Thin-Walled S uc u es 184 (2023) 110535
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The e ec o c oss-sec ion geome y on he la e al– o sional beha io o
hin-walled beams: Analy ical and nume ical s udies
Muhammad Z. Ha a a, Ma in Ho áčekb, Sándo Ádánya,∗
aDepa men o S uc u al Mechanics, Facul y o Ci il Enginee ing, Budapes Uni e si y o Technology and Economics, H-1111 Budapes , Műegye em kp.
3, Hunga y
bIns i u e o Me al and Timbe S uc u es, Facul y o Ci il Enginee ing, B no Uni e si y o Technology, Ve eří 331/95, 602 00 B no, Czech Republic
ARTICLE INFO
Keywo ds:
La e al– o sional buckling
Geome ically nonlinea analysis
Geome ic impe ec ions
Mono-symme ic c oss-sec ions
ABSTRACT
In his pape , he elas ic la e al– o sional beha io o simple beams is discussed by p esen ing a no el analy ical
solu ion and pe o ming nume ical s udies. The mo i a ion o he p esen ed esea ch is he obse a ion ha
classic analy ical p edic ion and ini e elemen p edic ion a e, ypically, signi ican ly di e en when he second-
o de nonlinea beha io o beams wi h ini ial impe ec ions is analyzed. To unde s and and explain he
obse ed di e ences, a no el analy ical model is wo ked ou o he geome ically nonlinea analysis o beams
wi h ini ial geome ic impe ec ions. The ad ancemen in he p esen ed analy ical solu ion is he explici
conside a ion o he changing geome y as he load inc eases. The mos impo an s eps o he de i a ions
a e summa ized, and he esul ing o mulae a e b ie ly discussed. The de i a ions a e done o gene al c oss-
sec ions, howe e , he bending is assumed o ac in one o he p incipal planes. Nume ical s udies a e also
p esen ed, ocusing on mono-symme ic c oss-sec ions. As pa o he nume ical s udies, i s , he esul s o
he new analy ical o mulae a e compa ed o hose om shell ini e elemen analysis. The esul s sugges
ha he new o mulae can cap u e he mos essen ial elemen s o he beha io obse ed in he shell ini e
elemen calcula ions, jus i ying ha he c oss-sec ion shape migh ha e a signi ican e ec on he nonlinea
la e al– o sional beha io o beams. Then he e ec o he la e al– o sional buckling is p edic ed by calcula ing
buckling educ ion ac o s, using he esul s o he geome ically nonlinea ini e elemen calcula ions. The
capaci y p edic ion esul s, again, jus i y ha he c oss-sec ion shape, as well as he sign o he assumed
geome ic impe ec ion, migh ha e a non-negligible e ec on he buckling educ ion ac o s, and on he
capaci y o he membe .
1. In oduc ion
I a s uc u e o s uc u al pa is slende , a po en ial ailu e mode
is buckling. Buckling may ake place in a ious o ms, bu he phe-
nomenon can usually be desc ibed as ollows: when he load in ensi y
is g adually inc eased on he s uc u e, he displacemen s a e slowly
inc easing (i.e. p ima y displacemen s), bu as he load app oxima es
a ce ain le el, he s uc u e s a s o de elop apidly inc easing dis-
placemen s (i.e., seconda y displacemen s) so ha he na u e and/o
he di ec ion o he seconda y displacemen s is dis inc ly di e en
om ha o he p ima y ones. I he s uc u e is a beam and i is
subjec ed o uniaxial bending in one o he p incipal planes, he p ima y
displacemen s a e in he plane o he loading, while he seconda y dis-
placemen s in ol e la e al ansla ion (i.e., ansla ions pe pendicula
o he plane o loading) and wis ing o a ion. In he case o beams, his
phenomenon is iden i ied as la e al– o sional buckling (LTB).
The e a e mul iple ways o analyze he buckling phenomenon. I
he s uc u e is ee om impe ec ions and i s ma e ial is pe ec ly
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (S. Ádány).
elas ic, he analysis is usually e med linea buckling analysis (LBA),
which leads o he buckling shapes and c i ical load alues (i.e., c i ical
momen s in he case o LTB). Closed- o m analy ical solu ions o he
c i ical momen s a e known and can be ound in ex books [1–3], a
leas o simple cases. Howe e , when he c oss-sec ion, loading, o
bounda y condi ions o he beam a e mo e complex, i is no easy o
ind analy ical solu ions; ha is why he opic is unde esea ch ill
now [4–7].
Ano he way o buckling analysis is when some geome ic impe -
ec ions a e di ec ly conside ed. I he analysis is elas ic, i is popula ly
abb e ia ed as GNI analysis o GNIA (i.e., Geome ically Non-linea
Analysis wi h Impe ec ions). The app oach was i s applied by Young
o columns [8]; i was shown ha he ini ial impe ec ion is ampli ied
due o he comp essi e o ce. The o mula o he ampli ica ion ac o
is known as he Young o mula and is widely used e en in design
calcula ions. I can be shown ha he o mula is alid o any elas ic
s uc u e wi h ini ial geome ic impe ec ion, a leas unde ce ain
condi ions [9]. GNIA can also be used o capaci y p edic ion, as i s
h ps://doi.o g/10.1016/j. ws.2023.110535
Recei ed 17 No embe 2022; Recei ed in e ised o m 5 Janua y 2023; Accep ed 5 Janua y 2023
A ailable online 17 Janua y 2023
0263-8231/©2023 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
M.Z. Ha a , M. Ho áček and S. Ádány Thin-Walled S uc u es 184 (2023) 110535
p oposed by Ay on and Pe y o comp essed columns [10]. This ap-
p oach is he basis o he Eu opean buckling cu es, p oposed by [11],
which a e s ill used in he cu en Eu ocode s anda ds [12,13], bo h
o column buckling and LTB. Fu he gene aliza ion o he app oach
is p oposed in [14].
A mo e ad anced e sion o buckling analysis conside s no only
ini ial impe ec ions bu also ma e ial nonlinea i y, known as GMNI
analysis o simply GMNIA. While GNIA can be pe o med analy i-
cally, a leas o simple cases, GMNIA equi es some nume ical ool,
which is usually he ini e elemen me hod. Using GMNIA he load–
displacemen pa h can be di ec ly es ablished, and he maximum poin
o he load–displacemen cu e iden i ies he capaci y o he s uc u e.
Bo h GNIA and GMNIA need some ini ial geome ic impe ec ion. In
he p ac ice, i is ei he assumed in a p ede ined shape, o as a buckling
shape om an LBA. The ad an age o using a buckling shape as ini ial
impe ec ion is ha his app oach is qui e gene al and au oma ic,
i.e., can be applied o i ually any s uc u e and du ing he p ocess,
i equi es no o li le enginee ing decision. The p ope de e mina ion
o he impe ec ion magni ude is a c ucial ques ion ha is no ully
answe ed ye . (In many s uc u al p oblems i is also a ques ion which
buckling shape is o be used, bu in he case o simple beams he i s
buckling mode is a p ope choice.)
The classic analy ical solu ion o GNIA is he Young o mula, e-
cen ly i was shown in [15,16], ha he e is a disc epancy be ween he
esul s p edic ed by he classic analy ical GNI solu ion (see e.g., [9,17])
and hose calcula ed by shell FEM GNI analyses. The disc epancies can
be impo an . Also, he disc epancies a e no limi ed o he di e ence
o ce ain nume ical alues, some basic ea u es o he beha io a e
a ec ed.
To e eal he easons o he expe ienced disc epancies, he au ho s
de eloped an ad anced analy ical model o sol e he GNIA p oblem
o beams subjec ed o la e al– o sional beha io . A simple e sion o
he analy ical model has been in oduced and discussed in [18], whe e
doubly symme ic I-sec ion beams we e conside ed. In his p esen
pape a gene alized e sion o he analy ical model is p esen ed. The
aim is, s ill, o ha e an analy ical solu ion o he nonlinea beha io
o simple beams wi h ini ial geome ic impe ec ions, howe e , he e
he c oss-sec ion is a bi a y. Since non-symme ic c oss-sec ions a e
a ely used in he p ac ice, he ocus is on mono-symme ic c oss-
sec ions. Mo eo e , i is discussed wha p ac ical consequences can
be expec ed due o he di e ences be ween he classic and he mo e
ealis ic solu ions (e.g., FEM o he he e-p esen ed ad anced analy ical
solu ions).
