Beam Elements

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25 Νοε 2013 (πριν από 3 χρόνια και 11 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
-
D Quadratic finite strain beam

Real Constants


Area


IZZ, IYY, IXX


TKZ, TKY (thickness)


Theta (orientation about
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

Distributed Loads


We can only apply loads to nodes in FE analyses


Hence, distributed loads must be converted to
equivalent nodal loads


With beams, this can be either force or moment loads

q=force/unit length

M

F

F

M

Determining Equivalent Loads


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
Equivalent Loads (continued)





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


Distributed load is constant


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


For adding distributed load, use
“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