Ci a ion: U uzola, J.; Ga mendia, I.
Imp o ed FEM Na u al F equency
Calcula ion o S uc u al F ames by
Local Co ec ion P ocedu e. Buildings
2024,14, 1195. h ps://doi.o g/
10.3390/buildings14051195
Academic Edi o s: Shaohong Cheng
and Haijun Zhou
Recei ed: 18 Ma ch 2024
Re ised: 12 Ap il 2024
Accep ed: 14 Ap il 2024
Published: 23 Ap il 2024
Copy igh : © 2024 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
buildings
A icle
Imp o ed FEM Na u al F equency Calcula ion o S uc u al
F ames by Local Co ec ion P ocedu e
Ja ie U uzola and Iñaki Ga mendia *
Mechanical Enginee ing Depa men , Enginee ing School o Gipuzkoa, Uni e si y o he Basque Coun y
UPV/EHU, Plaza de Eu opa, 1, E-20018 Donos ia-San Sebas ián, Spain; ja ie [email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : The accu a e calcula ion o na u al equencies is impo an o ib a ion and ea hquake
analyses o s uc u al ames. Fo his pu pose, i is necessa y o disc e ize each beam o column o
he ame in o one o mo e smalle elemen s. The equi ed numbe o elemen s pe membe inc eases
when he ame’s modal shapes ha e wa eleng hs simila o he beam leng hs. This pape p esen s
a me hod
ha educes he numbe o elemen s needed o a p ecise calcula ion. This is achie ed by
implemen ing a s aigh o wa d local co ec ion o he kine ic and elas ic ene gy o ce ain elemen s,
esul ing in a subs an ial dec ease in e o . The alidi y o his me hod is demons a ed h ough
a ange o examples, om simple canonical cases o mo e ealis ic ones. Addi ionally, he pape
discusses he unique ea u es o his me hod and examines i s ela ionship wi h o he app oaches.
Keywo ds: s uc u e; ame; mechanical; ib a ion; na u al equency; ini e elemen ; beam; column
1. In oduc ion
Modal analysis and na u al equency calcula ion by he FEM a e e y aluable
ools o s udy he dynamic beha io o building s uc u es [
1
]. Fo example, he Spanish
S uc u al Code [
2
], he Eu ocode [
3
] and he Ame ican Code ASCE [
4
,
5
] accep hei
alidi y o seismic analysis and wind load induced ib a ion analysis. The e o e, he mos
popula s uc u al analysis p og ams such as ETABS, Robo o S aad implemen hese
nume ical echniques.
F ames a e usually modelled using one elemen pe membe (beam o column), which
is accu a e enough o linea s uc u al analysis, bu i alls sho o ib a ion eigen alue
p oblems [
6
,
7
] because o he inadequacy o polynomials o ep esen localized modal
shapes. The e o e, he need a ises o de elop me hods o pe o m modal analysis in
a mo e
accu a e way wi h he leas nume ical cos and implemen a ion e o . Consequen ly, se -
e al app oaches ha e been p oposed in he scien i ic li e a u e o es ima e he incu ed e o
and possibly educe i , including co ec ion o mulas, he supe con e gen pa ch eco e y
echnique (SPR), he hie a chical FEM (HFEM), he smoo hed FEM (SFEM), he mass-
edis ibu ed FEM (MRFEM) and he use o a ious highe -o de beam ini e elemen s.
Co ec ion o mulas we e applied by Xie and S e en [
8
] o imp o e he accu acy o
he FEM calcula ion o na u al equencies in beam/column elemen s. Thei app oach
s ems om a p e ious s udy by Mackie [
9
] on he opic o nume ical dispe sion e o
educ ion.
A simila
echnique [
10
] can be applied o linea s uc u al buckling c i ical load
calcula ions. Thei me hod can also be applied o s uc u al ames by means o a weigh ed
a e age o single beam/column co ec ion e ms.
FEM e o es ima es [
11
,
12
] deal wi h he p oblem o nume ical inaccu acy induced
by he disc e iza ion o he con inuum o di e en ial equa ions. They a e mo e de ailed
han con e gence g aphs and can be used o e ine he mesh whe e necessa y o achie e
a ce ain le el o p ecision. Residual-based es ima o s measu e he e o on he exac
di e en ial equa ions [
11
] while eco e y-based es ima o s build a be e app oxima ion o
Buildings 2024,14, 1195. h ps://doi.o g/10.3390/buildings14051195 h ps://www.mdpi.com/jou nal/buildings
Buildings 2024,14, 1195 2 o 19
he displacemen o s ess ield [
13
–
15
] ha can be used o ob ain a mo e p ecise alue o
he na u al equencies.
