Kubíke al. Jou nal o Wood Science (2024) 70:42
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Jou nal o Wood Science
F ac u e p edic ions inimpac h ee-poin
bending es o Eu opean beech
Pe Kubík1, F an išek Šebek1* , Moj aba Hassan Vand2, Ma in B abec2 and Jan Tippne 2
Abs ac
Ha dwood has become widesp ead in Eu opean o es s. The s onges ac o is clima e change and damage o coni-
e s by he ba k bee le. The e o o s udy ha dwoods g ows wi h inc easing olume o applica ions. The e o e,
Eu opean beech wood was in es iga ed unde wo impac loads in wo ma e ial di ec ions, esul ing in ou unique
combina ions supplemen ed by he measu emen o he ic ion coe icien . Then, i was compu a ionally simu-
la ed o ep oduce he c acking, while he ma e ial model e lec ed he o ho opic beha iou in elas ici y, plas ici y
and ailu e. The model was coded using he use sub ou ine in Abaqus o ini ia e and p opaga e he c ack using
he elemen dele ion. The esul ing eac ion o ces we e in good ag eemen wi h hose om he expe imen s. C ack-
ing was nume ically simula ed in h ee o ou cases as expe imen ally obse ed, howe e , upon la ge de lec ions.
The e o e, he model is applicable o u he in es iga ions.
Keywo ds Aniso opy, Ba la , Dynamic, Explici , Flexu e, Rup u e
In oduc ion
Wood is a widesp ead na u al composi e, which has been
used as a cons uc ion ma e ial o cen u ies because i
is enewable, biologically deg adable, en i onmen ally
iendly, s ong, ligh weigh , easy o manu ac u e, elec i-
cally esis an , abso bing noise and aes he ic. The e o e,
models capable o p edic ing de o ma ion and ailu e
unde ope a ional o andom loading a e needed.
Bending is one o he common es s o wood. Jansson
[1] in es iga ed ailu e modes and s esses using s a ic
and impac bending es s showing dec easing impac
bending s eng h wi h dec easing ime o ailu e o he
Sp uce Pine Fi (SPF). Yoshiha a e al. [2] used a s a ic
h ee-poin bending es o de e mine he shea modulus
wi h he help o a co ec ion unc ion and a modi ied
Timoshenko beam o six wood species: Si ka sp uce,
wes e n hemlock, akama su, yellow popla , shioji and
balsa. Yoshiha a e al. [3] conduc ed a simila s udy o
asymme ic ou -poin bending es s. Yoshiha a and Oka
[4] ca ied ou he comp ession bending es o de e -
mine he elas ic modulus, p opo ional limi and bending
s eng h on specimens o Japanese i o a ious leng h-
o- hickness a ios. Then hey compa ed he esul s wi h
he con en ional bending es o con i m ha he co ec
lexu al p ope ies can be ob ained om he comp ession
bending es o a la ge leng h- o- hickness a io. Kubo-
jima e al. [5] used he impac bending es o es ima e
he elas ic modulus o Japanese ceda , hondo sp uce,
hiba a bo i ae, Japanese ed pine, paulownia, Manchu-
ian ash and Japanese e e g een oak. Polocoşe e al. [6]
in es iga ed he e ec o low- eloci y impac on ailu e
s esses and s i ness using he bending es consis ing o
a pendulum and specimens ha ing a leng h o 650mm,
wid h o 50mm and hicknesses o 20, 30 and 40mm
o ind ha ailu e was signi ican ly di e en compa ed
o quasi-s a ic es s o beech, la ch and pine. Polocoşe
e  al. [7] ound highe ene gy abso p ion in pine and
sp uce ein o ced wi h E-glass on he su ace loaded in
*Co espondence:
F an išek Šebek
[email p o ec ed].cz
1 Ins i u e o Solid Mechanics, Mecha onics and Biomechanics, Facul y
o Mechanical Enginee ing, B no Uni e si y o Technology, Technická
2896/2, 616 69 B no, Czech Republic
2 Depa men o Wood Science and Technology, Facul y o Fo es y
and Wood Technology, Mendel Uni e si y in B no, Zemědělská 810/3, 613
00 B no, Czech Republic
Page 2 o 12
Kubíke al. Jou nal o Wood Science (2024) 70:42
ension on a pendulum compa ed o specimens wi h-
ou ein o cemen . The ac o was 1.4 o pine and 2.5
o sp uce. Jacques e al. [8] ca ied ou ull-scale es ing
on indi idual ligh ame lumbe s made om SPF. Then,
he ou -poin bending es s wi h s ain a es o 6 × 10–6
o 4 × 10–1 s–1 se ed o p opose he s ess–s ain ela-
ionship applicable o he modelling o blas loadings.
