction is examined in this light with all the

dimensions on it, it will become apparent that a plan includes a very

great number of technical dimensions and some coordinating dimensions

only.

This realistic view is the antithesis of the myth of integral

modular coordination.

16. A further easiness has been found to make modular coordination

more flexible: number-pairs system. It is known that if elements of

length n.M and (n + 1) M are available, every modular length greater

than n (n - 1) M can be achieved by adding these elements, and too, a great number of smaller dimensions.

For instance, with partition panels of 3 and 40 cm, a partition can be made of 30, 40, 60, 70, 80 cm, etc. This gives ease to the user and

it is also very useful for the manufacturer.

7
The only limits to this device are that the system must operate with

an increased number of joints and that the appearance of these numerous

and irregularly spaced joints must be acceptable.

Attaching the Dimensions

17. Restricting the range of choice of dimensions is not enough to ensure that the aim of modular coordination is reached (to enable

assembling components produced independently of one another.) For such

assembly to be possible, there must be clear indications given as to how

these coordinating dimensions, to which conventional values are ascribed,

are to be attached.

18. Since the vital point is to enable assembly, where to attach the

coordinating dimensions is clearly dependent upon the technique of

assembly to be used. If we take a facade panel designed to form the

facade of a building with an already specified structure, the coordinating

dimension, i.e. the dimension affecting the assembly, will differ according

to whether the facade is external to the loadbearing structure or consists

of panels inserted into that structure.

In the former case, the panels will be bound to each other, and the

dimension characterizing this mode of assembly, i.e. the coordinating

dimension, will be the inter-axes span of the structure.

In the second case, where the facade panel is to be connected directly

to parts of the loadbearing structure, the distance between faces of

these structural elements is the coordinating dimension.

The coordinating dimension, whether it be inter-axes or between faces

is hence defined on the existing structure. The component to be erected

will be coordinated and called according to that structure (e.g. "panel

to be inserted into a bay of p. 3M.")

It may be seen that the construction grid idea is not the basis for

attaching the dimensions. The technical features have to be considered

for each case and the dimensioning deduced from them. This will then

lead to the drawing up of tables showing how the dimensions are attached

to structures and components according to the different techniques.

19. If a new element is worked up which is not included in this table,

the inventor may himself lay down how the coordinating dimensions are to be attached. This is until a universal convention is established to settle

this matter, or until a number of manufacturers have started production

of the new type of component.

It may happen that one and the same component may have several

dimensions which are technically significant. A facade panel, for example,

may simultaneously be inserted at its interior face and continuous at its

external face. In this instance, obviously the designer of the panel will

8
have had to make a choice between inter-axes and faces coordination.

One or other of the dimensions has, therefore, been designed to provide

take-up so that the other would then be the coordinating dimension.

If take-up is possible for both dimensions, the element has a dual

use. This is advantageous from the manufacturer's point of view.

20. Similar considerations apply to the vertical direction . Sometimes room height is the governing factor for the assembly of a given component and sometimes it is story height. Then this height will

be the coordinating dimension, thickness remaining a technical dimension.

It is futile to attempt to give one of these dimensions any preeminence over the other. No means of comparison is available presently

which could decide such an issue. Inserted panels, loadbearing internal

wall panels and partitions are bound to room height. Continuous external

flights of stairs, service ducts are connected to story height.

This comparison has no great value, because no open-prefabrication

yet exists and indeed it would be unwise to make a choice before

acquiring experience.

21. The fact that dimensions may be attached differently (between

axes or faces, for example, room heights or story heights) means that

for some components there must be two sets of modular dimensions.

(Only two sets will be needed so long as only inserted or continuous

facades are considered; possibly more if other components in other

technical systems are included.) Hence it has become necessary to have

two floor dimensions, one for floors manufactured to bear on walls

standing at distances between faces equal to P. 3M and another for floors

bearing on walls at P. 3M differences between axes.

This complication is a minor one since it is a simple matter to change the production-length of a floor slab. This costs less than

making all loadbearing walls 30 cm thick. On the other hand, it must

not be thought that one coordinating dimension means only one type of

floor. Floors bearing on walls at P. 3M between axes will vary

according to whether these walls are 12 cm thick, or 20, or 30, especially

in respect of the length given to protruding reinforcements.

22. The situation is rather different in the vertical direction. If

the room height is to be modulated, the flights of stairs will not be,

since their height is equal to the sum of the modular room height and

of the floor thickness which is not modulated. Similarly, if the story

height is modulated, then the partitions are not, and their dimensions

will not be fixed.
This situation is not causing any difficulties at the moment. This is

because a prefabricated flight of stairs can be erected with different

angles of slope, so that a discrepancy of ± 3 cm can easily be taken up. However, a prefabricated partition has a take-up capacity of the same order to facilitate erection and to compensate for inaccuracy in the

dimensioning of the loadbearing structure.

23. Some experiments have been made towards harmonizing the horizontal

dimensions (between axes and faces) and the vertical dimensions (room and

story heights)

.