In he pape , i s , he p oposed analy ical model is summa ized (in
Sec ion 2), hen he solu ion is discussed (in Sec ion 3). In Sec ion 4
nume ical solu ions om he in oduced analy ical model a e compa ed
o hose om FEM GNI analyses. In Sec ion 5, u he nume ical s udies
a e included, whe e capaci y p edic ions a e p esen ed o demons a e
he impo ance o he di e ences be ween he classic and mo e ad-
anced solu ions when GNIA o GMNIA is used o capaci y p edic ion.
Finally, conclusions a e d awn.
2. The analy ical model
2.1. Gene al
In Sec ion 2 he applied analy ical model is desc ibed. The model is
de eloped o simply suppo ed beams subjec ed o wo opposi e end-
momen s, see Fig. 1, hence he momen is uni o m along he leng h.
The c oss-sec ion is a bi a y, bu i is assumed ha he loading is in
a p incipal plane, i.e., he bending akes place a ound a p incipal axis
o he c oss-sec ion. The coo dina e sys em is adjus ed o he p incipal
axes, ha is Yand Za e he wo p incipal axes, and he Xaxis is
aligned o he mass cen e o he c oss-sec ion.
The mechanical ea u es a e as ollows: (i) he hin-walled mem-
be is modeled as an assembly o plane pla es ( e e ed o also as
Fig. 1. Simply suppo ed beam in uni o m bending.
s ips), (ii) global buckling is de ined as he modes wi h no c oss-
sec ion dis o ion, and wi h no in-plane ans e se ex ension and no
in-plane shea de o ma ions, (iii) he s ess–s ain ield is assumed
acco ding o Ki chho pla e heo y o he ou -o -plane beha io and o
a con en ional 2D s ess–s ain s a e o he in-plane beha io o each
pla e elemen , (i ) he ma e ial model ollows Hooke’s law, ( ) he end
momen s a e applied as dis ibu ed loading, linea ly a ying o e he
(end) c oss-sec ions.
I is o men ion ha mechanically iden ical analy ical models ha e
been success ully applied e.g., in [19,20].
2.2. Kinema ics
Since global buckling is de ined as displacemen s wi h no c oss-
sec ion dis o ion, he global displacemen s o he e e ence line o he
membe (de ined by he c oss-sec ion mass cen e s) a e exp essed as
ollows
𝑈=𝑈𝑚𝑓𝑈(𝑋)
𝑉=𝑉𝑚𝑓𝑉(𝑋)
𝑊=𝑊𝑚𝑓𝑊(𝑋)
∅=∅𝑚𝑓∅(𝑋)
(1)
whe e Um,Vmand Wma e global ansla ional displacemen ampli-
udes, ∅𝑚is he o sional displacemen ampli ude, and he unc ions
de e mine he displacemen s’ dis ibu ions in he longi udinal di ec ion.
Though he longi udinal displacemen s may ha e some e ec , as
discussed e.g., in [19], his e ec is ypically negligible o p ac ical
c oss-sec ions and p ac ical leng h anges, he e o e, he global lon-
gi udinal displacemen o he c oss-sec ions will be neglec ed he e,
i.e., Um= 0 is assumed. Mo eo e , since he loading is in he X-Z
plane, he p ima y displacemen is he W. E en hough he e migh
be an in e ac ion be ween he a ious displacemen componen s, i is
easonable o assume ha Wis due, mos ly, o p ima y loading. The
p ima y loading is a uni o m momen ; he e o e a quad a ic unc ion is
conside ed o W, in acco dance wi h he classic i s -o de solu ion.
Finally, o Vand ∅hal sinewa es a e conside ed, in acco dance
wi h classic linea buckling solu ions o LTB. The e o e, he assumed
displacemen unc ions a e as ollows:
𝑊(𝑋) = 𝑊𝑚
4
𝐿2(𝐿−𝑋)𝑋
𝑉(𝑋) = 𝑉𝑚sin (𝜋𝑋
𝐿)
∅(𝑋)=∅𝑚sin (𝜋𝑋
𝐿)(2)
whe e Lis he membe leng h. I is o unde line ha V,Wand ∅a e
he o al displacemen s, measu ed om he pe ec , s aigh posi ion.
The goal he e is o de i e an analy ical solu ion o GNIA, hence
ini ial geome ic impe ec ions a e conside ed, oo. In acco dance wi h
classic solu ions, hal sinewa es a e applied as ollows:
𝑉𝑖𝑛𝑖(𝑋) = 𝑉𝑚,𝑖𝑛𝑖 sin (𝜋𝑋
𝐿),∅𝑖𝑛𝑖(𝑋)=∅𝑚,𝑖𝑛𝑖 sin (𝜋𝑋
𝐿)(3)
2
M.Z. Ha a , M. Ho áček and S. Ádány Thin-Walled S uc u es 184 (2023) 110535
Fig. 2. Global/local coo dina es and displacemen s.
The global displacemen s o he membe ully de e mine he displace-
men s o he s ips, oo, because he c oss-sec ion is igid. By con-
side ing ha he global Xaxis is aligned o he mass cen e o he
c oss-sec ion, and he local and global Xand xa e pa allel wi h
each o he , he displacemen s o he s ips’ (longi udinal) mid-lines a e
exp essed as ollows:
𝑢𝑚,𝑖(𝑥)=−𝜕𝑉
𝜕𝑋 𝑌𝑚,𝑖 −𝜕𝑊
𝜕𝑋 𝑍𝑚,𝑖 +𝜕∅
𝜕𝑋 𝜔𝑚,𝑖
𝑣𝑚,𝑖(𝑥) = 𝑉 𝑐𝑜𝑠𝛼𝑖+𝑊 𝑠𝑖𝑛𝛼𝑖+ ∅ ((𝑌𝑚,𝑖 −𝑌𝑆)𝑠𝑖𝑛𝛼𝑖−(𝑍𝑚,𝑖 −𝑍𝑆)𝑐𝑜𝑠𝛼𝑖)
𝑤𝑚,𝑖(𝑥)=−𝑉 𝑠𝑖𝑛𝛼𝑖+𝑊 𝑐𝑜𝑠𝛼𝑖+ ∅ ((𝑌𝑚,𝑖 −𝑌𝑆)𝑐𝑜𝑠𝛼𝑖+(𝑍𝑚,𝑖 −𝑍𝑆)𝑠𝑖𝑛𝛼𝑖)
𝜑𝑚,𝑖(𝑥)=∅
(4)
whe e Y𝑚,𝑖 and Z𝑚,𝑖 a e he global coo dina es o he i h s ip’s mid-
poin , YSand ZSa e he global coo dina es o he shea cen e o he
c oss-sec ion, 𝜔𝑚,𝑖 is he sec o al coo dina e (wi h espec o he shea
cen e ) a he loca ion o he i h s ip mid-poin , and 𝛼𝑖o he local
y-axis o he i h s ip o he global Y-axis o he membe , see Fig. 2.
No e ha o mulae o he calcula ion o shea cen e and sec o al
coo dina es can be ound in ex books, as well as a good summa y
is gi en in he Eu ocode o cold- o med s eel, see Annex C o EN
1993-1-3:2006 [13].
