On he Insu moun able Size o T uss-like S uc u es
Olmedo Rojas, Ca los
1
; Vázquez Espí, Ma iano
2
; Ce e a B a o, Jaime
3
ABSTRACT
Galileo pos ula ed he exis ence o an insu moun able size o s one columns bea ing a use ul load as
he size o which he s uc u e is only able o esis i s sel -weigh . He ein a me hod o he
de e mina ion o he unsu moun able size o uss-like s uc u es is shown, gi en he o m o hese
s uc u es and he a io be ween he allowable s ess and he speci ic weigh o he ma e ial ( he
ma e ial s uc u al scope). Th ee ypes o ba s a e conside ed: s aigh ba s, wi h solid and hollow
ec angula c oss-sec ion, and ca ena y ba s wi h ci cula c oss-sec ion —a limi and heo e ical case
o es ima ing a meaning ul uppe bound o he s uc u al scope—. An app oxima e ule o es ima e
he s uc u al e iciency —he e named GA ule— is shown, and is compa ed wi h nume ical solu ions
using he p oposed me hod.
Keywo ds: s uc u al design, insu moun able size, s uc u al scope, usses, sel -weigh .
1. THE GALILEO PROBLEM AND THE AROCA RULE
In a i s app oxima ion, we can ep esen he physical cos o a s uc u e by i s sel -weigh , as many
cos du ing he manu ac u ing, bu no all, a e app oxima ely p opo ional o he sel -weigh o he
s uc u e: CO2 emissions, mine al esou ces consump ion, e c. Fo a gi en s uc u al p oblem, we
de ine he s uc u al e iciency as he a io be ween he use ul load and he whole load (i.e., he
use ul load plus he sel -weigh ) equi ed o sol e ha p oblem in a pa icula s uc u e.
Galileo [1] pos ula ed he exis ence o insu moun able sizes o s uc u es, as well as he ela ionship
be ween he size o a s uc u e and i s abili y o esis a use ul load: le us imagine a cylind ical s one
column, a he limi o i s esis ance only as a esul o i s sel -weigh ; we name s uc u al scope o
he column o he heigh (ℒ) o his column, which canno esis any addi ional load, hence being null
i s e iciency. A use ul column wi h he same base mus ha e he e o e a heigh 𝐿 smalle han ℒ.
This new column can esis an addi ional use ul load (𝑄), he alue o which is a mos he weigh
di e ence be ween he wo columns. Acco ding o he p e ious de ini ion o e iciency:
𝑟 ≤ ℒ−𝐿
ℒ=1−𝐿
ℒ=1−𝑡
(1)
1
Depa amen o de Es uc u as y Física de Edi icación. Uni e sidad Poli écnica de Mad id (Spain).
[email protected] (Co esponding au ho )
2
Depa amen o de Es uc u as y Física de Edi icación. Uni e sidad Poli écnica de Mad id (Spain).
ma iano. azq[email p o ec ed]s
3
Depa amen o de Es uc u as y Física de Edi icación. Uni e sidad Poli écnica de Mad id (Spain).
jaime.ce[email p o ec ed]
477
On he Insu moun able Size o T uss-like S uc u es
Thi d In e na ional Con e ence on Mechanical Models in S uc u al Enginee ing
Uni e si y o Se ille. 24-26 june 2015.
