DEPARTMENT OF COMMERCEBUREAU OF STANDARDSGeorge K. Burgess, Director

METHODS OF CALCULATING

HOSIERY SHIPPING CASE

DIMENSIONS

CIRCULAR OF THE BUREAU OF STANDARDS, No. 169

<

DEPARTMENT OF COMMERCEBUREAU OF STANDARDSGeorge K. Burgess, Director

CIRCULAR OF THE BUREAU OF STANDARDS, No. 169

METHODS OF CALCULATING HOSIERY

SHIPPING CASE DIMENSIONS

[August 12, 1924]

PRICE, 10 CENTS

Sold only by the Superintendent of Documents; Government Printing Office

Washington, D. C.

WASHINGTONGOVERNMENT PRINTING OFFICE

1924

METHODS OF CALCULATING HOSIERY SHIPPING CASEDIMENSIONS

ABSTRACT

Methods for calculating the dimensions of the most suitable arrangement of

hosiery boxes which will require a minimum surface area of the shipping case,

using both the proposed standard inside dimensions of hosiery boxes and boxes

of other dimensions, are given. The development of the equations of a minimumsurface of a hexahedron for a given volume is shown in the Appendix to be, when

2kwith sides a, b, and c, where k and c are constants, c— Use is made of this

in selecting the most economical case. The minimum areas are grouped and, in

addition, the most common sizes are listed for the use of the manufacturer of all

styles of hosiery. Considerations for the use of these dimensions are discussed

The saving resulting from the reduction of the surface area in the design of the case

is the feature brought out by this paper.

CONTENTS

I. Introduction

II. Purpose

III. Minimum surface for a definite volume

IV. Standard hosiery box dimensions used as basis for calculations.

V. Method of calculating most economical dimensions

VI. Utility and considerations for the use of these dimensions

VII» Summary and conclusions

VIII. Appendix

Page

1

2

2

2

4

32

33

33

I. INTRODUCTION

The recent adoption by the hosiery association of standard sizes

for hosiery boxes has encouraged similar efforts to establish dimen-

sions for economical hosiery shipping cases. Hosiery after boxing is

sent to the jobber or retail merchant in cases which contain 60, 100,

or 120 dozen pairs to the case. The 60 and 120 dozen cases are in

accordance with established practice, but the 100 dozen case results

from a more recent trade practice. These cases are made of wood,

plywood, or heavy corrugated fiber board, the thickness ranging

from three-eighths to seven-eighths inch for wood. Since the ma-terial used for the cases introduces a substantial item into the operat-

ing expenses of a hosiery mill, any means of reducing this is econom-ically desirable.

1

2 CIRCULAR OF THE BUREAU OF STANDARDS

An excellent basis for consideration of hosiery problems has been

established through the cooperation of the National Association of

Hosiery and Underwear Manufacturers. They have appointed a

research and standardization committee to cooperate and formulate

the problems which are of importance to their industry. In addition,

they have established a fellowship at this bureau to work on hosiery

problems. This cooperation has proven quite valuable in the solu-

tion of this and other hosiery investigations.

II. PURPOSE

The purpose of this investigation was (a) to establish a system of

arranging hosiery boxes in the case which will permit the most eco-

nomical size of packing case; (b) to list the dimensions of this system of

arrangement of boxes in layers for each type of standard hosiery

boxes which will enable the adoption of standard packing cases.

III. MINIMUM SURFACE FOR A DEFINITE VOLUME

Since hosiery is usually packed in quantities of a definite number of

dozens, little consideration need be given the fact that one large

case would have less surface than a number of smaller cases which

would hold the same number of boxes. The relation of the dimen-

sions of the case, however, should be given detailed study.

In considering the area of hexahedron cases (six sided, all sides

forming right angles), the minimum surface for a given volume is

obtained by using the perfect cube shape. 1 The relation of the sides

a,ka, ca, when they are not equal, for obtaining a minimum surface

of a case is, when k and c are constants

2kc~Jc+

1

A full development of this is given in the Appendix.

IV. STANDARD HOSIERY BOX DIMENSIONS USED AS BASISFOR CALCULATIONS

It was decided that the standard hosiery boxes as established bythe Bureau of Standards and the National Association of Hosiery

and Underwear Manufacturers should be used as the basis for all

calculations.2

These standards are:

1 See Appendix.J B. S. Tech. Paper No. 253, Standardization of Hosiery Box Dimensions.

HOSIERY SHIPPING CASE DIMENSIONS 3

Table 1.—Inside dimensions of boxes proposed as standards for ladies’ hosiery

[Accepted by National Association of Hosiery and Underwear Manufacturers]

THREE FOLD

Number Width Length HeightNumberof pairsin box

Numberof folds ofhosiery

Description of hosiery

1

Inches6

Inches

WsInches

X 3 3 All silk hose.2 6 Ws 1 3 3 Boot silk hose.3. 6 9H lVs 6 3 Mercerized and boot silk.

4 6 9% 12 3 Do.

HOSE FOLDED IN ANKLE

5 6 llVs IX 6 2 Sheer mercerized hose.

6 6 n7A m 6 2 Boot silk, mercerized, and light-weightcotton.

7 6 HVs IX 6 2 Medium-weight cotton.8 6 ny8 2X 6 2 Heavy-weight cotton.9 6 llVs 2X 12 2 Mercerized and boot silk.

10 6 nys 3 12 2 Light-weight cotton.11 6 11Vs 3H 12 2 Medium-weight cotton.12 6 nys 43A 12 2 Heavy-weight cotton.

