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DRAFT –1
Chapter 4 Beam Element
As derived in the case of the truss element, relationship between local and global stiffness
matrices is given as
[k
g
] = [T] [k] [T]
Where, [k
g
] = Global stiffness matrix of an element
[T] = Transformation matrix
[k] = Local stiffness matrix of the element
Substituting the values of [T] and [k], we get the global equation of a beam element
oriented arbitrarily at an angle
θ
as,
12S
2
-12SC
-6LS -12S
2
-12SC -6LS
12C
2
6LC 12SC -12C
2
6LC
k = EI/L
4L
2
6LS -6LC 2L
2
12S
2
-12SC
6LS
Symmetry
12C
2
4L
2
This is the equation of a beam element (without axial or torsional load, and oriented at an
angle
θ
.
Also, S = sin
θ
, C = cos
θ
in the above equation.
4.3 Beam Element with Combined Bending and Axial loads
First, we will derive the stiffness matrix in local coordinates and then convert it in to
global coordinates.
4.3.1 Stiffness matrix of a beam element with bending and axial loads in
local coordinates
The stiffness equation for the combined bending and axial load can be written by
superimposing the axial stiffness terms over the bending stiffness.
For axial loading, the structural equation is,
f
1x
1 -1
d
1x
= AE/L
3
f
2x
-1
1 d
2x
ME 273 Lecture Notes © by R. B. Agarwal
4-11
DRAFT –1
Chapter 4 Beam Element
And for bending loading, the structural equation is,
f
1y
12
6L
-12
6L
d
1y
m
1
6L 4L
2
-6L 2L
2
φ
1
= AE/L
3
f
2y
-12
-6L
12
-6L
d
2y
m
2
6L 2L
2
-6L
4L
2
φ
2
Therefore, the combined loading equation is
f
1x
C
1
0 0 -
C
1
0
0
d
1x
f
1y
0 12
C
2
6C
2
L 0
-12
C
2
6C
2
L
d
1y
m
1
0 6
C
2
L 4C
2
L
2
0
-6C
2
L 2C
2
L
2
φ
1
=
f
2x
-C
1
0 0 C
1
0 0
d
2x
f
2y
0 -12
C
2
-6C
2
L 0
12 C
2
-6C
2
L d
2y
m
2
0 6
C
2
L 2C
2
L
2
0
-6C
2
L 4C
2
L
2
φ
2
And,
[
−
−
−
−
−
−
−
−
=
2
2
2
2
2
2
2
2
2
2
1
1
2
2
2
2
2
2
2
2
2
2
1
1
4
6
0
2
6
0
6
12
0
6
12
0
0
0
0
0
2
6
0
4
6
0
6
12
0
6
12
0
0
0
0
0
]
L
C
L
C
L
C
L
C
L
C
C
L
C
C
C
C
L
C
L
C
L
C
L
C
L
C
C
L
C
C
C
C
k
Where, C
1
= AE/L, and C
1
= EI/L
3
ME 273 Lecture Notes © by R. B. Agarwal
4-12
DRAFT –1
Chapter 4 Beam Element
4.3.2 Transformation matrix for combined Bending and Axial loading.
For the axial loading, the relationship between the local and global coordinates was
derived earlier, as
S
d
C
d
d
S
d
C
d
d
d
d
y
x
x
y
x
y
x
x
2
2
2
^
1
1
1
1
1
^
sin
cos
+
=
+
=
+
=
θ
θ
Also, for bending load, derived previously,
φ
φ
φ
φ
=
−
=
=
−
=
^
2
2
2
2
^
^
1
1
1
1
^
S
d
C
d
d
S
d
C
d
d
x
y
y
x
y
y
Therefore, the relationship for the combined bending and axial loading can be written as
−
−
=
2
2
2
1
1
1
^
2
^
2
^
2
^
1
^
1
^
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
φ
φ
φ
φ
y
x
y
x
y
x
y
x
d
d
d
d
C
S
S
C
C
S
S
C
d
d
d
d
Or,
[ ]
{ }
d
T
d
=
^
ME 273 Lecture Notes © by R. B. Agarwal
4-13
DRAFT –1
Chapter 4 Beam Element
Where,
−
−
=
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
]
[
C
S
S
C
C
S
S
C
T
4.3.3 2-D Beam Element Equation for Combined Loading – Axial and
Bending – at an Arbitrary Orientation
θ
Substituting the values of
and [T] into the equation
[ ]
, we get
^
K
]
][
[
]
[
^
T
K
T
K
T
=
−
+
−
+
−
+
−
−
−
+
−
−
−
+
−
−
−
+
=
I
C
L
I
C
L
I
AS
symmetry
S
L
I
CS
L
I
A
S
L
I
AC
I
C
L
I
S
L
I
I
C
L
I
C
L
I
AS
CS
L
I
A
C
L
I
C
L
I
AS
S
L
I
CS
L
I
A
S
L
I
AC
S
L
I
CS
L
I
A
S
L
I
AC
L
E
K
4
6
12
6
12
12
2
6
6
4
6
12
12
6
12
6
12
12
6
12
12
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4.4 2-D Beam Element with combined loading Bending, Axial, and
Torsion (
θ
= 0)
The torsional loads are m
1x
and m
2x
, and the corresponding deflections are,
x
1
φ
and
x
2
φ
The torsional structural equation is:
m
1x
1
-1
φ
1x
=
JG/L
m
1x
-1
1
φ
2x
ME 273 Lecture Notes © by R. B. Agarwal
4-14
DRAFT –1
Chapter 4 Beam Element
These terms can be superimposed on the stiffness equation derived previously for the
combined bending and axial loads.
d
y
φ
y
3-D Beam Element: d
x
φ
z
φ
x
d
z
A 3-D beam element has 6 DOF at each node, and 12 DOF for each element. The
stiffness matrix can be derived by super-imposing the axial, bending, and torsion loadings
in the XY, XZ, and YZ planes. The equation is,
ME 273 Lecture Notes © by R. B. Agarwal
4-15