# Beam Elements

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25 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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Beam Elements

Jake Blanchard

Spring 2008

Beam Elements

These are “Line Elements,” with

2 nodes

6 DOF per node (3 translations and 3
rotations)

Bending modes are included (along with
torsion, tension, and compression)

(there also are 2
-
D beam elements with 3
DOF/node

2 translations and 1 rotation)

More than 1 stress at each point on the
element

Shape functions

Axial displacement is linear in x

Transverse displacement is cubic in x

Coarse mesh is often OK

For example, transverse displacement in
problem pictured below is a cubic function of
x, so 1 element can give exact solution

F

Beam Elements in ANSYS

BEAM 3 = 2
-
D elastic beam

BEAM 4 = 3
-
D elastic beam

BEAM 23 = 2
-
D plastic beam

BEAM 24 = 3
-
D thin
-
walled beam

BEAM 44 = 3
-
D elastic, tapered,
unsymmetric

beam

BEAM 54 = 2
-
D elastic, tapered,
unsymmetric

beam

BEAM 161 = Explicit 3
-
D beam

BEAM 188 = Linear finite strain beam

BEAM 189 = 3
-

Real Constants

Area

IZZ, IYY, IXX

TKZ, TKY (thickness)

X)

ShearZ
,
ShearY

(accounts for shear
deflection

important
for “stubby” beams)

Shear Deflection Constants

shearZ
=actual area/effective area resisting shear

Geometry

ShearZ

6/5

10/9

2

12/5

Shear Stresses in Beams

For long, thin beams, we can generally ignore
shear effects.

To see this for a particular beam, consider a
beam of length L which is pinned at both ends
and loaded by a force P at the center.

P

L/2

L/2

Accounting for Shear Effects



2
5
3
2
2
2
2
2
2
10
1
96
2
2
0
2
2
0
2
2
IGL
E
bh
EI
L
P
U
U
U
y
h
I
V
dV
G
U
L
x
Px
M
dx
EI
M
U
s
b
xy
xz
V
xz
xy
s
L
b

2
2
3
2
3
5
6
1
96
12
GL
Eh
EI
L
P
U
U
U
bh
I
s
b
Key parameter is height
to length ratio

We can only apply loads to nodes in FE analyses

Hence, distributed loads must be converted to

With beams, this can be either force or moment loads

q=force/unit length

M

F

F

M

Goal is to ensure equivalent loads produce same
strain energy

2
4
2
3
1
2
1
1
)
(
)
(
)
(
)
(
)
(

x
N
v
x
N
x
N
v
x
N
x
v

2
3
2
4
2
2
3
3
3
2
3
2
2
2
2
3
3
1
1
1
)
(
3
2
)
(
2
1
)
(
1
3
2
)
(
x
L
x
L
x
N
x
L
x
L
x
N
x
x
L
x
L
x
N
x
L
x
L
x
N

2
2
1
1
0
4
2
0
3
2
0
2
1
0
1
1
0
2
4
0
2
3
0
1
2
0
1
1
0
2
4
2
3
1
2
1
1
0
0
12
2
1
12
2
1
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(

L
v
L
v
qL
W
dx
x
N
dx
x
N
v
dx
x
N
dx
x
N
v
q
W
dx
x
N
q
dx
v
x
N
q
dx
x
N
q
dx
v
x
N
q
W
dx
x
N
v
x
N
x
N
v
x
N
q
W
dx
x
v
q
qdx
x
v
W
L
L
L
L
L
L
L
L
L
L
L

2
1
2
1

M
v
v
F
W

2
2
1
1
2
1
2
1
12
2
1
12
2
1

L
v
L
v
qL
M
v
v
F
W
12
2
2
qL
M
qL
F

M

F

F

M

Putting Two Elements Together

M

F

F

M

M

F

F

M

M

F

F

F

2F

M

An Example

Consider a beam of length D divided into 4
elements

For each element, L=D/4

192
12
8
2
2
2
qD
qL
M
qD
qL
F

qD/8

qD/4

qD/4

qD/8

qD/4

qD
2
/192

qD
2
/192

In
-
Class Problems

Consider a cantilever beam

Cross
-
Section is 1 cm wide and 10 cm tall

E=100 GPa

Q=1000 N/m

1.
D=3 m, model using surface load and 4 elements

2.
D=3 m, directly apply nodal forces evenly
distributed

use 4 elements

3.
D=3 m, directly apply equivalent forces (loads
and moments)

use 4 elements

4.
D=20 cm (with and without
ShearZ
)

EI
qL
v
8
4
max

Notes

“Pressure/On Beams”

To view stresses, go to “List
Results/Element Results/Line elements”

ShearZ

for rectangle is still 6/5

Be sure to fix all DOF at fixed end

Now Try a Frame

F (out of plane)=1 N

3 m

2 m

Cross
-
sections

6 cm

5 cm

m
v
I
J
I
R
R
I
xx
i
o
5
max
4
4
10
59
.
2
2
4