Japanese Journal of Structural Geology, No. .1, -1ῌ.2, ,**-

ArticleStress distribution and seismic faulting in the Nepal Himalaya :insights from ﬁnite element modeling
Md. Mahmudul Alam* and Daigoro Hayashi*
Abstract : In this study we use ,D ﬁnite element modeling to predict stress distribution
and fault development during N-S shorting in the Himalayas. The direction of
maximum compressive stress (s+) is horizontal within the shallow levels of both models.
Variation in the displacement boundary conditions results in changes of principal
stress directions in the deep crust while stresses in the shallow crust remain unchanged.
Mohr-Coulomb failure is observed along near-surface sections of the Siwalik, Tethys
and Granitic basement layers, with minor failure near the surface of the MBT, MCT and
STDS. Failure does not occur due to hydrostatic conditions in the deeper part of theHigher Himalaya, the Lesser Himalaya or the Granitic basement layer, despite changing the rheology of rock layers. Focal mechanism solutions of earthquakes in the
Himalayan region suggest the existence of thrust faults stretching along EW with one
plane dipping gently north beneath the Himalaya. The simulations predict a similar
distribution of thrust faults along the upper part of the Himalaya.
Key words : Finite Element Method (FEM), Mohr-Coulomb failure, Main Boundary Thrust(MBT), Main Central Thrust (MCT), South Tibetan Detachment System(STDS), Focal mechanism.

Introduction
Finite element modeling is an e#ective tool
for studying tectonic deformation in the crust.
The Himalayan mountain chain formed following collision of the Indian and Eurasian plates(Fig. + ; Dewey and Bird, +31* ; Windley, +32.,+33/). Following Eocene collision of the twocontinents, further convergence and crustal shortening was taken up in major intercontinental
subduction which led to the formation of theHimalayan chain (Pecher, +323).
The continent-continent collision model forthe uplift of the Tibetan Plateau has been modeled numerically by England and others (England and McKenzie, +32,, +32- ; England andHouseman +32/, +320). In these studies, they
assumed the Indian and Asian crusts to be an
incompressible non-Newtonian ﬂuid and the collision was analysed in --dimension by means ofReceived 2 April ,**, ; accepted +0 January ,*** Department of Physics and Earth Sciences, University of
the Ryukyus, Okinawa, Japana thin viscous sheet analysis and had the intention of explaining the structure of the interior of
the Asian continent.
The large negative gravity anomalies over the
Ganga Basin and the Himalaya are reproduced
by a model where the Indian crust underthrusts
Himalaya at shallow angles of +/῎῍/῎ along
several thrust zones (Indus Suture Zone, MCT
and MBT ; Warsi and Molnar, +311). In thismodel the Himalayan region is isostatically uncompensated and crustal shortening of -** to.** km is calculated to have occurred across theorogenic belt.
Hayashi (+321, +33,) numerically simulated theuplift of the Tibetan plateau and the Himalayas
by ﬁnite element modeling, assuming that thenortherly migration of the Indian plate deforms
the overlying Asian continental crust which
behaves as an incompressible Newtonian ﬂuid
above the rigid upper mantle. In these models,the upper mantle lithosphere is a rigid body forthe ﬁrst approximation and the deformablecrust overlies the upper mantle. The deforma-ῌ 37 ῌ

Md. Mahmudul Alam and Daigoro Hayashi

Simpliﬁed geologic map of the Himalaya showing the major tectonostratigraphic divisions. Modiﬁed from Gansser
Fig. +(+30.), Le Fort (+31/), Barnicot and Treloar (+331), Parrish and Hodges (+331), and Kaneko (+331).