HOSE FOLDED IN GORE OF HEEL

13 . 6 13K IX 6 2 Sheer mercerized hose.14 6 13X ix 6 2 Boot silk, mercerized, and light-weight

cotton.15 6 13X ix 6 2 Medium-weight cotton.16 6 13X 2X 6 2 Heavy-weight cotton.17 6 13X 2X 12 2 Boot silk and mercerized.18 6 13X 3 12 2 Light-weight cotton.19 6 13X 3X 12 2 Medium-weight cotton.20 6 13X m 12

1

2 Heavy-weight cotton.

Table 2.—Inside dimensions of boxes proposed as standards for men’s hosiery

[Accepted by National Association of Hosiery and Underwear Manufacturers]

Number Width Length HeightNumberof pairsin box

Description of hosiery

1

Inches

4X4X

Inches

13M13X13X

InchesIX2y23

6 Silk, mercerized, and light-weight cotton.Do.2 12

3... 12 Heavy mercerized and cotton.

Table 3.—Inside dimensions of boxes for proposed improved method of packingmen’s hosiery

[Accepted by National Association of Hosiery and Underwear Manufacturers]

Number Width Length HeightNumberof pairsin box

Description of hosiery

1__Inches

3Vs3Vs

Inches

&A8H

InchesIX2

6 Silk hose.2. 6 Mercerized and cotton hose.

4 CIRCULAR OF THE BUREAU OF STANDARDS

Table 4.—Inside dimensions of boxes proposed as standards for children's (ribbed

)

hosiery

[Accepted by National Association of Hosiery and Underwear Manufacturers]

BOXES DESIGNED FOR MEDIUM-WEIGHT HOSIERY

Number Size ofhose

Width Length Height Number of pairs to a box and type of box

Inches Inches Inches

1 4 234 5 1342._ 5 334 534 134

34

67

3X334

7X834

m2

3 pair in box; narrow box.

5 8 334 1034 2

6-._ 9 434 im 234

7 4 434 5 1348 5 534 5% 1349..10

67

6347

7348%

1342

6 pairs in box; 2 layers of 3 pairs; wide box.

1U 8 734 1034 2

12 9 834 liX 234

13 4 234 5 33414 5 234 5% 33415.16-.

67

334334

7X834

3344

6 pairs in box; narrow box.

17 8 334 10X 43418 9 434 11X 434

19_ 4 434 5 33420 - 5 534 534 33421 .

22 -, .

67

6347

3344

12 pairs in box; 2 layers of 6 pairs; wide box.

23 8 734 1034 43424 9 834 1134 434

The children’s boxes, because of their wide diversity of sizes, havenot been included in the present paper. However, the plan presented

in this paper may be followed for obtaining the dimensions of the

most economical case for each of these sizes.

V. METHOD OF CALCULATING THE MOST ECONOMICALDIMENSIONS

Since the proposed standards listed in Tables 1 to 4 are inside

dimensions of hosiery boxes, an addition was made to each dimension

to take care of the thickness of the cardboard and the covering

paper. This thickness is a variable, because of the different kinds

of cardboard in use and the difference of the covering paper. In

these calculations the maximum thickness of the boxes available has

been used.

The calculations in Tables 5 to 18, inclusive, have been made as

follows

:

A = width of box + 4 thicknesses of cardboard, etc.; Xthe numberof widths in a layer (0).

B = length of box + 4 thicknesses of cardboard, etc.; Xthe number of

lengths in a layer ([D)

.

C= number of widths of boxes in a layer.

D = number of lengths of boxes in a layer.

HOSIERY SHIPPING CASE DIMENSIONS 5

E=(CxD)= total boxes in a layer.

60 or]

dozen of hosiery in case100 or

120

E = number of complete layers if

boxes contain one dozenhosiery each (see G)

.

G= number of boxes necessary to hold one dozen hosiery.

H= (FxG) = number of complete layers.

60 or 1

/= 100 OT 'r — (3xE) = number of dozens of hose (see G to convert to

120J

boxes) remaining over the complete layers.

J ={IxG) = number of boxes remaining over the complete layers.

_ZT=-g= number of layers necessary to take care of remaining boxes.

L = number of boxes over the extra layer or layers, when division

is not a whole number.

M= extra layer necessary to take care of these extra boxes (see N).

N=(E—L) = number of empty boxes added to make the layer com-plete. These are known as “empties.”

0= height of box+ 2 thicknesses cardboard and covering paper.

P= (H+ K+ M) = total number of layers.

Q = height of box + 2 thicknesses of cardboard and covering paper Xnumber of layers.

D 2(ix5)+2(ix©+2(5x® t 4.^R= .- — = area oi surface of the arrange-

144 &

ment or pile of boxes in square feet.

Sa, Sb, Sc, etc., are the relative ratings of these areas based such that

the lowest area for that type is 100.

Groups I, II, III, IV, etc., in the tables show the combinations

which occur when the width and length are varied to form the layers

of different dimensions. Groups a, b, c, etc., show the calculations

for the various heights of boxes.

Table

5.—

Ladies’

hosiery

(8

folds),

method

of

yacking

60

dozen

to

case

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HOSIERY SHIPPING CASE DIMENSIONS 7

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Table

6.