The supe pa ch eco e y echnique (SPR) [
16
] uses a pa ch o neighbo ing elemen s o
adjus a highe o de polynomial o app oxima e he s ess in a ini e elemen using he
elemen alues as well as hose in he con enien ly weigh ed pa ch. I o igina es [
14
,
17
]
om he idea o i ing an imp o ed s ess dis ibu ion ield o a se o so-called supe -
con e gen poin s, when hey exis , whe e s esses a e calcula ed wi h a highe accu acy.
Wibe g e al. [
18
] i ed he polynomial o displacemen s ins ead o s esses (SPRD) in o de
o imp o e he calcula ed alue o na u al equencies, which depend no only on he
displacemen de i a i es, bu also on he displacemen s hemsel es.
The smoo hed ini e elemen me hod (SFEM) [
19
] uses a g adien smoo hing echnique
o educe he o e s i ening o he FEM. This me hod is based on he G space heo y [
20
] ha
makes i possible o use discon inuous shape unc ions in he elemen o mula ion while
main aining s abili y and con e gence o he exac solu ion. The node-based smoo hed
ini e elemen me hod (NS-FEM) [
21
] imp o es accu acy and gi es an uppe bound o he
elas ic ene gy, whe eas he edge-based smoo hed ini e elemen me hod (ES-FEM) [
22
]
p o ides a lowe bound.
The hie a chical FEM [
23
] employs nes ed polynomial shape unc ions o di e en
o de s o inc ease he accu acy o he elemen s when necessa y. The e o e, i can be used
o e o es ima ion and adap a i e mesh e inemen . Ea ly applica ion o he me hod o
dynamic analysis ocused on Be noulli–Eule beams [
24
]. Mo e ecen ly, he me hod has
been applied o a ious ypes o beams such as Timoshenko beams [
25
], h ee-dimensional
sandwich beams [26], e c.
Modi ying he elemen mass ma ix is ano he s a egy o imp o e na u al equency
calcula ions. F ied and Cha ez [
27
] used a weigh ed a e age o he consis en ma ix
and he lumped ma ix o model s ings and memb anes. A mo e economical al e na i e
was de eloped by F ied and Leong [
28
] using he consis en ma ix o he modal shape
calcula ion and a weigh ed mass ma ix o a Rayleigh quo ien co ec ion. Li and He [
29
]
changed he loca ion o he Gaussian poin s used o in eg a e he elemen mass ma ix.
Thei app oach s ems om p e ious wo k in acous ics by Guda i [30].
Highe -o de Eule –Be noulli elemen s [
31
] and he Timoshenko elemen [
32
–
34
]
p o ide imp o ed accu acy due o he be e ep esen a ion o displacemen s. This leads o
a educ ion in he numbe o elemen s equi ed o calcula e na u al equencies. Howe e ,
hei implemen a ion is mo e complex, and he esul ing equa ions ha e mo e unknown
a iables and will be wo se condi ioned [
35
]. Thin-walled beams [
36
] also equi e en iched
se s o modelling a iables because o hei complex geome ical and de o ma ion pa e ns.
The p esen pape shows a new me hod o imp o ing he accu acy o he calcula ion
o he na u al equencies o s uc u al ames when beams and columns a e modelled wi h
a small numbe o elemen s (possibly one o wo). Sway ames can o en be analyzed
accu a ely wi h one elemen pe membe bu non sway ones usually equi e a ine mesh
o a highe p ecision echnique like ou s. The algo i hm p oceeds in wo s ages. In he
i s s age, a coa se solu ion is calcula ed whe eas in he second one, local co ec ions a e
added a a ine le el. I necessa y, some elemen s a e subdi ided i he local co ec ion
excessi ely dis o s he modal shape.
Conce ning he no el y o his wo k, he au ho s ecen ly w o e a closely ela ed
pape [
37
] abou calcula ing he c i ical buckling loads o s uc u al ames using one
elemen pe membe . This la es wo k p esen s some undamen ally no el de elopmen s.