By using he abo e displacemen s in he mid-poin o he s ips, he
local displacemen unc ions o he s ips can be cons uc ed as ollows:
𝑢𝑖(𝑥, 𝑦, 𝑧) = 𝑢𝑚,𝑖 −𝑣𝑚,𝑖𝑦−𝑤𝑚,𝑖𝑧−𝜑𝑚,𝑖𝑦𝑧
𝑣𝑖(𝑥, 𝑦, 𝑧) = 𝑣𝑚,𝑖 −𝜑𝑚,𝑖𝑧
𝑤𝑖(𝑥, 𝑦) = 𝑤𝑚,𝑖 +𝜑𝑚,𝑖𝑦
𝜑𝑖(𝑥) = 𝜑𝑚,𝑖
(5)
Thus, he u(x,y,z), (x,y,z), w(x,y) and 𝜑(x) local displacemen unc-
ions a e exp essed by he Vm,Wmand ∅𝑚global displacemen ampli-
udes.
2.3. To al po en ial
Fo he solu ion he ene gy me hod is ollowed. The e o e, he
in e nal po en ial (i.e., accumula ed elas ic s ain ene gy) and ex e nal
po en ial (i.e., he nega i e o he wo k done by he loads) should be ex-
p essed. Howe e , i is assumed ha he load is applied in inc emen s,
hence he inc emen o he o al po en ial is necessa y, exp essed as
he sum o he in e nal po en ial inc emen and he ex e nal po en ial
inc emen , as ollows:
𝛥𝛱 =𝛥𝛱𝑖𝑛𝑡 +𝛥𝛱𝑒𝑥𝑡 (6)
A e ‘j-1’ inc emen s, he load alue is 𝑀𝑌 𝑎, and he co esponding
displacemen ampli udes a e 𝑊𝑚𝑎,𝑉𝑚𝑎 and ∅𝑚𝑎. This s a e is an equi-
lib ium s a e, and i is e e ed o as s a e ‘a’. The goal is o ind he
displacemen inc emen s 𝛥𝑊𝑚𝑗,𝛥𝑉𝑚𝑗 and 𝛥∅𝑚𝑗 in he 𝑗 h inc emen al
s ep as he load is u he inc eased by 𝛥𝑀𝑌, ha is when he membe
eaches he nex equilib ium s a e, e e ed o as s a e ‘b’. The load and
he displacemen s a he end o he inc emen al s ep a e, he e o e:
𝑀𝑌 𝑏 =𝑀𝑌 𝑎 +𝛥𝑀𝑌𝑊𝑚𝑏 =𝑊𝑚𝑎 +𝛥𝑊𝑚𝑗
𝑉𝑚𝑏 =𝑉𝑚𝑎 +𝛥𝑉𝑚𝑗 ∅𝑚𝑏 = ∅𝑚𝑎 +𝛥∅𝑚𝑗
(7)
2.3.1. In e nal po en ial inc emen
The s ain ene gy can eadily be exp essed in bo h s a es ‘a’ and ‘b’,
and hen he ene gy inc emen is simply he di e ence. Ma hema ically,
𝛥𝛱𝑖𝑛𝑡 =𝛱𝑖𝑛𝑡,𝑏 −𝛱𝑖𝑛𝑡,𝑎 (8)
wi h
𝛱𝑖𝑛𝑡,𝑎 =1
2
𝑛
∑
𝑖=1 ∫𝑉
𝐸𝑡𝑖(𝜕𝑢𝑖𝑎
𝜕𝑥 )2
+𝐸𝑡𝑖3
12 (𝜕2𝑤𝑖𝑎
𝜕𝑥2)2
+𝐺𝑡𝑖3
3(𝜕2𝑤𝑖𝑎
𝜕𝑥𝜕𝑦 )2
d𝐴
𝛱𝑖𝑛𝑡,𝑏 =1
2
𝑛
∑
𝑖=1 ∫𝑉
𝐸𝑡𝑖(𝜕(𝑢𝑖𝑎 +𝛥𝑢𝑖𝑗 )
𝜕𝑥 )2
+𝐸𝑡𝑖3
12 (𝜕2(𝑤𝑖𝑎 +𝛥𝑤𝑖𝑗)
𝜕𝑥2)2
+𝐺𝑡𝑖3
3(𝜕2(𝑤𝑖𝑎 +𝛥𝑤𝑖𝑗)
𝜕𝑥𝜕𝑦 )2
d𝐴
(9)
whe e 𝑢𝑖𝑎 and 𝑤𝑖𝑎 a e he displacemen unc ions o he i h s ip a
s a e ‘a’, and 𝛥𝑢𝑖𝑗 and 𝛥𝑤𝑖𝑗 a e displacemen inc emen unc ions o
he i h s ip in he j h load inc emen al s ep. These local displace-
men unc ions a e linked o he global displacemen unc ions h ough
Eqs. (4)–(5). Mo eo e , in Eq. (9) he in eg al, in ac , means double
in eg a ion wi h espec o xand y, o he whole su ace a ea o he
s ip, i.e., xis aken om 0 o L, and yis aken om (-b𝑖/2) o (+b𝑖
/2). Mo eo e , Lis he membe leng h, b𝑖and 𝑖a e he wid h and
hickness o he i h s ip, espec i ely, nis he numbe o s ips, and E
and Ga e he modulus o elas ici y and he shea modulus, espec i ely.
I is o obse e ha he s ain ene gy inc emen is exp essed wi h
espec o he global displacemen inc emen s 𝛥𝑉𝑚𝑗,𝛥𝑊𝑚𝑗 and 𝛥∅𝑚𝑗
(and ob iously, is also dependen on he cu en displacemen s).
2.3.2. Ex e nal po en ial inc emen
Fo he ex e nal po en ial, he gene ic exp ession is as ollows
𝛱𝑒𝑥𝑡 = −
𝑛
∑
𝑖=1 ∫𝑉
𝜎𝑥,𝑖𝜀𝑥,𝑖d𝐴(10)
In his exp ession 𝜎𝑥,𝑖 is he longi udinal no mal s ess unc ion o
he i h s ip, 𝜀𝑥,𝑖 is he co esponding s ain unc ion (i.e., longi udinal
no mal s ain). Hence, i is implici ly assumed he e ha he o he
s ess and s ain componen s a e negligibly small, which is a easonable
and classic assump ion o global buckling p oblems. S ill, he exac
meaning o 𝜎𝑥,𝑖 and 𝜀𝑥,𝑖 needs u he conside a ion.
Since we wan a geome ically non-linea analysis, we need o
conside linea and non-linea s ain e ms. A a ce ain s a e, e.g., s a e
‘a’ and ‘b’, he linea , i s -o de s ain unc ion o he i h s ip is
exp essed as:
𝜀𝑥,𝑖𝑎𝐼=𝜕𝑢𝑖𝑎
𝜕𝑥 𝜀𝑥,𝑖𝑏𝐼=𝜕𝑢𝑖𝑏
𝜕𝑥 =𝜕𝑢𝑖𝑎
𝜕𝑥 +𝜕𝛥𝑢𝑖𝑗
𝜕𝑥 (11)
Hence he s ain inc emen is:
𝛥𝜀𝑥,𝑖𝑗𝐼=𝜀𝑥,𝑖𝑏𝐼−𝜀𝑥,𝑖𝑎𝐼=𝜕𝛥𝑢𝑖𝑗
𝜕𝑥 (12)
Fo he non-linea pa he G een–Lag ange s ain enso is applied. The
s ain in he i h s ip can be exp essed as:
𝜀𝑥,𝑖𝑎𝐼𝐼 =1
2[(𝜕𝑣𝑖𝑎
𝜕𝑥 )2
+(𝜕𝑤𝑖𝑎
𝜕𝑥 )2]
𝜀𝑥,𝑖𝑏𝐼𝐼 =1
2[(𝜕(𝑣𝑖𝑎 +𝛥𝑣𝑖𝑗)
𝜕𝑥 )2
+(𝜕(𝑤𝑖𝑎 +𝛥𝑤𝑖𝑗)
𝜕𝑥 )2](13)
3
M.Z. Ha a , M. Ho áček and S. Ádány Thin-Walled S uc u es 184 (2023) 110535
om which he non-linea s ain inc emen can be exp essed as:
𝛥𝜀𝑥,𝑖𝐼𝐼 =1
2(𝜕𝛥𝑣𝑖𝑗
𝜕𝑥 )2
+𝜕𝛥𝑣𝑖𝑗
𝜕𝑥
𝜕𝑣𝑖𝑎
𝜕𝑥 +1
2(𝜕𝛥𝑤𝑖𝑗
𝜕𝑥 )2
+𝜕𝛥𝑤𝑖𝑗
𝜕𝑥
𝜕𝑤𝑖𝑎
𝜕𝑥 (14)
whe e 𝑣𝑖𝑎 and 𝑤𝑖𝑎 a e he displacemen unc ions o he i h s ip a
s a e ‘a’, and 𝛥𝑣 and 𝛥𝑤𝑖𝑗 a e displacemen inc emen unc ions o he
i h s ip in he j h load inc emen al s ep; and hese local displacemen
unc ions can be de i ed om he global displacemen unc ions by
using Eqs. (4)–(5).
