Compu e s and Geo echnics 171 (2024) 106341
A ailable online 27 Ap il 2024
0266-352X/© 2024 The Au ho . Published by Else ie L d. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/).
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Resea ch pape
Nume ical s udy on modeling he soil oo pad in e ac ion o luna so
lande s a ouchdown
T. Pucke
Ha enCi y Uni e si y, Henning-Vosche au-Pla z 1, Hambu g, 22605, Ge many
ARTICLE INFO
Keywo ds:
Luna lande
Geo echnics
So lande
Foo pad
Luna soil
Spacec a
ABSTRACT
So lande s a e a common sys ems o plane a y explo a ion and ha e been success ully landed on Moon, Ma s
and he come Chu yumo –Ge asimenko. In his s udy nume ical simula ions o he ouchdown p ocess o so
lande s on he moon su ace a e p esen ed. The ocus is on he soil oo pad in e ac ion and he esul ing loading
on he p ima y s u o he landing gea . Main in luence pa ame e s like slope inclina ion, ouchdown eloci ies
and soil p ope ies a e a ied o show each indi idual in luence on he s uc u al loading o he landing
gea . The nume ical simula ions a e done using he CEL-Me hod allowing o signi ican soil de o ma ion a
ouchdown.
1. In oduc ion
The so landing o obo ic spacec a on luna e ain has become
an in eg al pa o luna explo a ion. Recen ly, he Indian mission
Chand ayaan-3 success ully landed a so lande on he luna su ace
(Fig. 1). In con as , he Russian spacec a Luna 25 was des oyed upon
landing a ew days be o e, indica ion he need o be e unde s anding
o he isks associa ed wi h he landing p ocedu e. Focusing on he
geo echnical opics, he isks associa ed wi h he landing p ocedu e a e
unexpec ed g ound condi ions such as boulde s on o below he su ace,
c a e s and slopes. These condi ions can esul in s uc u al damage due
o unexpec ed loadings, uns able oo ings a e ouchdown o u ning
o e o he s uc u e.
The in e ac ion be ween he oo pad and he soil is o pa icula
in e es , because i de ines he s uc u al load on he landing gea a
ouchdown. In addi ion he oo pad ac s as a shallow ounda ion o
ensu e he s abili y o he s uc u e a e ouchdown. The mobilized
eac ion o ces in he e ical and ho izon al di ec ions and he pen-
e a ion o he oo pad a e c ucial in he design o landing gea s o
so lande s. To p e en se e e damage o he landing gea , ene gy
abso be s such as honeycomb ca idges a e in eg a ed in o he landing
gea . Such a sys em can p e en damage om high impac loads.
Pene a ion o he oo pad imp o es he ho izon al bea ing capaci y,
because he soil s eng h ypically inc eases wi h inc easing dep h. On
he o he hand, a minimum clea ance be ween he bo om o he lande
and he luna su ace needs o be ensu ed o p e en damage o he
lande and, i needed, o allow an engine o ascend he lande o a
e u n mission (e.g. he Apollo missions).
This s udy in es iga es he soil oo pad in e ac ion a he ouchdown
o a so lande . While he s udy ocuses on he moon en i onmen ,
E-mail add ess: [email p o ec ed].
so lande s a e also used o explo e o he plane s and objec s such
as come s. Famous examples a e he iking missions ha success ully
landed on Ma s and he Phliae lande ha success ully landed on he
su ace o he come Chu yumo –Ge asimenko.
The in e ac ion be ween he oo pad and luna soil su ace has
been in es iga ed in ensi ely by NASA (1968) o he p epa a ion o
he Apollo luna landings be ween 1969 and 1972. Fo he iking
missions a de ailed assessmen o he landing pe o mance including
se e al Mon e-Ca lo analyses was pe o med as desc ibed by Mu aca
e al. (1975). Mo e ecen in es iga ions including ull-scale es s o
ouchdown si ua ions a e documen ed by Wi e e al. (2010) and Wi e
(2015).
Zheng e al. (2018) analyze he ouchdown o a so lande using he
Fini e Elemen Me hod (FEM) wi h explici ime in eg a ion scheme.
The so wa e packages Abaqus and LS-Dyna a e compa ed o ensu e
simula ion accu acy and eliabili y. Al hough he componen s o he
so lande a e modeled in de ail, he luna su ace is modeled as a igid
plane su ace. This app oach implies a signi ican simpli ica ion o he
soil eac ions. The dynamic esponse o luna egoli h du ing landing
impac is s udied by Che e al. (2018). Model scale es s and FEM
simula ions a e pe o med using he so wa e package Abaqus Explici .