Olmedo e al. [9] ollowed he dynamic esponse o esh
s ems unde impac loading using he Mou on–Cha py
pendulum. Then hey analysed he load-bea ing capac-
i y o wooden cons uc ions made o elled ees se ing
as p o ec ion agains alling ocks. B anche iau e al. [10]
p oposed an analy ical ela ionship, which con e s he
elas ic modulus es ima ed om he h ee-poin bend-
ing es o he elas ic modulus appea ing in he ou -
poin bending es . The esul s we e expe imen ally
alida ed on specimens om six wood species wi h di -
e en densi ies. Di e en elas ic moduli o sp uce and
oak in h ee- and ou -poin bending es s we e also
analysed by Babiak e al. [11]. Ga e al. [12] examined
he e ec o he mal modi ica ion o Eu opean oak and
No way sp uce using he Cha py pendulum o ind ha
changes in basic chemical componen s and impac bend-
ing s eng h we e less a ec ed o oak han o sp uce.
The e ec o he mal modi ica ion on impac bend-
ing s eng h was also in es iga ed by Hassan Vand and
Tippne [13] o ind ha he mal modi ica ion caused a
dec ease in de lec ion and maximum longi udinal s ain
up o app oxima ely 50% acco ding o digi al image co -
ela ion (DIC) o h ee-poin bending es s o i e wood
species: ash, beech, la ch, oak and sp uce. Hassan Vand
e al. [14] also es ed he e ec o mois u e con en on
he beha iou o h ee wood species o beech, oak and
sp uce. They e alua ed he wo k equi ed o ini ia e he
c ack and b eak he specimens and ound, using DIC,
ha he maximum de lec ion and longi udinal ensile
s ain inc eased wi h inc easing mois u e con en .
The minimum o he li e a u e deals wi h he compu-
a ional modelling o wood unde impac . The e o e, his
wo k ocuses on he dynamic beha iou o Eu opean
beech (Fagus syl a ica L.) unde he h ee-poin bend-
ing es . All es s we e ca ied ou using a d op-weigh
impac es ing machine wi h wo hamme s o achie e
a ious impac eloci ies. Then, nume ical simula ions
ollowed o de elop a ma e ial model wi h good ac-
u e p edic abili y compa ed o expe imen s. I should
be no ed ha many wo ks [15, 16] use he o ho opic
yield c i e ion acco ding o Hill [17] o he desc ip ion
o plas ic beha iou . Howe e , some ma e ial pa ame e s
( h ee ou o six) o he equi alen s ess can be nega i e
due o signi ican ly dis inc yield s esses in espec i e
wood di ec ions [18–20]. The nega i e ma e ial pa ame-
e s can hen cause he equi alen s ess o be unde ined.
The e o e, he o ho opic yield c i e ion acco ding o
Ba la e al. [21] will be used o o e come his issue.
Expe imen s
Ma e ial
The ma e ial s udied was Eu opean beech belonging o
ha dwoods. The specimens we e made om ee g own
nea B no (Czech Republic). The wood was conside ed
o ho opic wi h a di ec ion pa allel wi h h ee p ima y
(heigh ) g ow h called longi udinal (L), a di ec ion pe -
pendicula o he ee p ima y g ow h passing h ough
he pi h o he ee called adial (R) and a di ec ion pe -
pendicula o ee g ow h and simul aneously as a an-
gen o annual ings called angen ial (T) as depic ed in
Fig.1. All specimens we e s o ed in a clima ic chambe a
a empe a u e o 20°C and a ela i e humidi y o 65% o
p oduce a uni o m equilib ium mois u e con en o 12%.