The s ess in he membe , he e o e he s ess in each s ip is
dependen on he loading, as well as on whe he we conside he
changed s ess s a e as he membe de o ms. I he membe is subjec ed
o uniaxial bending, he p ima y, i s -o de s ess is linea ly a ying
wi h Z. Fo he i h s ip, a s a e ‘b’, i can be exp essed as:
𝜎𝑥,𝑖𝑏𝐼= − 𝑀𝑌 𝑎 +𝛥𝑀𝑌
𝐼𝑌(𝑍𝑚,𝑖 +𝑦⋅𝑠𝑖𝑛 (𝛼𝑖)+𝑧⋅𝑐𝑜𝑠 (𝛼𝑖)) (15)
whe e 𝐼𝑌is he second momen o a ea calcula ed o he global Y-axis.
Since a load inc emen al p ocedu e is conside ed he e, whe e in
each inc emen al s ep he ac ual displacemen s a e calcula ed and
he s esses a e upda ed, we need o conside he e ec o changing
geome y, i.e., he second-o de s esses due o 𝑉and ∅. To de e mine
hese second-o de s esses, we u ilize classic di e en ial equa ions,
namely Eqs. (5–78) and (5–80) o [1].
Eqs. (5–78) and o [1] can be w i en (by using he no a ions o he
ac ual pape ) as ollows:
𝐸𝐼𝑧
𝜕4(𝑉−𝑉𝑖𝑛𝑖)
𝜕𝑋4−𝑀𝑌
𝜕2∅
𝜕𝑋2= 0 (16)
whe e 𝐼𝑍is he second momen o a ea calcula ed o he global Z-axis.
This equa ion exp esses he momen equilib ium in he la e al di ec ion
(i.e., in he plane pe pendicula o he plane o loading). The i s e m
ep esen s he s ess esul an , which, i an ini ial geome ic impe ec-
ion is p esen , is o be calcula ed om he (𝑉−𝑉𝑖𝑛𝑖)displacemen .
The second e m ep esen s he la e al componen o he M𝑌as he
c oss-sec ion is wis ed.
Conside ing he assumed displacemen unc ions o Eqs. (2)–(3), and
a e simpli ica ion, we ge :
(𝑉𝑚−𝑉𝑚,𝑖𝑛𝑖)𝜋2
𝐿2𝐸𝐼𝑍sin (𝜋𝑋
𝐿)+𝑀𝑌∅𝑚sin (𝜋𝑋
𝐿)= 0 (17)
The esul an o he s esses o he la e al bending can be exp essed
as:
𝑀𝑍=𝐸𝐼𝑍
𝜕2(𝑉−𝑉𝑖𝑛𝑖)
𝜕𝑋2(18)
o , wi h conside ing he shape unc ions o Eqs. (2)–(3):
𝑀𝑍= − (𝑉𝑚−𝑉𝑚,𝑖𝑛𝑖)𝜋2
𝐿2𝐸𝐼𝑍sin (𝜋𝑋
𝐿)(19)
Le us subs i u e he abo e equa ion in o Eq. (17), and we ge he la e al
bending momen due o he wis ing o a ion o he c oss-sec ion as
ollows:
𝑀𝑍=𝑀𝑌∅(20)
Thus, when he membe is wis ed, 𝑀𝑌gene a es 𝑀𝑍, i.e. he p ima y
bending gene a es la e al bending. F om he la e al bending he s ess
in he 𝑖 h s ip, a s a e ‘b’, is exp essed as:
𝜎𝑥,𝑖𝑏𝐼𝐼,𝑙𝑎𝑡𝑒𝑟𝑎𝑙 = − (𝑀𝑌 𝑎 +𝛥𝑀𝑌)∅𝑚𝑎
𝐼𝑍(𝑌𝑚,𝑖 +𝑦⋅cos (𝛼𝑖)
+𝑧⋅sin (𝛼𝑖))sin (𝜋𝑋
𝐿)(21)
Simila ly, i he membe is la e ally displaced, M𝑌gene a es bimomen .
Eq. (5–80) o [1] can be w i en (by using he no a ions o he ac ual
pape ) as:
𝐸𝐼𝑤
𝜕4(∅ − ∅𝑖𝑛𝑖)
𝜕𝑋4−𝐺𝐼𝑡
𝜕2(∅ − ∅𝑖𝑛𝑖)
𝜕𝑋2− 2𝛽𝑍𝑀𝑌
𝜕2∅
𝜕𝑋2−𝑀𝑌
𝜕2V
𝜕𝑋2= 0 (22)
whe e 𝐼𝑤is he wa ping cons an , 𝐼𝑡is he o sion cons an , and 𝛽𝑍is
a non-symme y cons an , de ined by Eq. (38).
Eq. (22) exp esses he equilib ium o o sional momen s. The i s
and second e ms ep esen he s ess esul an s om wa ping o sion
(i.e., esul an o wa ping no mal s esses) and Sain -Venan o sion
(i.e., esul an s o shea s esses), espec i ely. I ini ial geome ic
impe ec ion is p esen , hese e ms should be calcula ed om he
(∅ − ∅𝑖𝑛𝑖)displacemen . The hi d and ou h e ms ep esen he
o sional momen componen o he ex e nal M𝑌momen , due o he
displacemen s.
Bo h 𝑉and ∅a e assumed o ha e a hal sinewa e longi udinally,
and he same o he ini ial shapes. By conside ing hese unc ions we
ge :
𝐸𝐼𝑤
𝜋2
𝐿2(∅𝑚− ∅𝑚,𝑖𝑛𝑖)+𝐺𝐼𝑡(∅𝑚− ∅𝑚,𝑖𝑛𝑖)+ 2𝛽𝑍𝑀𝑌∅𝑚+𝑀𝑌𝑉𝑚= 0 (23)
om which:
(∅𝑚− ∅𝑚,𝑖𝑛𝑖)=−𝑀𝑌𝑉𝑚− 2𝛽𝑍𝑀𝑌∅𝑚
𝐸𝐼𝑤
𝜋2
𝐿2+𝐺𝐼𝑡
(24)
By in oducing:
𝐹𝑡=𝐺𝐼𝑡and 𝐹𝑤=𝜋2𝐸𝐼𝑤
𝐿2(25)
he Eq. (24) can be w i en as:
(∅𝑚− ∅𝑚,𝑖𝑛𝑖)=−𝑀𝑌𝑉𝑚− 2𝛽𝑍𝑀𝑌∅𝑚
𝐹𝑤+𝐹𝑡
(26)
I is known ha he bimomen is he o sional esul an o he wa ping
no mal s esses, he e o e:
𝐵=𝐸𝐼𝑤
𝜕2(∅ − ∅𝑖𝑛𝑖)
𝜕𝑋2(27)
om which, by using Eqs. (26)–(27), he bimomen ampli ude can be
exp essed as:
𝐵𝑚= −𝐸𝐼𝑤
𝜋2
𝐿2(∅𝑚− ∅𝑚,𝑖𝑛𝑖)= −𝐹𝑤(∅𝑚− ∅𝑚,𝑖𝑛𝑖)(28)
Subs i u ing Eq. (26) in o Eq. (28) we ge he bimomen ampli ude
which ep esen s he in luence o he second-o de displacemen s, as
ollows:
𝐵𝑚=𝐹𝑤
𝑀𝑌𝑉𝑚+ 2𝛽𝑍𝑀𝑌∅𝑚
𝐹𝑤+𝐹𝑡
=𝑀𝑌(𝑉𝑚+ 2𝛽𝑍∅𝑚)1
1 + 𝐹𝑡∕𝐹𝑤
(29)
F om he bimomen he s ess in he i h s ip can eadily be calcula ed,
e.g., a s a e ‘b’ i is:
𝜎𝑥,𝑖𝑏𝐼𝐼,𝑡𝑤𝑖𝑠𝑡 = −(𝑀𝑌 𝑎 +𝛥𝑀𝑌)𝑉𝑚𝑎 + 2𝛽𝑍∅𝑚𝑎
𝐼𝑤
1
1 + 𝐹𝑡∕𝐹𝑤
𝜔𝑖(𝑦, 𝑧) sin (𝜋𝑋
𝐿)
(30)
whe e 𝜔𝑖is he sec o al coo dina e unc ion o he i h s ip.