—

Ladies'

hosiery

(5

folds),

method

of

packing

100

dozen

to

case

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Table

6.—

Ladies’

hosiery

(8folds),

method

of

'packing

100

dozen

to

case

—

Continued

8 CIRCULAR OF THE BUREAU OF STANDARDS

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HOSIERY SHIPPING CASE DIMENSIONS 9

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•

Table

8.—

Ladies

’

hosiery

{2

folds),

folded

in

ankle,

60

dozen

to

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12 CIRCULAR OF THE BUREAU OF STANDARDS

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14 CIRCULAR OF THE BUREAU OF STANDARDS

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HOSIERY SHIPPING CASE DIMENSIONS 15

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16

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CIRCULAR OF THE BUREAU OF STANDARDS

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Table

11.

—

Ladies

’

hosiery

(2

folds)

,

folded

in

gore

of

heel

,

60

dozen

to

case

HOSIERY SHIPPING CASE DIMENSIONS 17

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Table

11.

—

Ladies’

hosiery

(2

folds),

folded

in

gore

of

heel,

60

dozen

to

case

—

Continued

OQ

CIRCULAR OF THE BURE;

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HOSIERY SHIPPING CASE DIMENSIONS 19

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Table

12.

—

Ladies'

hosiery

(2

folds)

,

folded

in

gore

of

heel,

100

dozen

to

case

—

Continued

20 CIRCULAR OF THE BUREAU OF STANDARDS

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HOSIERY SHIPPING CASE DIMENSIONS 21

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Table

13.

—

Ladies’

hosiery

(2

folds),

folded

in

gore

of

heel,

120

dozen

to

case

—

Continued

22 CIRCULAR OF THE BUREAU OF STANDARDS

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HOSIERY SHIPPING CASE DIMENSIONS 23

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S - M «

Table

14.

—

Men's

hosiery,

60

dozen

to

case

24 CIRCULAR OF THE BUREAU OF STANDARDS

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; ;:

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3 5J BOW BO

Table

15

.

—

Men's

hosiery,

100

dozen

to

case

HOSIERY SHIPPING CASE DIMENSIONS 25

OQ

) « CO i iO i i i O • ICO t i <N i i O i ' Oii, ii, ii. ii. ii. ii. ii. i i •

i i io i ioo i ioo • i o i i zo i i^ i1

1

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1

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l 1 i—( 1 i rH iit—

1

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&

1 l l l 1 1 1 1 1 i 1 l 1 1 i i

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101 i co i irt^i iC5« iCOi i i itJHi 1 od 1

l(Nl 1 O 1 It—II 1 o 1 rH 1 lOl 1 rH 1 'O'1 rH I I rH I IH > IH 1 i tH 1 1 rH I IH 1 1 rH i

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 l 1 l 1 l l i 1 l i i i i i 1

Cj

GQ121.8 103.5 115.3 100.0 113.8 101.

2111.9 104.2

ft?31.8 30.

836.

1

27.0 27.

1

31.1 30.

1

29.

234.

12&

125.5 28.

829.

728.9 33.5 26.4 26.

730.0 29.2 29.

233.

727.2 27.7 30.

8

o>67.

100

64.

800

77.

300

33.

550

33.

696

40.

196

53.

680

51.

840

61.

840

26.840

25.

920

30.

920

45.

628

44.

064

52.

564

22.814 23.328

27.

828

38.

918

38.

880

46.

380

20.

130

20.

736

24.

736

fti

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342

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342

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592

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092

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342

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592

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0921.342 2.592

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092

1.

342

2.

592

3.

092

1.

342

2.

592

3.

092

1.

342

2.

592

3.

092

te;

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ft?

13.

434

13.

434

13.

434

26.

868

26.

868

26.

868

13.

434

13.

434

13.

434

26.

868

26.

868

26.

868

13.

434

13.

434

13.

434

26.

868

26.

868

26.

868

13.

434

13.

434

13.

434

26.

868

26.

868

26.

868

17.

236

17.

236

17.

236

17.

236

17.

236

17.

236

21.

545

21.

545

21.

545

21.

545

21.

545

21.

545

25.854

25.

854

25.854

25.

854

25.854

25.

854

30.

163

30.

163

30.

163

30.

163

30.

163

30.

163

ill l| i ill til lit III i,i l,iIII l|

i III lii ill ill i,i i| iill i,i III III lii ill lii liiill i

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v-

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VI VIIVIII

Table

16

.

—

Men’s

hosiery,

120

dozen

to

case

OQ

CIRCULAR OF THE BUREAU OF STANDARDS

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Sb

1 © 1 • © • 1© • '© 1 ' 1 '© ' i © 1 1 00 1 1© ' 1© •

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l rH l i rH 1 1 rH l I rH l 1 rH 1 1 rH 1 IH 1 1 rH 1 • rH • • *"H 1

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OQ127.9

106.1

120.4 101.4 116.4 100.0 116.4 103.8 113.3 102.

4

ft?37.6 36.3 42.7 31.1 30.3 34.8 35.3 34.2 40.

1

29.7 28.9 33.0 34.

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138.6 29.3 28.6 32.3 34.1 33.9 39.3 30.4 29.7 33.3 33.2 32.3 37.3 30.0 30.5 34.0

i

o>80.

520

77.

760

92.

760

40.

260

38.

880

46.

380

64.

416

62.

208

74.

208

32.

208

31.

104

37.

104

53.

680

51.

840

61.

840

26.

840

25.

920

30.

920

46.

970

46.

656

55.

656

24.

156

23.

328

27.

828

40.

260

38.

880

46.

380

20.

130

20.

736

24.

736

fts

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3422.592 3.092 1.243 2.592 3.092

1.

342

2.

692

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0921.342 2.592

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0921.342

2.

592

3.

