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Beam Elements
Are they precise ?
Dr.-Ing. Casimir Katz
FENet Technology Workshops,
October 2004, Glasgow, Scotland.
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Cable Analysis
Cable
-H w(x) = p(x)
Solution for uniform load is aquadratic parabola or a sinh
H
p
HP1 P2Cable element
Linear between the nodes
Solution space: polygonal
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Cable Analysis
Solution Space
wh(x) = w1j1(x) + w2j2(x) + ... Solution is exact for:
Single forces
Other loadings:
Minimum of energy => equilibrium
Exact solution is obtained in the nodes!Why ?
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Greens Functions
The influence function (Greens function) for
the deformation of the cable at a point is thecable polygon created by a point load P=1 at
the point of interest.
If the FE-System is able to model this solution
exactly, the solution will be exact for this
value. In all other cases, i.e. if the Greens function
is not within the possible solution space,
one has an approximate solution only.
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What are Beam Elements ?
A 3D Continua with a length >> width / height
Simplification of the solution space(Bernoulli-Hypothesis, persistence of shape)
Simplification for manual analysis
(gravity centre, principal axis, shear etc.)
Superstition:
Beam elements are simple
Beam elements are exact
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Problematic continuous beam
Mechanic is not consistent
Plan view
Moments
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Sections
Normal stress
0 y z
y zx x
u u z y
u u E E z y
x x x x
= +
= = = +
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Sections Shear Stress
V is only valid if the normal
force and the section areconstant along the axis
Z is only valid if the section
is not multiple connected The shear stress is not
necessarily constant along
the width b For biaxial bending or non effective parts of
the section, the equation becomes morecomplex (Swains formula)
V Z
I b =
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An other strategy:
a deformation method
2 2
2 2
:
0
xx y
x
x z
x
x y y x z z
wG zy x
w
G yz x
w wG w G
y z x
B o u n d a r y C o n d i t i o n s
n n
=
= +
= + =
+ =
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Unit Warping
SOFiSTiK AG, 85764 Oberschleiheim, Bruckmannring 38, Tel:089/315-878-0UP (V11.05-21) 24.10.2002
M 1 :X
8.00 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00
Querschnitt Nr 12 VollquerschnittVerwlbung 1 CM = 50000.000 [cm2]
-20525.-12353.
12300.20471.
19815.
-9150.
15582.
24686.
33324.
41220.
13931.
-14264.
-25462.
-32880.
-26721.-16045.
16024.
267
00.
32
859.
254
41.
14243.
-13952.-41244. -33349.
-24712.-15608. 9123.
-19869.
12390.
-12446.
-21149.
-11843.
9088.
14050.
-14131.
-9162.
11781.
21093.
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Shear stress from Mt
M 1 : 68X
Z
8.00 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 m
Querschnitt Nr 12 Vollquerschnitt
Schub aus MT=1000.00 1 CM = 0.05 [MPa]
0.050.0
6
0.06
0.06
0.06
0.05
0.02
0.05
0.05
0.05
0.03
0.01
0.010.01
0.010.01
0.010.030.06
0.060.06
0.060.06
0.060.06
0.03
0.010.01
0.01
0.01 0.01 0.010.03
0.05
0.05
0.05
0.02
0.040.04
0.040.04
0.040.03
0.02
0.02
0.03
0.040.04
0.040.04
0.040.04
0.04
0.03
0.02
0.02
0.03
-0.04-0.03-0.020.020.030.04
-0.02
-0.03-0.03
0.030.03
0.02
-0.05-0.04-0.030.030.040.05
-0.06
-0.05
-0.04
0.04
0.05
0.06
-0.06
-0.05
-0.04
0.04
0.05
0.06
-0.06
-0.05
-0.04
0.030.040.05
-0.06
-0.05
-0.04
0.030.040.05
-0.06
-0.05
-0.04
0.04
0.05
0.06
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M 1 : 68XYZ
Y 8.00 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 m
Z
2.0
0
Querschnitt Nr 12 Vollquerschnitt
Schub aus VZ=1000.00 1 CM = 0.10 [MPa]
0.15
0.10
0.03
-0.03
-0.10
-0.15
-0.04
-0.2
0
-0.23
-0.22
-0.17-0
.09
-0.07
-0.05
-0.03
0.020.050.060.08
-0.09-0.05-0.020.020.050.09
-0.08-0.06-0.05-0.02
0.03 0.05 0.07 0.09
0.17
0.22
0.23
0.20
0.04
0.07
0.04
-0.04
-0.07
-0.14
-0.21
-0.23
-0.22
-0.19-0.12
-0.06
0.060.12
0.19
0.22
0.23
0.21
0.14
0.210.220.22
0.220.22
0.21
0.130.170.23
0.23
0.17
0.13
0.210.21
0.200.200.21
0.21
-0.08-0.08-0.08
0.13
0.150.14
-0.14
-0.15
-0.13
0.200.210.21-0.08
-0.07
-0.08
0.200.210.21
0.13
0.150.14
Shear stress from Vz
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Shear deformation Theory of Timoshenko/Marguerre (not exakt!)
