GEOPHYSICAL RESEARCH LETTERS, VOL. 25, NO. 20, PAGES 3903-3906, OCTOBER 15, 1998
The effect of inplane force variations on a faulted elastic
thin-plate, implications for rifted sedimentary basins
R o n a l d T. v a n Balen
Fac, der Aardwetenschappen, Vrije Universiteit, Amsterdam, The Netherlands
Yury Y. Podladchikov
Department of Earth Sciences, ETH-Zentrum, Ziirich, Switzerland
Abstract. Studies of flexural motions of lithosphere commonly
apply a differential equation based on the Lhin-plate approach. 111
this approximation, thc flcxural response to a changing horizontal
inplane force depends only on the curvature of the midplane of
the thin-plate representing the mechanical behaviour of the
lithosphere. However, in cases where an abrupt change of the
geomctry of the lithosphere occurs the midplane of the thin-plate
is offset. We demonstrate that the combinddun uf lhcufrhel with
an inplane horizontal force produces an additional, rheology
independent moment at the psi ti or^ ollhe gronrtly change. This
cffcct has been overlooked by previous studies of lithosphere
deflections, and thin-plate prohlems in general. In the presented
analysis, the thin-plate with an abrupt change in plate geometry
represents lithopshere with a mechanically healed, inactive fault.
However, the derived analytical solutions are general and can be
used to study problems with similar ahrupt geometry changes.
--- -- - --- -- - -A horizontal inplane force acting on a thin-plate causes
vertical deflection. For a thin-plate with acontinuou~midplane,
the effect of a changing inplane force on the deflection depends
on the curvature of the plate's midplane [Timoshenko and
Woinowsliv-Krieger, 19591. The deflections of such a thin-plate
due to changing inplane forces can be determined by applying a
differential equalion [e.g. Timoslrenko a t d Woinowchy-Krieger,
19591. However. if the thin-plate has an abrupt geometrical
change, its midplane is discontinuous. Therefore, in such a case
the differential equation can not he applied. In this paper
analytical solutions for this situation are presented. Here, the
abrupt change in plate geometry represents a prc-existing,
mechanically healed and inactive fault in the lithosphere. But, the
applied method and the derived analytical solutions can be used
in similar thin-plate flexural problems.
In lithospheric flexurc studies, the differentialelastic thinplate
equation is used to determine the flexural response of the
lithosphere to vertical loads and inplane forces [e.g. Cloriingh et
al., 1985; Karner, 19861. In these studies, the elastic thin-plate
represents the mechanical behaviour of the rheologically
complicated lithosphere (see Disc~ssion)The application ofthe
differential equation implies that the midplane of the thin-plate is
continuous. However, in many cases, especially in rifted
sedimentary basins, the lithosphere is deformed by faults, which
cause abrupt changes in the geometry of the midplane of the
adopted thin-plate. This has not been taken into account in the
existing studies. Below, we apply the derived analytical solutions
to typical settings in rifted sedimentary basins.
In our denvation oC the analytical solutions a vertical fault is
assumed, which cuts the entire thin-plate. Such a configuration
is a good representation for the situation in rifted sedimentary
basins as scisn~otcctur~ic
and dccy sdhnlic rcflcctioi~data suggest
that fundamental faults in the basement of rifted sedimentary
basins art. plarlar and arc restricted to the brittle, cold, topmost
part of the lithosphere, corresponding to the upper crust [King el
a/., 1988; K~~szniretnf.,
19911. Furthemre, the results offinite
element modelling demonstrate that the assumption of a vertical
fault plane does not significantly influence the predictions of
deflections, as long as the fault dip is equal to or more than 63"
[Van Rnlen eta/., 19981. The fundamental planar basement faults
must have thc largest effect on the deflection, as they cause most
of the pre-existing deformation of the uppercrustal competent
laycr in rifted basins. However, elastic solutions for faults not
cutting the entire upper crust also exist [Savage and Guohua,
19851 and possibly can be extended to include variations in
innlane force levels. In addition. in this ,naner
~ ~ onlv
r , flexural
rnitions of lithosphere for cases in which its behaviour can
described by a single elastic thin-plate are considered. The
generalimtion to multiple-layer thin-plate systems [ ~ c ~ uettal.,
t
1988: Burov and Diamenr.19951 is the subject of another paper
[van~~l~~ et al., 19981.
