Local Bending Stresses in Stay Cables with an Elastic Guide
Antonio Caballero, Dr., Head of Research and Development; Marcel Poser, CEO; BBR VT International Ltd, Schwerzenbach,
Switzerland. Contact: info@bbrnetwork.com
Abstract
Cable-stayed bridges, like no other civil engineering structure, combine functionality and exotic architectural design that are often used as a symbol of distinction
for the city or even for the region where they are situated. However, their attractive design requires special attention for multiple factors such as the occurrence
of bending stresses in the stay cables due to vibration phenomena, displacements
and rotations at the deck or pylon and construction tolerances. Therefore, in this
scenario, stay cables are normally subjected to bending stresses at the anchorages
or when passing through a saddle.
In this paper both fixed end and elastic guide are considered and a set of analytical
solutions is given. The development of the analytical solutions is accompanied by
a detailed description and thorough discussion of some particular observations.
Keywords: bending; stay cable; elastic-guide; guide deviator.
Introduction
In comparison with other structural
elements, stay cables are characterised
by possessing a very large structural
slenderness ~ A I , in which A is the
area and I is the inertia. This particularity makes them very flexible under the
action of distributed loading normal to
their axial configuration, and almost
precludes the appearance of bending
stresses under moderate rotations of
their cross section.
However, stay cables might exhibit local
bending stresses at certain points—at
anchorages—or when passing through
a saddle. In these situations, bending
stresses might be of the same order
of magnitude as the axial stresses and
require a specific analysis.
The fib bulletin1 and PTI2 specify
some recommendations with respect
to admissible angular deviations but
do not provide any tool to evaluate
bending stresses in stay cables. The
evaluation of the bending stresses is
the responsibility of the designer, who
for the simplest cases, such as that of
a cable attached to a fixed end or a
cable going through a saddle, might
find specific formulas in Refs. [3–5]. In
Peer-reviewed by international experts and accepted for publication
by SEI Editorial Board
Paper received: July 12, 2009
Paper accepted: March 27, 2010
254
Scientific Paper
the particular case of a cable attached
to a fixed end without a guide, bending
stresses are usually evaluated using the
expression, σ Bmax = 2α λ E fUTS , where
a represents the angular deviation, E
the elastic modulus of the stay cable, l
a loading parameter and fUTS the ultimate tensile strength of the stay cable
material.
In this context, it is very important to
note that, as shown in Fig. 1, a correct evaluation of the bending phenomenon in stay cable systems should be
performed taking into consideration
the deviation angle due to construction tolerances, deviation angle due
to loading and the most unfavourable
local deviation angle.
The stay cables to which the derived
theories are directly applicable are the
so-called parallel strand stay cables,
where the cables are made up of a
bundle of parallel seven-wire strands,
enclosed in a high-density polyethylene (HDPE) sheath of circular cross
section. The individual strands generally have a diameter of 15,7 mm
(0,62′′) and are of low-relaxation grade,
with a nominal cross-sectional area of
150 mm2 and a minimum guaranteed
ultimate tensile strength (GUTS) of
1860 MPa. The strands are individually
galvanised, waxed or greased and
individually sheathed with an HDPE
coating, providing each strand with an
individual multilayer protection system. Furthermore, the strands remain
independent of each other, as the
remaining free space inside the round
HDPE sheath is not injected.
However, the theories can be equally
applied to other types of stay cables,
such as parallel wire systems, stress
bar as well as bonded-type stay cables,
where the individual tensile elements
are bonded to each other. In the case of
bonded cables, the applicable moment
of inertia for the complete cable has to
be chosen appropriately.
Cable Attached to a Fixed End
Figure 2 shows the configuration of
an inclined cable anchored to a fixed
end. The stay cable is considered to be
loaded only in the axial direction T,
whereby its self-weight is neglected.
The displacements at the anchorage
are assumed to be fully restrained.
The reactions at the anchorage are the
horizontal force H, the vertical force
VFE, and the moment MFE, where the
suffix FE refers to the fixed end. Note
that the structure shown in Fig. 2 is
non-isostatic owing to the forced cable
inclination.
The solution for this configuration is
given in the Appendix, Eqs. (A1–A6).
