Civil Engineering Reference
In-Depth Information
9.9 Appendix - elastic analysis of joints
9.9.1 In-plane joints
Thein-planejointshowninFigure9.11aissubjectedtoamoment
M
x
andtoforces
V
y
,
V
z
acting through the centroid of the connector group (which may consist of
bolts or welds) defined by (see Section 5.9)
A
i
y
i
=
0
A
i
z
i
=
0
(9.37)
in which
A
i
is the area of the
i
th connector and
y
i
,
z
i
its coordinates.
Itisassumedthatthejointundergoesarigidbodyrelativerotation
δθ
x
between
its plate components about a point whose coordinates are
y
r
,
z
r
. If the plate com-
ponents are rigid and the connectors elastic, then it may be assumed that each
connectortransfersaforce
V
vi
,whichactsperpendiculartotheline
r
i
tothecentre
of rotation (Figure 9.11a) and which is proportional to the distance
r
i
, so that
V
vi
=
k
v
A
i
r
i
δθ
x
(9.1)
in which
k
v
is a constant which depends on the elastic shear stiffness of the
connector.
The centroidal force resultants
V
y
,
V
z
of the connector forces are
V
y
=−
V
vi
(
z
i
−
z
r
)/
r
i
=
k
v
δθ
x
z
r
A
i
V
z
=
V
vi
(
y
i
−
y
r
)/
r
i
=−
k
v
δθ
x
y
r
A
i
(9.38)
after using equations 9.37 and 9.1. The moment resultant
M
x
of the connector
forces about the centroid is
M
x
=
V
vi
(
z
i
−
z
r
)
z
i
/
r
i
+
V
vi
(
y
i
−
y
r
)
y
i
/
r
i
,
whence
A
i
y
i
+
z
i
,
M
x
=
k
v
δθ
x
(9.39)
after using equations 9.37 and 9.1. This can be used to eliminate
k
v
δθ
x
from
equations 9.38 and these can then be rearranged to find the coordinates of the
centre of rotation as
y
r
=
−
V
z
A
i
y
i
+
z
i
M
x
A
i
,
(9.40)
z
r
=
V
y
A
i
y
i
+
z
i
M
x
A
i
.
(9.41)
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