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# Transactions of the American Society of Civil Engineers (Volume 81) online

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tests on columns, and that analysis fails to show that any definite value can be ascribed
to it, when such analysis takes into account the necessity for toughness in all columns ;
he also believes that dependence on such reinforcement has led to much unsafe con-
struction and many failures. He would recognize as reinforced concrete columns only
such columns as have in addition to the longitudinal rods a complete system of close-
spaced hooping. He objects to the reading of Chapter VII, Section 9, Paragraph (b)
as being capable of interpretation that hooped columns are given an advantage in the
matter of unit stresses only below ten diameters in height. He recommends the
standardization of hooped columns, and suggests that columns be reinforced by a coil
or hoops of round steel having a diameter one-fortieth of that of the external diameter
of the column and eight upright rods wired to the same, the pitch of the coil being
one-eighth of the column diameter. He would consider available for resisting com-
pressive stress, the entire area of the concrete of a circular column or of an octagonal
column, but no part of the longitudinal rods or hooping. In a square column only 83%
of the area of concrete would be considered available. The compression he would
recommend on columns (for 2 000-lb. concrete) would be:

P=670 — 12 -T-
d

where P = allowable compression, in pounds per square inch ;

I = length of column, in inches ;

d = diameter of column, in inches.

1148 CONCRETE AND REINFORCED CONCRETE

CHAPTER X.

Appendix.

suggested formulas for
reinforced concrete construction.

These formulas are based on the assumptions and principles given
in the chapter on design.

1. Standard Notation.

(a) Rectangular Beams.

The following notation is recommended:

/., :^ tensile unit stress in steel;
fc = compressive unit stress in concrete;
Eg ^modulus of elasticity of steel;
Eg = modulus of elasticity of concrete;

M = moment of resistance, or bending moment in general ;

Ag = steel area;

d = depth of beam to center of steel ;

h = ratio of depth of neutral axis to depth, d ;

z = depth below top to resultant of the compressive stresses;

i ^ ratio of lever arm of resisting couple to depth, d;

jd = d — z ^ arm of resisting couple ;

J_

hcl'

p = steel ratio =

(h) T-Beams.

h = width of flange ;

6' = width of stem;

t = thickness of flange.

(c) Beams Reinforced for Compression.

A' = area of compressive steel ;

p' = steel ratio for compressive steel ;

fg' = compressive unit stress in steel ;

C = total compressive stress in concrete ;

C" = total compressive stress in steel ;

d' = depth of center of compressive steel ;

z = depth to resultant of C and C".

CONCKETE AND EEINFOECED CONCRETE

1149

(d) Shear, Bond and Weh Reinforcement.

V = total shear ;

y = total shear producing stress in reinforcement ;

V = shearing unit stress ;

u =bond stress per vinit area of bar;

= circumference or perimeter of bar ;

^0 = sum of the perimeters of all bars;

T = total stress in single reinforcing member ;

s = horizontal spacing of reinforcing members.

(e) Columns.

A = total net area ;
J.g=area of longitudinal steel;
^c^area of concrete;

2. Formulas.
(a) Rectangular Beams.
Position of neutral axis,

k = y/ 2pn + (pn)2 — pn

Arm of resisting couple,

j = l-jk (2)

TFory; = 15 000 to 16 000 and/^. =

7 -|

600 to 650, / may be taken at —.

Fiber stresses,

M M

(1)

./■.,=

/c =

AJd
2M

pjbd^

jkbd^ k

Steel ratio, for balanced reinforcement,
1 1

P =

(fe) T-Beams.

(5)

• ••

Fig. 4.

1150 CONCRETE AND REINFORCED CONCRETE

Case I. When the neutral axis lies in the flange, use the formulas
for rectangular beams.

Case II. When the neutral axis lies in the stem.

The following formulas neglect the compression in the stem.

Position of neutral axis,

, ^ 2 nclA, + ht^

''' = 0nA:-^2U ^'^

3kd — 2t t ^_^

z = . — (')

Position of resultant compression,

z

Arm of resisting couple,

jd = d — z (8)

Fiber stresses,

f=JL (9)

MM =A. * (10)

'^ 1 n 1 — k,

ht(M — — t)jd

(For approximate results, the formulas for rectangular beams may
be used.)

The following formulas take into account the compression in the
stem; they are recommended where the flange is small compared with
the stem:

Position of neutral axis.

2 udA^ + ( b — b') f / nA^ + (& — b') t \ '

nA^^ + (b—b')t

M.j^^^^^^^^^. c-^^r'i

(11)

Position of resultant compression,

(M«2 _ A (3^5 + [(fccz _ ty (t + ^ (kd — t))]b'

z ? ... (12)

t(2 kd — t)b + (kd — tf b' ^ ^

Arm of resisting couple,

id= d — z (13)

Fiber stresses,

i, = -i^ (14)

Ajd

2 Mkd
[(2 kd — t)U + (kd — tf b'-\jd'

fr = 7T,r^. :ttt^^73 ^^TTTT^ (15)

CONCRETE AND REINFOECED CONCRETE

(c) Beams Reinforced for Compression.

C*C'

1151

Position of neutral axis,

k^ J2n(p -{- 1)' Y^ + n''(p + p'f — n(p + p').

Position of resultant compression,

i kM + 2 p'ncV tk — -^^

e + ,,.n(k-£-)

Arm of resisting couple,
Fiber stresses,

jd = d — z.
6M

[»-*^+T('-f)0-f)]

bd'

_ M _ 1 — k

k

/: = n f

d'

(d) Shear, Bond, and Weh Reinforcement.
For rectangular beams,

_ V^

V

jd . ^0
For approximate results j may be taken at — .

.(16)

.(17)

(18)

• (1«)

• (20)
■(21)

.(22)
.(23)

1152 CONCKETE AND REINFORCED CONCRETE

The stresses in web reinforcement may be estimated by means of
the following formulas:

Vertical web reinforcement,

Vs

^ = la (2*>

Bars bent up at angles between 20 and 45° with the horizontal
and web members inclined at 45°,

-1^'

In the text of the report it is recommended that two-thirds
of the external vertical shear (total shear) at any section be taken
as the amount of total shear producing stress in the web reinforce-
ment. V therefore equals two-thirds of V.

The same formulas apply to beams reinforced for compression as
regards shear and bond stress for tensile steel.

For T-Beams,

' = ^1

Online LibraryAmerican Society of Civil EngineersTransactions of the American Society of Civil Engineers (Volume 81) → online text (page 97 of 167)