However, here the final solutions for
the forces at the anchorage are given:
Single strand/wire
Most unfavorable local deviation angle
Fixed end
Deviation angle due to tolerance
Guide
Deviation angle due to loading
Fig. 1: Typical configuration at an anchorage when the cable is conducted by a fixed guide
Structural Engineering International 3/2010
– Under the effect of loading that
causes small curvatures to the strand,
i.e. a small, wires do not slip with
respect to each other and, therefore,
the overall global stiffness approximately is the one defined by the
entire cross section as a single unit.
From now on, this will be referred to
as a strand configuration.
– Under the effect of loading that
causes large curvatures (i.e. near
anchorages), the wires might slip
with respect to each other. In this
situation, all wires behave independently of each other and the overall
global stiffness is the one defined
by the individual contribution of
all wires as independent elements.
From now on, this will be referred to
as a wire configuration.
T
y
y(x)
H
a
MFE
x
VFE
Fig. 2: Geometrical configuration and
forces for an inclined cable anchored to a
fixed end
H = T cos (α )
VFE = T sin (α )
MFE = EIT sin (α ) tan (α )
= EIT f (α )
(1)
All the force components are expressed
in terms of the inclination of the stay
cable a. In Fig. 3 (left), the normalised
moment at the anchorage is plotted
versus the inclination of the cable. The
diagram in the figure shows that MFE
varies linearly up to ~75°, after which
the moment at the anchorage increases
exponentially. In the linear region, the
increase of the moment can be roughly
estimated as 30% for every 15°. Figure 3
(right) shows the distribution of
moment normalised by MFE along the
length of the stay cable. The curve has
been obtained for a single wire loaded
up to 30% of its maximum tensile
strength and an inclination of 3°.
This change in the mechanical behaviour from strand configuration to
wire configuration leads to differences
in terms of the normal stress distribution and maximum bending stress. In
order to evaluate the effect on the
maximum bending stress, a seven-wire
strand cross section is analysed under
the effect of pure flexure. For the sake
of simplicity, the analysis only considers normal stresses caused by the
bending moment, whereas other possible sources such as the effect of the
helical curvature of the external wires
or the frictional contact between the
external and the centred wire have
been disregarded.6–8 The inertia of the
strand configuration, being hexagonlike, is known to be independent of the
helical angle q of the external wires
around the centred one. However, the
distance from the extreme external
fibre to the neutral axis varies with the
angle, leading to the scaling function
s(q), Fig. 4.
The bending moment is shown to
decrease exponentially with distance.
In particular, at a distance of 100 mm
from the anchorage, the moment is
only 1,5% of MFE, reflecting the local
nature of bending stresses in cable-like
elements.
Equation (1) also shows that the bending moment, and hence the bending
stress as well, is a function of the global
stiffness EI, which in a parallel strand
stay cable might also depend on the
deviation angle a:
The maximum bending stress for both
strand configuration and wire configuration are given by the expression in Fig. 4. In the expression, s(q)
is a scaling function that depends on
8
the local rotation of the cross section
with respect to the axis. The maximum
bending stress for the strand configuration, the red line, is shown to be
dependent on q, whereas the maximum bending stress for the wire configuration, the blue line, is shown to be
independent of it. Furthermore, for the
strand configuration, the helical rotation of the external wires around the
centred one leads to an increase of the
maximum bending stress compared to
wire configuration. In particular, for
the most unfavourable angle, q = 30º,
the maximum bending stress is shown
to be 13,5% higher. On the other hand,
it can be seen that the commonly used
expression given in the introduction
agrees with the wire configuration,
which according to the expression in
Fig. 4 might underestimate the real
bending stress for small deviation
angles. However, common stay cables
are often waxed or greased—and
are prone to slip with respect to each
other—which might make it difficult
to preserve the strand configuration
under the effect of even small/moderate deviation angles. Finally, establishing the limiting deviation angle at
which the transition from one configuration to the other takes place would
require very complex expressions
and is beyond the scope of this work.