Fi s , p e en ing s uc u al buckling equi es knowing jus he lowes c i ical load, bu in
o de o model s uc u al dynamics accu a ely, mul iple na u al equencies a e needed
and he algo i hm has o be modi ied acco dingly. Second, he in e play be ween mul iple
equencies coupled wi h he limi ed accu acy o indi idual elemen s leads o using wo
subelemen s ins ead o ou . Thi d, because o he same easons, some membe s will ha e o
be modelled wi h mo e han one elemen pe membe acco ding o a no el speci ic elemen
Buildings 2024,14, 1195 3 o 19
dis o ion c i e ion ha we will la e in oduce. Las ly, he de i a ion o he equa ions and
algo i hms has been op imized o cla i y, ease o implemen a ion and pe o mance.
The subsequen sec ions o his pape a e ou lined nex . Fi s , na u al equency
calcula ions a e ca ied ou o i e undamen al cases o one-ba (beam/column) s uc-
u al elemen s wi h he aim o assessing he e o associa ed wi h coa se meshes and he
p oblems a ising om mul iple modal in e play. Second, he ib a ion modes o hese
ba s a e modi ied by a local co ec ion p ocedu e and he elemen s a e subdi ided in wo
acco ding o a dis o ion c i e ion. Thi d, he me hod is ex ended o s uc u al ames made
up o mo e han one ba . Nex , he de ised algo i hms a e alida ed using 2D and 3D
cases ep esen a i e o ealis ic building s uc u es aken om [
10
,
37
]. Finally, he esul s,
discussion and conclusions a e p esen ed.
2. Na u al F equency Analysis o Some Fundamen al Cases Using One Elemen Pe Ba
We ha e selec ed a se o undamen al cases [
38
] (see Figu e 1) o es ou me hod
agains he s anda d FEM. Va ious suppo condi ions such as clamped (C), pinned (P) and
ee (F) a e conside ed. We ha e added he case o he second mode o he pinned–pinned
beam (PP2) because i will help us be e explain he algo i hm.
Figu e 1. Fi e undamen al beam ib a ion cases.
Na u al equencies o a s uc u al ame can be calcula ed by he FEM as he solu ion
o he eigen alue p oblem
K−ω2Mϕ=0 (1)
whe e
K
and
M
a e he s i ness and mass ma ices,
ω
is any na u al equency and
ϕ
is i s
co esponding modal shape.
Table 1shows he accu acy o he FEM calcula ion wi h
Nel
cubic elemen s.
The ela i e
e o s using a single elemen a e excessi e in all cases excep CF. E o s la ge han 1% a e
conside ed excessi e om a s uc u al enginee ing poin o iew [37].
Table 1. Rela i e e o 1in na u al equency compu a ion o some undamen al cases.
Nel 2CC CP PP CF PP2
1 - 32.92% 10.99% 0.48% 27.14%
2 1.62% 0.93% 0.39% 0.05% 10.98%
3 0.41% 0.20% 0.08% 0.01% 1.17%
4 0.13% 0.06% 0.03% 0.00% 0.38%
1
Rela i e e o e e s o “nea ly exac ” alues calcula ed wi h Abaqus and Nel = 10.
2
Nel: numbe o elemen s in
he disc e iza ion.
Looking a Figu e 1and by analogy wi h he column buckling p oblem, we can
in e p e ha a single elemen is no accu a e enough o model mo e han a qua e o
Buildings 2024,14, 1195 4 o 19
a sinusoidal
de o ma ion wa eleng h. We will see how o educe hese e o s in he
nex sec ion.
3. Co ec ed Calcula ion o Na u al F equencies in Some Fundamen al Cases Using One
Elemen Pe Ba
In his sec ion, we will imp o e he quali y o he displacemen s inside he s uc u al
elemen in wo ways: (1) we will use an auxilia y disc e iza ion o he ba elemen s (see
Figu e 2) wi h wo subelemen s and h ee nodes (1–3) o ob ain a local co ec ion o he
coa se mesh solu ion and (2) we will spli some elemen s in hal when necessa y (adap i e
mesh e inemen ). This app oxima ion esul s in accep able e o s nea hose ob ained in
Table 1wi h ou elemen s.
Figu e 2. Ba elemen “global” displacemen ug( om 1–3 o 1′–3′).
In ou p e ious wo k on buckling [
37
] we used an auxilia y disc e iza ion o ou
subelemen s ins ead o wo, bu we will see ha his app oach is no possible when
calcula ing mul iple eigen alues (o mul iple equencies) because o he o e ide p oblem
ha is la e explained.
We exp ess he nodal displacemen s
u
( : 1–3) as he sum o a “global” e m de i ed
om he coa se solu ion and a “local” co ec ion e m.
The global displacemen
ug
(see Figu e 2) esul s om a s a ic analysis in which
we ix he ex e nal displacemen s
ug
1
and
ug
2
and ob ain he alue o he in e nal nodal
displacemen ug
3by condensa ion [7].