We de ine s uc u al (o ela i e) size (𝑡) as he a io be ween he heigh 𝐿 o a column and i s scope
ℒ. This s uc u al size 𝑡 can ha e alues be ween 0 and 1. No e ha Eq.(1), ha we name Galileo’s
ule, is exac in he case o linea pieces wi h no angen ial s ess, bu i is no p o ed ha i would be
a gene al ule. We de ine cos (𝑘) as he in e se o e iciency, hence always highe han uni y. Then,
he sel -weigh o he column is:
𝑃=(𝑘−1)𝑄 wi h 𝑘=1/𝑟
(2)
Le us de ine he ma e ial s uc u al scope (𝒜), a cha ac e is ic leng h, as he a io be ween he
admissible s ess and i s speci ic weigh [2]. This amoun is he only in o ma ion ha we will need o
ob ain he s uc u al scope o he s uc u e i sel , besides i s geome y and he use ul load dis ibu ion
o be suppo ed. In he case o cylind ical columns, he scope o he columns is he scope o he
ma e ial: le 𝑓 be he admissible s ess o he ma e ial, and le 𝐴 be he a ea o he c oss-sec ion,
equa ing he s eng h capaci y o he column base o he weigh o he column o maximum heigh ,
we ob ain 𝑓 𝐴=ℒ 𝐴 𝜌 (whe e 𝜌 is he speci ic weigh o he ma e ial). Then:
ℒ=𝑓
𝜌= 𝒜
(3)
Maxwell [3] ound he way o compa e s uc u al cos s o a gi en s uc u al p oblem. We name [4]
Maxwell p oblem o he p oblem ha consis o de ining a s uc u e capable o suppo ing a
equilib a ed se o ex e nal o ces de ined bo h in posi ion and magni ude, and we name Maxwell
s uc u e o he s uc u e ha esol es a Maxwell p oblem and is buil ou o elemen s ha wo k
uniaxially, in ension o in comp ession. Maxwell showed ha o wo o hose s uc u es sol ing he
same p oblem, he di e ence in cos is p opo ional o he di e ence in he s ess olume 𝒱 o each,
de ined as:
𝒱= ∑|𝑒|ℓ
(4)
whe e 𝑒 is he alue o he in e nal o ce in each elemen o he s uc u e and ℓ i s leng h.
La e Michell [5] showed ha he sel -weigh o a s uc u e is minimal i i s s ess olume 𝒱 is also
minimal. Also he ound a necessa y c i e ion so ha a Maxwell s uc u e would be an absolu e
minimum, om which a e de i ed some op imal layou s o some speci ic p oblems. We ha e
de ined [6] Michell’s numbe (𝜈) o he dimensionless a io be ween he s ess olume o a s uc u e
and he p oduc o he o al use ul load imes he size o he p oblem ( he heigh in he case o
columns, he span o beams, e c.):
𝜈= 𝒱
𝑄𝐿
(5)
478
Ca los Olmedo Rojas1, Ma iano Vázquez Espí2 and Jaime Ce e a B a o3
Then a s uc u al o m is an absolu e minimum when i s Michell numbe is lesse o equal o any
o he s uc u e sol ing he same Maxwell p oblem.
The con ibu ions o Maxwell and Michell on he measu e o s uc u al e iciency do no ake in o
accoun he sel -weigh , as i can be seen in he de ini ion o Maxwell p oblem. Rica do A oca [2], [4],
[6], [7], [8] joined he heo ies o Galileo and he heo ies o Maxwell and Michell: gi en a Maxwell
p oblem and a s uc u e wi h s ess olume 𝒱 ha esol es i , he olume and he sel -weigh o
such s uc u e a e:
𝑉= 𝒱
𝜎 𝒱=𝜈𝑄𝐿 𝑃=𝑉𝜌= 𝒱
𝒜= 𝜈𝑄𝐿
𝒜
(6)
Bu p e ious exp essions a e only accu a e o he null size, since hey do no ake in o accoun ha
he sel -weigh mus also be equilib a ed. So o s uc u es wi h 𝑡>0:
𝒱≠𝜈𝑄𝐿 𝑃≠ 𝜈𝑄𝐿
𝒜
(7)
Usually he dis ibu ion o sel -weigh will be di e en o he use ul load, bu in many cases o
in e es ( o example, in bending s uc u es) bo h dis ibu ions can be ep esen ed by dis ibu ions o
simila o ces. We can hen es ima e he scope o he s uc u al o m as he size 𝐿=ℒ o which he
s uc u e jus esis s i s sel -weigh , wi hou possibili y o adding any addi ional load. I esul s hen
he ollowing app oxima e exp ession, subs i u ing he use ul load 𝑄 by he sel -weigh 𝑃 in he las
side o las exp ession in o Eq. (6):
𝑃≈ 𝜈𝑃ℒ
𝒜⇒ ℒ ≈ 𝒜
𝜈
(8)
These exp essions a e A oca’s ule. Subs i u ing he las exp ession in Eq. (1), we ha e he ollowing
ule o he e iciency (which we name GA ule, hono ing bo h Galileo and A oca):
𝑟≈ 1−𝜈𝐿
𝒜
(9)
This es ima e o he e iciency will be exac o linea s uc u es whose dis ibu ion o sel -weigh is
isomo phic o use ul load dis ibu ion and whose equilib ium equi es no angen ial s ess. In any
o he case i will be only an app oxima ion whose use ulness mus be p o ed.
