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THS

 

DEﬁfiié~i CURVES PM?
REiNI‘f-CSRCED CQNCRETE
ﬁﬂivﬁif‘éﬁﬁ FiﬁGT’EMGS ON SC‘E.

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0-189

-,____ ..._'___ J- l__4.. -..___:_._... ._ ‘J_'._._ _ A _;___.L..., _ L.--

This is to certify that the

thesis entitled

DESIGN CURVES FOR REINFORCED CONCRETE
COMBINED FOOTINGS ON SOIL

presented by

AVRAM KAZEZ

has been accepted towards fulﬁllment
of the requirements for

M. 8. CIVIL ENGINEERING

degree in

WWW

Major professglw

 

Date Mﬁmh 1.13 3 ﬁ1952

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L

 

DESIGN CURVES FOR REINFORCED
CONCRETE COMBINED FOOTINGS
ON SOIL

Thesis for the Degree of M.S.
Michigan State College
Avram Kazez

1952

DESIGN CURVES FOR REINFORCED CONCRETE
COMBINED FOOTINGS ON SOIL

By
AVRAM KAZEZ

A THESIS
Submitted to the School of Graduate Studies of.Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements

for the degree of
MASTER OF SCIENCE

Department of Civil Engineering

1952

THESlS

 

 

 

V/ E. g
IrI // 3’ a" '4 ‘V
l, \

.5 3'
1!

CONTENTS

Introduction . A. . . . .
Assumptions, Notations and Definitions.
Typical Shear and Moment Diagrams. .
Condition for Minimum Depth of Footing.
Derivation of Equations for Curves:
1. Length of Footing "L" . .
2. Effective Depth of Footing "d"
3. Width of Footing "B" . .
4, area of Steel Required "AS" .
5. Bond "21" . . . .
6. Shear "v" . . . .
Explanation of Curves:
1. Length of Footing "L" . .

2. Effective Depth of Footing "d"

3. Width of Footing "B" '. .
h. area of Steel Required"AS" .
5. bond "2." . . . . .
Numerical Example . . . . .
Summary . . . . . .
Appendix . . . . . .
Bibliography . . . . . .

10
«E
f-

Y

r" '4‘
My“.
Z3

10
12
12
13
1h

15
19
23
25
32
33
31.
35
1.1 _

INTRODUCTION

1. Introductory Statement

This study describes a simple graphical procedure for de-
signing reinforced concrete combined footings.

The design curves presented in this studv can be used by a
designer or checker.

Following the derivation of equations, the curves are drawn
and a typiCal example of their use is given to make the method
of procedure clear.

Derivation of the scale equations used to draw the charts
are given in the Appendix.

It is hoped that the use of the charts herein will eliminate

calculations thereby saving time.

2. Scope

With the use of the charts,the length and width of the footing,
the effective depth, the steel, bond required and the shear stress
are found.

This study is not concerned with the selection of the main

or web reinforcement bars, nor in their distribution.

-2-

3. Acknowledgement.

The author wishes to express his indebtedness to Dr. R. H.
J. Pian for his valuable advises and suggestions, and also to

acknowledge his thanks to Mr. W. A. Bradley who read the study.

ASSUMPTIONS, NOTATIONS AND DEFINITIONS

1. Assumptions

Adaptation of the nomograms will be made to design a combined
_ footing, which will meet the following requirements:
a) An area great enough to give the safe allowable soil

‘ pressure.
b) A shape such that a uniform net soil pressure is given.
c) The two loads acting on each column are equal.

d) The footing will have a constant depth.

2. Notations and Definitions

The symbols in this study, defined below conform essentially
to Letter Symbols for Structural Analysis prepared by the Americai
Standards Association, with ASCE participation, and approved by
the Association in l9h9. For their significance, where not confor-

ming to the standard, is made clear by Fig. l.

*1; P
Al:

.h

 

 

 

 

 

 

 

A
,..___-_ L a D B
*4.
h;

 

 

 

where

'0

21 T!

i

t“ {T w 0..

0

load on each column in kips.

fallowable soil pressure in 1b./ft.sq.

weight of footing in lb.