The nodal displacemen s o he elemen nodes in he local e e ence ame o he ba ,
ul
1and ul
2, can be exp essed as
ul
1=ϕ1ηul
2=ϕ2η(2)
whe e
ϕ1
and
ϕ2
a e he coa se modal shapes a nodes 1 and 2 exp essed in he local
e e ence ame and ηis a modal ampli ude a iable.
In o de o ind he inne nodal displacemen
ul
3
we will need he elemen s i ness ma-
ix. Assuming a uni o m beam, each subelemen (1-3 and 3-2) will ha e he same s i ness
and mass ma ices,
KS
and
MS
, which we can exp ess in e ms o hei
nodal subma ices:
KS=KAA KAB
KBA KBB MS=MAA MAB
MBA MBB (3)
Buildings 2024,14, 1195 5 o 19
Assembling hese subelemen ma ices, we ob ain he s i ness and mass ma ices o
he e ined elemen , K and M :
K =
KAA 0KAB
0KBB KBA
KBA KAB KAA +KBB
M =
MAA 0MAB
0MBB MBA
MBA MAB MAA +MBB
(4)
The e o e, he sough in e nal displacemen esul s in
ul
3=(KAA +KBB)−1(KBAul
1+KABul
2=ϕ3η(5)
whe e we de ine ϕ3as
ϕ3=(KAA +KBB)−1(KBAϕ1+KABϕ2(6)
The local displacemen e m
∆ul
(see Figu e 3), inc eases he in e nal node displace-
men (
∆ul
3
) wi hou modi ying he ex e nal nodal displacemen s ( o economy o no a ion,
we g oup nodal o a ions and displacemen s in one e m).
∆ul=
0
0
∆ul
3
(7)
Figu e 3. Elemen inc emen al local displacemen s ∆ul( om 1–3 o 1′–3′).
Figu e 4shows he o al nodal displacemen o he e ined elemen
u
esul ing om
bo h he global and he local e m.
Figu e 4. Re ined elemen o al displacemen s
u
(global om 1–3 o 1
′
3
′
and local om 1’–3’
o 1”–3”).
Buildings 2024,14, 1195 6 o 19
Now, we de ine he e ined elemen nodal displacemen s
u
as a unc ion o he modal
coo dina e ηand he in e nal inc emen al displacemen ∆ul
3using a p ojec ion ma ix P
u =
ϕ10
ϕ20
ϕ3I
η
∆ul
3=Pη
∆ul
3(8)
whe e
I
is he iden i y ma ix. As a esul , we can ob ain a co ec ed na u al equency
ωp
by sol ing he p ojec ed eigen alue p oblem
PTK Pϕp=ω2
pPTM Pϕp(9)
We summa ize he p ocedu e o calcula e he co ec ed na u al equency in
Algo i hm 1.
Algo i hm 1. Co ec ion o he na u al equency o a one-elemen ba
E alua e K and M as 2-subelemen e inemen s o Keand Meusing Equa ion (3)
E alua e he p ojec ion ma ix P wi h Equa ion (8)
Calcula e ωpas he lowes na u al equency in Equa ion (9)
A e he co ec ion p ocess he e o s in na u al equencies change as shown in Table 2.
Table 2. Rela i e e o in co ec ed na u al equency calcula ion (Nel = 1).
Nel CC CP PP CF PP2
1 1.61% 0.93% 0.39% 0.05% 42.42%
Looking a hese esul s, we can see ha co ec ing he one elemen pe membe
model in he CC and PP2 cases does no educe he e o up o a le el ha is accep able in
enginee ing. We can unde s and wha is happening i we s udy wha we will designa e
as he dis o ion ac o : he maximum change in V o T a e applying he co ec ion (see
Table 3), whe e V and T a e he s i ness and mass quad a ic o ms, espec i ely.
Table 3. Dis o ion ac o (maximum pe cen age change in V o T) a e co ec ion (Nel = 1).
Nel CC CP PP CF PP2
1 -% 211.33% 49.66% 1.73% 1.4 ×1029%
Wha we can see he e is ha he local co ec ion has la gely dis o ed V and/o T in
bo h cases (mos no ably o PP2).
Fi s , we will s udy he cause o he mos oubling case, PP2. We can see a g aphical
depic ion o i s dis o ion wi h al e ed scales in Figu e 5. The so e CC mode (K = 22.4)
has almos comple ely o e idden he s i e PP2 mode (K = 39.5) he eby supp essing an
exis ing mode and eplacing i wi h a ough duplica e o a p e iously calcula ed one (CC).