tion of the crust is then expected to form plateaus and mountain ranges. Under such anassumption, isostatic readjustment becomes impossible and the calculated values of surface
uplift tend to be overestimated. Further, therigid plate assumption prohibits simulation for aperiod longer than *.+ Ma.
The aim of this study is to simulate the stress
distribution and patterns of fault developmentin the Nepal Himalaya by means of twodimensional plane strain FEM. In the FEM analysis, it is assumed that geological material ishomogeneous and elastic, although it is likelythat crustal rocks are visco-elastic (Makel andWalters, +33- ; Bott and Stern, +33, ; Ghose etal., +33*). This assumption is simple, but themodeling is instructive and contributes to ourunderstanding of the Himalayas.
Calculated principal stresses and failure elements are used to identify the fault pattern. We
focus our investigation on the following threepoints : (i) intensity and direction of principal
stresses along the Nepal Himalaya under various boundary conditions, (ii) development offaults, and (iii) comparison of the simulated
stress distribution with focal mechanism solu-

tions from the Himalaya region.
Geological Setting
The Himalayan mountain chain is divided longitudinally by principal thrusts into four tectonic zones ; the Sub-Himalaya (SH), the Lesser
Himalaya (LH), the Higher Himalaya (HH), andthe Tethys Himalaya (Gansser, +30.). The principal thrusts are the MBT, MCT and STDS (Fig. +).
Sub Himalaya (Siwalik)
Forming the southernmost belt of the Himalayan range, the Siwaliks are the lowest and narrowest range in the Himalayan system, with anaverage elevation of about 3**ῌ+,** m and, awidth of as little as +0 km. The Sub-Himalayanzone is bounded by the MBT to the north along
the southern foot of the Mahabharat Range andthe Main Frontal Thrust (MFT) to the south at
the southern edge of the Siwalik (Churia) Group(Kizaki, +33.).
,. Lesser Himalaya
The Lesser Himalaya has been variously
named the Midland meta-sediments (Arita, +32-)
and Lesser Himalayan Sequence (Kaneko, +33/,+.

ῌ 38 ῌ

Structural Geology No. .1

Stress distribution and seismic faulting in the Nepal Himalaya : insights from ﬁnite element modeling+331). Lying between the HH region in the northand the Siwaliks to the south, the LH region
forms the intermediate zone of the Himalayan
mountain chain (Sakai, +32/). Lower than the
HH, this region has an average altitude of -1**ῌ
./** m.
-. Higher Himalaya
The Higher Himalaya is bounded to the southby the MCT, and to the north by the STDS. The
STDS is a series of normal faults on the northern
ﬂank of the Himalaya, and forms the contact
between Tibetan Tethys sediment and underlying Higher Himalayan Sequence (Hashimoto etal., +31- ; Burg and Chen, +32. ; Burchﬁel et al.,+33,). The HH has been previously named theHimalayan Gneiss (Arita, +32-), Higher Himalayan Sequence (Kaneko, +33., +33/). The Higher
Himalayan belt forms the backbone in the Himalayan system. With an average elevation of
about 0+** m, the HH dominates the northern
frontiers of India and the entire northern boundary of Nepal.
.. Tethys Himalaya
The Tethys Himalaya is seperated from the
HH by STDS to the north. In the Tibetan Himalayan zone, the Palaeozoic to early Cenozoic
Tethys sediments are distributed along the sou-thern most margin of the Tibetan Tethys basin,such as the Sagarmatha region, the LanguManang basin, the Saipal - Amlang basin, theSpiti basin and the Kashimir basin which isunderlain by granite (Kizaki, +33.).
Finite Element ModelingIn our modeling, the signiﬁcant parameters
are the rock properties including cohesion andangle of internal friction. Cross sections of twomodels are shown in Figs. ,ῌ-. All simulations
were conducted using the rock properties andboundary conditions shown in Table + and ,,and Figs. .ῌ/.
In this simulation, we investigated how stress orientation and fault pattern are inﬂuenced by rock properties and boundary conditions of two-dimensional FEM.
First we conducted test calculations using a standard model in order to determine the range ofthe key parameters, such as Young’s modulus,
Poisson’s ratio, strength and angle of internal
friction of rock formations. Then taking into
account the results of the test, we simulated theprincipal stresses and failure blocks under most
reasonable boundary conditions.

Fig. ,
Cross section (Aῌ-A in Fig. +) of the central Himalaya. Modiﬁed from Seeber et al. (+32*) and Gansser (+30.). MCT :Main central thrust, MBF : Main boundary front.

