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195
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 2
R. M. F. CANHA |
G. M. CAMPOS |
M. K. EL DEBS
Also, at the column bottom, there is the internal transverse force
V
cb
and the internal eccentric normal force N
cb
.
The problem thus involves four unknowns (
t
M
,
t
N
, V
cb
and N
cb
). Be-
sides the three static equilibrium equations, there is an additional
equation related to the normal reaction
N
cb
at the column bottom.
This force is considered as the
d
N
reduced by ratio of the column
cross section
c
A
to the external surface area of the pedestal
A
cp
,
according to the following equation:
(17)
cp
c
d
cb
A
AN N
=
Based on the structural outline of this element, the diagrams of the
internal forces were determined with the corresponding expres-
sions given below.
Considering a pressures variation along the embedded length
l
emb
,
for practical applications, the column transverse reinforcement
area is determined for maximum internal forces.
Hence the
'
p
topf
and
'
p
topr
values are given, respectively, by
equations 18 and 19:
(18)
f
emb
csf
emb
f
topf
tan
R2
H2 '
p
b ×
×
=
×
=
l
l
(19)
r
emb
ssf
emb
r
topr
tan
R2
H2 '
p
b ×
×
=
×
=
l
l
The shear force
V
cb
at the column bottom and the shear stress
t
N
are determined from the equilibrium equations of forces in the
horizontal and vertical directions and are given, respectively, by:
(20)
f
r
d
cb
HHV V
- + =
(21)
emb
cb
d
N
)b2h2(
NN
l
×
+
-
=t
From the moment equilibrium relative to point O, the shear stress
t
M
can be defined by equation 22:
(22)
(
)
2
h bh
e N HH
3
2 V M
2
emb
nb
cb
f
r
d
emb
d
M
+ ×
×
- - + +
= t
l
l
The internal forces along the column base are calculated from the
three static equilibrium equations in plane.
The bending moment
y
M
acting at a given distance
y
from the
column bottom is calculated by the following equation:
(23)
(
)
3
2
emb
f
r
f
r
emb
d
nb
cb
nb
cb
y
y
3
HH y
3
HHM e N e N M
l
l
-
- ×÷ ÷
ø
ö
ç ç
è
æ
-
+
- ×
+ ×
-=
Also, the shear force
y
V
at a given distance
y
from the column
bottom is given by:
(24)
(
)
2
2
emb
f
r
f
r
d
y
yHH HHV V
l
-
- - + =
The internal normal force
y
N
at a given distance
y
from the col-
umn bottom is:
(25)
y N N N N
emb
d
cb
cb
y
×÷÷
ø
ö
çç
è
æ -
+ -=
l
The Bending Moment Diagram has a cubic shape. The absolute
maximum value
max
M
at the column top and absolute minimum
value
min
M
at the column bottom are given, respectively, by:
(26)
d
max
M M
-=
(27)
nb
cb
min
e N M
×
-=
On the other hand, a parabolic diagram is adopted for the Shear
Force Diagram with the maximum value
max
V
at the column bot-
tom and the minimum value
min
V
at the column top are calcu-
lated, respectively, by:
(28)
f
r
d
max
HHV V
- + =
(29)
d
min
V V
=
And lastly, the Axial Force Diagram has a trapezoidal shape with
its maximum
max
N
value at the column top and the minimum