'soil pressure against base of footing resisting column

loads in lb./ft. of length.

effective depth of the footing in inches.

width of the rectangular footing in feet.

height of base of footing in inches.

length of footing in feet.

distance between column loads in feet.

distance between outside edge of column and width of
footing in feet.

size of column along the length of footing in inches.
distance between outside edge of the columns in feet.
distance between center line of footing and column load

in feet.

 

 

 

lll\l:'|lll|l|\ ('1 l‘ (I || r

Typical Shear and Moment Diagrams.

3

 

 

 

 

 

 

 

o. l‘b
(I)
. +
Shear /\ b g _i
Diagram é \\;//’/,
(II)
Pu
Na
+
+
Moment 9 .

 

Diagram

 

(III)

Fig. 2.

CONDITION FOR MINIMUM DEPTH OF FOOTING

From the moment diagram in Fig.2(II), the moment at point a;

Ma - r2? - BKd2
T
but, A - BL
therefore, f2P - AKd2 . . . . . . . _ (a)

Keeping P, A and K constant, the variation of "f" with "d"
will be investigated. Then '
d - elf where if is a constant.
The equation above represents a family of straight lines
through the origin with positive slopes. Plotting "d" as ordinate

and "f" as abscissa, the family of lines below is obtained.

d‘}

 

Fig. 3.

Similarly theexpression for the moment at point b willbe

Mb - Py - 12w - BKd2
- . 8

Similarly, the expression for the moment at point b can be written;
Mb- Py - L2w/8 - BKd2
Simplifying terms and substituting, B - A/L
hyL - L2 - hAKdz/P . . . . . . (b)
but, y - (c-b)/2 and L-2febsc
therefore,
2(c-b)(2f+b+c) - (hf2+hfb+ufc*b2+2bc+c2) - AAKdz/P
simplifying and transposing terms,
-c2+4hr.2b2+ur2+arb+b2+2bc+uAKd2/P - 0
Keeping P, A, K,b, and c constant the variation of "f" with "d"
will be investigated. Then
kldzo 4r2+ 6bf + k2 - o
where k1 and k2 are constants.
The equation above represents a family of ellipses with

center at (-h, 0). Plotting "d" as ordinate and "f" as abscissa

the family of ellipses below is obtained:
d4L

 

 

-8-

Superimposing a typical curve of each family shown above,

at Painf, 0..

‘__-__~
ﬁx

or PO;an b'

 

 

Fig.5.

Investigating the variation of "d" with "f" in Fig. 5 it is
concluded that; the depth "d" of the footing is to be designed
for the largest value of "d" required at any point. Therefore
the smallest value of "d" required for the footing will be when
the required depth at points "a" and "b" are equal.
From equations (a) and (b), and Fig. 2(III), it is obvious

that for the above condition

 

 

ﬁlaJ = II‘ 1 b

DERIVATION OF EQUATIONS FOR CURVES

1. Length of Footing "L"

An expression for the length of the footing "L" will be

derived for the condition IMa|- IMb

 

From the moment diagram in Fig. 2(II); the following equa-

tion can be written.

lMal ' £2!
and ‘Mb' ' Elgihl “'ééﬂ
hence Eéé ' P§n+b2 _ Lg!
or Egg * Egg . Ejggbl
from Fig. l y - 2&2
therefore Egg + 3%: . py
also w - é:
therefore ggéi-e 23%: . py

simplifying terms 4r2+ Lz- AyL

From the above equation it is concluded that the length

"L" of the base is independent of the loads "P" on the column.

-10-

 

Substituting f = L-§-C

and I 2y = c in the above equation
(L-b-c)2 + L2 - 2cL
(L2+b2+c2-2Lb-2Lc+2bc) + L2 = 20L
2L2 + (b2+2bc+02) - 2Lb -th = o
L2 + (b20)2 - L(22+Ac) = O

L2 — (b+2c)L + (b+c)2 = 0
__§___

Solving the quadratic equation above for "L" in terms of "b"

 

 

and "c",

or, L = b + 2c + J2c2 — b2 . . , . (1)
2

8.150, f g L "' b - C o e o e o o (2)

 

2. Effective depth of footing "d"

The value "d" is usually calculated by assuming the weight
of the footing, then checking the weight.