092

1.

3422.592 3.092 1.342

2.

5923.092 1.342

2.

5923.092 1.342

2.

592

3.

0921.342 2.592

3.

092

tei

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434

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26.

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13.

434

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434

13.

434

26.

868

26.

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26.

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13.

434

13.

434

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434

26.

868

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13.

434

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434

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236

17.

236

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25.

854

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30.

163

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34.

472

34.

472

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i'i lil ill III III ill li‘. i,i lit ii

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1 III 1 | lIII 1 | 1 III III III III lit 1 |1 III 1 | IIII 1 | 1 III III III III l|l 1 | 1 III 1 | 1ll 1 | 1 III l|l III III l|l l|l lit 1 | 1III 1 | 1 III l|l III III 1 | 1 1 | l l|i l

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tig

HOSIERY SHIPPING CASE DIMENSIONS 27

loo '© ' CO l CO lO 100 IH ICS lO l »C l »0 lO 100 1 © iTf 105 IH 105 I CS ICS i co

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*1l 1 1 1 1 1 1 1 1 1 1 1 1 ( 1 1 l 1

1 1 1 1 1 1 I 1 1 1 1 1 1 1 l i 1 l 1 1 1

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l l 1 1 1 1 i 1 1 1 l l 1 1 1 1 1 i | i i

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tq 338S222SSS33 S3°®^^22 wco tD(0SSSS'0<0^

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fc. 8SS 2 '°'Dl0 ‘0 3 S t^ ^-ocoeocscscoco^ coeooo-orocococscs oo co^ cs cs^ co co

fqcocococoooosos^ooco SS0 ,0SS3 a;SS!i

<3

rH.-HOSCSCOCO'tf-^T-l.-ICSCS COeO-tf'^T-C.-HCSCSCOCO tf^rHT-CCSCSCOCO'<t<'<*l i-HrHCSCSC0COr-<i-(CSCS

. o10*0©©©©©©©© l> 1> t>. t>. 1> CO 00 00 00

cq iiiiiissISi! lllgilllll gllllllllg Illllllill""aass^s^^aa ssss"“*aa8s ss““sassss ““aasa^aa

aaaaaaaallii lliliallal 8832333333 2§§sasg£Kg

asaasadd^ss ssssasaass sasasassss aaaaaa^ssssllllllllllll Illllllill illllllill Illllllillllllllllllll Illllllill illllllill Illllllillllllllllllll Illllllill illllllill Illllllillllllllllllll Illllllill Illllllill illllllillllllllllllll Illllllill Illllllill Illllllillllllllllllll Illllllill illllllill Illllllillllllllllllll Illllllill Illllllill Illllllillllllllllllll Illllllill Illllllill 1 1 J

* * * * 1 * l

e.c5!3.o!3.oe.oeioero e« e.o esa cj« e.o_a.©„e.o. so t3*B« ea.e*

i : :

J ri 0 fc P P P a

:

d B B B £ B> > G M MX M X X X

Table

18

.

—

Men's

hosiery,

improved

method

of

packing,

100

dozen

to

case

28 CIRCULAR OF THE BUREAU OF STANDARDS

ICO 1 O ICO i CO i CQ >03

is ig id ig ig ig

lie ICO 1 CT> |»0 1 ois ig ig is is

i o i o i ico ieo

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ig is is ig ig

GQ

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co io !« !

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r-4 IH * r-i IH • rH IH I

1 1 l 1 1 1

l l l i i i

O ' O i »o i O i iO i

s is ig is is i

i i i i i

O i iO i i iO i iO i

s ig ig is id i

ie lie i io i o i io i

g ig is ig ig i

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o>llgglssSIlll sSSlIIllss llSISSilll 1111111111sssss^gssssgss sa^ssssaaa oisssssasSa

SSSSSSSSSSSS 2SSSS522 0505 S8222SSS22

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fei

HH^^^N^tJIOOOO Tji^XOOOOOOOO OO ->*1 Hf rtf CO CD CO CO COCOOOOOOOoOOO

>i

HHHHHHHHOOOO rH rH rH rH O O ©O rH rH OOhhhhhhhh HHHHHHOOHH

NNNNWNOOXOOOO WOOOOOOOOOOU5U5 OO<NC<I00 00<NIM00 00 ^ Tt< <# HH O O 00 00

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tuSSSSS^SSSSSS SS22S3SS22 2SSSSS2S 0000

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fcl

cocococooocc^oooo «g2o^ooSS oocDcDCCoooo^g; nnh;^hh»ooSS

Cl

HHCC«M^in<HHCC MW^TjlHHCCMM TX-KtHHCCCOCO^Hf rH rH C C CO CO rH rH C C

o eo eo eo co eo co co eo Tt< Tt< rt< rt< rf< ^ >c *o *o *o o *o *o *o co co co co co co co co i> r- «> i> tr oo oo oo oo

«liiiiiiggiiii iiggiiiin ggmiiigg mining“i '<is£:aas5s““£;s; aa^s”“6saa ss^^^saaa^

EsBEEEsSgggg gggggiiggg 18111*1111 3§g§22§KRgasadasssssss ssssaaaaaa aasssssaaa aaaaaasssss^

• i i I i i i i i i i i i i i i i i i i i i i i i i i i i i i i S i i ! ! i ! « !

i i i t i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i iiii,iiiiiiii i i i i i i i i i i i i i i i * i * i i i i i i i i i i i i

i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i

i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ie^ero^ro^rc^ee

n

i

ni

ii

vi VIIVIII “ XI

1

xn

1

XIIIXIV xv

-

XVIXVII

XVIII

XIX XXXXI

Table

19.