ῌ 39 ῌ

Md. Mahmudul Alam and Daigoro Hayashi

Fig. Geological cross section (A-Aῌ in Fig. +) of the Himalayan Metamorphic belt in central Nepal. Modiﬁed from
Kaneko (+331). LHT : Lesser Himalayan thrust, STDS : South Tibetan detachment system, HHS : Higher Himalayansequence, LHS : Lesser Himalayan sequence, IT : Indus suture, TTS : Tibetan Tethys sequence.

Methods
Convergence rate
The Indian plate is currently moving northward at a rate of /* mm/yr and is underthrusting Tibet (Thakur et al., ,***). Of the total convergence, ῍,* mm/yr is accommodated acrossthe Himalaya (Bilham et al., +331 ; Lave andAvouac, +332) and the remaining convergence istaken up farther north (Peltzer and Saucie,+330 ; Yeats and Thakur, +332). Lyan-Caen and
Molnar (+32/) suggested a thrusting rate of between +* and +/ mm/yr during the last +/ to ,*Ma. An estimated -ῌ.** km of shortening has
occurred in the Himalaya (Gansser, +300 ; Warsi
and Molnar, +311) and assuming that underthrusting began in the Oligocene or late Eocene(Gansser, +30.), the average rate of shortening isabout +* mm/yr. This rate is consistent with therate of +* to ,* mm/yr inferred by Molnar et al.
(+311), and from seismic moments of earthquakes (Chen and Molnar, +311). Over geologicaltime scale (/ Ma) the ῍,* mm/yr estimated shortening rate (Cattin and Avouace, ,***) acrossthe range is accommodated by localized thrustfaulting along the Main Himalayan Thrust(MHT). From the above discussion we may
conclude that the most reasonable convergence
rate is +*῍,* mm/yr, on the average +/ mm/yr
that is taken in our simulation.
+.

,. Constraints of model geometry and boundary condition
We have selected two representative geological sections (Fig. ,, Seeber et al., +313, and Fig. -,Kaneko, +331) across which we simulate stress
distribution and fault development.
Bothmodels are between 3* to +*/ km long and ,* to,/ km in depth.
The Himalayan proﬁle, model + is composedof four tectonic zones ; Siwalik (sandstone, siltstone, mudstone), Lesser Himalaya (sandstone,
limestone), Higher Himalaya (gneiss and granite), and Tethys Himalaya (sandstone, impure
limestone). For model ,, we added a “Granitic
basement layer”, which is composed of gneissand granite. The rock properties of the models
are shown in Table + and ,.
We have used 1/ m (for /*** years at +/ mm/yr) and -1/ m (for ,/*** years) as the displacement imposed at the right edge of cross sections.
In all models the upper surface is a free surface.
The lower boundary is restricted vertically but
allowed to move horizontally. The nodes at theleft boundary can move only vertically (zerohorizontal displacement). The node at the junction of the lower and the left boundary is ﬁxed.
Horizontal shortening of up to 1/ m (or -1/ m) is
applied proportionally from the southern edge
(1/ m or -1/ m) to the north (* m) over the crosssection (Figs. .ῌ/).
-. Proximity to rock failure under Mohr - Coulomb criterion

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Structural Geology No. .1

Stress distribution and seismic faulting in the Nepal Himalaya : insights from ﬁnite element modeling

Table +

Rheology of rock units for model +.

Table ,

Rheology of rock units for model ,.

Mohr-Coulomb failure criterion is a condition
that helps to identify the location of failure
within the crust or at the surface. Since wecompute a two-dimensional stress ﬁeld on planestrain state, the third principal stress (s*) can beobtained from the equationῌs *ῑn῍s+῏s,῎where n is Poisson’s ratio (Timoshenko and Goodier, +31* ; Hayashi and Kizaki, +31, ; Lu andHayashi, ,**+), and s+ and s, are the principal
stresses. After evaluating the values of s+, s, ands * for each element, we compute which principal
stresses, s+, s,, and s * are the maximum compressive, intermediate and minimum compressive.
According to the order of magnitude and thesign, we deﬁne the new principal stresses, s+, s,
and s-.
The proximity to failure (Pf) is deﬁned as theratio of the radius of the Mohr circle (calculateds+ῐs, and the radius at failure (yield,s+ῐs-
stress)  , failure as follows.
stress) s+ῐs-
 
