Transcript Finite Element Primer for Engineers

Design and strength assessment of a
welded connection of a plane frame
Structural connections
•
Structural connections of a plane frame must
be able to transfer
1) internal forces between beam and beam
2) internal forces between beam and column
3) reaction forces between column and ground
•
These are typical permanent connections and can be riveted, bolted
or welded
•
The basic criterion in the design of connections include
- assessment of their static strength and endurance
- assessment of the right transfer of the internal forces
The welded connection at point B must be designed
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Reaction and Internal moments
•Reaction forces and internal moments can be evaluate:
1) using handbook formulae for a similar structure loaded with distributed
load or concentrated load, and then applying the superposition principle.
2) applying the Principle of Virtual Work
3) by means FEM model of the frame
The suggestion is to evaluate reaction forces and internal moments by
means of handbook formulae and to compare results with results obtained
by Principle of Virtual Work or FEM analysis
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Moment for distributed load: handbook formulae*
*Manuale for mechanical engineer, Hoepli edition 1994, (in italian).
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Moment for concentrated load: handbook formulae
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Principle of virtual work for frames
 P    N d   M d   T d   M
1
all appliedloads


d
all beams s
s
s
s 


external work
internal work
1
1
1
t ,1
where
s is thecurvilinear coordinatealong theaxis of each beam
P is theappliedload on theauxiliarystructure
 is thedisplacement,on thereal structure, of thepoint where theload is applied
N1 , M 1 , T1 , M t ,1 are theinternalforceson theauxiliarystructure
d , d , d , d are therelateddeformations on thereal structure
For plane frame Mt=0 and the deformations due to the axial and the
shear forces are negligible, only the internal bending must be taken
into account
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The examined plane frame
• The plane frame is symmetric only half of the frame have to be
considered
Q/2
Q/2
p
p
B
E
B
RE
E
ME
h
RE and ME: hyperstatic
unknown
A
A
l/2
The structure is two times hyperstatic
The internal moment M(x) on the real structure is
M(x)=M0(x)+RE*M1(x)+ME*M2(x)
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Internal moment M(x) on the real structure
Q/2
p
B
E
B
Isostatic
structure
RA
A
MA
1
E
B
E 1
Auxiliary
structure n. 2
Auxiliary
structure n.1
A
1
1*h
M1(x)
M0(x)
A
1
M2(x)
M(x)=M0(x)+RE*M1(x)+ME*M2(x)
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PVW for the auxiliary structure n. 1
h
1  0   M 1, AB ( x)d AB 
0
l/2
M
1, BE
( x)d BE
0
where
1  appliedforceon theauxiliarystructure
0  displacement of point E on thereal structure
M 1, AB(x) and M 1, BE ( x)  momentson theauxiliarystructure
M AB(x)
EJ 
  rotationson thereal structure
M (x)
  BE 
EJ 
d AB  
d BE
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PVW for the auxiliary structure n. 2
h
l/2
0
0
1  0   M 2, AB ( x)d AB   M 2, BE ( x)d BE
where
1  appliedmomenton theauxiliarystructure
0  rotationof point E on thereal structure
M 2, AB(x) and M 2, BE ( x)  momentson theauxiliarystructure
M AB(x)
d AB  
EJ 
  rotationson thereal structure
M (x)
d BE   BE 
EJ 
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Hyperstatic unknown
• The system
h
l/2

1  0   M 1, AB ( x)d AB   M 1, BE ( x)d BE

0
0

h
l/2
1  0  M ( x)d  M ( x)d
2 , AB
AB
2 , BE
BE



0
0

allows the calculation of RE and ME
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FEM Analysis
20000 N/m
20000 N
Constrains:
Point A U1=U2=UR3=0
Point E U1=UR3=0
Model
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Deformed shape
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Example of results
31420 Nm
The same cross section
IPE 330 has been
used for the beam
and for the column
Moment at nodes
10370 Nm
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The cross section of the beam and of the column
• Cross sections can be choose on the basis of the bending moment only
• On each cross section act the bending moment due to the distributed
load constant and the bending moment due to the concentrated load Q
varying sinusoidally with time
Mb,tot  Mb, p  M p,Q sin wt
y
x
E
z
Mb,p
Mb,Qsinwt
Maximum of Mb,p
and Mb,Qsinwt
z
x
y
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Bending stress on the cross section at point E
• We consider the cross section at point E where both Mb,p and
Mb,Qsinwt are maximum.
• The bending stresses result linearly varying with the distance from
the neutral axis:
 b,tot, max
Aa
 b,tot 
a a a
a
a
A
A
 b,tot
 b,tot
a,max
a
m
M b, p
M b,Q sin wt
m
a
y
y
J xx
J xx

 

a
• and sinusoidally varying with time
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t
15
• It is maximum at the points that are most distant from the neutral
axis of the section
M b , p h M b ,Q sin wt h
 b ,tot , max 

J xx 2
J xx
2

 

 m, max
• The condition
 a , max
 b,tot , max  lim
where  is the safety factor and lim can be obtained from the Haigh diagram of the
material
allows the calculation of Jxx of the beam section.
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Bending Haigh diagram
a
a,f
lim
a,max
m,max
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UTS
m
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Design of the structural node
• The node between the column
and the beam, realized with a
double T section, must be
designed in order to realize a
clamped constrain.
M2
• The aim is to transfer the
boundary moment M1, from the
transverse beam to the vertical
column
ht
M1
M1  M 2  M 3  0
hc
M3
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The welded connection
• The end of the horizontal beam, upper plate, lower plate and
web are welded to the upper plate of the column
• The weld is a fillet weld type
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Moment transfer
• In the double T sections, if subjected to
flexural moment in the plane of the
web, the axial forces that originate
from the flexural moment are
transmitted by the upper and lower
plate.
• As a consequence the upper plate of
the column receives the normal forces
of the flexural moment from the
transverse beam, and deflects, except
close to the web.
• An overview of the deformations of the node is given in the figure, as result
of a finite element analysis.
• The level of deformation, in absence of any reinforcement, is quite high,
and not acceptable.
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Reinforcements
• From the previous considerations, it is intuitive that local reinforcements
are needed, to correctly transfer the flexural moment to the upper and
lower plate of the column.
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Adopted solution
•
In the adopted solution, the node is considered as a group of four beams, plus
a diagonal member, all hinged at their ends.
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• Let M be the moment to be transmitted to the column.
• Axial load on the upper and lower plate
of the beam, transferred to the
reinforcement results
M
St 
ht
• Axial load on the upper and lower plate
of the column
M
Sc 
hc
• On the diagonal AD acts the force:
S d  St
Sc
D
St
Sd
Sc
E
C
A
Sc
Sc
hc2  ht2
hc
• If the contribution of the web of the column
2
2
h

h
t
is take into account, by means of the
1   
S d  St c
coefficient :
hc
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• The comparison between the reinforced node (a) and the one
without reinforcement (b) allow to visualize their different behavior.
(a)
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(b)
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Verification of the beam-column welded joint
• In the following the verification of the welding is reported. Let the two
profiles be a IPExxx for the beam and for the column.
• The reference sections of the fillet of the welding are place as
shown in figure below.
TB
MB
• J is the moment of inertia of the resistant section of the welding
• MB and TB are the bending moment and the shear that must be
transmitted by the welded joint
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• At the edges of the fillets welding along the web, where bending
 T and LT
and shear are present, the stresses are
• So that the reference stress results:
2
 *   T2   LT
  T2
• The corresponding safety coefficients then results:

K LIM
*
• At point A (top of the horizontal fillet) only T due to bending is
present and he corresponding safety coefficients results:

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K LIM
T
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