~
~
~
Derivation of analytical solutions of deflections
caused a
Faulting of a thin-plate causes an offset of the plate's midof ail inplane
force
to a thinplane (
~ 1). qIe
i
~
~
~
~
subsidence
.. ._... ..........:......:. ... .. .
. . . . .
.. ..... ..... ..... ..... ..... ..... ..... ..... ...
.. . .. . ., . ., . .. . .. .., ... ... .
. . . . . . . . .
flexed
thin-plate
Copyright 1998 by the American Geophysical Union.
Paper number GRL-1998900005.
0094-8276198lGRL-1998900005$05.00
~
faulted
area
flexed
thin-plate
Figure 1. Moment occuring at the fault due to application of an
inplance force to a thin-plate with a displaced midplane.
3903
3904
VAN BALEN AND PODLADCHIKOV: THE EFFECT OF INPLANE FORCE VARIATIONS
plate with an offsct midplanc causcs a momcnt, M, at the position
of the fault. This moment equals the product of the inplane force
and the magnitude of the offset:
with F = inplane force, defined positive for compression, u =
displacement of the midplane across the fault, defined positive if
thc down thrown block is on the left side (x < 0) and negativc
otherwise. This equation is rheology independent.
From the general solution of the elastic thin-plate flexurc
equation [Hetinyi, 19461, a solution for the case in which the
rigidity D is constant and deflections and bending moments are
zero at both ends of the thin-plate (at infinity) can be derived. It
consists of two parts, for positive and negative x:
w+ = e
with ~
=
J
~
- (c;~ c ~o s ~p + c; sin @)
,
~
=
J
,
/
~
3
c,', cl-, c;, c; = constants, Ap = density d~fferencebetween
infilling and compensating material, w'= deflection for x > 0,
w = deflection for x < 0. The flexural displacement around and
at a fault due to a changing inplane force can be found from
equations (2) by joining the two solutions at the origin. The
values for the four constants can be found from conditions for the
joint. By constraining the magnitude of the differences for
negative and positive x for the deflection, slope, bending moment
and vcrtical shcaring forcc at thc origin, the constants can be
determined from the zeroth, first, second and third derivatives of
cquations (2). For thc casc of an inplanc forcc induccd faultmoment the appropiate boundary conditions are:
For this case the values of the four constants are:
Typical prnfiles predicted hy the resulting equations are presented
in the next section.
Flexural profile for a half-graben and a rift
flexural rigidity, D, is calculated fmm Te and Young's modulus,
E, and Poisson's ratio, v. The adopted values for E and v are
7 x l 0 " ' ~ aand 0.25, respectively. The applied density contrast,
A , is taken to be equal to 900 kgni3, based on a sediment
density of 1800 kgm? (assuming a porosity of 50% for sediment
deposited or eroded during the flexural motions) and a lower
crustal density of 2700 kgni3. Tnc prcdicted flcxural profile for
the half-graben is characterized by a maximum uplift of about 75
m at thc footwall and an cqual amount of subsidence at the
hangingwall (Figure 2a), which is relatively small compared to
the fault offset. The two maxima are separated horizontally by 5 0
km. Inplane force-induced flexural profiles for multiple fault
systems can be obtained by applying the method of elastic
superposition [Hefenyi, 19461, i.e. by adding-up the flexural
profiles for the single fault solution. The differential subsidence
profile predicted for a system consisti~~g
of two master 11or111al
faults dipping in opposite directions and spaced 50 km apart,
delineating a rift, shows almost 150 m subsidence in the graben
and 75 m uplift at the flanks (Figure 2b). The larger amount of
subsidence
in
+
F
/ the~graben
D is caused by positive interference ofthe
two flexural profiles induced by the fault-moments. The flexural
deflections resulting from tensional force changes instead of
compressional have exactly opposite values.