The soil is modeled as a linea elas ic, ideal plas ic ma e ial. The s udy
ocuses on he wa e p opaga ion in he soil.
A de ailed ocus on he soil eac ion a ouchdown is p esen ed
in Yuncheng e al. (2020). The au ho s pe o med a nume ical s udy o
he soil oo pad in e ac ion using he Disc e e Elemen Me hod (DEM).
The size and p ope ies o he DEM pa icles a e chosen o ep esen
he p ope ies o he g ains o luna egoli h. Because o he ex emely
h ps://doi.o g/10.1016/j.compgeo.2024.106341
Recei ed 10 Oc obe 2023; Recei ed in e ised o m 9 Janua y 2024; Accep ed 12 Ap il 2024
Compu e s and Geo echnics 171 (2024) 106341
2
T. Pucke
high numbe o soil pa icles a ec ed by he ouchdown unde nea h
one oo pad, he s udy has been pe o med in 2D using a small sec ion
o soil.
Ji and Liang (2021) model he ouchdown p ocess o a so lande
by combining di e en nume ical app oaches. The lande is modeled
wi h FEM using shell and beam elemen s. The mo ion cha ac e is ics
o he lande a e calcula ed using mul ibody dynamics (MBD) and he
soil is modeled using DEM. Since he soil olume ha needs o be
conside ed in ol es a la ge numbe o soil g ains, he DEM pa icles a e
inc eased in size up o 45 cm. The e o e, he DEM domain allows o
la ge de o ma ions; howe e he pa icles do no ep esen he ac ual
soil g ain beha io .
Ano he nume ical s udy using he Abaqus Explici so wa e pack-
age is p esen ed by Liang e al. (2022). The s udy models he soil in a
FEM domain wi h a linea elas ic, ideal plas ic ma e ial. This app oach
does no allow o la ge soil de o ma ions unde nea h he oo pad
wi hou isking mesh dis o ion p oblems. Addi ionally, only a qua e
o he so lande , including one leg, is modeled in he s udy. The e o e,
he in luence o an inclined soil su ace is no in es iga ed. Wang
e al. (2023) p esen a 6-DOF heo e ical dynamic model o simula e
he ouch down o a so lande . This dynamic model is compa ed o
a mul i-body model calcula ed using MSC Adams. Consequen ly, he
impo ance o accu a ely modeling he ic ion in e ac ion be ween he
oo pad and soil is highligh ed.
Yin e al. (2023) in es iga e he impac o a lande oo pad on he
as e oid egoli h and de elop a mac o model o calcula e he no mal
o ce on he oo pad. The mac o model is calib a ed using a pa ame ic
s udy based on he Disc e e Elemen Me hod (DEM). The DEM allows
he modeling o la ge de o ma ions and dynamic impac s. Howe e , he
DEM pa icles used in his s udy do no e lec he ac ual size and shape
o he as e oid egoli h and can only be seen as a s ongly simpli ied
soil model.
A nume ical simula ion o he ouchdown p ocess ocusing on he
soil oo pad in e ac ion and he esul ing o ces in he landing gea o
he spacec a was used in his s udy o in es iga e he in luence o
ouchdown eloci y, soil p ope ies and su ace slope. The p esen ed
app oach allows o a de ailed in es iga ion o he oo pad soil in-
e ac ion because he soil is modeled as a con inuum. The Coupled
Eule ian–Lag angian-Me hod (CEL) applied in his s udy allows o
dynamic loading and la ge soil de o ma ion, which is expec ed o occu
when he oo pads pene a e he luna egoli h. The quali y o he
soil esponse depends on he applied cons i u i e model and can be
enhanced depending on he e ec s ha shall be in es iga ed. Because
he s i ness o he landing gea and he s i ness o he soil a e modeled
accu a ely, i s in e ac ion can be calcula ed mo e ealis ically han
in p e ious s udies whe e ei he he soil o he landing gea was
conside ed o be igid.
2. So lande model
A so lande wi h ou legs was in es iga ed. The so lande mod-
eled in his s udy is aken in analogy wi h he s udies o Wi e (2015)
(Fig. 2). The main legs a e a ached o he uppe ing o he lande
and suppo ed by wo seconda y s u s o each leg. The seconda y
s u s a e a ached o he lowe ing o he lande and a e connec ed a
app oxima ely 55 cm abo e he oo pad. The e o e, he lowe pa o
he leg, called he p ima y s u , expe ience a bending momen when
he oo pad in e ac s wi h he soil. The leg has an ou e diame e o
80 mm and a wall hickness o 5 mm. The ou e diame e o he p ima y
s u s is 68 mm and i s wall hickness is 9 mm while he seconda y
s u s’ ou e diame e is 34 mm and he wall hickness is 4 mm. In
his s udy he ma e ial p ope ies o s eel a e assumed using a Young’s
modulus o 210 GPa and a Poisson’s a io o 0.2 o he legs and p ima y
and seconda y s u s.
Fig. 1. Pic u e om Chand ayaan-3 on he luna su ace (ISRO,2023).
Fig. 2. LEM body- ixed coo dina e sys em and p incipal dimensions de ini ion (Wi e,
2015).
The lande has a diame e o 120 cm and a heigh o 58 cm.
I is modeled as a igid body and he e o e, canno de o m du ing
he analysis. The geome ical p ope ies a e p o ided in Table 1. The
nume ical model o he lande is illus a ed in Fig. 3.
The oo pad geome y ma ches ha o a sphe ical segmen as shown
in Fig. 4. The adius o he sphe ical segmen is 36 cm and he p ojec ed
Compu e s and Geo echnics 171 (2024) 106341
3
T. Pucke
Fig. 3. Lande geome y and soil mesh a ouch down on a slope wi h an inclina ion
o 5◦including he naming o he legs.
Table 1
Geome y and mass p ope ies o he so lande .
P ope y Value Wi e Value s udy
mass 𝑚[kg] 310.7 311
cen e o
mass 𝑥𝐶𝑂𝑀 [mm] 115 117
g ound clea ance
𝑥𝑔𝑐 [mm] 806 800
landing gea
oo p in 𝑑𝐹 𝑃 [mm] 2400 2400
Fig. 4. Foo pad geome y.
diame e o he oo pad is 30 cm. The hickness o he oo pad is aken
o 1 cm al hough he oo pad is modeled as a igid body and he e o e
canno de o m. The oo pads a e ully connec ed in all ansla ional
di ec ions o he p ima y s u s while i can eely o a e a ound he
connec ion poin .
An aluminum honeycomb ca idge a e included in he p ima y and
seconda y s u s o eal so lande s o p e en he lande om being
damaged du ing ouchdown. The ca idges abso b he impac ene gy
due o plas ic de o ma ion once a p ede ined load limi is eached. In
his nume ical s udy he ca idges we e no aken in o accoun because
he ac ual impac load on he legs is in es iga ed and he e o e should
no be limi ed.
The bes case in e ms o leg loading occu s when all oo pads ouch
he soil su ace simul aneously. In u n he wo s case occu s when one
oo pad ouches he soil su ace be o e he o he s, o example when
he lande descends on a slope. In his s udy only he wo s case is
conside ed by o a ing he lande such ha always one single leg, Leg
1, is loaded i s du ing ouchdown. Fig. 3 shows he nume ical model
o a lande pene a ing he su ace o a slope wi h an inclina ion o
5◦. Due o he inclina ion, Leg 1 ouches he su ace be o e he o he
legs a e in con ac . The la ges dis ance be ween he oo pad and soil
su ace occu s a Leg 3, which in u n ouches he su ace las .
Table 2
Soil p ope ies o he h ee di e en soils used in his s udy.
soil 𝜑′[◦]𝑐′[kPa] 𝐸[kPa] 𝜈[−]𝜌[ /m3]
1 42 0, 5 180 0, 25 1, 5
2 30 0, 1 55 0, 33 1, 3
3 55 3, 5 500 0, 15 1, 9
3. Soil
3.1. Soil p ope ies
Samples o luna soil ( egoli h) we e e u ned o Ea h by he
Ame ican Apollo missions and he So ie space s a ions Luna-16, −20
and −24, see Slyu a (2014). Simulan s ha e been de eloped because
he quan i y o hese soil ma e ials is limi ed. These simulan s we e
designed o ep oduce he mine alogy, g ain-size dis ibu ion, and
soil mechanical p ope ies o luna egoli h (Weiblen and Go don,
1988,S u e (2006)). Many s udies ha e in es iga ed hese simulan s.
(e.g. Pe kins and Madsen (1996); Klosky e al. (2000); Alshibli and
Hasan (2009); Kobayashi e al. (2009); A slan e al. (2010); V e os
(2012); Venugopal e al. (2020)). The mechanical p ope ies o luna
egoli h we e ob ained by combining hese esul s wi h hose o he
o iginal soil ma e ial. Summa izing ecen in es iga ions and based
on Ca ie e al. (1991), Slyu a (2014) p esen ed a ange o mechanical
p ope ies ha can be expec ed o he luna egoli h. In his s udy
Soil 3 ep esen s he uppe ange and Soil 2 he lowe ange o
he mechanical p ope ies p esen ed in Slyu a (2014) o in es iga e
he expec ed ange o he oo pad soil in e ac ion. In addi ion he
mechanical pa ame e s o luna egoli h a he su ace o in e c a e
a eas ob ained by Ca ie e al. (1991) a e used as Soil 1. The p ope ies
o luna egoli h change signi ican ly wi h dep h, see Ca ie e al.
(1991). Ne e heless, cons an soil pa ame e s a e chosen in his s udy
o demons a e he in luence o he soil pa ame e s on he loading o
he landing s uc u e. The soil pa ame e s o Soil 1 o 3 a e lis ed in
Table 2.
The a ia ion in soil pa ame e s shows unce ain y wi h espec o
he soil p ope ies in ex a e es ial applica ions. Looking a he moon,
a e y limi ed olume o eal sample ma e ial exis s, while o o he
objec s, no ma e ial is a ailable o soil es ing a all. Addi ionally, his
s udy deals wi h he landing o explo a ional space ships, meaning ha
he e will be no soil in es iga ion be o e landing.
The soil is modeled using a linea elas ic, ideal plas ic cons i u i e
model wi h ailu e su ace acc. o Moh –Coulomb. The modeled soil
body has a dep h o 10 m and an addi ional heigh o 2 m abo e he
soil su ace. The olume abo e he soil su ace is called he oid a ea
and consis s o emp y elemen s ha do no con ain any ma e ial, (see
Sec ion 4.1). The oid a ea allows he soil o mo e in o his a ea du ing
he impac e en o he oo pads. The ou e dimensions o he soil a ea
is 50 ×50 m o ensu e ha he bounda y condi ions ha e no in luence
on he ac ual impac e en .
3.2. Slope inclina ion
The ini ial soil s ess s a e is applied as he ini ial condi ion o
he nume ical model. The soil s ess s a e is de ined as he e ical
s ess s a e depending on he dep h 𝑧and he ho izon al s ess s a e
is calcula ed by he e ical s ess and ea h p essu e coe icien 𝐾0.
G a i y is aken as 1.62 m∕s2 o conside he luna en i onmen . To
model an inclined slope, he ini ial s ess s a e needs o be calcula ed
using he nume ical model in a sepa a e calcula ion s ep. Al e na i ely,
he ini ial s ess s a e can be de ined o e e y in eg a ion poin wi hin
he model, see Dassaul Sys èmes (2022).
Because he calcula ion o he ini ial s ess s a e by he nume ical
model i sel in ol es an addi ional calcula ion s ep and accompanying
de o ma ions, a di e en app oach is used. Ins ead o inclining he
Compu e s and Geo echnics 171 (2024) 106341
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T. Pucke
Fig. 5. Dimensions o he nume ical model.
soil su ace ela i e o he global coo dina e sys em, he so lande
i sel is o a ed o ensu e co ec o a ion be ween he soil su ace
and he lande . Hence, he di ec ion o he lande ’s eloci y as well
as he di ec ion o g a i y mus also be o a ed. In doing so, he
e ical di ec ion o soil s ess is pe pendicula o he slope su ace. To
e alua e he calcula ions esul s, he o a ion o he sys em needs o be
conside ed o ensu e a compa able coo dina e sys em.
4. Nume ical me hod
4.1. CEL-me hod
The Coupled Eule ian–Lag angian-Me hod (CEL) can o e come
mesh dis o ion p oblems in he simula ion o la ge de o ma ion p o-
cesses. The CEL-me hod is b ie ly in oduced in his sec ion. De ailed
in o ma ion can be ound in Dassaul Sys èmes (2022). I combines
he ad an ages o he Lag angian and Eule ian o mula ions. Bo h
o mula ions di e in hei desc ip ion o he mo emen o small
olume ic elemen s as a unc ion o ime.
The Lag angian o mula ion desc ibes he mo emen o a con inuum
as a unc ion o ma e ial coo dina es and ime. Each node o he
Lag angian mesh mo es along wi h he ma e ial du ing he simula ion.
Thus, he elemen s de o m du ing he simula ion and p oblems due o
mesh dis o ion can occu . The su ace o he con inuum is speci ied
p ecisely using his o mula ion. Fu he mo e, each Lag angian elemen
is dedica ed o one ype o ma e ial only. Lag angian o mula ion is
used in classical Fini e-Elemen analyses and is o en applied in solid
mechanics.
In con as Eule ian o mula ion desc ibes he mo emen o a con-
inuum as a unc ion o spa ial coo dina es and ime. The nodes o he
Eule ian mesh a e ixed du ing he simula ion, such ha he elemen s
canno de o m. To ealize he mo emen o a con inuum, he ma e ial
lows h ough he Eule ian mesh. The e o e, an Eule ian elemen is no
dedica ed o only one ype o ma e ial. The elemen has o be illed wi h
any ma e ial a all. Wi h his o mula ion, no mesh dis o ion occu s
and he simula ion o la ge de o ma ions is possible.
The Coupled Eule ian–Lag angian-Me hod ealizes an in e ac ion
be ween Lag angian elemen s and Eule ian ma e ial using an Eule ian–
Lag angian con ac o mula ion, see Dassaul Sys èmes (2022). The
Eule ian ime in eg a ion is ealized by applying he “Lag ange-plus-
emap” o mula ion. Fi s , a adi ional Lag angian phase is calcula ed
o each ime inc emen and he nodes o he Eule ian mesh a e em-
po a ily ixed wi hin he ma e ial. The e o e, he Eule ian elemen s can
de o m. Second, in he Eule ian phase, he so-called anspo phase,
he elemen s a e es ed o signi ican de o ma ion. These elemen s a e
au oma ically emapped and he ma e ial low h ough hese elemen s
is calcula ed. The imes ep has o be su icien ly small, so ha no
elemen dis o ions occu in one imes ep.
The applica ion o he CEL-Me hod o geomechanical p oblems
in ol ing la ge de o ma ions was app o ed e.g. by Qiu e al. (2009,
2010), Bienen e al. (2011), Pucke and G abe (2012)o Wang e al.
(2015).
4.2. Disc e iza ion
The soil domain co e s an a ea o 50 ×50 m and a dep h o 10 m,
see Fig. 5 The soil olume is disc e ized using C3D8R elemen s: a
linea b ick elemen wi h educed in eg a ion and hou glass con ol,
see Dassaul Sys èmes (2022). The elemen size is 0.1 ×0.1 ×0.1 m a
he soil su ace in he nea ield o he lande . Wi h inc easing dis ance
om he lande , he elemen size inc eases up o 1 m in heigh a
he soil bo om and 4 ×0.5 m a he su ace. The bounda ies a e
ixed in no mal di ec ion a he sides and bo om o he soil domain.
To educe he in luence o wa e e lec ions a he bounda ies, he
dis ance be ween he lande and he bounda ies is chosen o be 20
imes he diame e o he lande ’s gea oo p in . This app oach has
been success ully used e.g. in Henke (2012) and G abe e al. (2013).
4.3. Con ac
The Eule ian–Lag angian con ac o mula ion is an ex ension o
he gene al con ac o mula ion in Abaqus/Explici (Dassaul Sys èmes,
2022). The con ac algo i hm au oma ically iden i ies he in e ace be-
ween he Lag angian s uc u e and Eule ian ma e ial. The Lag angian
s uc u e pushes he ma e ial ou o he Eule ian elemen s, and hus,
he oid a eas e ol e. Eule ian elemen s illed wi h ma e ial a e no
allowed in he egion in which he Lag angian s uc u e is placed.
The e o e, he Eule ian ma e ial is p e en ed om lowing in o he
elemen s unde lying he Lag angian s uc u e. The Lag angian s uc u e
can occupy Eule ian elemen s, such ha he Eule ian mesh can be
c ea ed independen ly om he con ac in e ace.
Con ac in he no mal di ec ion is de ined as ha d con ac ; he e o e,
no in e sec ion is allowed. The angen ial con ac model is a Coulomb
ic ion model wi h a ic ion coe icien o 𝛿= 0.5.
4.4. Damping
In Abaqus Explici a bulk iscosi y damping is associa ed wi h
he olume ic s aining, see Dassaul Sys èmes (2022). This app oach
imp o es he modeling o high-speed dynamics e en s, such as impac
e en s. Based on ex ensi e s udies by Kelm (2004) on he in luence o
bulk iscosi y on he soil esponse in dynamic compac ion p ocesses, a
bulk iscosi y o 0.42 and a quad a ic bulk iscosi y o 1.6 is chosen
o his s udy.
Compu e s and Geo echnics 171 (2024) 106341
5
T. Pucke
Table 3
Pa ame e a ia ions calcula ed in his s udy.
No. 𝑣𝑣[m/s] 𝑣ℎ[m/s] soil no. 𝛼[◦]
1 2 0 1 0
2 2 0 1 5
3 2 0 1 15
4 2 0 2 0
5 2 0 2 5
6 2 0 2 15
7 2 0 3 0
8 2 0 3 5
9 2 0 3 15
10 1 0 1 5
11 1 0 2 5
12 1 0 3 5
13 3 0 1 5
14 3 0 2 5
15 3 0 3 5
16 2 0, 5 1 5
17 2 0, 5 2 5
18 2 0, 5 3 5
19 2 1, 0 1 5
20 2 1, 0 2 5
21 2 1, 0 3 5
4.5. Loading
The simula ion is di ided in o h ee calcula ion s eps. In he i s
s ep, he ini ial s esses o he soil and luna g a i y wi h 𝑔= 1.62 m∕s2
is applied. The lande is ixed a a dis ance o 1 cm be ween he
bo om o he closes oo pad and soil su ace. Hence, any ini ial con ac
be ween he oo pad and soil is a oided.
In he second s ep, he ixa ion o he lande is emo ed and eloc-
i ies 𝑣𝑣in he e ical and 𝑣ℎin he ho izon al di ec ion a e applied o
all elemen s o he lande . This allows he in es iga ion o p ede ined
ouchdown eloci ies.
The eloci y bounda ies a e emo ed in he hi d s ep, immedia ely
be o e he oo pad comes in o con ac wi h he soil. The lande s ill
mo es wi h he ini ia ed eloci ies om s ep wo whe eas he eloci ies
change owing o he in luence o g a i y and in e ac ion wi h he soil.
5. Pa ame ic s udy
In his s udy he in luence o ouchdown eloci y in he e ical
and ho izon al di ec ions, su ace slope and soil p ope ies was in-
es iga ed. The ange o ouchdown eloci ies is aken in he e ical
di ec ion o 𝑣𝑣= [1.0,2.0,3.0] m∕s and in he ho izon al di ec ion o
𝑣ℎ= [0.0,0.5,1.0] m∕s, as desc ibed in Wi e (2015). The in luence o
he su ace slope is in es iga ed by a ying he su ace slope angle
wi h 𝛼= [0.0,5.0,15.0]◦whe eby he inclina ion o 15◦co esponds
o he maximum alue conside ed o he Viking lande , see Mu aca
e al. (1975). Soil p ope ies we e a ied by applying h ee se s o soil
pa ame e s (see Sec ion 3). A o al o 21 calcula ions we e pe o med
in his s udy, see Table 3.
6. Resul s
The calcula ed eloci ies o oo pads 1 and 3 (a ached o Leg 1
and Leg 3) a e shown in Fig. 6. I can be seen, ha he ini ial impac
eloci y o 𝑣𝑧= 2.0 m∕s in e ical di ec ion and 𝑣ℎ= 0.0 m∕s in
ho izon al di ec ion is applied co ec ly a 𝑡= 0.0s. Foo pad 1 ouches
he soil su ace i s i a slope inclina ion 𝛼 > 0◦is conside ed. Fo
he case wi hou slope inclina ion, as indica ed by he g een lines in
Fig. 6, he e ical eloci y dec eases a e he oo pad is in con ac
wi h he soil su ace. The e ical eloci y eaches i s minimum alue
o −1 m/s ha is he co esponding maximum ebound eloci y o he
lande . The ho izon al eloci y o bo h oo pads emain 0 m/s. Looking
a he esul s ob ained o he slop inclina ion, he blue and g ay cu es
in Fig. 6 show ha he e ical eloci y o oo pad 3 inc eases be o e
dec easing. This is because o he dis ance be ween oo pad and soil
is la ge han he dis ance be ween oo pad 1 and soil due o he
su ace inclina ion. G a i y s ill ac s on he s uc u e causing addi ional
accele a ion on he lande and oo pad 3, and he e o e inc easing i s
e ical eloci y un il ouchdown occu s. A e con ac wi h he soil
su ace, he e ical eloci y o oo pad 3 also s a s dec easing. A e
app ox 0.7 s, he e ical eloci y o bo h oo pads is app oxima ely
ze o and he lande eaches a s able condi ion.
Once oo pad 1 is in con ac wi h he soil su ace he ic ion esis-
ance is mobilized and he lande s a s o o a e i a slope inclina ion
is conside ed. This o a ion causes he leg o mo e ho izon ally in he
upwa d di ec ion o he slope (Fig. 6 bo om le ). Due o g a i y and
he mobilized ic ion esis ance, his uphill mo emen comes o a s op
a app ox. 0.1 s o he slope inclina ion o 𝛼= 5◦. Subsequen ly, he
lande s a ed o mo e in he downhill di ec ion un il he sliding esis-
ance eached su icien mobiliza ion. Fig. 7 shows he co esponding
shea s ess a he con ac a ea be ween he soil and oo pad.
6.1. In luence o su ace slope on he s uc u al loads
The in luence o he su ace slope inclina ion on he ime his o y o
he ouchdown o ces o he p ima y s u s in Leg 1 and 3 is shown in
Figs. 8 and 9. Subsequen ly, he e m leg is used o add ess he loading
o he p ima y s u s. The esul s a e ob ained om simula ions using
Soil 3 (Va ia ion No. 7–9). Foo pad 1 ouches he soil su ace i s and
he ini ial axial loading cha ac e is ic o Leg 1 is nea ly independen
o he slope inclina ion (Fig. 8 op). A e he ini ial impac he lande
s a s o o a e, and oo pad 3 ouches he soil a a la e poin in ime
depending on he slope inclina ion (Fig. 8 bo om). I can be obse ed
ha he ampli ude o axial loading o Leg 3 depends on he slope
inclina ion.
The co esponding bending momen s a e shown in Fig. 9. The legs
a e inclined, and he e o e poin away om he main lande body
(Fig. 3). Due o his geome ical bounda y, he bending momen s o
he legs a e signi ican ly in luenced by he slope inclina ion. In he case
wi hou su ace inclina ion whe e all legs ouch he soil su ace a he
same poin in ime, he legs a e sp ead apa du ing ouchdown and a
bending momen is induced (see g een cu es o 𝛼= 0◦in Fig. 9).
Depending on he o ien a ion o he local coo dina e sys em o he
leg, a posi i e bending momen occu s. I he su ace is inclined, he
same e ec can be obse ed o Leg 1 a he beginning o ouchdown
(Fig. 9 op). Once oo pad 1 pene a es he soil and he lande s a s
o a ing, he oo pad mobilizes he ho izon al bea ing capaci y. Due o
he o a ion o he lande and he ho izon al esis ance o oo pad 1,
he bending momen o Leg 1 changes i s sign and becomes nega i e,
as shown by he blue lines a he op o Fig. 9. Ideally, his e ec can
lead o a educ ion in he absolu e maximum bending momen as in he
case o an inclina ion o 5◦.
The in luence o he su ace slope inclina ion on he maximum axial
comp ession load o he p ima y s u s in Leg 1 and Leg 3 is shown
in Fig. 10. The axial comp ession load shows only a mino in luence
o he su ace inclina ion (Fig. 10 op). Only o he high inclina ion
o 𝛼= 15◦, he p ima y s u o Leg 3 expe iences a highe axial
comp ession load han he p ima y s u o Leg 1.
In addi ion, he maximum absolu e bending momen o he p ima y
s u s in Leg 1 and Leg 3 shows a mino in luence o he su ace
inclina ion (Fig. 10 bo om). Due o he e ec o he change in sign o
he momen and a co esponding educ ion in he maximum absolu e
bending momen o he slope inclina ion o 5◦, some imes he p ima y
s u in Leg 1 is highe loaded and some imes he one in Leg 3.
Ne e heless, he absolu e maximum bending momen s o bo h legs
emain a a simila le el.

Compu e s and Geo echnics 171 (2024) 106341
6
T. Pucke
Fig. 6. Time his o y o he oo pad eloci ies o Leg 1 (le ) and Leg 3 ( igh ) in e ical ( op) and ho izon al (bo om) di ec ion o he ouchdown p ocess on a Soil 3 su ace
wi h di e en slope inclina ions 𝛼, an impac speed o 𝑣𝑧= 2.0 m∕s in e ical di ec ion and 𝑣ℎ= 0.0 m∕s in ho izon al di ec ion.
Fig. 7. Shea s esses a he oo pad su aces o s simula ion ime 𝑡= 0.10 s o Soil
3 wi h di e en slope o 𝛼= 5◦, an impac speed o 𝑣𝑧= 2.0 m∕s in e ical di ec ion
and 𝑣ℎ= 0.0 m∕s in ho izon al di ec ion.
6.2. In luence o e ical impac eloci y on he s uc u al loads
The a ia ion in he e ical eloci y shows a clea in luence on
he axial comp ession load o he p ima y s u s in Leg 1 and Leg 3,
see Fig. 11. All esul s a e shown o a ho izon al impac eloci y o
𝑣ℎ= 0 m∕s and a slope inclina ion o 𝛼= 0◦. As expec ed he e is a
clea inc ease in he axial comp ession load o he s u s wi h inc easing
e ical impac eloci y (Fig. 11 op). This e ec is also shown in he
inc ease in he maximum absolu e bending momen wi h inc easing
e ical impac eloci y (Fig. 11 bo om).
Fig. 8. Time his o y o axial loading o Leg 1 ( op) and Leg 3 (bo om) depending on
he su ace slope inclina ion ob ained o Soil 3.
Compu e s and Geo echnics 171 (2024) 106341
7
T. Pucke
Fig. 9. Time his o y o bending momen o Leg 1 ( op) and Leg 3 (bo om) depending
on he su ace slope inclina ion ob ained o Soil 3.
Fig. 10. Maximum axial comp ession load in ( op) and maximum absolu e bending
momen o (bo om) he p ima y s u s o Leg 1 and Leg 3 depending on he su ace
inclina ion 𝛼 o h ee di e en soils a e ical impac eloci y o 2 m/s and ho izon al
impac eloci y o 0 m/s.
6.3. In luence o ho izon al impac eloci y on he s uc u al loads
The in luence in he a ia ion o he ho izon al impac eloci y on
he axial comp ession load o he p ima y s u s in Leg 1 and Leg 3
Fig. 11. Maximum axial comp ession load in ( op) and maximum absolu e bending
momen o (bo om) he p ima y s u s o Leg 1 and Leg 3 depending on he e ical
impac eloci y 𝑣𝑧 o h ee di e en soils a ho izon al impac eloci y o 0 m/s and
a su ace inclina ion o 𝛼= 5◦.
is shown in Fig. 12. These esul s a e ob ained o a e ical impac
eloci y o 𝑣𝑧= 2 m∕s and a slope inclina ion o 𝛼= 5◦. The ho izon al
impac eloci y poin s downwa ds om he slope. The e o e, he axial
comp ession o ce in he p ima y s u o Leg 1, which ouches he
su ace i s , dec eases wi h inc easing ho izon al eloci y (Fig. 12
op). Howe e , he axial comp ession o ce in he p ima y s u o
Leg 3 inc eases wi h inc easing ho izon al eloci y. These e ec s do
no coun e ac each o he , and he maximum comp ession loads o
bo h s u s inc ease wi h he ho izon al eloci y. The e a e also some
a ia ions in he bending momen in he p ima y s u s o he legs wi h
inc easing ho izon al impac eloci y (Fig. 12 bo om); howe e , his is
no as signi ican as ha obse ed o he axial comp ession loads.
6.4. In luence o soil p ope ies
The in luence o he soil p ope ies on he maximum axial com-
p ession load and he maximum absolu e bending momen o he
p ima y s u s o Leg 1 and Leg 3 a e shown in Figs. 10–12. I can be
clea ly obse ed ha he soil wi h he highes s eng h and s i ness
pa ame e s, Soil 3, esul s in he highes impac loads. In u n he soil
wi h he lowes s i ness and s eng h pa ame e s, Soil 2, esul s in he
smalles impac loads.
7. Conclusion
Nume ical simula ions o so lande ouchdown a e p esen ed. The
main in luencing pa ame e s slope inclina ion, impac eloci ies and
soil p ope ies we e a ied in his s udy. O e all, he in luence o
h ee di e en e ical and ho izon al impac eloci ies, h ee slope
inclina ions and h ee soils was in es iga ed, esul ing in a o al o
21 calcula ions, The esul s show a signi ican in luence o he impac
eloci y and he soil p ope ies while he slope inclina ion only has
a mino impac . Ne e heless, he p esen ed app oach s ill implies
impo an simpli ica ions. The linea elas ic, ideal plas ic cons i u i e
Compu e s and Geo echnics 171 (2024) 106341
8
T. Pucke
Fig. 12. Maximum axial comp ession load in ( op) and maximum absolu e bending
momen o (bo om) he p ima y s u s o Leg 1 and Leg 3 depending on he ho izon al
impac eloci y 𝑣ℎ o h ee di e en soils a e ical impac eloci y o 2 m/s and a
su ace inclina ion o 𝛼= 5◦.
model canno accu a ely accoun o s ess dependen s i ness o small
s ain e ec s. In addi ion, he po en ial e ec s o he luna en i onmen
such as acuum, elec os a ic cha ge and empe a u e a e neglec ed.
E hical use disclaime
In acco dance wi h p inciples o esponsible esea ch and o p e en
po en ial mili a y applica ions, he au ho o his publica ion eques s
ha he indings and in o ma ion con ained he ein no be used o any
mili a y o de ense- ela ed pu poses.
CRediT au ho ship con ibu ion s a emen
T. Pucke : Concep ualiza ion, Da a cu a ion, Fo mal analysis, In-
es iga ion, Me hodology, P ojec adminis a ion, Resou ces, Valida-
ion, Visualiza ion, W i ing – o iginal d a .
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
Da a a ailabili y
Da a will be made a ailable on eques .
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