Th ee‑poin impac bending es
All specimens we e manu ac u ed wi h ib es o ien ed
along he leng h o 300mm ha ing a squa e c oss sec-
ion wi h an edge o 20mm. Be o e es ing, specimens
we e weighed in analy ical balance wi h 1mg eadabil-
i y, which esul ed in an a i hme ic mean o densi y o
727kg/m3 wi h a s anda d de ia ion o 33kg/m3. The
symme ic h ee-poin bending was ca ied ou in wo
ma e ial di ec ions R and T (Fig. 1), espec i ely. The
span- o-dep h a io was 12, esul ing in a suppo span o
240mm. Bo h he suppo s and he hamme had a adius
o 15mm and a su ace oughness Ra o 0.2μm.
The es s we e pe o med on he d op-weigh impac
es ing machine DPFes 400 om Labo ech ( igh Fig.2)
a oom empe a u e. Two di e en hamme s we e
d opped om wo ini ial heigh s o ob ain wo impac
eloci ies as gi en in Table1 o p o ide a di e se ma e ial
o nume ical simula ions. The es ing machine au oma -
ically measu ed he ue impac eloci ies (Table1). Fo
each o hese wo condi ions, 6 specimens we e es ed in
he R and T di ec ions, esul ing in 4 da ase s. The ham-
me displacemen was measu ed wi h 0.01 mm p eci-
sion, while he o ce was measu ed using a piezoelec ic
o ce ansduce CFT + 50 kN om HBM a ached o he
hamme .
The es s we e eco ded wi h Fas cam SA-X2 om
Pho on. The high-speed came a had a cell size o 20μm
and was equipped wi h a Nikon Mic o-Nikko G lens
wi h a ocal leng h o 105mm and a Nikon Z TC-2.0 × .
I was placed app oxima ely 0.9m om he la e al speci-
men su ace, which was pa allel o he came a senso .
Addi ional ligh ing was ensu ed by wo Mul iLed QT
s andalone lamps (le Fig.2). The ield o iew was i -
ed o a cen al pa o la e al specimen’s su ace wi h a
hamme . Images wi h a esolu ion o 1024 × 672px we e
Page 3 o 12
Kubíke al. Jou nal o Wood Science (2024) 70:42
cap u ed wi h a ame a e o 20,000 ps. The image pos -
p ocessing was pe o med in Vic-2D 2010 so wa e om
Co ela ed Solu ions. In o de o de e mine he con-
e sion ac o , a simple calib a ion was done using he
one-dimensional scale de ined by he known dis ance.
The displacemen ield was calcula ed om he ield o
3 × 3p . The Lag ange s ain ield was de e mined wi h
he lowes possible s ain il e size o 5 × 5p so ha he
maximum possible spa ial esolu ion was achie ed. In
20
T
LR
L
R
T
20
240
300
R15
R15
R15
Fig. 1 Loading in he R ( op) and T di ec ions (bo om) wi h all ma e ial di ec ions du ing impac h ee-poin bending es s (all dimensions in mm)
Impac es
i
ng mac
hi
n
e
H
igh
-spee
d
came a
L
ED
ligh
s
D
a a ac
q
u
i
s
i
i
on s
y
s e
m
Fig. 2 High-speed came a wi h accesso ies (le ) and d op-weigh impac es ing machine ( igh )
Page 4 o 12
Kubíke al. Jou nal o Wood Science (2024) 70:42
addi ion, he op ical measu emen se ed o he analysis
o he ac u e (i any).
