Precast Prestressed Concrete Horizontally Curved Bridge Beams
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Curved Origami beams January 2012 Curved Origami beams 24 April 2012
Curved Origami beams
Laurent HUMBERT Statuseminar SAH 2012
24 April 2012
Hani BURI
Yves WEINAND
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1) Origami beam geometry
2) Experimental investigation
Outline
3) Numerical study
- Origami principles
- Parameterization
Conclusion
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1) Curved prototypes geometry
Origami principles :
Reverse fold as a reflection about a plane
a) with planar elements
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Curved Origami beams January 2012 Curved Origami beams 24 April 2012
1) Curved prototypes geometry
Origami principles :
Reverse fold as a reflection about a plane
a) with planar elements
IBOIS Laboratory for Timber Constructions
Curved Origami beams January 2012 Curved Origami beams 24 April 2012
1) Curved prototypes geometry
Origami principles :
Reverse fold as a reflection about a plane
b) with curved elements
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Curved Origami beams January 2012 Curved Origami beams 24 April 2012
1) Curved prototypes geometry
Origami principles :
Reverse fold as a reflection about a plane
b) with curved elements
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Geometric Design: Construction by two Profile Curves
a) Folded plate structures b) Curved origami figure
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Curved origami beam
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Roofing structure composed of several elements
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• Slowly varying height along arc E of constant curvature (plane x-z), similarly
for the beam width
Geometrical considerations:
• Beam axis x directed along the line of centroïds
0 x L
cross section at x=L/2
cross section at x= 0
• Cross sections parallel to the y-z plane, delimited by the vertices a, b, c, d at
the mid-thickness of the panel
Beam length L [mm]
Global coordinate system xyz
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h0, h1, h2 : beam heights at x = 0, (L-S)/2, L/2 respectively
Elevation (plane x-z):
E
2 0max
2z
h hf
Radius r of arc E:
2
22
max4
z
Lr r f with
2 2
max
max
4
8
z
z
f Lr
f
solving,
E
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Half-height evolution :
max sin 1z zf f r x
with the variable part,
0
2z
hh x f x
max
0 0
2
z
z z
f
f L f
such that
0
0 max 2
0 2
2 2 2z
h h
h L h f h
Not at scale !
2
acosL x
xr
where
L
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Half-width evolution :
tan2
y zf x f x
max max tan2
y zf f
22 max
2y y y
ww x f f f x
with
2 22 tan cotan
2 2 2y
h hf
max
0 0
2
y
y y
f
f L f
such that
2 2 max
2 2
0 2
2 2
y y
y
w w f f
w L w f
max max 2, 02
y z yf f f when
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y
z
a
a
a
Vertices coordinates:
t : panel thickness [mm]
y
z
b
b
by
z
c
c
cy
z
d
d
d
where,
y
z
a x w x
a x h x
2 cotany y z
z z
b x a x a x
b x a x
y y
z z
d x a x
d x a x
y y
z z
c x b x
c x a x
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• Parametrization of the curved beam:
Li
S
- length/span ratio :
- height at end / height at mid-length ratio :
0
2
hq
h
- height / width ratio at mid-length:
2
2
hp
w
- Reflection angle
(loading consideration)
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- “Aspect” ratio :
max2 2 max,
hk h w h
S half the perimeter at mid-length
2
2
hp
w
Same value of k for the two configurations
Cross section at x=L/2 (=90°)
1p 3p
2w
2h
square rectangular
maxh
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k=0.2
p=1
q=1/4
= 90 °
p=1
q=1/4
= 90 °
k=0.1
Illustration:
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20
max
hp
halternatively,
p=3
q=1/4
= 90 °
p= 3
p0=0.75 where 0
01
pp
p
k=0.1
p=3
q=1
= 90 °
k=0.1
constant cross section :
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Limiting cases :
q=0
0 1
p
p
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2) Experimental investigation
• Curved prototypes of length 2m :
OB1, OB2, OB3
Prototypes:
• Screwed and glued connection between the panels
• Okoumé plywood panels (three plies) of thickness t = 6 mm
1 2
3
4 6 7
mid-length
8
5
9 10
• 10 Stiffeners - Okoumé plywood of thickness 22 mm :
4-5-6-7 linked together
2, 9 placed at support position
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mid-surfaces of stiffeners and panels depicted
S = 1778 mm
L = 2000 mm
h0 = 33.33 mm
w0 = 340.71 mm
h2 = 200 mm
w2 = 170.59 mm
Parameters values:
1.1249i
0.166q
1.172p
78 44
0.208k
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Experimental setup:
Load cell 1
Load cell 0
S=1778mm
LVDTs
LVDT
W+B 300kN loading system Four point bending test :
transducers locations (i=1..7)
(2)
F0 F1
(3) (4) (5) (6)
(7) (1)
290 290 309 290 290 309
three specimens OB1, OB2, OB3
Load cell 1 Load cell 0
S=1778mm
LVDTs
LVDT LVDT
OB1 and OB2
OB3
LVDTs
transducers locations (i=1..5)
(1)
F0 F1 (2) (3) (4) (5)
444 445 445 444
S/3 S/3
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F0 F1 F0+F1
OB1 6.7 6.4 13.1
OB2 8.5 8.1 16.6
OB3 5.8 5.7 11.5
Maximum load (KN) :
- Loading process:
- Imposed grip displacement at 0.05mm/s
OB1
OB2
OB3
Failure :
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Measurements of deflection :
Increasing load
OB3 OB2
OB1
F0+F1 (KN)
(1) (2) (3)
OB1 2.02 4.10 8.08
OB2 1.99 4.11 8.04
OB3 2.07 4.00 7.96
(1)
(2)
(3)
Three levels of loads
examined
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3) Numerical study – 1D simplified model
• Frame work : bending analysis of elastic beams with general cross section variation :
assumptions : - classical Euler beam theory
- homogeneous material
- rigid connections between the panels
• Model implemented in Matlab :
- Finite Element formulation not used here
- handling problems with - depth/width beam variations
- any number of concentrated/linearly varying distributed loads
- general end conditions
- interior supports
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0
0
0 0
0n
n
x xx x
x x x x
• Implementation
-Use of singularity functions of order n (=-1,0,1):
satisfying,
1
0 0
1
1
xn n
o
x x dx x xn
n= -1 : appropriate for describing a concentrate load
Nf concentrated loads Fj acting at positions x fj
Nr ramped loads varying (linearly) between j jp x q
-Prescribed (general) external loads :
1 0 0 1 1
1 1
f rN N
j j
e j j j j j j j j
j j j j
Q Pw x F x f P x p Q x q x p x q
q p
x
F1
F2
F3 P1 Q1 P2 Q2
0 x L
L : beam length
L
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Moment & deflection:
'm x v
Shear & load:
1
0
1
0
0
1
.
'
',
s
s
N
e j j
j
x
e j e
N
j
j
e
v x q x w x R x r
v x v v xd
v x w x dxdx
R x r
NS reaction forces Rj (to be determined) at the support positions x r
j
- System of differential equations :
1
0
1 0
0 ,sN
j e ee
x
j
j
m mx m v x m x R x x vr x dx
+ four end conditions imposed at x=0 and x=L
x
R1 R2
R3
''
m xy x
E I x
varying cross section moment of inertia I(x) ,
Young modulus E
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a= S/3
L
F1 F0
a a
S
• Loading configuration (four-point bending) :
• Simulated geometry:
L (mm) 2000
i 1.125
q 0.166
p 1.172
78°44
k 0.208
Young modulus E=2398 MPa
(from Thebault company)
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Normal and shear maximum stresses evolution :
Simulation for F0+F1=2KN
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Deflection curves
Comparison between experimental & numerical results
S
Numerical model
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Conclusion
• Efficient parameterization of the origami beam geometry
• Experimental & numerical characterization of the mechanical
behavior of (small) origami beams under static bending load
• Application of origami principles to curved beams with
potential architectural interest
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• Prospects:
Parameters values:
i =1.1
q=1
p=5.666
=90
k=0.05
i =1.1
q=0.176
p=5.666
=90
k=0.05
i =1.1
q=1
p=1.609
=90
k=0.05
S = 4000 mm
L = 4400 mm
t = 15 mm
Larger prototypes with different section profiles and material
Introduction of semi-rigid dove-tail joints (no stiffeners)
Numerical investigation of the joints mechanical
behavior