Thus, wi h all he abo e conside a ions, he ex e nal po en ial in-
c emen can be exp essed as ollows:
𝛥𝛱𝑒𝑥𝑡 = −
𝑛
∑
𝑖=1 ∫𝑉(𝜎𝑥,𝑖𝑏𝐼)(𝛥𝜀𝑥,𝑖𝐼+𝛥𝜀𝑥,𝑖𝐼𝐼 )d𝐴−
𝑛
∑
𝑖=1 ∫𝑉(𝜎𝑥,𝑖𝑏𝐼𝐼,𝑙𝑎𝑡𝑒𝑟𝑎𝑙
+𝜎𝑥,𝑖𝑏𝐼𝐼,𝑡𝑤𝑖𝑠𝑡)(𝛥𝜀𝑥,𝑖𝐼𝐼 )d𝐴
(31)
The i s e m shows he wo k o he p ima y loading, while he second
e m is due o he seconda y s esses, i.e., he s esses induced by he
seconda y displacemen s. I is o obse e ha he wo k inc emen ,
again, is exp essed wi h espec o he global displacemen inc emen s
𝛥𝑉𝑚𝑗,𝛥𝑊𝑚𝑗 and 𝛥∅𝑚𝑗, hence he inc emen o he o al po en ial is
exp essed by he displacemen inc emen s.
4
M.Z. Ha a , M. Ho áček and S. Ádány Thin-Walled S uc u es 184 (2023) 110535
2.4. Equilib ium equa ion
In equilib ium he o al po en ial is s a iona y, he e o e:
𝜕𝛥𝛱
𝜕𝛥𝑊𝑚𝑗
= 0 𝜕𝛥𝛱
𝜕𝛥𝑉𝑚𝑗
= 0 𝜕𝛥𝛱
𝜕𝛥∅𝑚𝑗
= 0 (32)
Eq. (32) is a sys em o h ee equa ions, which can be summa ized in o
one single ma ix equa ion as ollows:
𝐊𝐞𝜟𝐝+𝑀𝑌 𝑏𝐊𝐠𝜟𝐝=𝜟𝐟(33)
whe e he i s ma ix is he 𝐊𝐞elas ic s i ness ma ix, while he second
one is he 𝐊𝐠geome ic s i ness ma ix o he p oblem:
𝐊𝐞=⎡⎢⎢⎢⎢⎢⎣
𝐹𝑌
64
𝜋2𝐿0 0
0𝐹𝑍
𝜋2
2𝐿0
0 0 𝐹𝑋
𝜋2
2𝐿
⎤⎥⎥⎥⎥⎥⎦
𝐊𝐠=⎡⎢⎢⎢⎢⎢⎣
0 0 −∅𝑚𝑎
2
𝐿
0 0 𝜋2
2𝐿
−∅𝑚𝑎
2
𝐿
𝜋2
2𝐿
𝜋2
𝐿𝛽𝑍+ ∅𝑚𝑎
4𝜋
3𝐿𝛽𝑌
⎤⎥⎥⎥⎥⎥⎦
(34)
Mo eo e , he displacemen ec o con ains he displacemen inc e-
men s o he gi en j h load s ep
𝜟𝐝=⎡⎢⎢⎢⎢⎣
𝛥𝑊𝑚𝑗
𝛥𝑉𝑚𝑗
𝛥∅𝑚𝑗
⎤⎥⎥⎥⎥⎦
(35)
and he load ec o a he igh -hand-side o he equa ion is de ined as:
𝜟𝐟=⎡⎢⎢⎢⎢⎢⎢⎢⎣
−𝑀𝑦𝑏
8
𝐿+𝑀𝑦𝑏∅2
𝑚𝑎
2
𝐿−𝑊𝑚𝑎𝐹𝑌
64
𝜋2𝐿
−𝑀𝑦𝑏∅𝑚𝑎
𝜋2
2𝐿− (𝑉𝑚𝑎 −𝑉𝑚,𝑖𝑛𝑖)𝐹𝑍
𝜋2
2𝐿
−𝑀𝑦𝑏𝑉𝑚𝑎
𝜋2
2𝐿−𝑀𝑦𝑏∅𝑚𝑎
𝜋2
𝐿𝛽𝑍−𝑀𝑦𝑏∅𝑚𝑎
4𝜋
3𝐿𝛽𝑌
+𝑀𝑦𝑏𝑊𝑚𝑎∅𝑚𝑎
2
𝐿− (∅𝑚𝑎 − ∅𝑚,𝑖𝑛𝑖)𝐹𝑋
𝜋2
2𝐿
⎤⎥⎥⎥⎥⎥⎥⎥⎦
(36)
In he abo e ma ices and ec o s, he 𝐹𝑥,𝐹𝑦and 𝐹𝑧symbols a e
de ined as ollows:
𝐹𝑌=𝜋2𝐸𝐼𝑌
𝐿2, 𝐹𝑍=𝜋2𝐸𝐼𝑍
𝐿2, 𝐹𝑋=𝐹𝑡+𝐹𝑤=𝐺𝐼𝑡+𝜋2𝐸𝐼𝑤
𝐿2(37)
whe e non-symme y cons an s a e de ined as:
𝛽𝑌=𝑦𝑆− 0.5∫𝐴(𝑦2+𝑧2)𝑦d𝐴
𝛽𝑍=𝑧𝑆− 0.5∫𝐴(𝑦2+𝑧2)𝑧d𝐴
(38)
whe e 𝑦𝑆, 𝑧𝑆a e he coo dina es o he shea cen e ela i e o he
cen oid.