Measu emen o  ic ion coe icien
T ibological expe imen s we e pe o med o collec inpu
o nume ical modelling. The es s we e ca ied ou using
he Uni e sal Mechanical Tes e (UMT) T iboLab om
B uke (Fig.3). The pin o 5mm in diame e was made
o AISI 52100 alloy s eel ha ing a su ace oughness Ra
o 0.2μm, he same as he suppo s and he hamme in
he h ee-poin impac bending es . I was o ced agains
wooden specimens by 196 N, which was measu ed by a
dual o ce senso DFH-100 wi h a sampling equency
o 100 Hz and co esponded o a con ac p essu e o
10MPa. This con ac p essu e was chosen as i eached
ens o MPa in he ollowing nume ical simula ions. The
pin oscilla ed wi h a equency o 2Hz and ampli ude
o 5mm o 120s. Six specimens wi h dimensions o
40 × 20 × 10mm we e es ed. These specimens we e man-
u ac u ed om he same ee as hose o he h ee-poin
impac bending es and s o ed in he clima ic chambe
o each he same uni o m equilib ium mois u e con en
o 12%. The pin was o ced agains h ee LR and h ee LT
su aces o 40 × 20mm mo ing in he R and L di ec ions,
espec i ely. The e o e, he measu emen was wi hin
one annual ing o he L di ec ion and app oxima ely 5
annual ings o he R di ec ion wi h an a e age annual
ing wid h o 2.5mm.
The pin ajec o y was 10mm, bu he esul s we e p o-
cessed in MATLAB R2024a o a 6-mm-long po ion o
he specimen in o de o omi he dead ends whe e he
pin decele a ed and accele a ed. Then, he conside ed
ic ion was kine ic and a ea o e alua ion app oxima ely
50 mm2. Also, he e alua ion was no conduc ed o he
beginnings o he es s whe e a ploughing was p esen .
Nume ical simula ions
The calcula ions we e pe o med in Abaqus/Explici
comme cial code based on he explici o mula ion o he
ini e elemen me hod. The ma e ial model was imple-
men ed using he VUMAT use sub ou ine, as i is no
a s anda d one. The ailu e was modelled by dele ing ele-
men s ha eached c i ical damage, which is a simple
echnique ha does no equi e emeshing, con a y o
he node sepa a ion me hod.
Model o ma e ial
The hamme and suppo s we e made o s eel, which is
much s i e han he es ed wood specimens. The e o e,
he s eel pa s we e modelled as igid, sa ing some com-
pu a ional ime. On he con a y, wood was modelled
as a homogeneous o ho opic con inuum. The he e o-
genei y was neglec ed o sa e some compu a ional ime
again h ough ma e ial model simpli ica ion, including i s
calib a ion.
The o ho opic elas ici y was desc ibed by he gene al-
ised Hooke’s law as
whe e
εi
,
Ei
and
σi
a e he no mal s ain, elas ic modulus
and no mal s ess, espec i ely,
εij
,
νij
,
Gij
and
σij
a e he
enso ial shea s ain, Poisson’s a io, shea modulus and
shea s ess o
i,j=L,R,T
, espec i ely. All elas ic con-
s an s (Table2) we e aken om [20], whe e he disin e-
g a ion o he same wood was ca ied ou unde a ious
s ain a es in a ious ma e ial di ec ions.
The o ho opic plas ici y was desc ibed by he model
o Ba la e al. [21], who p oposed he yield condi ion as
(1)







εL
εR
εT
εLR
εRT
εTL







=










1
EL
−
νLR
EL
−
νLT
EL
000
−
νRL
ER
1
ER
−
νRT
ER
000
−
νTL
ET
−
νTR
ET
1
ET
000
0001
2GLR
00
0 0 001
2GRT
0
0 0 0001
2G
TL

















σL
σR
σT
σLR
σRT
σTL







(2)
σB−σy=0,
Table 1 Con igu a ion o he impac h ee-poin bending es s
Hamme
weigh
(kg)
Ini ial
heigh
(mm)
A e age ue
impac eloci y
(m/s)
Numbe o
specimens
(–)
Ma e ial
di ec ion
(–)
9.05 458.87 2.70 6 R
9.05 458.87 2.70 6 T