An expression for "d" in terms of the already known values
will be derived, thus eliminating the ambiguous method of trial

and error.

As before,

but;

hence

also;

where,

For a balanced design;

or,

'14

and

-11..

M = f2w/2

w . 2P/L

M - fZP/L
- BKd2

B is in feet.
f is in feet.
d is in inches

f2P/L - BKd2

BL a f2P/Kd2 o e o

(3)

Two eXpressions for the weight of the footing "N" can be written,

and,

where,

Equating equations

Substituting equation

multiFlying by

VJ‘BBPL'ZP o o o

 

(h)

w - 150(%+L)LB . 12.5(d+h)LB . . (5)
1

L is in feet.
P is in lbs.

(A) & (5);
BpL - 2P = l2.5(d+h)LB .

(3) in (6);

f2.P - 2P = 12.2(d&4)f2P
EEZ“ Kdz

d2
sz;

(6)

-12-

2

p - 2K? = 12.5(d+h)
or, 2Kd2 + 12.5(d+t) = p . . . (7)

"d" can be calculated from equation (7), as all other variables

are know.

3. Width of footing "B"
The value of "d" once known, the width of the footing "B"
will be evaluated.
As before M = fZP/L
also, M = KBd2
for a balanced design, KBd2 = M = f2P/L . . . . (11)

"B" can be calculated.from equation (11), as all other variables.

are known.

L. Area of steel required "AS"
The area of steel "AS" will be calculated from equation below

PI 2: JdeAS e o e e o (12)

-13-

5 . Bond " 20"

The shear force "V", at the outside edge of the column from

Figs. 1 a 2 is: v a 2Pf/L . . . . . . (13)

and at the inside edge is:

v = 2Py/L = P(c-b)/L . . . (14)

the larger value of V above is of interest,

P(c-b) #7 22; . . . . . (15)
L L

dividing both sides by P/L and substituting equation (2) for"f"
c-b f L-b-c
or, c # L-c . . . . . . (16)

from chart A the.following approximate equation can be written

L -(%%;%)c = 1.75c

substituting the latter value of L into equation (16);

v0 f 1.75c-c
c K 0.75c
hence; c>0.750
therefore (C - v/Z > f e o o o o o (153)

The shear force "V" expressed by equation (It) will have the

larger value, which will be used to evaluate"2k"
also, V 3 g‘d°u‘200 o o e o e (17)

equating equations (14) and (17),

2Py/L = V = %-d~u-ZO . . . . (18)

-1g-

6. Shear "v"

The shear "v" will be calculated from equation below

V 3 VJdB o o o o o e o (19)

-15-

EXPLANATION OF CURVES

1. Length of footing "L"

The first curve will be drawn to evaluate the length of the

footing "L" by equation;

L - h122_:_122E;£E1 . . . . . . (l)

2

Owing to the fact that the variables in the above equation
cannot be separated, a value of "b" will be assigned and thus
eliminating one variable. For the same value of "b" different
values of "c" will be given and the corresponding values of "L"
calculated.

The L- and c— scales will be at right angles to each other
and will have uniform scales. Hence the nomogram will have the

form shown below:

 

 

Fig.6.

-16..

c will vary from 5 to 25 ft. Its scale: 5 ft. = 1 inch.
b will vary from 12 to 36 inches.
1 will vary from 9 to A5 ft. Its scale: 10 ft. = 1 inch.

(See chart A.)

The second curve will be draWn to evaluate "f", by equation
L a 2f + b + c . . . . . (2)

alignment charts will be used.
let L - c = T
then L - c a T = 2f + b

Let the modulus of L, mL be equal to 1/10, which is the same
value as before.
The scale length of "c" will be about A inches.
m s h" 2 1/5
C "F
3-1...

hence, mT = mg . mc

mL + me

= 1/15

consider equation T = 2f + b
ms= .1." = 1/3
3
Inf 8 ml . mb
mb - HIT
= 1/12

The relative distances of the scales to each other will

be calcuhated:

-17-

mL/md = 5/10 = 1/2
similarly mf/mC - 3/12 = l/A
and the scale equations will then be:

XL = L/lO (L is measured in feet)

Xc = c/S (c is measured in feet)

Xb - b/3 (If b is measured in feet)

= b/36 (If b is measured in inches)

and Xf = f/6 (f is measured in feet.)