The abo e equa ion sys em can be sol ed analy ically. By in oduc-
ing

𝑀2
𝑐𝑟 =𝑀𝑐𝑟2− 2𝑀𝑐𝑟𝐹𝑍𝛽𝑍(39)
he esul ing displacemen inc emen s a e gi en as in Box I
3. Discussion o he analy ical solu ion
3.1. LBA
Based on he abo e de i a ion, i is easy o ge he LBA p oblem,
by simpli ica ion. In LBA he e is no ini ial impe ec ion, only he
p ima y s esses and only he second-o de s ain e ms a e o be
conside ed. Mo eo e , he i s equa ion in Eq. (33) is no necessa y,
since in he solu ion he p ima y displacemen s ha e no ole. Finally,
no load inc emen s a e necessa y, i.e., i is enough o ocus on he
i s s ep o he inc emen al p ocedu e; consequen ly: he 𝛥𝑉𝑚𝑗 and
𝛥∅𝑚𝑗 displacemen s a e simply he o al displacemen s 𝑉𝑚and ∅𝑚, and
𝛥𝑀𝑌is simply he o al 𝑀𝑌. The equa ion sys em in Eq. (33) he e o e
simpli ies o he ollowing one:
[𝐹𝑍0
0𝐹𝑋].[𝑉𝑚
∅𝑚]+[0𝑀𝑌
𝑀𝑌2𝑀𝑌𝛽𝑍].[𝑉𝑚
∅𝑚]=[0
0](43)
The solu ion exis s i 𝑀𝑌 akes a speci ic alue, he 𝑀𝑐𝑟 c i ical momen ,
which is exp essed as:
𝑀𝑐𝑟 =𝐹𝑍𝛽𝑍±√(𝐹𝑍𝛽𝑍)2+𝐹𝑋𝐹𝑍(44)
By back-subs i u ion, we can ind he buckled shape. Ob iously, he
𝑉(𝑋)and ∅(X)displacemen unc ions ha e a hal -sinewa e shape
longi udinally. Since he 𝐊𝐞+𝐊𝐠ma ix is ank-de icien in he c i ical
s a e, i.e., when 𝑀𝑌=𝑀𝑐𝑟, he 𝑉𝑚and ∅𝑚ampli udes a e dependen
on each o he , o he wise he ampli udes a e a bi a y. The ela ionship
be ween 𝑉𝑚and ∅𝑚is as ollows
𝑉𝑚= −∅𝑚
𝑀𝑐𝑟
𝐹𝑍
(45)
3.2. Classic GNIA
Classic GNIA solu ion can also be ob ained by simpli ying Eq. (33).
Only he p ima y s esses and only he second-o de s ain e ms a e o
be conside ed. Mo eo e , he i s equa ion in Eq. (33) is no necessa y.
I he ini ial geome ic impe ec ion is he buckling shape om he LBA,
𝑉𝑚,𝑖𝑛𝑖 and ∅𝑚,𝑖𝑛𝑖 a e dependen on each o he , see Eq. (45). Finally, no
load inc emen s a e necessa y, i.e., i is enough o ocus on he i s s ep
o he inc emen al p ocedu e. He e, o dis inguish be ween he ini ial
displacemen and he displacemen inc emen , we keep he no a ion
o he 𝛥𝑉𝑚and 𝛥∅𝑚displacemen inc emen s, howe e , he momen
inc emen 𝛥𝑀𝑌is simply he o al 𝑀𝑌. Finally, he equa ion sys em o
Eq. (33) simpli ies o he ollowing one:
[𝐹𝑍0
0𝐹𝑋].[𝛥𝑉𝑚
𝛥∅𝑚]+[0𝑀𝑌
𝑀𝑌2𝑀𝑌𝛽𝑍].[𝛥𝑉𝑚
𝛥∅𝑚]
=[−𝑀𝑌∅𝑚,𝑖𝑛𝑖
−𝑀𝑌𝑉𝑚,𝑖𝑛𝑖 − 2𝑀𝑌𝛽𝑍∅𝑚,𝑖𝑛𝑖](46)
The solu ion o he displacemen inc emen s:
𝛥𝑉𝑚=𝑉𝑚,𝑖𝑛𝑖
1
𝑀𝑐𝑟∕𝑀𝑌− 1
𝛥∅𝑚= ∅𝑚,𝑖𝑛𝑖
1
𝑀𝑐𝑟∕𝑀𝑌− 1
(47)
The o al displacemen is he sum o he ini ial one and he inc emen ,
which leads o he well-known o mulae exp essing he displacemen
ampli ica ion (also known as he Young o mula):
𝑉𝑚=𝛥𝑉𝑚+𝑉𝑚,𝑖𝑛𝑖 =𝑉𝑚,𝑖𝑛𝑖
1
1 − 𝑀𝑌∕𝑀𝑐𝑟
∅𝑚=𝛥∅𝑚+ ∅𝑚,𝑖𝑛𝑖 = ∅𝑚,𝑖𝑛𝑖
1
1 − 𝑀𝑌∕𝑀𝑐𝑟
(48)
I is obse ed ha he a io o Vand ∅ emains he same du ing he
analysis.
3.3. Doubly symme ical c oss-sec ions
The exp essions o he displacemen inc emen s, see Eqs. (40)–
(42) can be simpli ied i he c oss-sec ion is doubly symme ic, i.e., i
𝛽𝑌=𝛽𝑍= 0. This also means ha :

𝑀𝑐𝑟 =𝑀𝑐𝑟 = ±√𝐹𝑋𝐹𝑍(49)
I is o obse e ha he second-o de la e al ansla ion and wis ing
o a ion a e di e en om hose p edic ed by he Young o mula.
5

M.Z. Ha a , M. Ho áček and S. Ádány Thin-Walled S uc u es 184 (2023) 110535
𝛥𝑊𝑚=
−𝑀𝑌 𝑏
𝜋2
8𝐹𝑌[
𝑀2
𝑐𝑟
𝑀𝑌 𝑏2− 1]−𝑉𝑚,𝑖𝑛𝑖∅𝑚𝑎
𝜋2𝐹𝑍
32𝐹𝑌+ ∅𝑚,𝑖𝑛𝑖∅𝑚𝑎

𝑀2
𝑐𝑟
𝑀𝑌 𝑏
𝜋2
32𝐹𝑌− ∅𝑚𝑎
𝜋𝐹𝑍𝛽𝑌
3𝐹𝑌−𝐹𝑍𝛽𝑍𝜋2
4𝐹𝑌

𝑀2
𝑐𝑟
𝑀𝑌 𝑏2− 1 + ∅𝑚𝑎
8𝐹𝑍𝛽𝑌
3𝜋𝑀𝑌 𝑏 +2𝐹𝑍𝛽𝑍
𝑀𝑌 𝑏 − ∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌
−𝑊𝑚𝑎 (40)
𝛥𝑉𝑚=
𝑉𝑚,𝑖𝑛𝑖

𝑀2
𝑐𝑟
𝑀𝑌 𝑏2+𝑉𝑚,𝑖𝑛𝑖
2𝐹𝑍𝛽𝑍
𝑀𝑌 𝑏 + ∅𝑚𝑎
𝑀𝑌 𝑏
2𝐹𝑌− ∅𝑚,𝑖𝑛𝑖
𝐹𝑋
𝑀𝑌 𝑏 +𝑉𝑚,𝑖𝑛𝑖∅𝑚𝑎
8𝐹𝑍𝛽𝑌
3𝜋𝑀𝑌 𝑏 −𝑉𝑚,𝑖𝑛𝑖∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌

𝑀2
𝑐𝑟
𝑀𝑌 𝑏2− 1 + ∅𝑚𝑎
8𝐹𝑍𝛽𝑌
3𝜋𝑀𝑌 𝑏 +2𝐹𝑍𝛽𝑍
𝑀𝑌 𝑏 − ∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌
−𝑉𝑚𝑎 (41)
𝛥∅𝑚=
−𝑉𝑚,𝑖𝑛𝑖
𝐹𝑧
𝑀𝑌 𝑏 − ∅𝑚𝑎
𝐹𝑧
2𝐹𝑦+ ∅𝑚,𝑖𝑛𝑖
𝑀𝑐𝑟2
𝑀𝑌 𝑏2

𝑀2
𝑐𝑟
𝑀𝑌 𝑏2− 1 + ∅𝑚𝑎
8𝐹𝑍𝛽𝑌
3𝜋𝑀𝑌 𝑏 +2𝐹𝑍𝛽𝑍
𝑀𝑌 𝑏 − ∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌
− ∅𝑚𝑎 (42)
Box I.
Mo eo e , he o mulae o 𝑉and ∅a e di e en . Independen ly o
whe he he ini ial shape is buckling shape o no , he ampli ica ion
o 𝑉and ∅a e di e en .
An impo an cha ac e is ic o hese o mulae is ha i he sign o
he ini ial geome y is e e sed hen (i) he e ical 𝑊displacemen is
unchanged, (ii) he sign o he la e al inc emen is e e sed, and (iii)
he sign o he wis ing o a ion inc emen is e e sed. This also means
ha a symme ic bi u ca ion is p edic ed (as he ini ial displacemen
con e ges o ze o).