4.55 815.77 3.25 6 R
4.55 815.77 3.25 6 T
D i in
g
mechanism
Sp
ec
i
men
h
o
ld
e
Sp
ecimen
Pin
F
o
ce
se
n
so
L
oa
di
n
g
mec
h
an
i
sm
Fig. 3 UMT T iboLab used o measu emen o ic ion coe icien
Page 5 o 12
Kubíke al. Jou nal o Wood Science (2024) 70:42
whe e
σy
is he yield s ess, which is dependen on he
equi alen plas ic s ain (Fig.4), while
σB
is he equi a-
len s ess acco ding o Ba la e al. [21] as
(3)
σ
B
=1
m
√2
m
|
K1
−
K2
|
m
+|
K2
−
K3
|
m
+|
K3
−
K2
|
m
,
whe e
m
is he exponen (in luencing he shape o he
yield su ace),
K1,K2
and
K3
a e he p incipal alues o
he linea ly ans o med de ia o ic s ess enso
whe e
a
,
b
,
c
,
,
g
, and
h
a e he ma e ial pa ame e s,
which we e i ed (Table3; Fig.5) so ha he yield su ace
co esponded o ha in [20], whe e he model o Hill [17]
was used wi h a nega i e ma e ial pa ame e
(H<0)
. A
change in he yield c i e ion was sough due o he elimi-
na ion o a nega i e ma e ial pa ame e esul ing om a
signi ican di e ence in he yield s esses in he L di ec-
ion compa ed o o he s (R and T in Fig.5), which can
be p oblema ic in some cases. The yield c i e ion was
aken om [20], whe e i was no necessa y o model di -
e en yield s esses in ension and comp ession in ini e
elemen s. Then, he yield c i e ion was ecalib a ed o
Eu opean beech, s ill neglec ing he s eng h di e en ial
e ec . Howe e , mo e sophis ica ed yield c i e ia should
be conside ed when signi ican di e ences in ensile and
comp essi e yield s esses we e obse ed.
(4)
K
=



c
(σL
−
σR)
−b
(σT
−
σL)
3hσLR gσTL
hσLR a(σR−σT)−c(σL−σR)
3 σRT
gσTL σRT b(σT−σL)−a(σR−σT)
3


,
Table 2 Elas ic cons an s o Eu opean beech wood [20]
Ma e ial di ec ion L R T LR RT TL
Elas ic modulus (MPa) 13,000 3500 3000 – – –
Poisson’s a io (–) – – – 0.302 0.362 0.318
Shea modulus (MPa) – – – 1608 460 1059
00.2 0.4 0.6 0.8 1.0
Equi alen plas ic s ain [–]
0
10
20
30
40
50
Y
i
e
ld
s ess [MPa]
Fig. 4 Flow cu e o he Eu opean beech wood [20]
Table 3 Plas ici y- ela ed ma e ial pa ame e s o Eu opean beech
m
(–)
a
(–)
b
(–)
c
(–)
(–)
g
(–)
h
(–)
2 2.357 0.862 0.009 0.456 0.642 0.525
15
30
37.5
22.5
7.5
–7.5
–22.5
–37.5
–75 –45 –15 15 45 75 –75
–45
–15
15
45
75
R[MPa]
2
3
(
(
0,5
T[MPa]
2
3
(
(
0,5
R[MPa]
L[MPa]
2
3
(
(
0,5
L
[MPa
]
LR [MPa]
Fig. 5 Yield locus in he Haigh–Wes e gaa d space (le ) and he space o wo no mal and one shea s esses ( igh ) o he Eu opean beech wood

Page 6 o 12
Kubíke al. Jou nal o Wood Science (2024) 70:42
The o ho opic cumula i e damage was desc ibed by
he damage pa ame e as:
whe e
ε
i
,
εD
i
and
εp
i
a e he ac u e s ain, plas ic s ain
o a gi en loading pa h and plas ic s ain, espec i ely.
As men ioned abo e, he elemen is emo ed when he
c i ical alue is eached, speci ically when
max (Di)=1
.
Fu he mo e, ac u e s ains a e dependen on s ess
iaxiali y (apa om he a e dependence in oduced
la e ) o inco po a e he ension/comp ession ailu e
asymme y (which is simple han he ac u e models in
[15, 22–24])
whe e
σm
is he mean s ess
and
σ
is he equi alen s ess acco ding o on Mises
The dependence o ac u e s ains on s ess iaxial-
i y was based on [20] and was u he e ined using he
ial and e o me hod using ens o nume ical simula-
ions so ha he esul s ma ched expe imen al obse a-
ion in impac h ee-poin bending es s. F ac u e s ains
dependen on he s ess iaxiali y,
ε
i
(η
)
, a e shown in
Fig.6 o he e e ence equi alen plas ic s ain a e o
1 s–1.