Second, we u n ou a en ion o he o igin o he sligh ly unaccep able e o o he
CC case. We can a ibu e i o he ac ha he co ec ion p ocess can ne e su pass he
accu acy o doubling he elemen a he coa se le el.
In o de o sol e bo h p oblems, we p opose spli ing elemen s in hal when he
dis o ion ac o ( om now on called
γ
) su passes he 100% h eshold. The PP2 spu ious
modal o e ide p oblem will be sol ed because he hal elemen co ec ions canno oughly
ep esen he CC mode in isola ion. In u n, he CC sligh ly unaccep able e o will be
educed because o he supe io quali y o he e ined coa se mesh.
I Is clea now ha using a ou subelemen disc e iza ion o local elemen co ec ion
is no accep able because i will be plagued by he o e ide p oblem in he same way as he
wo-subelemen one.
Buildings 2024,14, 1195 7 o 19
Figu e 5. Locally co ec ed PP2 modal shape using one elemen and wo subelemen s.
The e o e, we will allow one o wo elemen s pe membe a he coa se le el and wo
subelemen s a he local le el, which allows o a o al o ou subelemen s pe membe ,
oughly equi alen o wha we did o buckling in [37].
In he nex sec ion, we ex end his co ec ion/ e inemen p ocess o gene al s uc u al
ames made up o mul iple ba s and one o wo elemen s pe ba .
4. Co ec ion o Na u al F equencies o Mul iple-Elemen S uc u es
Simila ly o wha we did in [
37
], we a e going o gene alize he p ocedu e o single
elemen s based on ou main ideas:
1.
The local elemen co ec ions can be combined addi i ely in o an o e all modal co ec ion.
2.
When calcula ing local co ec ions o an elemen , he es o he s uc u e can be
su icien ly ep esen ed by he ame modal shape ϕand an ampli ude a iable η.
3.
The co ec ed na u al equency o he whole ame can be calcula ed using Rayleigh’s
quo ien wi h he co ec ed modal shape.
4.
Local co ec ions o di e en na u al equencies can be calcula ed in isola ion om
each o he once he dis o ion ac o has been in oduced o sol e he o e ide p oblem.
Le us examine how he whole p ocedu e would wo k o ou mos p oblema ic case,
PP2. Figu e 6shows he modal shape o he PP2 case beam disc e ized a he coa se le el
wi h wo elemen s pe membe . In o de o imp o e he quali y o he modal shape, we
will ix he end nodal displacemen s o each coa se elemen and subsequen ly co ec i s
inne displacemen s, and, as a esul , we will ob ain he co ec ed ib a ion shape in he
same igu e.
Figu e 6. PP2 modal shape using wo elemen s a e local co ec ion.
Buildings 2024,14, 1195 8 o 19
The local co ec ions o each elemen will be calcula ed sepa a ely (as shown in
Figu e 7). Fo his pu pose, we main ain all he elemen s in he ame mesh excep he one
o be co ec ed, which is eplaced wi h wo subelemen s (see Figu e 7).
Figu e 7. S uc u al disc e iza ion used o co ec he uppe pa o wo-elemen PP2 modal shape.
The local modal shape co ec ion is he solu ion o a p ojec ed eigen alue p oblem
ha is de i ed below.
The s uc u e s i ness quad a ic o m calcula ed wi h he coa se mesh can be exp essed as:
V=uTKu (10)
whe e uand K a e he nodal displacemen ec o and s i ness ma ix o he ame.
We can modi y Vby eplacing elemen e con ibu ion wi h i s e ined coun e pa
V=uTKu −uT
eKeue+uT
K u (11)
whe e
ue
and
Ke
a e he nodal displacemen ec o and s i ness ma ix o elemen e, while
u
and
K
a e hei e ined e sions (using wo subelemen s and h ee nodes) calcula ed in
he local e e ence ame o he elemen .