Thin-plate solutions including a variable rigidity
Differences in rigidity across a fault can arise from plate
thickness changes or from pure-shear thinning in the lower crust
underlying the fault, resulting in a thermal and associated
rheological perturbation. The amplitude and wavelength of
flcxural deformation is controlled by the rigidity of the plate.
Therefore, a changing rigidity along the profile influences
predictions of deflection. One particularly interesting aspect of a
chan~inr
- .riridity
- . is that the fault itself will undergo u~liftor
subsidence in response to a changing farfield inplane force.
Although a systematic variation of rigidity complicates the
analytical solution considerably, a sudden change in rigidity
across the fault can be relatively easily incorporated into our
solution scheme.
For the derivation of the analytical solution the dependence of
a and pin equation (2) on the inplane force is neglected. Both a
and V depend only m~ldlyon the magnitude of the inplane force
[Hetinvi, 19461 The simplification causes a and P to be equal.
However, they are different for positive and negative x:
- .
with at and Di= flexural parameter and rigidity for x > 0, and aand D- = flexural parameter and rigidity for x < 0. The joint
conditions are:
The flexural profile for a nloment occuring at a single norrrral
fault, representing for example a half-graben in a rifted
sedimentary basin, is obtained by applying an elastic t h i c h ~ e s ~
(Te) of 5 km,a fault throw u = 2 km and an inplane compression
of I00 MPa throughout the whole plate, resulting in an
inplane force of 5x10" ~ m ~This
' . force is within the limits
defined by the yield strength of the lithosphere [Kusznir and The resulting values for the four constants become:
Park, 1984; Savage and Guohua, 19851 and values deduced
from studies on folding of oceanic [Cloerin~h
and Worfel, 1985;
c: = c ; = 2 F u
(a+)'(a-)'(at - a - )
(6)
Aeekman cr al., 19961 and continental lithosphere [Lambeck,
( a ' + a ) ( ( a ++)(*L ? ) ~ ) A ~ ~
1983; Cloetingh and Burov, 19961. The elastic thickness is
adopted from effective elastic thickness estimates based on
deflections around nonnal faults in rift settings [King et al.,
1988; Kusznir et al., 19911, see Discussion for explanation. The
3905
VAN BALEN AND PODLADCHIKOV: 'ITE EFFECT OF INPLANE FORCE VARIATIONS
-100
-50
0
50
100
-100
distance (km)
-50
0
50
100
distance (km)
Figure 2. Differential subsidence profiles (continuous line), or relative sea level change, for a half-graben and a rift. Faults have throws
of 2 km. The elastic plate thickness is 5 km and is subjected to a 100 MPa compressive stress. Dashed lines indicate the faulted
basement profile, including fault throw. For the basement profile the vertical scale has to be multiplied by 10.
a) Resulting profile for a single fault, representing a half-graben. Compression induces 75 m uplift of the footwall and an equal amount
of suhridence of the hangingwal.
b) Rift with two master faults spaced 50 km apart. Predicted footwall uplift (flank) is about 75 m, stress-induced subsidence in the
graben center equals 150 m.
The resulting equations predict that the fault zone experiences a
deflection in the same direction as that part of the plate which has
the least rigidity. For example, if a system where the downthrown
block has a lower rigidity is compressed by a farfield inplane
force, the fault itself subsides (Figure 3). Furthermore, upon
decreasing rigidtty, the amplitude of the subsidence of the
downthrown block increases whereas its wavelength decreases.
Discussion and conclusions
As shown by several studies, horizontal inplane force changes
cause differential vertical motions in sedimentary basins
[Cloetingh er a/., 1985; Karner, 1986; Kooi and Cloetingh,
1992; Karner et al., 1993: Van Wees and Cloeringh, 19961.