Finally, ac u e s ains we e addi ionally depend-
en on he equi alen plas ic s ain a e as expe imen s
we e conduc ed a a ious impac eloci ies. Con a y
o he linea a e dependence in [20] o signi ican ly
g ea e s ain a es, an exponen ial dependence on he
equi alen plas ic s ain a e based on he equa ion p o-
posed by Johnson and Cook [25] was ecalib a ed o i
he expe imen s as ollows (using
ε
i
(η
)
ha has al eady
been calib a ed p e iously o he e e ence equi alen
plas ic s ain a e o 1 s–1)
whe e
C
is he ma e ial pa ame e , calib a ed as 0.1 a e
se e al addi ional nume ical simula ions o e e ence
(5)
D
i=
ε
D
i

0
1
ε
i

dεp
i
,
(6)
η=
σm
σ
,
(7)
σ
m
=
σ
L
+σ
R
+σ
T
3,
(8)
σ=
1
√2
(σL
−
σR)2
+
(σR
−
σT)2
+
(σT
−
σL)2
+
6σ2
LR
+
6σ2
RT
+
6σ2
TL
.
(9)
ε
i=ε
i(η)

1+Cln

˙
ε
p
˙
ε0
,
equi alen plas ic s ain a e
˙
ε
0=1s
−1
. Finally,
˙
εp
is he
equi alen plas ic s ain a e
whe e
˙εp
is he plas ic s ain a e enso .
Model o geome y andbounda y condi ions
As men ioned abo e, he ad an age o he elemen dele-
ion echnique is i s simple implemen a ion. Howe e ,
i has i s d awbacks as i depends on he size o he ele-
men , which is usually kep as small as possible o eal-
is ically p opaga e he c ack using one o wo elemen s.
The e o e, he elemen s had a size o 0.1mm in a eas o
wood ailu e as well as in con ac egions (Fig.7). O he
a eas we e meshed wi h elemen s o a size o 1.82mm.
Only a 0.1-mm-wid h specimen was modelled as a plane
s ain o ep esen he inne laye o wood ins ead o
using 20mm o sa e conside able compu a ional ime.
Geome y was meshed wi h 8-node linea b ick ini e
elemen s wi h educed in eg a ion and hou glass con-
ol (labelled C3D8R in Abaqus). The mesh consis ed o
128,774 nodes and 63,850 elemen s o which 4 we e addi-
ionally 6-node linea wedge ini e elemen s wi h educed
in eg a ion and hou glass con ol (labelled C3D6R in
Abaqus) nea he con ac egions whe e he ine mesh
changed in o coa se mesh (highligh ed in ed in Fig.7).
As men ioned ea lie , he hamme and suppo s we e
modelled as igid, he e o e, as su aces which come
in o con ac wi h he specimen. The wid h o he igid
(10)
˙
ε
p
=
2
3˙
εp
:˙
εp
,
S ess iaxiali y [–]
–0.5 00.5 1.0
0
0.5
1.0
1.5
2.0
F ac u e s a
i
n [–]
0
L
R
T
Fig. 6 F ac u e s ain o all ma e ial di ec ions and he e e ence
equi alen plas ic s ain a e o 1 s–1
Page 7 o 12
Kubíke al. Jou nal o Wood Science (2024) 70:42
bodies, 0.3mm, was g ea e han he wid h o he spec-
imen, 0.1mm, and we e cen ed wi h each o he . The
hamme and suppo s we e meshed wi h 4-node bilin-
ea quad ila e al igid ini e elemen s (labelled R3D4
in Abaqus) ha ing a size o 0.1mm, he e o e, 3 ele-
men s pe wid h. One suppo , modelled he same as
a hamme , had 1892 nodes and 1416 elemen s. Mo eo-
e , an elemen was added o he e e ence poin o he
hamme (highligh ed by a blue pen ag am in Fig.7) in
o de o inco po a e he olume load (labelled MASS
in Abaqus), equi alen o he hamme weigh .