Nex , he nodal displacemen s can be exp essed as a unc ion o he modal coo dina e
η
and he inc emen al inne nodal displacemen o he elemen being co ec ed
∆ul
3
, simila ly
o wha we did o a single-elemen ba in Equa ion (8).
u=ϕη (12)
ue=ϕeη(13)
u =
ϕ1η
ϕ2η
ϕ3η+∆ul
3
(14)
and he same ope a ions can be pe o med on he mass quad a ic o m:
T=uTMu −uT
eMeue+uT
M u (15)
As a esul , we ob ain a p ojec ed eigen alue p oblem whose solu ion con ains he
local modal shape co ec ion:
Kpϕp=ω2
pMpϕp(16)
whe e
Kp=V−Ve+V 0
0KAA +KBB(17)
Buildings 2024,14, 1195 9 o 19
Mp=T−Te+T ϕT
1MAB +ϕT
2MBA +ϕT
3(MAA +MBB
symme ic MAA +MBB (18)
ϕ =
ϕ1
ϕ2
ϕ3
(19)
V =ϕT
1KAAϕ1+ϕT
2KBBϕ2+ϕT
1KABϕ3+ϕT
3KABϕ2(20)
T =ϕT
1MAAϕ1+ϕT
2MBBϕ2+ϕT
3(MAA +MBB)ϕ3+2ϕT
1MABϕ3+2ϕT
3MABϕ2(21)
We can pa i ion he p ojec ed mode in e ms o i s modal ampli ude pa
ϕp0
and i s
in e nal co ec ion ϕp3
ϕpe =ϕp0
ϕp3(22)
and di iding he igh -hand side by ϕp0, we gene a e a p ojec ed mode
ϕ∗
p=(1
ϕp3
ϕp0)(23)
which ep esen s he sum o he o e all modal shape
ϕ
plus he local in e nal co ec ion
e m ∆ϕc3
∆ϕc3=ϕp3
ϕp0
(24)
and his modal shape can be used o calcula e he co ec ed mass and s i ness quad a ic
o ms o he elemen .
Vce =V +∆ϕT
c3(KAA +KBB∆ϕc3(25)
Tce =T +2∆ϕT
c3(MBAϕ1+MABϕ2+∆ϕT
c3(MAA +MBB(2ϕ3+∆ϕc3)(26)
We show in Algo i hm 2 he comple e p ocedu e o compu ing Vce and Tce.
Algo i hm 2. Compu a ion o he co ec ed quad a ic o ms o an elemen Vce and Tce
O e all ame inpu s: ϕTKϕ,ϕTMϕ
F ame elemen inpu s: ϕT
eKeϕe,ϕT
eMeϕe,ϕe
Subelemen inpu s: KAA,KAB,KBB,MAA,MAB,MBB
E alua e K and M as 2-subelemen e inemen s o Keand Meusing Equa ion (3)
Con e ϕe o local elemen coo dina es by he ollowing ope a ions:
ϕ1=RT
eϕe1ϕ2=RT
eϕe2(Re: elemen o a ion ma ix)
Calcula e ϕ3,ϕ in Equa ions (6) and (19)
Calcula e Kp,Mpin Equa ions (17) and (18)
Ob ain ϕpas he i s eigen ec o o Equa ion (16)
E alua e Vce,Tce by applying Equa ions (24)–(26)
The elemen -co ec ed quad a ic o ms o he whole s uc u e can be collec ed in
Rayleigh’s quo ien o ob ain an imp o ed alue o he s uc u e’s na u al equency ωc
ω2
c=∑eVce
∑eTce (27)
The comple e p ocedu e o calcula e N na u al equencies is gi en in Algo i hm 3.
I should be poin ed ou ha he denomina o s o he dis o ion ac o
γe
ha e been
modi ied o cope wi h he possibili y o elemen s wi h e y small
Ve
o
Te
, which would
lead o nea di ision by ze o. The e o e, one hund ed h o he ame
V
o
T
is dis ibu ed
equally among all elemen s when measu ing ela i e change, while he o he 99% comes
om he elemen i sel .
Buildings 2024,14, 1195 16 o 19
Table 12. Co ec ed calcula ion s a is ics (3D b aced building s uc u e wi h 2 elems./membe ).
Mode # Exac
ω( ad/s)
2-Elem.