As he bending momen inc eases, he denomina o o he o mulae
can dec ease o ze o, which iden i ies singula i y. The co esponding
momen can be exp essed as
𝑀𝑌 ,𝑠𝑖𝑛𝑔 =𝑀𝑐𝑟
√1+∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌
(50)
F om he o mulae, i can be seen ha he singula i y belongs o a
bending momen smalle han 𝑀𝑐𝑟. The dis ance o he singula i y o
𝑀𝑐𝑟 is la gely dependen on he wis ing o a ion. Howe e , since he
whole heo y assumes ha he o a ions a e small, he bending momen
whe e singula i y happens is only ma ginally smalle han 𝑀𝑐𝑟. S ill,
an impo an di e ence is ha while he classic analy ical solu ion sug-
ges s ha singula i y happens only a in ini ely la ge displacemen s, he
new analy ical o mulae p edic he singula i y a ini e displacemen s.
3.4. Mono-symme ic c oss-sec ions, bending in he symme y plane
Now we conside a mono-symme ic c oss-sec ion. The axis o sym-
me y is he Z-axis, i.e., 𝛽𝑌= 0, and he bending is ac ing in he
symme y plane. The exp essions in Eqs. (40)–(42) o he displacemen
inc emen s can sligh ly be simpli ied. By analyzing he o mulae o
he la e al ansla ion and o he wis ing o a ion, i can again be
obse ed ha hey a e di e en om he Young o mula and om each
o he . I can also be unde s ood ha he sign o he ini ial geome y has
no eal e ec , i.e., a symme ic bi u ca ion is p edic ed (as he ini ial
displacemen con e ges o ze o)
Singula i y happens when he denomina o is equal o ze o. The
bending momen which causes singula i y is he solu ion o a quad a ic
equa ion as ollows:
𝑀𝑌 ,𝑠𝑖𝑛𝑔2(1+∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌)−𝑀𝑌 ,𝑠𝑖𝑛𝑔2𝐹𝑍𝛽𝑍−(𝑀𝑐𝑟2− 2𝑀𝑐𝑟𝐹𝑍𝛽𝑍)= 0 (51)
om which he bending momen can be exp essed as:
𝑀𝑌 ,𝑠𝑖𝑛𝑔 =
2𝐹𝑍𝛽𝑍±√(2𝐹𝑍𝛽𝑍)2+ 4 (1+∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌)(𝑀𝑐𝑟2− 2𝑀𝑐𝑟𝐹𝑍𝛽𝑍)
2(1+∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌)(52)
Howe e , i he wis angle is small, he (1+∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌) e m is close o
one. Wi h his app oxima ion, he abo e o mula can be simpli ied o:
𝑀𝑌 ,𝑠𝑖𝑛𝑔 ≅𝑀𝑐𝑟 (53)
Tha is, he singula i y belongs app oxima ely o 𝑀𝑐𝑟.
3.5. Mono-symme ic c oss-sec ions, bending pe pendicula o he symme y
plane
Now we conside mono-symme ic c oss-sec ions whe e he axis o
symme y is he Y-axis, i.e., 𝛽𝑍= 0. The bending is s ill in he e ical
plane, ha is in a plane pe pendicula o he symme y plane. The
exp essions o he displacemen inc emen s in Eqs. (40)–(42) can be
simpli ied. The c i ical momen o mula is iden ical o ha o doubly
symme ic c oss-sec ions, see Eq. (49).
Pa ly simila obse a ions can be made as abo e. Howe e , now
he e is a signi ican di e ence: he exp ession in he denomina o s o
he o mulae is dependen on he sign o he ini ial displacemen ; and
no only he sign o he exp ession, bu also i s absolu e alue. This
means ha he magni udes o bo h he p ima y ( e ical) displacemen
and he seconda y displacemen a e in luenced by whe he he ini ial
impe ec ion is conside ed wi h posi i e o nega i e sign. This iden i ies
asymme ic bi u ca ion i he ini ial impe ec ion con e ges o ze o.
As he bending momen inc eases, he denomina o o he o mulae
can dec ease o ze o, which iden i ies singula i y. Ma hema ically, he
co esponding bending momen is he solu ion o a quad a ic equa ion:
𝑀𝑌 ,𝑠𝑖𝑛𝑔2(1+∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌)−𝑀𝑌 ,𝑠𝑖𝑛𝑔∅𝑚𝑎
8𝐹𝑍𝛽𝑌
3𝜋−𝑀𝑐𝑟2= 0 (54)
om which he bending momen is
𝑀𝑌 ,𝑠𝑖𝑛𝑔 =
∅𝑚𝑎
8𝐹𝑍𝛽𝑌
3𝜋±√(∅𝑚𝑎
8𝐹𝑍𝛽𝑌
3𝜋)2
+ 4 (1+∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌)𝑀𝑐𝑟2
2(1+∅2
𝑚𝑎
𝐹𝑍
8𝐹𝑌)(55)
E en i he ∅2
𝑚𝑎 e m is neglec ed, we ha e:
𝑀𝑌 ,𝑠𝑖𝑛𝑔 ≅ ∅𝑚𝑎
2𝐹𝑍𝛽𝑌
3𝜋±√(∅𝑚𝑎
2𝐹𝑍𝛽𝑌
3𝜋)2
+𝑀𝑐𝑟2(56)
The posi i e solu ion can be ob ained by using he posi i e sign in
he abo e o mula. Mo eo e , i ∅𝑚𝑎 = 0, hen 𝑀𝑌 ,𝑠𝑖𝑛𝑔 =𝑀𝑐𝑟. I
∅𝑚𝑎 >0,𝑀𝑌 ,𝑠𝑖𝑛𝑔 is clea ly la ge han 𝑀𝑐𝑟; he la ge he ∅𝑚𝑎, he
la ge he di e ence be ween 𝑀𝑌 ,𝑠𝑖𝑛𝑔 and 𝑀𝑐𝑟. I ∅𝑚𝑎 <0, hen 𝑀𝑌 ,𝑠𝑖𝑛𝑔
can be la ge o smalle han 𝑀𝑐𝑟, depending on he magni ude o
∅𝑚𝑎. The dependence o singula i y on he sign o he wis ing o a ion
again jus i ies ha he bi u ca ion p oblem is asymme ic in he case o
mono-symme ic c oss-sec ions loaded pe pendicula ly o he plane o
symme y.
6
M.Z. Ha a , M. Ho áček and S. Ádány Thin-Walled S uc u es 184 (2023) 110535
Fig. 3. Displacemen inc emen s o a ious ini ial impe ec ions shapes.
3.6. The e ec o he impe ec ion shape
The abo e p esen ed solu ion is alid o any impe ec ion shape
(wi h a hal -sinewa e longi udinal dis ibu ion). I is wo h looking a
he e ec o impe ec ion shape. Howe e , o make he compa ison o
he a ious impe ec ion shapes, we ocus on he i s inc emen al s ep
only, i.e., 𝑉𝑚𝑎 =𝑉𝑚,𝑖𝑛𝑖 and ∅𝑚𝑎 = ∅𝑚,𝑖𝑛𝑖 and 𝑀𝑌 𝑏 =𝛥𝑀𝑌.
Mo eo e , o u he simpli y he inal o mulae, we assume a
doubly symme ic c oss-sec ion, and we neglec he highe -o de dis-
placemen e ms in he o mulae. Fou cases a e conside ed, and he
ob ained displacemen inc emen o mulae a e summa ized in Fig. 3.
The impo an obse a ions a e as ollows:
•The o mulae, e en in his simpli ied case, a e (wi h one excep-
ion) di e en om he Young o mula.
•E en i he buckling shape is used as an impe ec ion, he dis-
placemen ampli ica ions a e di e en o he la e al ansla ion
and he wis ing o a ion.