Displacemen s and o a ions o e e ence poin s o
he suppo s we e es ic ed. The same applies o he
hamme excep o he e ical displacemen . The
ini ial eloci y was p esc ibed in his di ec ion, co -
esponding o he a e age ue impac eloci y o he
expe imen (Table1). The specimen had cons ained
displacemen s only in he wid h di ec ion o simula e
he plane s ain condi ion ( he geome y o he speci-
men was disc e ised by 1 elemen along he wid h).
The accele a ion due o g a i y o 9.807 m/s2 was
applied on he whole geome y in he di ec ion o
impac .
In addi ion o he de ini ion o he con ac be ween
he specimen, hamme and suppo s, sel -con ac was
applied o all elemen edges in he a ea o expec ed
ailu e (whe e he elemen s wi h 0.1mm edges we e).
I ensu ed he con ac o new ee su aces, which we e
no ini ially p esen in he geome y, eme ging a e
he elemen dele ion ha simula ed he c acking. The
no mal beha iou o he con ac s was se as ‘ha d’ o
allow any p essu e when he su aces a e in con ac ,
while he ic ion coe icien o 0.15 ob ained om he
expe imen s in he p e ious sec ion was se in he an-
gen ial di ec ion.
Resul s anddiscussion
Figu es8 and 9 summa ise all he o ce esponses agains
he hamme displacemen s ob ained om he impac
es ing machine o hamme s ha weighed 9.05 and
4.55kg, espec i ely. The DIC did no se e o ob ain
he displacemen s. The do ed lines highligh he es s in
which he specimens emained in ac . The co espond-
ing specimens bended o wa d and back ( he hamme
e u ned app oxima ely o he e e ence posi ion co e-
sponding o he displacemen o 0mm—i is ze o de lec-
ion). The dashed-do ed lines highligh he es s in which
he specimens pa ially b oke (did no b eak h ough he
whole heigh ). The co esponding specimens exhibi ed
signi ican sp ingback, he e o e, he hamme e u ned
app oxima ely o he e e ence posi ion. The solid lines
highligh he es s in which he specimens ailed (b oke
h ough he en i e heigh ). The co esponding specimens
emained ben a e he es (all in Fig.8). Mo eo e , he
momen o c ack ini ia ion is highligh ed by a ci cle. The
a e age alue o he hamme displacemen co espond-
ing o he momen o c ack ini ia ion is highligh ed by a
hick e ical dashed g ey line. This line is no plo ed in
le Fig.9, whe e only 2 ou o 6 specimens pa ially b oke
and none ailed o he hamme weighing 4.55kg loading
in he R di ec ion. Only 1 specimen ailed o he ham-
me ha weighed 4.55kg in he T di ec ion ( igh Fig.9).
O he specimens pa ially b oke, while he specimen co -
esponding o Tes 6 in igh Fig.9 was c acked along
almos he whole heigh unde hese condi ions, he e-
o e, he hamme s a ed e u ning la e .
