Civil Engineering Reference
In-Depth Information
Performing the matrix multiplications of Equation 9.31 gives the element stiff-
ness matrix as
1
+
2
1
1
2
1
+
C
2
1
1
2
+
C
1
2
1
3
+
C
1
3
1
2
+
C
2
1
1
3
+
C
3
1
1
+
2
2
2
2
2
2
2
+
C
2
3
+
C
2
3
2
1
+
C
1
2
2
3
+
C
3
2
Et
4
A
(1
1
+
2
3
3
[
k
]
=
2
3
+
C
2
3
3
1
+
C
1
3
3
2
+
C
2
3
−
2
)
SY M
2
1
+
C
2
1
1
2
+
C
1
2
1
3
+
C
1
3
2
2
+
C
2
2
2
3
+
C
2
3
2
3
+
C
2
3
(9.32)
where
C
Equation 9.32 is the explicit representation of the stiff-
ness matrix for a constant strain triangular element in plane stress, presented for
illustrative purposes. In finite element software, such explicit representation is
not often used; instead, the matrix triple product of Equation 9.24 is applied
directly to obtain the stiffness matrix.
=
(1
−
)
/
2
.
9.2.3 Distributed Loads and Body Force
Frequently, the boundary conditions for structural problems involve distributed
loading on some portion of the geometric boundary. Such loadings may arise
from applied pressure (normal stress) or shearing loads. In plane stress, these dis-
tributed loads act on element edges that lie on the global boundary. As a general
example, Figure 9.3a depicts a CST element having normal and tangential loads
p
n
and
p
t
acting along the edge defined by element nodes 2 and 3. Element thick-
ness is denoted
t
, and the loads are assumed to be expressed in terms of force per
unit area. We seek to replace the distributed loads with equivalent forces acting
at nodes 2 and 3. In keeping with the minimum potential energy approach, the
concentrated nodal loads are determined such that the mechanical work is the
same as that of the distributed loads.
(
p
)
f
3
y
3
3
(
p
)
f
3
x
3
p
t
n
p
y
p
n
p
x
(
p
)
f
2
y
(
p
)
f
2
x
1
1
1
2
2
2
(a)
(b)
(c)
Figure 9.3
Conversion of distributed loading to work-equivalent
nodal forces.