—

Men's

hosiery

,

improved

method

of

packing,

120

dozen

to

case

HOSIERY SHIPPING CASE DIMENSIONS 29

ja

GQ

1 1 O i 03 1 »C 1 03 IH IH i 1

1

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:s ;2 :s :a :S :S :3 :s :s :S :S :S :s :s :3 :s :s .’2 :si i i i i i i i i i i i i i i i i i i i i

i i i i i i i i t i i i i i i i i i i i i

dGQ

i i t i i i i i i i i i' i i i i i i i i© 1 © iH i C5 i i CO i t>- iiO CC i O 1 O i C i O 1 GO i © i CO i CO i i i O iN i

id • co id * oo ioo 1 c4 i^* id ico id ! co i c4 - i co id i c4 id id id id i

iO i i i r—i i i—i i 1 0 i © 1 O iCO i O 1 O i O ICO »o i O i O i CO 1 0 1 0 id i o i

r—1 IH IH IH It—i I t—I 1 t-H i H IH i H iH i H iH « t—i iH ii—1 1 r—1 iH i H iH IH l

l 1 1 1 1 1 1 l 1 1 l 1 1 1 I l 1 l 1 l l

NHiOOOONiOCONCOOlO NiOHOdOOCOHoO OQOCOOiOdOM»ON CO 00 CO O CO H iO « N iO

Orggllgllillll iillliSslI 1111111111 IlssIlilsSagsa^sagsgs's aasssgsssts ussssssssas sgasasssa^

223338:2322 ££22228822

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-

i ei c<i

te!

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>!

OOOO^tHOOOOOO OOOOOOOOOO OOOOOO^^OO ^HT-HT-HTH^HrHOOOO

MOOOOOCOOOOOOO OOOOOOOOOO OOOOOOOOOO NNNNOJOOOOO

NOOOOOOOOOOOO o O r-1 r-l o O O O O O OOOOOOrHr-HOO OO^tHt-ItHOOtHtH

s 00000*000000 00*0000000 000000^^00 ccooooooo*

-OOOOCOCOOOOCOO OOOOOOOOOOOO OOOOOONNOO HHCOOO>0|OOOOOCO

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C5

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3 222*° lo2222 88'° 5e222288 t~ t~3288 Q00022

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aEBEBBssasga lillissiSI ggSlIIHII =§333§£§Ssaaaaaaaasisss sssssasssis ssddaasaaa

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1 1 1 1 1

ii

i i 11

l I 1 i 1 l I1

1' 1 i 1 1 l i

i i Ii I l I i l

|I | I l I ' I

I

1 | 1 | l ' 1 i 1 | 1 | 1 1 11

1 | 1 1 I 1 1 1 1 1 1 1 1 1 1 | 1 |1

11

11

11 1

i : i : ! i

:i

! : : : i i i : ! : ! i ! i i i i i i i i i i : !: i !!!!!! !

-o. ee -o 0 .00 -o_o -o. 0 -o__o e -0.

s-c.eo.e-o eo.e-o. 0 -0.

0

-o_o -o_o -o.es -0.

^

1

»

1

111 IV

-

V

<

VI

1

VIIVIII

IX x XI xnXIII

XIV XVXVI..

XVIIXVIII

XIX XXXXI

Table

20

.

—

Summary

of

minimum

areas.

Area

of

the

surface

and

the

dimensions

of

the

arrangement

of

boxes

in

layers

which

will

'permit

a

minimum

surface

area.

Arrangement

dimensions

given

in

nearest

one-eighth

of

an

inch.

For

more

accurate

figures

,

see

tables

30 CIRCULAR OF THE BUREAU OF STANDARDS

120

dozens

per

case

Dimensions

of

arrange-

ment

of

boxes

in

layers

(nearest

one-eighth

inch) o* MHNM* hNh(mh|mh|mlOHO CUOOJN

CO CO CO CO M M CO

Hi®®*

aaMh^COhK HHM«HleM<'XH|«H|« wH< M« H» H®OO to MHO^NHNHOOH iO COCOH CO H(NCO 00 CO CN CO CO CO fO <N CO CO l?5 (M CO OH CH

eqOOM

>o|aow)|aoio|<» H|0OHl«OHtcOH|cO

00 00 00 o CO COMMM M CO CO COCOM CO

H|cOf-t|oo M«H®hH h|-#H®t-|«H®hNH®hN H»H®H® h|-#m*CO <0 CDCDOOCDCOCDOCOO COCOO COCOCO MM^^WMW^M^ MMM M CO

CO CO CO CO CO CO CO CO co coHH HNHHHH D?t-H L? rH XO xO XT? cTt*?COCO CO CO CO CO CO CO CO CO CO CO MMM MM

Arrange-

ment

32 32 16 8 24 16 16 16 M 00'

0000 *0<0<OMM O 00 O 00 OOO COOM M rH rH rH r-H r—< rH M rH rH HH

C) CO COCOCO M CO CO CO M CO COCO MMCOCOMMMCOMCO MMM CO ^

Ci xo iO lO lO xo xo xo xo xO XO XQ XO XOOXOXO<OXO<OXO<OXO CO CO CO xO CO

Area Sq.ft.

34.6 41.2 38.4 37.8 35.0 39.3 43.0 50.5 34.