, 
pfῑ
s+ῐs-, failure῍

s+ῐs-
s+ῐs-
where  , failureῑccosf῏ , sinf, here
c and f are cohesive strength and angle of internal friction respectively. One can evaluate whether faulting develops in a certain element according to the value of the parameter Pf . If
Pfῒ+, the stress state is within the failure envelope (i.e., no fault develops), but if Pf+, failure
occurs.
Modeling Results
The numerical modeling of the Himalayan

ῌ 41 ῌ

Md. Mahmudul Alam and Daigoro Hayashi

Fig. .

Boundary conditions of Model +. The model contains -/, nodes and /0. elements.

Fig. /

Boundary conditions of Model ,. The model contains -/- nodes and 0,. elements.

system has revealed two principal stress regimes, characterized by tension and compression.
Stress distribution and fault patterns are controlled by two parameters ; displacement boundary conditions, and rheology. The numericalexperiments show that realistic deformation ofrock layers are calculated for the speciﬁc sets ofdisplacement boundary condition (1/ m//***yr,+/* m/+****yr, ,,/ m/+/***yr, -** m/,****yr,and -1/ m/,/***yr for +/ mm/yr) and also forthe values of cohesion and angle of internal
friction. The occurrence Mohr-Coulomb failure
depends on the values of displacement boundary condition, cohesive strength (c) and angle ofinternal friction (f). An important advantage ofnumerical modeling compared to scale modelingis that the former is able to calculate the stressﬁeld at any time anywhere in the structure.
Figs. 0ῌ3 show the magnitude and orientation ofprincipal stresses in the rock layer during pro-

gressive displacement from 1/ m to -1/ m displacement. s+ and s- are the maximum andminimum compressive stresses respectively.
For both the models, maximum value of compressive s+ and s- are /** MPa and -0* MPa, respectively in the case of the displacement boundary condition 1/ m. While maximum valuesof compressive s+ and s- are 2** MPa and 0**MPa, respectively in the case of the displacement boundary condition -1/ m. Maximum tensile s- is 3/ MPa. Directions of s+ are shown in
Tables - and ..
Discussion+. Pattern of stress and failure in the Himalaya
Modeling result shows that in the upper partof each model, direction of s+ is horizontal but
vertical direction is found in the deeper partῌ 42 ῌ

Structural Geology No. .1

Stress distribution and seismic faulting in the Nepal Himalaya : insights from ﬁnite element modelingStress distribution in Model +. 1/ m displacement boundary condition is applied to the base of the model during a
Fig. 0/*** year interval.

Stress distribution in Model +. -1/ m displacement boundary condition is applied to the base of the model during a
Fig. 1,/*** year interval. Red arrows indicate an extensional stress ﬁeld.

Stress distribution of Model ,. 1/ m displacement boundary condition is applied to the base of the model during a
Fig. 2/*** year interval.

Stress distribution of Model ,. -1/ m displacement boundary condition is applied to the base of the model during a
Fig. 3,/*** year interval. Red arrows indicate an extensional stress ﬁeld.

ῌ 43 ῌ

Md. Mahmudul Alam and Daigoro Hayashi

Table -

Table .

Direction of s+ in model +. H : Horizontal, V : Vertical, Hydr : Hydrostatic, Sub H : Sub-Horizontal.