Principal Axis of shear deformation
zy
z
y y
VG A
w
x
=
= +
=
z
y
zyz
yzy
z
y
V
V
GAGA
GAGA*
11
11
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Shear in haunched beams
h =100 cm
p = 10 kN/m
L = 10.0 m
h = 50 cm
h =100 cm
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FE-Analysis
tau-xy
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FE-Analysis
tau-xy in vertical section
10.62
57.90
52.10
53.96
0
0
0
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Shear in haunches
sig-x
tau-xySchnitt senkrecht
zur SchwerachseSchnitt senkrecht
zur Referenzachse
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FE-Analysis
tau-nq in perpendicular section
-28.
-53.
-32.
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Shear in haunched beams Example for a shear reduction along the
provisions of design codes Shear force V 40.0 kN
Yielding a faulty shear stress 63.2 kN/m2
Reduction of shear by M/d* tan 5.6 kN Reduced shear stress 54.9 kN/m 2
FE-shear stress 53.0 kN/m 2
Although the engineering approach has a
completely different view, the maximum stress
is quite good, distribution is faulty however.
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FE of a haunched beam
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Reference of forces ? Gravity axis with N + V
Results suited for
design Gravity axis with D + T
Similar as for 2nd
order Theory
Superposition offorces
General reference axis
Superposition offorces if acomposite sectionchanges by in situconcrete ortendons.
N Q
D
T
D
T
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Excentricities at the end points
Interpolation u0 linear, v,w,x,y cubic
Displacements within section
Strains from derivative (position of gravity centre is not constant!)
Principles of the FE-Beam
u u z yu u j z y
i i yi i zi i
j yj j zj j
0
0= + = +
( ) ( )u u z z y y y s z s= + 0
( ) ( )
x o y s z s
y s z s
u
xu z z y y
z y
= = +
+
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Example of a haunched beam
h =100 cm
p = 10 kN/m
L = 10.0 m
h = 50 cm
h =100 cm
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Results for haunched beam
w[mm] Ne[kN] Nm[kN] Mye[kNm] Mym[kNm]
Inclined axis
C-Beam (1 element) 0,397 -80,50 -78,00 -73.58 31,91
C-Beam (8 elements) 0.208 -46,30 -43,80 -94,87 19,17
FE-Beam (1 element) 0,172 -39,80 -37,30 -93,65 22,00
Fe-Beam (8 elements) 0.206 -45,80 -43,30 -95,02 19,14
Horiz. reference axis
FE-Beam (1 element) 0.168 -37,90 -37,90 -93.01 22,52
FE-Beam (8 elements) 0.204 -44,20 -44,20 -94.85 19,10
- 3 7 . 8 8 - 3 7 . 8 8
- 9 3 . 0 1
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The precision of the
FE beam (bending)
19.287615.430119.2876Cent. deflection
41.14741.14741.147End Rotations
281.25281.25281.25Max. Moment
2 Elements1 ElementExact
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Bending precision
The theoretical solution is a polynom of 4th order
The shape functions are cubical splines The highest
symmetric ansatz function is the quadratic parabula
As the influence function for the nodal displacementsare cubic functions, the deformations and rotations in
the nodes are exact. The moments and shear forces are also exact.
The only values which are not are the displacements
between the nodes (which are of less interest) For a buckling analysis there are the same principles,
but know the buckling force (which has a high
importance!) is not ok for a simple element.
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The precision of the
FE beam (Buckling analysis)
52879
53249
53550
-
Euler IV
Numerical
132134 Elements
132193 Elements
133122 Elements
5284816065133121 Element
Euler IV
Exact
Euler II
Numerical
Euler II
Exact
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Warping Torsion and Shape
deformation
> 6. Degrees of freedom per node
More forces and moments:secondary torsional moment Mt2 etc.
Pure bending = 84.3 N/mm2
Warping / 2nd Order Torsion = 136.1 N/mm2
240 kN2.625 kN/m
l = 16 m
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Conclusion
Beam elements are not simple in theory
Beam elements are easy to use
There are a lot of modelling errors possible