Hence, for the sake of simplicity and
also to present the results in an objective manner, from now on bending
stresses will always be referred to as a
single wire. Subsequent to this assumption, the bending stress in a wire is:
s Bmax = 2 lEfUTS sin α tan α
(2)
in which fUTS is the steel’s ultimate
tensile strength and λ is the axial loading parameter:
s
T
λ= T =
(3)
fUTS A fUTS
Note that for a single wire s(θ) is constant and is equal to 2. From Eq. (2), it
can be seen that the maximum bending
1
7
0,8
6
M(x)
MFE
MFE
EIT
5
4
3
2
0,6
0,4
0,2
1
0
0
15
30
45
a
60
75
90
0
0
0,05
0,1
0,15
0,2
x (m)
Fig. 3: Evolution of the normalised moment versus inclination of the stay cable (left); normalised moment distribution (right)
Structural Engineering International 3/2010
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255
2,2
y’
θ x’
s(q)
2,1
2
sBmax = s(q) lEs fUTS
1,9
sin a tan a
Strand configuration
Wire configuration
1,8
0
15
30
45
60
75
90
(°)
Fig. 4: Representative strand cross section
stress is independent of the size of the
wire and only depends on the loading
parameter and on the deviation angle.
Similar to Eq. (3) , the bending loading
parameter can be defined as:
σ max
β= B
fUTS
(4)
By definition, g = λ + β ≤ 1 (normalised
total stress). Figure 5 plots the evolution of the normalised total stress
versus the factored axial loading (Lf
× l) for a set of deviation angles, a ∈
[0°–3°] according to Eq. (2), where Lf
is the load factor that, depending on
the national standards and on the final
load combination, typically ranges
between 1,35 and 1,50.
Axial service loads for a stay cable
range typically between 30 and 50%
of the ultimate tensile strength, with,
for example, 50% being the service
limit states (SLS) stipulated by fib.
When verifying the ultimate limit state
(ULS), typical factored axial loads
range between 40 and 70%. fUTS for
the tensile element can typically be
considered as the characteristic tensile strength of the stay cable, and
From Fig. 5, it can be seen that γ
increases very quickly for relatively
small angles. Angles that are beyond
~1,0° (for Rf = 1,50) or ~1,5° (for Rf
= 1,35) lead to an inadmissible stress
level if no action is taken (the use of
a cable guide). Note that this graph
is universal for any stay cable consisting of a set of circular wires that work
separately.
Since the admissible stress region leads
to very small angles, Eq. (2) can be further simplified :
(5)
σ Bmax = 2α l E fUTS
From this equation, the maximum possible angle can be easily obtained:
α max ≤
γ −λ
2 λ E fUTS
(6)
Limiting the axial loading parameter
from the range of 65% (Rf = 1,5) to
75% (Rf = 1,35), Eq. (6), leads to the
range of maximum angles. Figure 6
plots the maximum rotation angle for a
wire in terms of the axial stress at which
the cable is loaded. The lowest safety
factor, Rf = 1,35, might be interpreted
as the delimiting boundary between
admissible and inadmissible rotation
angle region. Accordingly, the maximum rotation angle for a fixed-end
configuration might be established as
~1,25° to 1,5°, depending on the axial
stress and on the safety factor chosen
at the project design stage.
Cable through an Elastic Guide
In the previous section, it was shown
that, when attaching an inclined stay
cable to a fixed end, the bending
moment at the anchorage grows
quickly, with the inclination of the
cable generating a concentration of
bending stresses near the anchoring
zone. To reduce these bending stresses,
the cable might be passed through a
guide system. Generally, guide systems
have been considered to have infinite
stiffness (i.e. the guide system does not
experience any displacement because
of the interaction with the stay cable).
However, this assumption may not
be correct in certain circumstances in
which the guide, coupled to a damper,
is placed in a steel pipe and far from
the anchorage. In such a scenario, the
flexibility of the entire set-up might
lead to non-negligible movements,
which may affect the performance of
the guide.
In this section, the entire guide system—guide + support (steel pipe)—is
analysed. In particular, the behaviour
of the whole system is assumed to be
always elastic and conceptually similar to a vertical spring of stiffness S.
Figure 7 shows the configuration of
5
Rf = 1,35
4,5
Rf = 1,50
4
Inadmissible
3,5
3
amax
resistance factors in accordance with
national standards shall be applied
to find the limiting design strength.
Such resistance factors (Rf) depend
on national regulations, but usually
take values between 1,35 and 1,50. The
combination of both boundaries leads
to an admissible stress region, which is
depicted with blue shading in Fig. 5.