One simple solution would be to limit the variety of wall and floor

thicknesses by standardization. So far, the evidence does not show that

the restriction thus imposed on technical dimensions has been balanced by

any material advantage. It has further been suggested that sub-modules,

i.e. fractions of the 10 cm module, should be used for this purpose.

Again, the use of these fractions has not yet proved profitable in any

way, especially since the dimensions of these sub-modules are practically

the same as or even smaller than the tolerance values, which effectively

precludes any advantage from the use of these sub-modules for dimensioning.

In fact, at the present stage of development of modular coordination

considerations relating to the use of sub-modules seem to be academic

exercises from the total lack of any practical experience.

Tolerances

24. When a builder himself constructs a house, a number of components

designed and manufactured by him, the problem of productive accuracy of

components, in other words the problem of permissible tolerances, is an internal one to the builder. However, when the contractor for a part of

a structure comes to incorporate new components into an already existing

structure, it is absolutely essential that he should know with what

degree of accuracy the coordinating dimensions are appearing in the

actual structure. There has been no discussion of tolerances until now

because the dimensions under discussion have been theoretical. On the

contrary, all actual manufacturing or constructional dimensions are indeed assigned tolerances.

25. Up to now, it has been usual for many prefabricated components

to be assembled on/or into structural carcasses erected by traditional

methods. Hence, the tolerances of the traditional loadbearing structures

have to be communicated to the manufacturer. These tolerances are large.

A commonly accepted value is ± 2 cm over a length of the order of

magnitude of the dimensions of one room. We should prefer to say that the

true outline of a bay is contained between two rectangles drawn one centimeter inside and one centimeter outside the precise design of the bay?

§For a more general theory of tolerances, see "Note on the Problem of

Tolerances in Building" (G. Blachere) , Cahiers du C.S.T.B. no. 48, vol. 380.

10
To use the language of point deviation? this comes down to saying that

the point deviation over a part of the structure of one room dimension

is 1 cm

.

It is possible to manufacture in advance components to be used in

monumental type buildings (built on more careful lines), but this would

entail a diminution in the potential market. Clearly, in some types of

wholly-prefabricated construction, a much higher degree of accuracy than

the point deviation value of 1 cm can be achieved.

Nevertheless it must be remembered that in large concrete-panel

construction the point deviation actually recorded is not less than 1 cm.

It is, therefore, up to the manufacturer to decide whether he wishes

to construct components to fit traditional structures or structures of

similar accuracy (in which case, he will take into account the tolerance

of 1 cm) or whether to choose to work only on more accurate structures.

This choice will have a vital influence on jointing techniques and on

the provision made for taking-up.

It may be noted in passing that when components are manufactured for

assembly with traditional structures, where i 2 cm may need to be taken

up over a given length, the fact "technical dimensions" are non-modular

seems to be of little significance.

Convention on Jointing

26. As noted earlier, if a prefabricated unit is to be assembled with

a prefabricated or built-to-order component or structure, there is no need to refer to any generally accepted convention on jointing. In such

a case the manufacturer will be told what configuration of joints he is

to provide.

Of course, this does not apply when prefabricated catalog units from

different sources have to be assembled together.

Very little progress has been made in establishing a general

convention on jointing. There exists a convention whereby a component must

stand within its modular space. Actually, it is merely a specialized

form of a general convention on jointing, though it is commonly regarded

as a major principle of coordination. It is a simple and effective

convention, but an extremely limiting one from the technical point of

view. In practice most prefabricated elements do not conform to this

rule. There is, consequently, a need for a true general convention which

will take into account actual jointing technology.

§Cf. French Standard NF P 01.101.

11
The establishment of a general convention on jointing remains, in our view, the most important task for us at present. This should provide

builders, designers as well as manufacturers with an adequate set of

conventions, enabling them to erect structures by assembling industrially

prefabricated components selected from catalogs.
/ — building dimensions

^i L2 = coordinating dimensions

Figure 1

L,= E 4 2o

Joint with Rebate Insertion

Joint with Butt Projections

I = building dimensions

\~i L, = coordinating dimensions Figure 2

* U. S. GOVERNMENT PRINTING OFFICE: 1 972 —i»8l -33l/l 85
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEETS

1. PUBLICATION OR REPORT NO. NBS TN-710-12. Gov't Accession No.3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Account of the Principles of Modular Coordination,

Industrialization in Building

5. Publication Date March 1972

6. Performing Organization Code

7. AUTHOR(S)

G. Blachere

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. Project/Task/Work Unit No. NATIONAL BUREAU OF STANDARDS DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

S ame as No . 9

13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) This paper is translated from the French original and is published under the

Building Research Division/Centre Scientifique et Technique du Batiment information

exchange program.

Modular coordination is indispensable to the industrialized production of units

for assembly in buildings. It must be subject to the limitations which are strictly necessary to achieve this end. A large module must be chosen, the

reference for the coordinating dimensions must be fixed, and the tolerance

problem solved. This is the theme of the report.

17. KEYWORDS (Alphabetical order, separated by semicolons) r ,,„ 4-,-„ j' • -\ j • ' v " Conventions; dimensional coordination;

industrialized production; modular coordination; tolerance.

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21. NO. OF PAGES

15

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