138.6

COOO t-H xO^ xO CO CO l>- l>- CO CO CO Hb.MOO NNCO*<OHHxOxO(N 05 CO <N <N 05^ rH CO rH ^ xo CO rH rH rH rH »0 M M CO MM

100

dozens

per

case

Dimensions

of

arrange-

ment

of

boxes

in

layers

(nearest

one-eighth

inch) O*

«WHO|0DH|eqrfcqH« H®H®H«H®«j*(NOOacCO COHCDOiClOH<N<M<MCO<N <M CO CO CO CO <M CO

COr? COH<O<O<OHCO rH M CO CO ^ M CO

COCO

H» M« N®M®rH COCOH OQHrH M M CO MM

oqOO OOOO oooo<M <M M <M <M <M<M<MCOCO<M<M aa o’M

H® M®N«H® wlaowHiCO CO CO CO t>- rHM MMM rH CO

«H< mH< nHihhh^hhhCO M CO M CO CO CO CO <M CO CO CO COCO CO CO CO CO CO CO CO

H^HiCSr^N «l««|C0

CO MMM ^ M

Arrange-

ment

Ah OOON^OO<MCO<MH <M<M<MHHHH OO ooooooHH M M M M rH rH

o s sss ®s

0 COCOCOCOCO <MC<IMCOCO<MM MM MMMMMM M M MMM M*

Ci XO iO h* xo iO xo xo xO xo xO xo xO »0 xO xO xO xo xo xO xo xo

Area Ah Sq.ft.

31.

1

31.1 36.7 34.734.6

:30.9 34.7 38.5 46.

1

46.

1

30.2 34.0 37.8 44.5 33.1 37.

141.

149.

1

32.3 36.3

CO

©rH

rH rH XO 00 O XQ

60

dozens

per

case

Dimensions

of

arrange-

ment

of

boxes

in

layers

(nearest

one-eighth

inch) 0»

hNM

H«io|oohHi h|«H«»o|coh|oo

ass £££8HlMHlemW r+N

CO 1-t 00 H to O CO t'- CO CO GOCHWrtNCH NNNCHNH M«H« iHlWHiMHkS Hhd«|0Oco O xo 00 I> *4M M rH rH rH rH

sq05

M»lQ0W5lX>O|Q0 fH|«Hl«Hlfl0pH|00

00 00 GO Tt* ^M M M M M M Mi-t|cOHl®H|®H|®H|® *-|®H®H®H®H®H®-H rfl rf rH tH CO CO CO CO CD COMMMMM MMMMMM 8

H® H®H«H® eo|oocH®

CO CO CO co u- t>-M MMM rH rH

coi*hHMMM MMMCO oooohhh -^**hoohH i—l CO CO CO M M M CO H CO CO

HHiHcMW r<Hi«|cO

rH NHH O rhCO rH M M MM

Arrange-

ment

Ah 30 00>0 xo VO xo M O O CO CO CO lOWlOINO® co CO xo CO CO MO

cq M COCOCO M M M M MMMMM MMMMMM M M MMM MM

o Hjl Tt< -rt< Tt< CO CO xO xO xO ^ XO CO XQ XO xo tJ< xo xo xo CO

Area A? Sq.ft.21.9 26.

1

24.4 24.

1

21.9 24.5 27.0 31.8

0X0X0 00 i ocoooco*a a a c§ i a si a §§

28.8

|33.0 18.8 18.

520.5 13.9 18.5

Heightof boxes O“0 <M o? CHI r- <M CHNlMCHNMlNNCHN NNNNNCHNNNt- <N (M (N

-tt< -tt* 05 CO 05 -cfl T}< 03 05 05 05 CO 05 r(H 05 05 05 05 05 CO -cH 05 05 CO 00>§000000500 CO 50 00 CO CO >000 50 CO‘OOOW>OOOiOiO'f C0*00 rH

jS © ’hhm n H H oi <n' oi eo co coH H H H oi <m' co eo eo co H H c<i co H ei

Type

Ladies’

hosiery,

three-fold

Ladies’

hosiery,

two-fold,

folded

in

ankle

Ladies’

hosiery,

two-fold,

folded

in

gore

of

heel.

Men’s

hosiery

Men’s

hosiery,

improved

method

of

packing.

Table

21

.

—

Sizes

for

use

when

manufacturer

ships

all

styles

HOSIERY SHIPPING CASE DIMENSIONS 31

32 CIRCULAR OF THE BUREAU OF STANDARDS

VI. UTILITY AND CONSIDERATION FOR THE USE OF THESEDIMENSIONS

These calculations have been based on the standard hoisery boxdimensions. It must be remembered that the calculated dimensions

given are not the inside dimensions of the packing case, but are the

outside dimensions of the assembly of empty hoisery boxes arranged

as indicated.

In the summary of minimum areas, Table 20, there have been col-

lected all the arrangements which resulted in a minimum surface area.

Table 20 will be useful to any manufacturer who ships only a few of

these styles, for be can select from the list the arrangement resulting

in the greatest economy.

In Table 21 there has been prepared a list of the dimensions of the

arrangements of boxes for the use of manufacturers shipping all styles.

These dimensions were obtained by selecting the length and width

dimensions which would permit the smallest area when all styles of

hosiery of a certain type were considered. To use these dimensions

it is necessary in a few instances to use a case slightly larger than the

minimum given in Table 20, but it is thought that the saving resulting

from the cutting of lumber or the manufacture of these boxes on a

large production basis will warrant this.

Since there is a tendency among the more progressive manufacturers

toward the use of the standard hosiery boxes exclusively, the measure-

ments given herein will serve as a means of supplying them with the

dimensions for the most efficient hosiery case.