2,3
2,5
2
1,5
1
0,5
Admissible
0
0
1
0,1
0,2
0,3
0,4
Lf ⋅l
0,5
0,6
0,7
0,8
Fig. 6: Maximum angle versus the loading parameter for a given maximum normalised stress
0,8
Rf = 1,3
Rf = 1,5
0,6
0°
g
0,1°
0,2°
0,4
T
y
0,5°
1°
0,2
2°
3°
H
0
min
0
0,2
max
0,4
0,6
0,8
VBG
1
Lf ⋅l
Fig. 5: Evolution of the normalised total stress versus the loading parameter for different
angles
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y(x)
MEG
MBG
a
S
ex
x
VEG
Fig. 7: Geometrical configuration and
forces for an inclined cable passing
through a guide
Structural Engineering International 3/2010
the cable and the location of the elastic
guide system. The stay cable has a fixed
inclination far from the guide, and the
displacements at the anchorage are
fully impeded. The vertical displacement of the stay cable at the guide is
unknown and controlled by the stiffness of the fixing system used to attach
the guide. The cable, however, is still
free to rotate and to move axially
through the guide.
Under these considerations, the reactions at the anchorage and at the guide
are H, VBG, MBG and VEG, respectively.
Suffixes BG and EG refer to the beginning and end of the guide.
The solution for this configuration is
given in the Appendix, Eqs. (A7–A13),
where finally the forces at the anchorage and at the guide, Eqs. (A11) and
(A13), have been expressed in terms
of the forces at the anchorage for the
fixed-end configuration and the scaling
functions β ey , β MS and β MS
′ .
The evolution of the relative shear
forces in the stay cable at the location of the anchorage VBG VFE and at
the guide VEG VFE is shown in Fig. 8
(left and right, respectively). Plots are
obtained for a single wire of diameter
5,2 mm, l = 0,3 and a = 3°. Both the diagrams are plotted in terms of the horizontal distance between the anchorage
and the guide ex for different stiffness
relations between the guide support
and the cable, S/EI = [0, 1e3, 1e4, 1e5,
5e5 and 1e8].
In general, the vertical reaction at the
location of the anchorage decreases
as the distance between the guide
and the anchorage increases. The
opposite behaviour is noticed at the
location of the guide. In terms of the
stiffness of the support of the guide,
the presence of the guide does not
have any influence when the stiffness is set to zero. In this specific
case, the shear force at the anchorage
is equal to that in the fixed-end configuration, and at the location of the
guide there is no force. With subsequent increase of the values of S/EI,
the relative vertical reaction at the
anchorage decreases but increases
at the guide. For very high stiffness,
1,2
S/EI > 5e5, the values of the reaction
at the anchorage might reverse its
sense of actuation.
The bending moments in the stay cable
at the location of the anchorage and
the guide are given in Fig. 9, left and
right, respectively. Similar to the beha–
viour explained above, the presence of
the guide reduces the moment at the
anchorage. All the curves show that
the bending moment at the anchorage
decreases as the distance between the
guide and the anchorage increases. This
reduction is very pronounced when the
distance is short and tends to stabilise
for long distances.
The final value of the bending moment
at the anchorage also depends on the
stiffness of the guide’s support. In
general, the stiffer the support, the
smaller the bending moment at the
anchorage. In particular, for very
stiff supports, the sign of the moment
at the anchorage might reverse.
However, for very flexible supports,
the efficiency of the guide might be
very poor.