ω( ad/s)
Rela i e
E o (%)
Co ec ed
ω( ad/s)
Rela i e
E o (%)
Dis o ion
Fac o γ(%)
Dis o ed
Elemen s #
1 124.49 124.54 0.03 124.50 0.00 2.35 0
2 126.75 126.79 0.03 126.75 0.00 4.65 0
3 146.27 146.33 0.04 146.28 0.00 3.94 0
4 151.65 151.71 0.04 151.66 0.00 4.33 0
5 184.03 184.13 0.05 184.04 0.00 6.87 0
6 212.50 212.66 0.08 212.52 0.01 4.66 0
7 245.42 245.67 0.10 245.44 0.01 7.20 0
8 276.69 277.13 0.16 276.73 0.01 17.03 0
9 366.95 368.89 0.53 367.08 0.04 9.62 0
10 384.71 387.27 0.67 384.89 0.05 16.12 0
11 392.67 395.59 0.74 392.87 0.05 3.82 0
12 403.95 407.28 0.82 404.18 0.06 4.83 0
A e examining he esul s, we can conclude ha ou models achie e he same le el o
accu acy as Abaqus s anda d FEM models wi h wice as many ba s (on he condi ion ha
dis o ion ac o s lie below 100%), he e o e hey can wo k wi h s i ness and mass ma ices
wice smalle . In addi ion, we ha e a ained calcula ion imes 15% smalle measu ing he
main componen s o equi ed p ocessing powe , i.e., he whole ame eigen alue p oblem
and he local beam eigen alue p oblems.
6. Discussion
The me hod by Xie and S e en [
10
] p o ides local na u al equency upda es o
single ba s a he han local modal shape co ec ions, which leads o using a weigh ed
c i e ion wi h a weake physical ounda ion han Rayleigh’s quo ien . In addi ion, s uc u al
membe s a e disc e ized wi h ou o i e elemen s while in ou case one o wo (in a ew
ba s) elemen s a e needed.
The SPRD echnique by Wibe g e al. [
18
] bea s some simila i ies wi h ou app oach.
I elies on a polynomial i ing o he exis ing mode a some supe con e gen poin s while
ou p ocedu e comple ely upda es he mode a inne poin s wi hou being cons ained by
he coa se calcula ion on he inside. Plus, he whole s uc u e a he han a single elemen
and i s neighbo ing pa ch pa icipa es in he adjus men by means o he coa se modal
ampli ude
η
. I can also be no ed ha SPR echniques use an ex e nal pa ch o elemen s
while ou me hod elies on an inne se o e ined elemen s.
G adien smoo hing me hods such as [
19
] also ely on neighbo ing elemen s o im-
p o e he quali y o he solu ion bu hey do i be o e sol ing he sys em o equa ions
o he whole s uc u e, he eby inc easing connec i i y and he enhanced elemen ma ix
compu a ion ime. In addi ion, he e is no s aigh o wa d way o applying he smoo hing
concep o beam elemen s o di e en sec ions and o ien a ions sha ing a node.
Modi ied s i ness and mass ma ices [
29
], while imp o ing he accu acy o dynamic
analysis, a e limi ed by he ac ha hey do no depend on he na u al equency being
s udied (like in he case o dynamic s i ness me hods) o on he ac ual modal shape, as
happens in ou me hod.
Highe -o de ini e elemen s [
31
–
34
] p o ide be e accu acy bu a e mo e complex
o implemen , ha e o sol e la ge sys ems o equa ions and lead o wo se condi ioned
ma ices. In con as , ou me hod wo ks well wi h he s anda d ini e elemen me hod and
could wo k wi h highe -o de ini e elemen s as well o imp o e hei accu acy. As o
hin-walled beams [
36
], hey equi e highe -o de models in o de o ep esen complex
de o ma ion pa e ns, bu hey a e ully compa ible wi h ou co ec ion algo i hm.
The hie a chical FEM [
23
–
25
] elies on e o es ima o s o e ine he s uc u al mesh
and imp o e accu acy. In con as , ou co ec ion does no need a ull eanalysis o in-
c ease p ecision bu an a ay o concu en elemen -cen e ed co ec ions. I necessa y,
Buildings 2024,14, 1195 17 o 19
he dis o ion ac o indica es wha ba s equi e wo elemen s ins ead o one. Like he
highe -o de elemen s discussed abo e, hie a chical elemen s o highe o de can be used
o he co ec ions ins ead o ou wo-subelemen se .
As men ioned in he in oduc ion, he au ho s ecen ly w o e a closely ela ed
pape [37]
abou calcula ing he c i ical buckling loads o s uc u al ames using one elemen pe
membe . Fo his pu pose, a local co ec ion p ocedu e was applied using ou subelemen s.
This la es wo k p esen s some undamen al di e ences. Fi s , p e en ing s uc u al buckling
equi es knowing jus he lowes c i ical load, bu in o de o model s uc u al dynamics
accu a ely, mul iple na u al equencies a e needed. Second, he in e play be ween mul iple
equencies coupled wi h he limi ed accu acy o indi idual elemen s makes i necessa y o
use wo subelemen s ins ead o ou . Thi d, because o he smalle numbe o subelemen s
in ol ed in he local co ec ions, some membe s ha e o be spli in o hal beams acco ding o
a no el elemen dis o ion c i e ion which measu es he ela i e change in kine ic and elas ic
ene gy caused by he co ec ion o modal shapes.