The a i hme ic mean ic ion coe icien o 0.15 wi h
a s anda d de ia ion o 0.005 was measu ed i espec-
i e o he ma e ial di ec ion. Then, he expe imen ally
and compu a ionally ob ained esul s a e compa ed
in Figs.10 and 11. The p edic ed o ces we e in good
Fig. 7 Fini e elemen mesh wi h de ails
Page 8 o 12
Kubíke al. Jou nal o Wood Science (2024) 70:42
ag eemen wi h he expe imen s be o e he maximum
de lec ion. Fo ces oscilla ed mo e in nume ical simula-
ions han in impac h ee-poin bending es s. Howe e ,
his is an in insic ea u e o explici dynamics. C ack-
ing was p edic ed la e in nume ical simula ion han in
expe imen s o a 9.05kg hamme in Fig.10. No c acking
0246810 12 14
Displacemen [mm]
0
1000
2000
3000
4000
Fo ce [N]
Tes 1
Tes 2
Tes 3
Tes 4
Tes 5
Tes 6
0246810 12 14
Displacemen [mm]
0
1000
2000
3000
4000
Fo ce [N]
Tes 1
Tes 2
Tes 3
Tes 4
Tes 5
Tes 6
R, 9.05 kg T, 9.05 kg
Fig. 8 Fo ce–displacemen esponses o a 9.05 kg hamme in he R (le ) and T di ec ions ( igh ), espec i ely ( o in e p e a ion o he e e ences
o colou in his igu e legend, he eade is e e ed o he web e sion o his a icle)
0246810 12 14
Dis
p
lacemen [mm]
0
1000
2000
3000
4000
Fo ce [N]
T, 4.55 kg Tes 1
Tes 2
Tes 3
Tes 4
Tes 5
Tes 6
0246810 12 14
Dis
p
lacemen [mm]
0
1000
2000
3000
4000
Fo ce [N]
R, 4.55 kg Tes 1
Tes 2
Tes 3
Tes 4
Tes 5
Tes 6
Fig. 9 Fo ce–displacemen esponses o a 4.55 kg hamme in he R (le ) and T di ec ions ( igh ), espec i ely ( o in e p e a ion o he e e ences
o colou in his igu e legend, he eade is e e ed o he web e sion o his a icle)
0246810 12 14
Dis
p
lacemen [mm]
0
1000
2000
3000
4000
Fo ce [N]
R, 9.05 kg Expe imen s
Simula ion
0246810 12 14
Dis
p
lacemen [mm]
0
1000
2000
3000
4000
Fo ce [N]
T, 9.05 kg Expe imen s
Simula ion
Fig. 10 Compa ison o o ce–displacemen esponses om nume ical simula ions and expe imen s o a 9.05 kg hamme in he R (le ) and T ( igh )
di ec ions, espec i ely
Page 9 o 12
Kubíke al. Jou nal o Wood Science (2024) 70:42
was achie ed o he R di ec ion wi h a 4.55kg hamme
(le Fig. 9) compu a ionally o ep oduce he expe i-
men s (le Fig.11). Howe e , pa ial c acking was no
achie ed compu a ionally o he T di ec ion wi h a
4.55kg hamme as in he expe imen s ( igh Fig.9). The
elas ic s ain ene gy was so high in nume ical simula ions
o ha case ha led o he p edic ion o c ack p opa-
ga ion along he whole heigh , con a y o he impac
h ee-poin bending es s whe e specimens b oke jus
pa ially ( igh Fig. 9). The homogenei y o he model
also con ibu ed o he lack o p edic ion o c ack a es ,
which occu ed in expe imen s a e some delamina ion
because, among o he s, he wood is s ongly inhomoge-
neous. This should be aken in o accoun by modelling
he geome y based on da a om compu ed omog aphy
speci ically o each specimen, which migh no be p ac-
ical o indus ial applica ions.
The p edic ed con ou s o he damage pa ame e a e
displayed in Figs.12 and 13 a e he impac h ee-poin
bending es s compa ed o expe imen s app oxima ely
in scale. Howe e , i should be no ed ha he p edic ed
de lec ions a e g ea e (app oxima ely 1.5mm), as is e i-
den om Figs.10 and 11. The damage pa ame e s
DR
and
DT
eached hei c i ical alue in espec i e R and T
0246810 12 14
Dis
p
lacemen [mm]
0
1000
2000
3000
4000
Fo ce [N]
R, 4.55 kg Expe imen s
Simula ion
0246810 12 14
Dis
p
lacemen [mm]
0
1000
2000
3000
4000
Fo ce [N]
T, 4.55 kg Expe imen s
Simula ion
Fig. 11 Compa ison o o ce–displacemen esponses om nume ical simula ions and expe imen s o a 4.55 kg hamme in he R (le ) and T ( igh )
di ec ions, espec i ely
R, 9.05 kg T, 9.05 kg
38.029.000.185.076.057.033.024.005.080.071.052.00.00
Fig. 12 Expe imen s ( op) compa ed o p edic ed con ou s o he damage pa ame e (bo om) o he 9.05 kg hamme and he R (le ) and T ( igh )
di ec ions (app oxima ely in scale)