1,6
1,E+04
1,E+03
1,E+05
5,E+05
1,E+08
0
1,0
1,4
0,8
1,2
0,6
0,2
VFE
VEG
VBG
VFE
1,0
0,4
0,6
0,0
0,4
−0,2
−0,4
−0,6
0,00
0,8
0,20
0,40
0,60
0
1,E+04
1,E+03
1,E+05
5,E+05
1,E+08
0,80
0,2
0,0
0,00
1,00
0,20
0,40
0,60
0,80
1,00
ex (m)
ex (m)
0,8
1,0
0,7
0,8
0,6
0,6
0,5
MEG
1,2
0,4
0,2
0
1,E+04
5,E+05
1,E+03
1,E+05
1,E+08
0,4
0,3
0,0
0,2
−0,2
−0,4
0,00
MFE
MBG
MFE
Fig. 8: Evolution of the relative reactions at the anchorage location (left) and guide (right)
0,20
0,40
0,60
0
1,E+04
1,E+03
1,E+05
5,E+05
1,E+08
0,80
1,00
0,1
0,0
0,00
0,20
0,40
ex (m)
0,60
0,80
1,00
ex (m)
Fig. 9: Evolution of the relative bending moments at the anchorage location (left) and guide (right)
Structural Engineering International 3/2010
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257
1,2
0,5
0,8
0,4
0,6
0,3
MEG
MFE
MBG
MFE
0,6
ex = 0,25 m
ex = 0,50 m
ex = 1,0 m
1
0,4
0,2
0,2
0,1
0
0
−0,2
1,0E–09
1,0E–06
1,0E–03
1,0E+00
1,0E+03
1,0E+06
1,0E+09
−0,1
1,0E–09
ex = 0,25 m
ex = 0,50 m
ex = 1,0 m
1,0E–06
1,0E–03
Relative stiffness (S/EI)
1,0E+00
1,0E+03
1,0E+06
1,0E+09
Relative stiffness (S/EI)
Fig. 10: Evolution of the relative moments at the anchorage (left) and at the guide (right)
In particular, for a guide located 1 m
from the anchorage and a relative stiffness of S/EI = 1e3, the final bending
moment in the cable at the location of
the anchorage will still be 60% of that
for the fixed-end configuration.
The bending moment at the location
of the guide exhibits a tendency to
increase with increase in both distance
ex and S/EI. For very stiff supports
(S/EI > 1e5) and conventional guide
installation distances (ex ≥ 0,5 m), the
bending moment at the location of the
guide is shown to be always not less
than 50% of the bending moment for
the fixed-end configuration. A more
flexible support might lead to smaller
bending moments at the guide, but
there will be an increase of bending
stresses at the anchorage. Note that
the moment at the location of the
guide, for very flexible supports, might
increase when ex is very small (<100
mm) owing to the bending moment at
the anchorage.
Finally, although not shown here, the
evolution of the previous diagrams
seems to be independent of the loading parameter l, and identical tendencies and similar values are obtained for
normal stress values in the range of 0,4
≤ l ≤ 0,5.9
Figure 10 plots the evolution of the
normalised bending moment at the
anchorage (left) and at the guide
(right). The evolution is plotted against
the relative stiffness S/EI. Curves
were obtained for the parameters l =
0,3 and a = 3°. In both the diagrams,
three different locations of the guide
were considered: 0,25, 0,50 and 1,0 m.
In general, the following tendencies
might be observed:
– When S/EI ≤ 1e2, it can be considered that the guide system is fully
flexible and the solution of the
fixed end without guide is valid. The
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Scientific Paper
error of this approximation, for the
examples presented here, is less than
1,4%.
– When S/EI ≥ 1e7, it can be considered that the guide system behaves
as a fixed support. The error of this
approximation is always less than
1,3%.
– The behaviour between the above
limits might be considered linear in
a logarithmic plot as seen in Fig. 10.
Regression analysis has shown that
all curves exhibit a similar slope
(positive or negative values are
the same), with slight differences
depending on the horizontal position of the guide. Normal values of
the slope for a logarithmic plot in
the linear region are ~0,18 ± 0,02.
Discussion and Conclusions
The first section of this paper, for the
fixed-end configuration, showed the
relevance of bending stresses in stay
cables when no action is taken. It has
been shown that small angular deviations of 1° or 1,5° are sufficient to
cause yielding of the material.
A new formulation to evaluate bending stresses at a stay cable with an elastic support at the guide is also given.
The results obtained from this new
formulation may be summarised as
follows:
– When the stiffness of the support is
very high (S/EI→ ∞), the bending
stresses in the cable at the anchorage might be reduced to below
5% of that for the fixed-end configuration. Simultaneously, local
bending stresses in the cable
start to emerge at the location of
the guide. The values of the new
bending stresses decrease with
the distance between the anchorage and the guide, but they were
shown to have a minimum near
to 50% of the bending stresses
obtained at the fixed-end configuration. Therefore, the installation of
a stiff guide reduces the bending
stresses in a stay cable by up to 50%
at most. The above results are independent of the deviation angle and
of the axial loading in the cable.
– If the guide is assumed to have a
finite stiffness, the bending moment
at the guide might be smaller than
50%, but always at the cost of an
increase of the bending stresses at
the anchorage.