There is a practice among packers of hosiery to make a slight addi-

tional allowance in the width and height of the hosiery case, so that

more rapid packing may be obtained. This should not he carried to

the point where the boxes are packed loosely in the case, for it is this

constant shifting of the boxes which causes the greatest amount of

breakage.

When the standard hosiery box dimensions were investigated, it

was decided not only to make the hosiery fit snugly in the length and

width, but to make the height slightly less, so that a slight compres-

sion would result. The reason for this was that the hosiery wouldact as a support for the box and, in addition, insure the presentation

to view of unrumpled hosiery when the box was opened.

This compression feature will be nullified if the case is made with

too large a tolerance in the height. It may be well to sacrifice a slight

loss in time of packing to obtain better conditions of the finished prod-

uct at the receiving end. However, the necessity for some slight

tolerance is obvious to take care of variations in the thickness of the

HOSIERY SHIPPING CASE DIMENSIONS 33

cardboard and covering paper and variations in the dimension of the

box. It should be remembered that the thickness of box and cover

paper which was added in these calculations was the maximumthickness of available boxes.

The actual tolerance for any type is a matter of individual experi-

ence and depends somewhat on the method of placing the boxes for

packing. This can be taken care of individually by each manu-facturer.

VII. SUMMARY AND CONCLUSIONS

There has been presented here a method of calculating the mosteconomical case for packing hosiery boxes, and by using this methodthe most desirable arrangement for each type of hosiery has been

selected. These calculations have been based on the standard

hosiery box dimensions, but a suggestion for obtaining the dimensions

of a case when other than the standard boxes are used is given.

The most desirable feature which recommends the use of these

standard cases is the reduction of waste material in the construction

of the case. The case with the smallest surface is most economical

where other packing considerations are similar. Considered from the

standpoint of a single case, the saving may be small, but the large

quantity of hosiery shipped in cases makes this saving of primary

importance.

The adoption of the suggestion regarding the use of the dimensions

in Table 21 by manufacturers of all styles of hosiery will promote

better and more efficient packing and shipping systems by those manu-facturers, since the lengths and widths can be readily determined bythis method, large quantities of this lumber can be made up and

stored for convenient use. This plan might also be followed by the

manufacturers of one or two styles.

The adoption of uniform sizes of cases should be welcomed by the

jobber, for he could then be better enabled to plan his storage space

to the best advantage. Manufacturers of special types of cases

would benefit also by the general adoption of these cases, for not

only would the cost per case be reduced by any reduction in the

surface area, but the parts could be made up on a large quantity

production basis for the market would be certain.

VIII. APPENDIX

Ratios of the sides of a hexahedron to give minimum surface for a given volume:

Let the sides of the hexahedron be a, ka, ca; k and c being ratios and a the

largest side.

34 CIRCULAR OF THE BUREAU OF STANDARDS

Then the volume (7) and surface (S) are, respectively,

V=a?Jcc

S — 2a? (Jc-\-c-\-kc)

By substituting for a from (1) into (2) there is obtained:

k+c\-kcS =2V2

/3

(kc) 2/z

(ir( 1+c-2l)’ ff

=2'3 (e)!/s

(1+i -2i)

for V fixed, S is a minimum at

8S5k

( 1 )

(2)

(3)

(4)

SS=8S=8k 8c

0

The zero values of the partial derivatives determine, respectively (fc, c, ^o),

2ck= 1+c

2k

1+k

(50

(52)

2kHence, for. V and k fixed, minimum S determines which gives by sub-

stituting in (2)

S=6ka? (6)

For V above fixed, minimum S determined by simultaneous solution of (5f)

and (52)

k ~c = 1 (7)

Therefore, the cube is the hexahedron of minimum surface.

In order to make use of values for a, ka, and ca, different values have been as-

signed to k, and the ratios of the sides have been found which will give minimumsurfaces for the conditions when one side is equal to the other multiplied by a

constant. These ratios are given in Table 1.

Table 1.—The ratio of the sides for different values of k, whenk--2c

1+c

Condition Ratios

1__ ; a:Tia:ca— 1:1: 1= 30:30:302. a:ka:ca= 2:l:4/3= 60:30:40

a:ka:ca= 3:l:3/2= 90:30:45a:ka:ca= 4:1:8/5=120:30:48a:ka:ca= 5:1:5/3=150:30:50a:ka:ca= l/2:l:2/3= 15:30:20

34.56

In order to show what the effect would be if this side were increased or de-

creased from the value which provides a minimum surface condition, this wascalculated in terms of surface per unit volume, the results of which are given in

Tables 2 to 7. These results are plotted in Figure 1. The less the surface perunit of volume the more economical the box. The value of ca has been taken in

the tables such that the curves for all cases would be comparative. The curveholds for any multiple or division of the values for ca, since the surface per unit

of volume is a unit quantity.

HOSIERY SHIPPING CASE DIMENSIONS 35

Eig.