The general form of the Alignment Chart for the equation,

L = 2f + b + c, is shown below:

 

 

 

 

Fig. 7.

Since the modulus of L is the same in both charts, Fig. 6
and 7 will be combined.
The complete chart thus, and drawn to scale is drawn on

next page, (chart A).

 

 

 

an»: E 0
5 m 5 m- 5
i 2
FLILILIL21L)h(-gla)L a - - - - _ - - - - _
3...: v:~ C. ON
5 o 5 x
.5. 3 2 K- M
.w — * r—- '—
T u
-
7-).) (_1.(_IJJ((4|.I.)1 4. 414(14)- A
mob7s54oalo .
“v.* Zak
+l
r
a
.h
If; C- J
0
b H b h L 5
2
a
(O
2
Y
T
#5
y f .
a,
.m
C
1.0

 

 

-19-

2. Effective depth of footing "d"

The value of "d" will be found from equation

2kd2 + 12.5(d+l) - p . . . . . (7)

let z .

substituting equation (8) in (7);

zd2 + 12.5(d+t) = p . . . . . (9)

First a curve for equation (8) will be drawn. The values of "K"
can be taken from "Reinforced Concrete Design Handbook".

f varies from 1 to 10 feet.

K varies from 125 to #23.

therefore 2 varies from 5 to 500.

The scaIe lengths are to be approximately 5 inches.
equation(8) can be written in the form;
log 2k = 2 logf + log 2
-2 log f = - log 2k + log 2
then, mK = 6" = 11.3
log 8A6 - logV25O

A more convenient value mK= 10 will be used.

and, m = " = 3
Z log SOO - log 5

 

therefore, mf = mKi. ng = 3x10 =

~20-

The relative distances of the scales to each other will be cal-

culated: mg

 

= 10
3—

the scale equations will then be;

XK

X
z

Xf

The general form of the Alignment

 

z = 2K , is shown below:
T?
A
/—4
K /-4k/
/// ,c 2
(V/
() ’
Fig. 8.

10 (log2K

3 (log 2 -

30x2 logf
13

 

 

 

log 250)

= 10 log (K/125)

log 5) = 3 log (2/5)
A.62 log (f)

Chart for the equation,

The value of "d" will now be evaluated. substituting (8) in (9);

zd2

or; zd2

solving for d; d

+ 12.5d + (so-p) = o

+ 12.5(d+u) - p

 

- 12.5 +~Jl12.5)2 - 4.2 (50 - p)1. .

2.2

(10)

-21..

A value of "p" will be assigned and the corresponding values
of "d" will be calculated for different values of "2".

Same procedure will be repeated for different values of "p"
assigned.

The general form of the nomogram is shown below:

A

 

Fig. 9.

The modulus of "z" will have the same value and hence the
latter two curves will be combined to give the complete nomogram

drawn to scale on next page.

 

 

 

 

 

 

 

Q5 R
a
.(
[3)_1 4)
I
I
—< ( Z
i x
1 Y 3
_J I
i.
4 4'
.1
(hm—a 8“
_,
i Z
—< k 4
4-
,, ’1
a U A
V- —-1
-J
7; 1 C5—(
l -
d x.
E A “
x <
\. ﬂ _.}_”4
500——(
3—1
4 h
a b_‘
l -\ I
.4
——(
- /——( \.
a \s» ——
—< 8*
400—4 t]-——.
.4 i
i ’0‘
425
l
E
jTIITTllTjTerr,)lellllTrlllT‘l
o 5 IO (5 80 25 30

d )n inches

Charf B

   

-23-

3. Width of footing "B".

First the curve for equation, M - f2P/L will be drawn

:2. l
M P

or,
f varies from 0 to about 10 feet.

L varies from O to about 50 feet.

P varies from 0 to about 1300 kips.
M.

therefore,

[I] I " I 0.05
f TEEECZG

m - 5" - 0.1
L “UT—5

mP - ". - 0.00385, Use mp - 0.00h
o -

’

varies from 0 to about 3000 ft-kips.

therfore, mM - m£.mz m 0.002
mL
The scale equations are;
X

X

- 0.051“2
L - 0.1L

XP - 0.00hP
XM - 0.002M .
The general form of the chart for the equation.M - f2P/L is

f

shown below:

 

Fig. 10.

 

The value of "B" will be evaluated next from,

or,

therefore

then,

the scale equations

The general form of the chart for the equation M =

below:

-24-

M = KBd

q a
32

B
I7K

p-a

varies from

ﬂy

0 to 3,000,000 ft-lb.

d varies from O to 30 inches.

K varies from 125 to 423 psl.

B varies from O to 10 feet

a 6"
3,000,000

a l/500,000

 

m. 4‘
N.

m = A N

900

. 6.1“!
l 125

= 0.004,ll

mK - 800

mB - u.m5 - 0.36

md

are:

a:
ll

M/500,000

0.00t,ttd2

N
O.
u

800/K

>4
\
ll

>4
I

_ 0036.8

KBd2 is shown

-25-

 

 

Fig. 11.

The M - scale lengths are the same in both equations and hence

the twoczharts foreequation (11), will be combined. (See chart B)

A. Area of steel required "AS"

The value of "as" will be evaluated next from,

lbo-ino o o o o (12)

I511: 7ofSodoA
g S

or M = 1 x .f -d-A lb.-ft.
’ rigs“;
or, g = 7A 96
d

M varies from O to 3,000,000 ft.- lb.

d varies from 0 to 30 in.
fsvaries from 8,000 to 30,000 psi.

Asvaries from 0 to 100 in.

mg; "’ " = 1 00 00
W00— /5 ’ O

-26-

md = h.2" = 0.1h
lﬂfs I: 6" = 14,92,000
172000"
m.S = 48,000 x 1/500,000 = 2a/35

11

 

0.14
the scale equetions are:
xM= m/500,000
id = 0.1hd

xfsa 48,000/1‘S

X.
As

As/20

The general form of the chart for the eqution M =

is shown below:

 

 

 

 

As A k
f
/
/ A
/ ’iﬁ»
M [##I.’
«V’if/ / d
v /
/
J =
1“,
Fig. 12.

The modulus of "M" will have the same value and hence the
latter three curves will be combined to give the complete nomg-

gram drawn to scale on next page.

 

   

feet

in

7—4

_.__.. 1-1. __ A

400
, L,__,l g

300

 

 

 

P H)
900 800
_;*l -__1 ___-l l -1.
/
’9
I
I j I 7' T T I I: I
do (’5
L in her

1

I

Y 7

, 1 L i.
r T I I
3:5

U ' Z .5 3 5 b 7 8 V If
4L. _J_.TI-1 lr_.L.l,__L TL _ 1T _ TL . ‘T l r ex__!i ,_A.,,--L T l- ' I' i -_l,_.l
\\ IO 20 JO 4O 50 I 60 I”) ﬁx 1 ' ‘00

 

 

,\\\\
/
\\\
\
\\ /
r \\
i \
i \
i
I
i , _
i “ \
l , \ t
f \ \
i \
f ‘..
s
-\
\
\\
,\
i.
l
a f; 1") K]; Ln,
5 to r2 ;4 ' 0 :3 a0 22 24 27 30
I 1 I 1 f ,____,‘J___V_ _ 1cm, _‘L___,l._--,1a+ 33—1—4
l25 130 2"; U 10(1 400
K m psi
(:f)ar"f C

inchis
1%

_50

m

d

25~m

20~~¢5

~L20

 

inches

in

d

 

5. Bond '1 20"

The value of "V" will be evaluated from equation V = 2P1. .(lh)

L'"

or,

y varies from 0 to 12 feet.
L varies from 0 to 50 feet.

P varies from 0 to 1,300,000 lb.

6": 1/2

m
y I?“

III

5" = 1/10
30‘

m = 5.2" = 1/500,000
P 2,600,000

 

therefore mV = mg . mz = l/100,000

NIL

the scale equations are:

X y/2

Y

KL L/10

P/250,000

KP
v/100,000

Y
4L

V

The general form of the chart for the equation V = 2Py/L is

shown below:

-29-

 

 

 

 

A A, P
‘ W
\
A \
1K\\\ \ V
, \\\ \_
\ “‘-\~lﬂ
\
\ V
\
L . —.‘:
Fig. 13.

The value of "2;" will be evaluated next, from,

v = 7ud2 . . . ‘ . . . (17)

“8'“

or , V = 20
7d73 175

d varies from 0 to 30 inches.

u varies from 140 to 350

m . 1/100,000

 

v
m = 2.5" = &_ use m = l/7
d 30 x 773 21 ’ d -
m = 5" = 700
u T7TZU‘
therefore m2°= 700 x 1/100,000 = 0.0h9
1/7

XV = v/100,000

' x = d/8

 

-30-

Xu = 700/u
therefore X20= 0.01.9 2.

The general form of the chart for equatiai V = 7udzois shown

below:

 

 

Fig. IA.
The V - scale length are the same in both equations and
the two charts for eqdation (18) will be combined. (See chart D)
6. Shear "v"
The value of "v" will be evaluated from equation
V g VBjd o o o o o o o (19)

or, V = 12.B
‘ 7d7h I7v
mV = 1/100,000

md = A" x 8 = 2
50 x 7 13 use, md = 1/7

 

m

m =

205
10 x 12

mn.md

mv

The scale equations are:

Xv = v/100,000

[x 3

>4
II

P";
II

The general

shown below:

 

d/8
B/z.

297.5/v

fOrm of the chart for the equation V

-31-

1/48

297.5

 

ijd is

 

ban-b,"

P In kip: £0 ”7 I
woo Q00 I100 1000 000 300 700 too 500 400 .300 200 :00 o 0 x) a. :u\ 4() L0 70 50 90 ’00
I L 1 l A 1 a I l I 1 L 1 l l 1' l l - l 1 I l 1 ' I

-«L
—<
_.,

g 33
3

 

 

Icef

'7

Sir—Ci}.
U:
1

inches

In

 

T ’U‘ in pm.

 

 

 

 

0 r1 1 Y‘IY v 1 [1"ij [I 1 1111': r1 T! % rvrr I] l T111 111 1 1 11 4— 1 I 1—
0 IO 20 30 40 M0 :0 Am) 250 300 550
L m chf a m PSI

Chorf D

   

-33-

NUMERICAL EXAMPLE

an example of the use of the charts will be given to make

the method of procedure clear.

Given:

Solution:

P = 600 kips

c = 15 ft.

b = 2A" = 2 ft.

r = 30,000 psi.

f'= 2500 psi.

p = 6,000 #/ft?

u = 250 psi. (deformed bars, A. C. I. 1951 Code)

From chart A. and with values of c and b:

L = 27 ft. and f = 5 ft.

K 2 156 ( Reinf. Conc. Handbook)

From chart B and values of K and f: d = 21 in.

From chart 0 and values of P, f, L, d, K, f5:

B = 8.1 ft. and AS = 12.1 in2.

From chart D and values of P,y,L,d,u,v,B,

2: 63 in. and v = 163psi.

-34-

S UI‘IH‘II A RY

The curves are more flexible and cover a wider range of
design conditions than the most comprehensive tables. Their
use entails a minimum of calculation and shows clearly the de-
sign procedure and the relation between the design conditions.

The resultant design will be within the practical limits

of loading and soil bearing determination.

-35..

APPENDIX

An alignment chart in its simplest form consists of three
parallel scales so graduated that a straight line cutting the
scales will determine three points whose values satisfy the
given equation.

In general, alignment charts may consist of three or more
straight-line scales, of curved scales, or of combinations of

both.