However, none of these studies have included the permanent
deformation of the lithosphere caused by pre-existing crustal
faults. The presented results indicate that fault-moments
contribute considerably to inplane force-induced vertical motions,
leading to lower force levels required to model observed
deflections [Van Ralen et a/., 19981. The predicted flexural
motions are of the same order of magnitude as relative sea level
changes inferred from the stratigraphic record of sedimentary
basins [Haq et al., 19871.
The presented and discussed deflections for inplane forceinduced fault moments are based on effective elastic thicknesses
which are comparable to those of rift-settings. Due to the spaceand timescales of the loads acting on the continental lithosphere
I
1 km
during upper crustal faulting, flow of lower crust causes isostatic
compensation for these loads at a lower crustal level [King et al.,
1988; Kusnzir et al., 19911, leading to Tck in thcsc scttings
-100 -50
0
50 100
which
am in the range of 2-1 0 km. Rifting processes occur at the
distance (km)
same time scale as inplane force magnitude changes, generally a
Figure 3. Deflect~onsfor a model including a rigidity change Tew Ma (in fact, rifting is caused by an inplane force magnitude
across the fault caused by 100 MPa compression. In this model, which exceeds the tensile strength of upper crustal lithosphere).
the footwall has a constant elastic thickness of 5 km, the fault Furthermore. both faulting and fault-moments cause concentrated
throw equals 2 km and the applied inplane stress is 100 MPa.
Results are plotted for hanging wall Te's of 5 , 4 , 3 , 2, and 1 km. loading of the lithosphere. Therefore, flexural bending due to
The predicted profiles show that the fault, located at x = 0, normal faulting occurs at the same spatial scale as fault-mnomnent
subsides and that the amplitude of the subsidence of the bending. As a result, normal faulting during rifting and flexural
downthrown block increases whereas its wavelength decreases. bending due to inplane force-induced fault-moments are
controlled by the same crustal Te. As shown by Burov and
See text for discussion.
_____ __
l l l I l l l - J
3906
VAN BALEN AND PODLADCHIKOV: THE EFFECT O F INPLANE FORCE VARIATIONS
Diament [1995], inferred largescale Te's for continental
lithosphere show a bimodal distribution, corresponding to singleand multiple layer thin-plate systems. The single layer systems
represent lithosphere rheologies dominated by t h e uppercrustal
competent layer. The multiple layer systems arise from the
flexural strength o f the mantle of the lithosphere. lherefore, for
modelling of large-scale, hasinwide inplane-force induced
deflection in rifted basins, the potentially important flexural
strength and pre-existing deformation of the upper part of the
lithospheric mantle should also be taken into account. This topic
is dealt with in a separate paper lVan Balen er al.. 19981, in
which the theory presented in this paper is applied to the southern
North Sea basin, the Pannonian Basin and the Norwegian marsin.
Acknowledgements. The authors want to thank Fred Beekman,
Sierd Clenngh, Ritske Hutsmans, He& Kmi, Tore Skar and Marlies tm
Voarde for helpful discussions and three anonymous reviewers for
thoughtful reviews.. Ronald van Balen was sponsored by Norsk Hydro
a.r., Yury Podladchlkov by the "Universitair Stimulerings Fonds" of the
Vrije Univasiteit. Netherlands Research School of Sedimentary Geology
publication no. 980802.
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~~-~~~~~~~~~
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~~~
~
R.T. van Balen, Structural Geologymedonics Group, Fac. der
Aardwetenschappen, Vrije Universiteit, De Boelelaan 1085, 1081 HV
Amsterdam, The Netherlands. (=-mail:balr@geo.vu.nl)
Y.Y. Podladchikov, Department of Earth Sciences, ETH-Zcntrum,
Sonneggstrasse 5. CH-8092 Z"rich, Switzerland. (e-mail:
yury@erdw.ethz.ch)
(Received March 9, 1998: revised June 29, 1998;
accepted August 14, 1998.)