Appendix
Cable Attached to a Fixed End
To solve this structure, first the distribution of moment M (x) is formulated in
terms of the forces at the anchorage:
M ( x ) = Hy ( x ) − VFE x + MFE .
(A1)
Exploiting the relationship between
the moment and the curvature Eq.
(A1) can be rewritten as follows:
M ( x)
V
M
H
y ( x ) − FE x + FE
= y′′ ( x ) =
EI
EI
EI
EI
(A2)
and also obtains a second-order differential equation (A3):
y′′ ( x ) = Ay ( x ) + Bx + C
(A3)
Equation (A3) leads to the following
general solution in terms of the constant coefficients, A, B and C:
( Ax) +
C
B
b cosh ( Ax ) − x −
A
A
y ( x ) = a sinh
(A4)
where a and b can be obtained by
imposing the boundary conditions:
a=
M
−VFE
; b = FE
H
H
H
EI
(A5)
Structural Engineering International 3/2010
Finally, the forces at the anchorage can
be recovered:
H = T cos (α )
where aL, bL, aR and bR can be obtained
by imposing the corresponding boundary conditions:
VFE = T sin (α )
aL =
(A6)
MFE = EIT sin (α ) tan (α )
= EIT f (α )
aR =
Cable through an Elastic Guide
M A cosh K − VBG sinh K
;
A AEI (sinh K − cosh K )
[3] SETRA. Cable Stays. Recommendations
of French Interministerial Commission on
Prestressing. Bagneux Cedex: France, 2002.
(A10)
in which, in order to simplify the
expressions, K = λ x H EI . Finally,
the forces at the anchorage can be
recovered:
Following a similar strategy as in the
previous section, the moment distribution is first formulated in terms of
the forces at the anchorage and at the
guide. Note that, because of the presence of the guide, M(x) is split into
two parts ML(x) and MR(x) corresponding to the left and right side ends
of the guide.
(
)
H = T cos (α ) ; VBG = VFE 1 − Sβey ;
MBG = MFE βMS
(A11)
)
M L ( x ) = T cos (α ) y ( x ) − T sin (α ) − S ey x + MBG
M R ( x ) = T cos (α ) y ( x ) − T sin (α ) x + MBG + S ey ex
As in the previous section, two second-order differential equations can
be obtained in terms of the constants
A, BL, CL, BR and CR:
yL ″ ( x ) = Ay ( x ) + BL x + CL
yR ″ x = Ay x + B x + C
( )
( )
R
(A8)
R
Equation (A8) leads to the following
general solution:
( Ax) +
B
C
+ b sinh ( Ax ) −
x−
A
A
y ( x) = a sinh ( Ax ) +
B
C
+ b sinh ( Ax ) −
x−
A
A
yL ( x) = aL sinh
L
L
L
R
R
R
R
R
(A9)
[1] fib. Acceptance of Stay Cable Systems Using
Prestressing Steels. International Federation for
Structural Concrete: Lausanne, 2005.
[2] PTI. Recommendation for Stay Cable
Design, Testing and Installation. Post-Tensioning
Institute: Phoenix, Arizona, 2001.
bR = − aR
The reactions at the anchorage and at
the guide are H, VBG, MBG and VEG,
respectively. Suffixes BG and EG
refer to the beginning and the end of
the guide.
(
−VBG
M
; bL = BG ;
H
H
H
EI
References
(A7)
and at the end of the guide:
[4] Gimsing NJ. Cable Supported Bridges:
Concept and Design, 2nd edn. Chichester: John
Wiley and Sons, 1997.
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VEG = VFE S βey ; MEG = MFE β MS
′ (A12)
Forces at the anchorage and at the
guide (A11) and (A12) have been
expressed in terms of the forces at
the anchorage for the fixed-end
configuration. The new scaling functions are given by the following
expressions:
βey [ K ] =
(1 − K ) ( cosh K + sinh K ) − 1
((2 − K )cosh K + (1 − K )ssinh K − 2 ) S − A
cosh K + sinh K − 1
βey
cosh K + sinh K
⎛H
⎞
β ′ MS [ K ] = βMS + K ⎜ + S⎟ βey − K
e
⎠
⎝ x
AEI ( cosh K + sinh K )
βMS [ K ] = 1 − S
Structural Engineering International 3/2010
(A13)
Scientific Paper
259
Marcel Poser