•I one o he componen s o he ini ial geome y (i.e., ei he he
la e al ansla ion o he wis ing o a ion) is ze o, s ill, non-
ze o inc emen s a e esul ed e en in he i s load inc emen al
s ep. The e o e, a single impe ec ion componen (i.e., ei he he
la e al ansla ion o he wis ing o a ion) gene a es bo h la e al
ansla ion and wis ing o a ion du ing he non-linea analysis.
I is o no e ha he abo e obse a ions a e alid o mo e gene al
cases, oo, wi hou he applied simpli ica ions, (i.e., gene al c oss-
sec ions, and/o conside ing highe -o de displacemen e ms in he
o mulae,) howe e , he esul ed o mulae a e much mo e complex,
ha is why no shown he e.
4. Nume ical s udies: compa ison o analy ical and shell FEM
esul s
4.1. Summa y o he nume ical s udies
Nume ical s udies a e comple ed by using he abo e-desc ibed new
analy ical model, compa ing he esul s o hose om shell FEM. Fou
c oss-sec ions a e conside ed, CS1 o CS4, as shown in Fig. 4, whe e
he pla e dimensions a e gi en in mm, and he wid hs a e in e p e ed
as middle-line dimensions. The membe leng h is he same in each case:
Fig. 4. C oss-sec ions.
L= 5 m. The buckling shape om LBA is applied as ini ial geome ic
impe ec ion.
The analy ical model is used in a e y simple way. The load is
imposed in small inc emen s. The p ocedu e s a s om 𝑊𝑚𝑎 = 0,
𝑉𝑚𝑎 =𝑉𝑚,𝑖𝑛𝑖 and ∅𝑚𝑎 = ∅𝑚,𝑖𝑛𝑖, hen 𝛥𝑀𝑌is applied, and he displace-
men inc emen s a e calcula ed by using Eqs. (40)–(42), o ha e he
displacemen alues 𝑊𝑚𝑏,𝑉𝑚𝑏 and ∅𝑚𝑏, using Eq. (6). In he nex inc e-
men al s ep hese alues a e he s a ing alues, and new inc emen s
a e calcula ed, and hen hese s eps a e epea ed. Since he o mulae
a e simple, a la ge numbe o inc emen al s eps can be applied. The
expe ience is ha 𝛥𝑀𝑌=𝑀𝑐𝑟∕100 is enough, bu in he p esen ed
examples 𝛥𝑀𝑌=𝑀𝑐𝑟∕200 load inc emen has been applied.
The shell FEM calcula ions ha e been pe o med by using he com-
me cial so wa e Abaqus [21]. Linea 4-sided shell elemen s ( ype S4R)
based on he Reissne –Mindlin pla e heo y a e used. A egula mesh is
gene a ed wi h a mesh size o 10 mm. S anda d iso opic s eel ma e ial
is used, 𝐸= 210 GPa,𝐺= 80.8 GPa.
The bounda y condi ions applied a he end sec ions o he beam
simula e simple suppo s, i.e., globally and locally hinged suppo s bu
wi hou es ic ing he wa ping. P ac ically, he langes we e suppo ed
e ically only, he web was suppo ed ho izon ally only. A one end o
7
M.Z. Ha a , M. Ho áček and S. Ádány Thin-Walled S uc u es 184 (2023) 110535
Fig. 5. Sample de o med shapes om shell FEM GNIA. (Only hal o he membe is shown.).
he beam, he membe was suppo ed in he longi udinal di ec ion a
he cen e o g a i y.
The c oss-sec ions a e de ined in a speci ic way so ha he 𝑀𝑐𝑟∕𝐹𝑍
alues – o he gi en leng h – a e he same. The impo ance o his is
ha he a io o he wo componen s o he buckling shape is cons an :
i.e., 𝑉𝑚,𝑖𝑛𝑖∕∅𝑚,𝑖𝑛𝑖 is cons an , as an immedia e consequence o Eq. (45).
This means ha i , o example, he buckling shape is scaled so ha
he ini ial wis would be he same in each case, his au oma ically
gua an ees he equi alence o he ini ial la e al ou -o -s aigh ness
alues, oo. Hence, exac ly he same ini ial impe ec ion can be applied
o all he analyzed membe s; i di e ences a e ound in he nonlinea
beha io , hese di e ences canno be caused by he di e ences in
ini ial shapes. In ac , we ha e conside ed an impe ec ion alue wi h a
la e al ou -o -s aigh ness ampli ude equal o 20 mm, i.e., 𝐿∕250, which
can be conside ed as a easonable alue o an equi alen geome ic
impe ec ion in a design si ua ion.
4.2. Resul s
S esses and displacemen s a e calcula ed o he whole load–
displacemen pa h o each c oss-sec ion. Sample de o med shapes om
FEM analyses a e shown in Fig. 5, whe e he hal o each beam is shown
a a load le el app oxima ely equal o he c i ical momen . Though he
displacemen s a e la ge, he igu es p o e ha c oss-sec ion dis o ion
is negligible, hence he ob ained esul s ep esen la e al– o sional
beha io .
Fig. 6. Load–s ess plo s, doubly symme ic I-sec ion (CS1).
The s esses a e plo ed in Figs. 6,8and 10, o loca ions o he
c oss-sec ions: BL is he bo om lange le side, BR is he bo om lange
8
M.Z. Ha a , M. Ho áček and S. Ádány Thin-Walled S uc u es 184 (2023) 110535
Fig. 7. Load–displacemen plo s, doubly symme ic I-sec ion (CS1).
Fig. 8. Load–s ess plo s, mono-symme ic I-sec ions (CS2 and CS3).
igh side, TL is he op lange le side, TR is he op lange igh side.
The displacemen s a e plo ed Figs. 7,9and 11. I is o no e ha in
he e ical axis o each plo he momen is no malized by he c i ical
momen , and o his no maliza ion he c i ical momen calcula ed by
he shell FEM model is employed.
4.3. Obse a ions om GNI esul s
The e ical displacemen 𝑊is plo ed in Fig. 7 o he doubly
symme ic I sec ion. As can be obse ed, he new analy ical model
p edic s a nea ly linea load–displacemen ela ionship o he en i e
ange o loading. Fo lowe loads he calcula ed e ical ansla ion
alues a e p ac ically iden ical o hose om classic i s -o de heo y,
bu e y simila e en o la ge loads. On he con a y, he shell
FEM alues a e sligh ly di e en , and he load–displacemen plo is
clea ly nonlinea . The small di e ences a lowe loads a e no due
o nonlinea i ies; hey show ha he global lexu al s i ness o he
shell model is sligh ly di e en om a classic beam model (mos ly
due o shea de o ma ions which a e neglec ed in he beam models
bu conside ed in he shell models). These obse a ions on he p ima y
displacemen s a e li le a ec ed by he c oss-sec ion shape, which is
why 𝑊is shown only o one c oss-sec ion.
By looking a he s ess plo s, one can obse e ha he s esses
a e qui e simila o all he cases, up o app ox. 80%–90% o he
c i ical momen . This p o es ha he analy ical model easonably well
conside s he seconda y s esses due o he seconda y displacemen s,
see Eqs. (20) and (29).
Simila ly, he load–displacemen pa hs calcula ed by he new ana-
ly ical model and by he shell FEM analysis a e simila in all he cases.
Though non-negligible di e ences exis , he analy ical esul s and shell
FEM esul s show simila and signi ican de ia ion om he classic
analy ical solu ion (which is shown by he ‘Young’ cu es).
As he o mulae o he new analy ical model sugges (see e.g. Fig. 3),
he ampli ica ion o 𝑉and ∅a e no iden ical. The di e ence in he
ampli ica ions clea ly depends on he c oss-sec ion shape. P ac ically,
his di e ence can be ega ded as small, bu s ill, he nume ical esul s
p o e ha he de o med shape du ing he geome ically non-linea
analysis is no me ely he ampli ica ion o he ini ial impe ec shape,
e en i he impe ec ion is aken as he buckling shape.
I is o unde line ha pe ec ag eemen be ween he analy ical
model and he shell FEM canno be expec ed due o se e al easons.
9