2. THE GALILEO PROBLEM FOR TRUSS-LIKE STRUCTURES
He e we p opose a simila app oach o ha desc ibed abo e, now o he case o usses wi h use ul
load a anged as o ces applied a he nodes, o ob ain he s uc u al scope o hese s uc u es,
subjec ed o he ollowing limi a ions: (i) ba s a e o cons an c oss-sec ion; (ii) iden ical ension-
comp ession pa e ns on ba s due o bo h use ul load and sel -weigh (i.e., equal sign o he in e nal
o ce in each ba in hese wo independen load condi ions); (iii) he ma e ial has he same absolu e
alue o allowable s ess in ension han in comp ession; and (i ) buckling o comp essed ba s is no
479
On he Insu moun able Size o T uss-like S uc u es
Thi d In e na ional Con e ence on Mechanical Models in S uc u al Enginee ing
Uni e si y o Se ille. 24-26 june 2015.
aken in o conside a ion. The aim is o ob ain a i s es ima e o he s uc u al scope o a gi en
s uc u al o m, and he s uc u al e iciency.
The me hod p oposed he e aim o include he cos o ansmi ing he dis ibu ed sel -weigh along
he leng h o each ba o i s ex emes, because in la ge s uc u e his cos will gene ally be impo an
in ela i e e ms. Roz any [9] p oposed an op imiza ion me hod ha include sel -weigh bu ha
equi e ba s wi h a iable c oss-sec ion —excluded in ou app oach, limi a ion (i)—, in ac adop ing
an exponen ial unc ion. Such o m o ba s (close o he so named ’cons an maximum s ess
design’) equi e a s ess enso ha does no ul il he di e en ial equa ions o equilib ium [17], so
he solu ions ob ained canno be conside ed easible solu ions o he p oblem. O he au ho s [10],
[11], [12], ci cum en he p oblem o ha ing o conside any bending e ec s including only hal o
he ba weigh in each o i s nodes, bu hen sma algo i hms, as Simula ed Annealing, will choose
solu ions wi h e y la ge leng h, as he bending is ee o cos [10]. The di icul ies o ackling wi h
sel -weigh in uss-like s uc u es disappea in con inuous s uc u es using o example FEM [13],
[14]. The selec ed limi a ions a e jus i ied o se e al easons. (i) has a p ac ical meaning. Wi h (ii),
we a oid special cases whe eby he sel -weigh o a s uc u e can be equilib a ed by he ex e nal
loading, as poin ed ou by Bendsoe [15]. Wi h (iii) and (i ), we keep he model simple, bu (iii) i is no
di icul o o e come and (i ) maybe be supp essed in u u e esea ch applying new esul s on his
subjec [16] .
2.1. Equilib ium equa ions wi h use ul loads and sel -weigh
Le be 𝐍𝐐 he in e nal o ces in a Maxwell s uc u e unde he ac ion o he use ul load 𝐐 (he ea e
bold capi al deno e ec o o a ays). Suppose ha we ha e sol ed he p oblem o designing wi h
ba s ha include hei sel -weigh . Such ba s ep esen an addi ional load due o i s sel -weigh ha
we can in oduce using s a ically equi alen o ces a hei ends, P. The local equilib ium o he sel -
weigh a in e io poin s o he ba depends on he ype o ba : o s aigh beams, on i s bending; o
cables o ba s wi hou bending s i ness, on he cu a u e o he ba i sel . The Maxwell s uc u e will
ha e o de elop addi ional in e nal o ces o hese new loads, NP . We can conside ha he design
p oblem is sol ed i i esul s ha o each ba an axil in e nal o ce in he 𝜒 di ec ion de ined by i s
ends is de eloped, 𝑁𝜒= 𝑁𝑄+𝑁𝑃, and he ba is dimensioned o s ic ly esis he esul ing s esses.