1—

The

relation

of

surface

per

unit

volume

to

value

of

side

c

36 CIRCULAR OF THE BUREAU OF STANDARDS

Table 2.—Condition 1,

varying ca when k—1 the ratio a:ka:ca—l :1 :l—30:30:30=minimum

ca F FV3 F2/3 S S

1

F2/8

4, 500 16.51 272.3 2, 400 8. 809, 000 20.80 432. 6 3, 000 6. 94

13, 500 23. 81 566.4 3, 600 6. 3618, 000 26.21 686.4 4, 200 6. 14

25 22,500 28.24 795.2 4, 800 6. 0427,000 30.00 900.0 5, 400 6. 0031, 500 31. 60 998. 6 6, 000 6. 0236,000 33. 02 1, 089. 0 6,600 6. 07

40,500 34. 38 1, 183. 4 7, 200 6.08

50 - - 45, 000 35.57 1, 267. 4 7, 800 6. 16

55 - - 49, 500 36. 78 1, 354. 2 8, 400 6. 2060... - 54, 000 37. 80 1, 428. 8 9,000 6.31

S/ F2/3= surface per unit of volume.

Table 3.—Condition 2, varying ca when k=2 the ratio a:ka:ca=6:3:4=60:30:40

ca F Fl/3 F2/3 S s/vy8

5... 9,000 20.80 432.6 4, 500 10.410„_ 18, 000 26. 20 686.4 5, 400 7. 8615 - 27, 000 30.00 900.0 6, 300 7.0020... 36,000 33. 02 1,089 7,200 6.61

25 — 45,000 35.57 1, 267 8, 100 6. 4130- 54,000 37. 80 1,429 9, 000 6. 3035 63, 000 39.79 1, 584 9, 900 6. 2540 72,000 41. 60 1, 731 10,800 6. 23

45... 81,000 43. 27 1, 875 11, 700 6.2450. _ - 90, 000 44.81 2,007 12, 600 6. 2655 - 99,000 46. 26 2, 144 13, 500 6. 2860 108,000 47. 62 2, 266 14, 400 6.34

S/ F2/3=surface per unit of volume.

Table 4.—Condition 3, varying ca when k—S the ratio a:ka:ca=6:2:3=90:30:45

ca F Fi/3 F2/3 S s/vy8

5 13, 500 23. 83 566.4 6,600 11.710 27,000 30.00 900 7, 800 8. 6715-.- 40,500 34.39 1, 183 9, 000 7. 6020 54,000 37.80 1,429 10, 200 7. 1525 67, 500 40.80 1, 664 11, 400 6. 86

30- 81, 000 43. 27 1, 875 12,600 6. 7335-. 94, 500 45. 58 2,079 13, 800 6. 6540- 108, 000 47. 62 2, 266 16,000 6. 6245 121, 500 49. 58 2,460 16, 200 6. 5550.. 135, 000 51.30 2,632 17, 400 6. 62

55 148, 500 53. 00 2, 809 18,600 6.6460— 162, 000 54. 51 2, 970 19,800 6. 6765 175, 500 56.00 3, 136 21, 000 6.7070- 189,000 57.39 3,283 22,200 6.76

S/ F2/3= Surface per unit of volume.

HOSIERY SHIPPING CASE DIMENSIONS 37

Table 5.—Condition 4, varying ca when k—4 the ratio a:ka:ca=20:5:8=120:80:48

ca F FV3 F*/3 S si F2/3

5 18,000 26.20 686 8,700 12.710— - 36,000 33.02 1,089 10,200 9.3815... 54,000 37.80 1, 428 11, 700 8. 2120— 72,000 41. 60 1, 730 13, 200 7.6325 90,000 44 81 2, 007 14, 700 7. 30

30 108,000 47.62 2, 268 16, 200 7.1435-. 126,000 50. 13 2,510 17, 700 7.0540-. 144, 000 52.41 2, 746 19,200 6. 9945.. 162, 000 54 51 2, 970 20, 700 6.9748. 172, 800 55. 70 3, 102 21, 600 6.95

50 180,000 56.46 3,192 22,200 6.9655 — 198,000 58. 29 3, 399 23, 700 6. 9860 216,000 60.00 3, 600 25, 200 7.0065 234,000 61. 62 3, 795 26,700 7.04

S/ F2/3=surface per unit of volume.

Table 6.—Condition 5, varying ca when k=5 the ratio a:ka:ca=15:3:5=150:80:50

ca

1

y Fi/3 F2/3 S S/ F2/3

5 22,500 28. 25 795.2 10,800 13.610 — 45,000 35.57 1,267 12,600 9.9515. 67,500 40.80 1, 665 14, 400 8.6520 — —

.

90,000 44.81 2, 007 16, 200 8.0525... - 112, 500 48. 20 2,323 18,000 7. 74

30-. 135,000 51. 30 2,632 19, 800 7.5435. 157, 500 54 01 2,916 21, 600 7.4340 180,000 56. 46 3, 192 23, 400 7.3445 202, 500 58. 80 3, 451 25, 200 7. 3050- 225,000 60.82 3, 710 27,000 7.28

55... 247,500 62.85 3, 956 28, 800 7. 3060... 270,000 64.63 4,173 30,600 7.3465.. 292, 500 66. 49 4,422 32,400 7.3470— 305,000 67.31 4, 529 34, 200 7.5375... 327, 500 68. 97 4,761 36,000 7.56

Sj V2P=surface per unit of volume.

Table 7.—Condition 6 ,varying ca when k=\ the ratio a:ka:ca=8:6:4=15

:

80:20

ca F FV3 F2/3 S s/vy3

5 2,2504, 500

6, 7509,000

11, 250

13.11 171. 6 1, 3501, 8002,2502, 7003, 150

7.8810— 16. 51 272. 3 6. 6215 18. 92 357. 2 6.3320 20. 80 432. 6 6. 2525 22. 41 501. 8 6. 2830 13, 500 23.83 566.4 3, 600

1

6. 36

S/F2/3=surface per unit of volume.

Washington, February 25, 1924.