ALIGNMENT CHARTS FOR EQUATIONS OF THE FORM
rl(u) + r2(v) - f3(w)

Suppose we have three parallel scales (Figure 16), A, B, and C,
so graduated that lines (iSOp1eths) 1 and 2 cut the scales in

values which satisfy the equation fl(u) + f2(v) s f3(w). Now,

\V‘ ,
Q1
)

Xu
Xw
X

 

 

 

 

 

 

Fig. 16.

 

-36-

>4
I

u - mu fl(ul) - fl (no)

v mv f2(V1I ' f2IV0I

><
I

Xw = mw f3(wl) - f3(w0)

If no, v0, "0’ represent zero values of the function, and

if line 2 is any line, we may drop the subscripts and write simply

‘7
4\

u mufl(U) o o o o o o (1)

>4
u

mvf2(v) . . . . . . (2)

KW

Let us agree further that the Spacing of the scales is in the

mwf3(W) o o o o o o (3)

ratis a/b. If we graduate the scale of f1(u) and f2(v) in
accordance with their scale equations (1) and (2), reSpectively,
what will be the modulus for the scale equation of f3(w) and
what will the ratio a/b equal if the chart satisfies this relation
f1(u) + f2(V) = f3(W)?

In figure 16 draw lines through points W1 and v1 parallel

to line uovo. The shaded triangles are similar by construction,

 

hence,
X - X = a
,u___gﬂ 3
AW ' Av
Kub + Xva = Xw(a + b)
Xub + Xva = X (a + b)
”11‘. ab

ab ab

-37-

 

 

or XI + E! = Kw
a b ab

a + b
since in = mufl(u)
Xv = mvf2(v)

f

- mwf3(w)

mufl(u) + mvf2(v) = m f2(w)

a b ab
a + b

 

 

2
mu = a, m = b
therefore % - mu
mv
and m = ab = m
m m‘”
m + m

-33-

PROPORTIONAL CHARTS OF THE FORM

fl(u) == f3(w)

f2(v) fh(QI

 

 

This equation can be solved in a manner similar to that
used in the preceding problem. This simply means transforming
the above equation to the form

log f1(u) - log f2(v) = log f3(w) — log fh(q)

In many cases, however, where the functions are linear,
the prOportional type alignment chart has a distinct advantage

in that the scales

D hplltiL

 

 

 

 

 

1’.
A 2
,/ ‘EJL
/
r
m
g B
u. ‘01 C
w' x.
Fig. 17.

are uniform thus permitting more accurate readings and also simp-

lifying the construction of the scales.

__A_.l’__ #;

_. '_.- .‘L—

_'__'__‘.&. .3—_n_' __

-39-

Consider the figure shown in Figure 18.
Scales A and B are parallel to each other, and graduated in

accordance with the scale equations:
Xu a mufl(u) and 1v = mvf2(v), respectively

In a similar manner, scales 0 and D are parallel to each

other, and are graduated in accordance with the scale equations:

X = m f w and X = m f c
w W3” q qt.”

The angle between scales A and C may be of any convenient magnitude.

Triangles u uOT and vlvOT are similar, hence

l
X11 3 . 110T

Likewise, triangles w Twl and qOT l are similar, hence
q

0
Kw - wOT

 

But lenghts uOT . w T; and vOT = qOT. Therefore

0

or muf (u) = mwf3(w)

Since ' f1(u)= f3(W)

 

 

-40-

it follows that m =
u

S
Elia

V q
This means that three moduli may be determined from the given
data, but the fourth modulus will be dependent upon the first

three.

 

-41-

BIBLIOGRAPHY

"heinforced Concrete Structures" by Dean Peabody, Jr.

"Reinforced Concrete Design Handbook" of the American

Concrete InstitutiOn..

The ACI Building Code 1951. (ACI 318-51)

"Recommended Practice and Standard Specifications for Concrete
and Reinforced Concrete" Joint Committee on Standard Speci-

fications for Concrete, and Reinforced Concrete, l9h0.

"Nomography" by A. S. Levens.

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