The abo e (ii) limi a ion can now be exp essed saying ha sgn(𝑁𝑃)=sgn(𝑁𝑄), and hence
sgn(𝑁𝜒)=sgn(𝑁𝑄).
The equilib ium equa ions a e he same o bo h se o loads:
Q= HNQ P=HNP Q + P= HN𝜒
((10)
Le 𝜔𝑖 be he a io be ween he equi alen weigh due o sel -weigh in he e ex 𝑖 o he ba and
he in e nal o ce, i.e., 𝜔𝑖= 𝑃𝑖𝑁𝜒
⁄. Then:
P= 𝛀𝐿N𝜒
(11)
480
Ca los Olmedo Rojas1, Ma iano Vázquez Espí2 and Jaime Ce e a B a o3
In his exp ession 𝛀𝐿 is he ma ix o coe icien s 𝜔𝑖𝑗 o each ba 𝑗 o each componen 𝑖 o P,
depending on
L
, he size o he s uc u e. O cou se, each column o 𝛀𝐿 has only wo non-null
componen s. The e o e,
Q+ 𝛀𝐿N𝜒 = HN𝜒 Q = (H − 𝛀𝐿)N𝜒
(12)
No e ha hese equa ions a e nonlinea , as 𝛀𝐿 depends on he size 𝐿 o he s uc u e and on he
sign pa e n o N𝐐, which is gi en.
When Q →0, i.e., when he s uc u e canno esis mo e han i s sel -weigh (and i s size 𝐿 is hen
equal o he scope ℒ o i s o m), esul s:
(H − 𝛀ℒ)N𝜒 =𝟎
(13)
And he alue o ℒ is de e mined as he lowes alue o 𝐿 o which (H− 𝛀𝐿) is singula , excluding
𝐿=0.
2.2. Beams (s aigh ba s)
Le ℓ be he leng h o he ba , and 𝛽 he angle o med by he ba and he ho izon al, see Fig. 1. The
exp essions o he in e nal o ces along he ba ( ension is posi i e, comp ession is nega i e) a e:
𝑁(𝑠)=𝑁𝜒+ 𝜌𝐴sin𝛽(𝑠−1
2ℓ); 𝑀(𝑠)=1
2𝜌𝐴cos𝛽·𝑠(ℓ− 𝑠 )
(14)
Figu e 1. Beam.
We will use he ollowing design condi ions: he ba s ha e de ined he dep h ℎ and he adius o
gy a ion 𝑖 as ac ions o he leng h ℓ o he ba : ℎ=𝑘1·ℓ 𝑖=𝑘2·ℎ=𝑘3·ℓ
The maximum no mal s esses depending on 𝑠 (posi ion on he 𝜒 axis o he ba ) a e:
481

On he Insu moun able Size o T uss-like S uc u es
Thi d In e na ional Con e ence on Mechanical Models in S uc u al Enginee ing
Uni e si y o Se ille. 24-26 june 2015.
𝜎(𝑠)=𝑁(𝑠)
𝐴±𝑀(𝑠)
𝑊
(15)
𝜎(𝑠)=𝑁(𝑠)
𝐴±ℎ𝑀(𝑠)
2𝐴𝑖2=𝑁(𝑠)
𝐴±𝑘1𝑀(𝑠)
2𝐴𝑘3
2ℓ
(16)
𝐴𝜎(𝑠)=𝑁(𝑠)±2𝑘4𝑀(𝑠)
ℓ; 𝑤𝑖𝑡ℎ 𝑘4=1
4𝑘1
𝑘3
2=1
4𝑘1𝑘2
2
(17)
𝐴𝜎(𝑠)=𝑁𝜒+𝜌𝐴𝑠((1−ℓ
2𝑠)sin𝛽±𝑘4(1−𝑠ℓ)cos 𝛽)
(18)
The s ess is maximum o 𝑠=𝑘5𝑖ℓ, wi h
𝑘5𝑖= 𝑘4𝑐𝑜𝑠 𝛽±sin𝛽
2𝑘4cos𝛽
(19)
𝑘5𝑇=1
2 (1+1
𝑘4 an𝛽 ) , in he si ua ion wi h lowe comp ession o wi h highe ension
(20)
𝑘5𝐶= 1
2 (1−1
𝑘4 an𝛽 ) , in he si ua ion wi h lowe ension o wi h highe comp ession
(21)
which means ha o 𝛽 such ha an𝛽≥𝑘4, he s ess is maximum a he ends o he ba .
Taking 𝑘6𝑇=min{1,𝑘5𝑇}:
𝐴𝜎𝑇=𝑁𝜒+𝜌𝐴 𝑘6𝑇(sin𝛽 (1− 1
2𝑘6𝑇)+𝑘4cos𝛽(1−𝑘6𝑇)) ℓ
(22)
By g ouping he pa ame e s co esponding o he ba , wi h he ollowing de ini ion o a new cons an
𝑘𝑖:
𝑘𝑖= 𝑘6𝑇(sin𝛽 (1− 1
2𝑘6𝑇)+𝑘4cos𝛽(1−𝑘6𝑇))
(23)
we can w i e:
𝐴𝜎𝑇=𝑁𝜒+𝜌𝐴𝑘𝑖ℓ
(24)
and simila ly:
𝐴𝜎𝐶=𝑁𝜒−𝜌𝐴𝑘𝑖ℓ
(25)
482
Ca los Olmedo Rojas1, Ma iano Vázquez Espí2 and Jaime Ce e a B a o3
Being 𝑓 he allowable s ess o he ma e ial; making 𝜎𝑇= −𝜎𝐶= 𝑓 o selec he app op ia e
cons an c oss-sec ional a ea 𝐴, i esul s, depending on he sign o 𝑁𝜒:
𝐴
𝑁𝜒= ± 1
𝑓−𝜌𝑘𝑖ℓ
(26)
The coe icien s 𝜔𝑖 o he ma ix 𝛀 a e:
𝜔𝑖= ±𝜌ℓ
2𝐴
𝑁𝜒=sgn(𝑁𝑄)1
2(𝒜
ℓ−𝑘𝑖)
(27)
As he Eq. (18) only conside s he componen 𝜎𝜒 o he s ess enso wi h he model based on he
hypo hesis o Na ie only app op ia e o e y slende beams, o which he e ec s o Sain Venan ’s
p inciple can be neglec ed, he exp ession in Eq. (27) is a good app oxima ion o he sel -weigh o
e y slende beams. We keep us on his simple model in his ini ial wo k o he sake o simplici y.
Mo e accu a e models will be used in u u e esea ch.
2.2.1. Ba s wi h ec angula sec ion
Wi h ec angula sec ion, 𝑘2=1/√12, being he wid h 𝑏 he ee design pa ame e . The c oss-
sec ional a ea is =ℎ·𝑏=𝑘1𝑏·ℓ . Using his alue o 𝑘2 o calcula e 𝑘𝑖, om Eq.(27) we can
calcula e he alues o 𝜔𝑖 o gi en alues o 𝑘1.
2.2.2. Ba s wi h hollow ec angula sec ion
The adius o gy a ion o a hollow ec angula sec ion wi h dep h ℎ, wid h 𝑏 and hickness 𝑡 is:
𝑖= √12ℎ𝑡3−6ℎ2𝑡2−6𝑏ℎ𝑡2+3𝑏ℎ2𝑡−8𝑡4+4𝑏𝑡3+ℎ3𝑡
−24𝑡2+12ℎ𝑡+6𝑏𝑡
(28)
Elimina ing e ms wi h powe s o 𝑡, we ob ain he alue o 𝑖 when 𝑡→0:
𝑖= √3𝑏ℎ2+ℎ3
12(ℎ+𝑏)
(29)
Taking o his case as an addi ional design decision ha he wid h 𝑏 is p opo ional o he dep h
(𝑏=𝑘𝑏ℎ), esul ing he alue o 𝑘2:
𝑘2=1
√12√3𝑘𝑏+1
𝑘𝑏+1
(30)
being he hickness 𝑡 he ee design pa ame e . Now o gi en alues o 𝑘1 and 𝑘𝑏 , we can ake he
a ea o he sec ion as 𝐴=2(ℎ+𝑏)𝑡=2(1+𝑘𝑏)𝑘1𝑡·ℓ, and using his alue o 𝑘2 we can calcula e
he alues o 𝜔𝑖.
483
On he Insu moun able Size o T uss-like S uc u es
Thi d In e na ional Con e ence on Mechanical Models in S uc u al Enginee ing
Uni e si y o Se ille. 24-26 june 2015.
2.3. Ca ena y ba s o cons an c oss-sec ion
Among all he al e na i es o design wi h cons an c oss-sec ion and sel -weigh , he ca ena y a c,
see Figu e 2(a), is he be e known in espec o e iciency, because he e is no angen ial s ess
in ol ed [17].
Figu e 2. Ca ena y a c.
O cou se his is a heo e ical solu ion, e y di icul o build in p ac ice. Bu as he op imal solu ions
om Michell’s heo y –in ac , unicula s uc u es and hence in insic ins able ones– hei s udy lead
o heo e ical limi s ha no o he solu ion wi h cons an c oss-sec ion can exceed.
Le 𝑝,𝑛 be axes such ha axis 𝑝 ollows di ec ion 𝜒, see Figu e 2(b). The g a i y axis is 𝑔, so 𝛽 is he
angle o med by ho izon al axis and 𝑝. The cho d o he ca ena y a c is 𝑐, i s heigh is 𝑣, and i s base
is ℎ (ℎ=𝑐cos𝛽; 𝑣=𝑐sin𝛽).
The basic equa ion esul om he equilib ium o a di e en ial a c d𝑠: he a ia ion o he in e nal
o ce plus he weigh mus be null:
d 𝑁
󰇍
󰇍
+𝜌𝐴(−sin𝛽,−cos𝛽)d𝑠=0
󰇍
(31)
and i can be in eg a ed as:
𝑁
󰇍
󰇍
=𝜌𝐴(𝑠sin𝛽+ 𝐾1,𝑠cos𝛽+𝐾2)
(32)
Le 𝑠0 be he poin o he cu e wi h pa allel angen o 𝑝 axis:
𝑁
󰇍
󰇍
(𝑠0)=𝑁(𝑠0)·(1,0)=𝜌𝐴(𝑠0sin𝛽+𝐾1,𝑠0cos𝛽+𝐾2)
(33)
484
Ca los Olmedo Rojas1, Ma iano Vázquez Espí2 and Jaime Ce e a B a o3
Fu he mo e, 𝑁(𝑠0) is he oblique componen o he in e nal o ce, 𝑂 he ea e , i.e., he componen
in he cho d di ec ion o he oblique 𝑝,𝑔 axes, i.e. 𝑁𝜒. Hence:
𝑁
󰇍
󰇍
(𝑠)=(𝑁𝑝,𝑁𝑛)=𝜌𝐴((𝑠−𝑠0)sin𝛽+𝑘,(𝑠−𝑠0)cos𝛽), wi h 𝑘= 𝑂
𝜌𝐴
(34)
Now, changing o he 𝑙,𝑡 axes o he igu e, 𝑠0=0, 𝑙(0)=0, y 𝑡(0)=0:
𝑁
󰇍
󰇍
(𝑠)=(𝑁𝑙,𝑁𝑡)=𝜌𝐴(𝑠sin𝛽+𝑘,𝑠cos𝛽), wi h 𝑘= 𝑂
𝜌𝐴
(35)
No e he de ini ion o he 𝑠 coo dina e: 𝑠 g ows wi h he conca i y on he le , i.e., in he g owing
di ec ion o d𝑡/d𝑙. In he igu e, 𝑠 g ows o he igh and he conca i y poin s up. As a esul , 𝑐 will be
nega i e o a comp ession a ch, acco dingly wi h he de ini ion o ℎ and 𝑣 as unc ion o 𝑐 and 𝛽
abo e.
De ining an𝛼 as he slope o he angen o he cu e,
an𝛼=𝑁𝑡
𝑁𝑙 cos𝛼= 𝑁𝑙
√𝑁𝑙2+𝑁𝑡2 sin𝛼= 𝑁𝑡
√𝑁𝑙2+𝑁𝑡2
(36)
and he pa ame ic equa ions o his cu e can be ob ained by in eg a ion :
𝑙(𝑠)=∫cos𝛼 d𝑠 𝑡(𝑠)= ∫sin𝛼 d𝑠
(37)
De ining
𝜙1(𝑠)=𝑘sinh−1 (𝑠+𝑘sin𝛽
𝑘cos𝛽 ) 𝜙2(𝑠)=√𝑘2+𝑠2+2𝑘𝑠sin𝛽)
(38)
The pa ame ic equa ions a e
𝑙(𝑠) =(𝜙1(𝑠)− 𝜙1(0))cos2𝛽+(𝜙2(𝑠)− 𝜙2(0))sin𝛽
(39)
𝑡(𝑠) =−(𝜙1(𝑠)− 𝜙1(0))sin𝛽cos𝛽+ (𝜙2(𝑠)− 𝜙2(0))cos𝛽
(40)
No e ha he limi o 𝑙(𝑠) o ±𝜋/2 is ±(𝑠−|𝑘|)+𝑘, and he limi o 𝑡(𝑠) is null (a e ical line).
The modulus o he axial o ce is:
|𝑁
󰇍
󰇍
(𝑠)|=𝜌𝐴 √𝑘2+𝑠2+2𝑘𝑠sin𝛽 =|𝑂
𝑘|𝜙2(𝑠)
(41)
The coo dina es 𝑠1 and 𝑠2 a e de e mined by he ac ha he ends o he a c mus be he poin s
(𝑙1,𝑡1) and (𝑙2,𝑡2) wi h he condi ions 𝑙2−𝑙1=𝑐 and 𝑡1=𝑡2, i.e.:
𝑙(𝑠1)=𝑙1, 𝑡(𝑠1)=𝑡1; 𝑙(𝑠2)=𝑙1+𝑐, 𝑡(𝑠2)=𝑡2;
(42)
485
On he Insu moun able Size o T uss-like S uc u es
Thi d In e na ional Con e ence on Mechanical Models in S uc u al Enginee ing
Uni e si y o Se ille. 24-26 june 2015.
Le us ou line ha he abo e p ocedu e consis in a ew and e y simple calcula ions a he i s s ep
o he design p ocedu e, p o iding we ha e a ou disposal he kind o da a o he Table 1 o he
Maxwell p oblem unde conside a ion. The eaching o his p ac ice and he associa ed design heo y
a e included in o he cu iculum o g adua e s uden s o he School o A chi ec u e o Mad id since
h ee decades ago.
5. CONCLUSIONS
In his pape we ha e p esen ed a nume ical me hod o deal wi h he sel -weigh load be o e he
c oss-sec ions o i s ba s a e known. When applied o well de ini e Maxwell p oblems he me hod can
be used o esol e conc e e s uc u al o m de e mining he hickness o i s ba s, i s o al sel -weigh
and i s s uc u al e iciency. Besides, he me hod can be used o explo e he space o solu ions
associa ed o he Maxwell p oblem and o a de ini e s uc u al schema, de e mining he ela ionships
be ween size, slende ness, insu moun able size, and e iciency o each solu ion in he sea ch space.
Wi h his me hod we ha e checked he GA ule (i.e., Galileo-A oca ule), an app oxima ed bu simple
o mula ion o de e mine he insu moun able size and e iciency o s uc u al o ms o schema a,
ou lining i s ad an ages and d awbacks, and sugges ing u he esea ch o esol e he la e ones.
The p oposed me hod could be imp o ed in se e al ways: (i) inco po a ing a mo e ealis ic model o
he s ess dis ibu ion o beams (s aigh ba s wi h bending); (ii) managing di e en alues o
ension and comp ession allowable s ess; (iii) conside ing he local buckling o beams and
comp essed ca ena y a cs. In his way, his wo k can be conside ed as a econside a ion o he
seminal p oblem o a heo y o s uc u al design, o mula ed by Galileo in 1638, and he app oxima e
solu ion en isaged by Rica do A oca o i s ime in he eigh ies o he las cen u y. We hope ha his
wo k can con ibu e o new and p omising esea ches in he u u e.
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