Direction of s+ in model ,. H : Horizontal, V : Vertical, Hydr : Hydrostatic, GL : Granitic Basement Layer.

under the boundary condition 1/ m. Under the
displacement boundary condition -1/ m, in mostof the regions of all the models, s+ is horizontally
directed. The simulation shows that the principal stresses are stronger in the deeper part than
the shallower one. Principal stresses are compressive along all the region but some elementson the shallower region of the HH (red arrows in
Figs. 1 and 3) showing tensile stress (less than+** MPa) ﬁeld.
The values of cohesive strength (c) and angle
of internal friction (f) are shown in Tables + and,. For displacement of less than 1/ m, no failure
occurs in any of the models. As displacement
increases to ,,/ m failure occur at the shallow
levels. At -1/ m of displacement, failure occurs
in the shallow levels of all the models, and mostof the failure occurs at the SH, the Tethys Himalaya and the Granitic basement layer (Figs. +*ῌ++). Also some failure occurs along the upperboundaries of MBT, MCT, and STDS (Figs. +*ῌ++). Even after varying controlling parameters,
we did not obtain failure in the deeper parts ofthe HH, LH and Granitic basement layer. This isdue to hydrostatic conditions in these areas.
The elements of the greatest proximity to fail-ure occur along surface and number of the element decreases progressively with depth due tothe material properties of rock formation. After-1/ m of convergence, the uppermost parts ofboth the models are failed over the entire length(Figs. +*ῌ++). As the failure proximity ratio
varies within a layer, the lower parts of eachlayer are out of failure but upper parts are underfailure (Figs. +*ῌ++).
,. Comparison of faulting pattern with focal
mechanism data
The Nepal Himalaya consists of nappes andthrust blocks that are thrust southward onto theIndian shield (Gansser, +30. ; LeFort, +31/ ;Pecher, +323 ; Valdia, +32* a, +32* b, +32+ ; Baranowski et al., +32.). The Indian shield is warped
under the Himalaya and underthrusts the LH toa gentle northerly or northeasterly dipping
thrust fault (Seeber et al., +32+). Lyon-Caen and
Molnar (+32-) reached the same conclusion froma study of gravity anomalies over the Himalaya,assuming that the Indian plate is ﬂexed down by
the weight of the Himalayan range thrust on topof it.
Focal mechanisms of shallow depth earthquakes show compressive stress pattern alongῌ 44 ῌ

Structural Geology No. .1

Stress distribution and seismic faulting in the Nepal Himalaya : insights from ﬁnite element modelingFig. +*

Failure elements of Model +. Green blocks indicate failure.

Fig. ++

Failure elements of Model ,. Green blocks indicate failure.

Fig. +,
Focal mechanism solutions. Single arrows show presumed direction on underthrusting. Opposing arrows forevent , show approximate direction of maximum compression and diverging arrows for event 1 show approximate directionof least compression. Circles show lower hemisphere plots of focal sphere - dark areas represent quadrant withcompressional ﬁrst arrivals (Molnar et al., +311).

ῌ 45 ῌ

Md. Mahmudul Alam and Daigoro Hayashi

the Himalayan region. Several published studies of focal mechanism solutions of earthquakes in the Himalayan region give the samegeneral pattern of thrust faulting, with one
plane dipping gently beneath the Himalaya(Fitch, +31* ; Ichikawa et al., +31, ; Molnar et al.,+31- ; Rastogi et al., +31. ; Banghar, +31. ;Tandon and Srivastava, +31/ ; Armbruster et al.,+312 ; Chen and Molnar, +311 ; Chandra, +312 ;Baranowski, et al., +32. ; Fig. +,). The present
study also shows the same pattern of thrustfaulting in the shallower region of the Himalaya,whatever the boundary condition might be.
-. Limitation of our analysis(+) We assume that geological body is elastic,
though the body in nature has to be elastoplastic when we consider about faulting.
(,) Each tectonic zone is composed of several
rock layers. The composite tectonic zone isassumed to have the average physical propertiesof several rock layers, while each rock layer has
distinct physical properties. This estimation of
average properties of tectonic zone is not fully
justiﬁed, though we believe this is the betterway at present

simulation predicts the same distribution ofthrust faults along the upper part of the models
as shown by focal mechanism solutions.
Acknowledgments
We would like to express gratitude to the
Ministry of Education, Science and Culture,
Japan (MONBUSHO) for ﬁnancial support
during the study. We thank Aaron Stallard ofuniversity of Canterbury, New Zealand for critical review.
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(+) Variation of the displacement boundary
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