As a as he e iciency o ou me hod is conce ned, mos o wha was s a ed in [
37
]
emains applicable: he algo i hm’s g ea es sou ce o e iciency comes om i s ully
pa allelizable na u e and he local eigen alue p oblems can be sol ed wi h minimal com-
pu a ional esou ces by he powe me hod. In addi ion, a e sh inking he submodel o
wo elemen s, all he ma ices ha appea in he local eigen alue p oblem can be easily
p og ammed wi h scala ope a o s and unc ions, he eby educing he cos o sol ing he
eigen alue p oblem.
Ou echnique can be applied o enhance he s anda d FEM analysis o any s uc u e
made up o beam/column elemen s. The s uc u e could also con ain shell, pla e o lumped
elemen s bu he gain in accu acy would only occu o he ba elemen s. Ou app oach
can be used o calcula e na u al equencies wi h e o s accep able in enginee ing (below
1%) using one o wo elemen s pe membe o o inc ease he accu acy o a calcula ion
wi h any numbe o elemen s pe membe . The e o e, ou app oach o e s he po en ial
o a educ ion in memo y equi emen s and calcula ion speed when compa ed wi h he
s anda d FEM a he cos o some addi ional coding. In o de o ully exploi he ad an ages
o he algo i hm, i is ad isable o dis ibu e he co ec ion calcula ions o he GPU.
Conce ning u u e esea ch de elopmen s based on he p esen wo k, we ha e selec ed
a ew a eas o in e es . Fi s , he e is always a signi ican disc epancy be ween nume ical
ib a ional p ope ies and expe imen al measu emen s [
39
,
40
] because o app oxima e mod-
elling, nonlinea i ies, empe a u e e ec s, e c., which could be add essed ad an ageously
wi h he p oposed nume ical echnique o an enhanced e sion o i . Likewise, in he ield
o heal h s uc u al moni o ing, he e is also he need o deal wi h disc epancies caused
by s uc u al ailu e o de e io a ion, and o diagnose hei na u e and loca ion [
41
,
42
].
Howe e , he p esen s udy is only conce ned wi h nume ical e iciency and has no di ec
applica ion in hese a eas in i s p esen o m.
7. Conclusions
A new me hod o imp o ing he accu acy o he s anda d FEM na u al equency
calcula ion o s uc u al ames made up o beam/column elemen s has been p esen ed.
The algo i hms a e based on p e ious wo k by he au ho s on s uc u al ame buckling,
bu signi ican no el modi ica ions ha e been made o imp o e e iciency and ease imple-
men a ion, and o accoun o he challenges o calcula ing se e al eigen alues ins ead
o he lowes one. The ully pa allel na u e o he me hod makes i e y con enien o
ake ad an age o he cu en end owa ds GPU-based a chi ec u es. Fo his pu pose,
he main calcula ion cos d i e is a small indi idual nodal cen e ed eigen alue p oblem
sol able wi h a ew powe i e a ions.
S uc u al membe s a e modelled wi h one o wo elemen s ollowing a no el subdi-
ision c i e ion based on he deg ee o dis o ion caused by he co ec ion o he o iginal
modal shape. As a esul , enough accu acy o enginee ing applica ions is achie ed wi h
a modes inc ease in compu a ion ime and s o age equi emen s. The app oach is e y
Buildings 2024,14, 1195 18 o 19
lexible and can accommoda e di e en beam ypes, highe -o de models and ine meshes
and a ge p ecision le els, e en hough i has been demons a ed wi h simple cubic el-
emen s. Algo i hm inpu s a e eadily a ailable da a on FEM codes such as ame and
elemen s i ness and mass ma ices, o a ions and modal shapes ha can be p ocessed
wi h scala unc ions and ope a o s be o e he local eigensol e co ec ion s ep.
Au ho Con ibu ions: Concep ualiza ion, J.U.: me hodology, J.U.; so wa e, J.U.; alida ion, J.U.
and I.G.; w i ing, J.U. and I.G. All au ho s ha e ead and ag eed o he published e sion o
he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
Da a A ailabili y S a emen : The o iginal con ibu ions p esen ed in he s udy a e included in he
a icle. Fu he inqui ies can be di ec ed o he co esponding au ho .
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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Disclaime /Publishe ’s No e: The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
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people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .
