Numerical study on modeling the soil footpad interaction of lunar soft landers at touchdown

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/). Con en s lis s a ailable a ScienceDi ec Compu e s and Geo echnics jou nal homepage: www.else ie .com/loca e/compgeo 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 4 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 . 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