Corso di TECNICA DELLE COSTRUZIONI
POLITECNICO DI TORINO ’ (Vercelli) Docente: Rosario Ceravolo
Dip. Ingegneria strutturale e geotecnica
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PARTE 3: STRUTTURE IN ACCIAIO
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COMPOSIZIONE COMPOSIZIONE STRUTTURALE La struttura in acciaio deriva dall’assemblaggio di elementi monodimensionali, i profilati, e/o e/o bidimensionali, le lamiere, prodotti in officina. officina. Acciaieria
Carpenteria
Cantiere
Produzione profilati e lamiere
Trasformazione in elementi o complessi strutturali
Montaggio in opera
C ern ie ra
Vincolo mutuo tendenzialmente debole
In ca stro
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Nonostante il materiale presenti un legame simmetrico simmetrico sia a trazione, che compre compressi ssione one,, i proble problemi mi di stabil stabilità ità locale locale e di insiem insiemee posson possono o render renderee non simmetrica la risposta degli elementi strutturali (aste compresse, travi inflesse, pannelli e anima anima delle travi, zone zone compresse delle sezioni). asta teorica asta industriale indefinitamente elastica Ncr =π2·E·A/λ 2 asta industriale elasto-plastica λ=l0/ρ N f yA
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La necessità di controllare accuratamente la presenza di eventuali labilità e di valutare corre orrett ttaamente ente le ris risposte oste dei dif differe feren nti elem lementi nti strut trutttura urali nei nei rig riguar uardi dell’instabilità richiede di concepire la struttura nello spazio a tre dimensioni. Le strutture in acciaio presentano elevata deformabilità in presenza delle azioni di eser eserci cizi zio o a causa causa degl deglii elev elevat atii live livell llii tens tensio iona nali li ragg raggiu iung ngib ibil ilii e del del modest odesto o contributo del peso proprio. La verifica a deformazione deformazione può diventare determinante determinante nel dimensionamento, dimensionamento, più che quella di resistenza. Limitazioni di freccia dell’ordine di 1/500 l risultano più condizionanti dei limiti tensionali.
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Edifici multipiano Colonne, travi principali e secondarie in acciaio. Soletta collaborante in calcestruzzo
I carichi verticali vengono raccolti dagli impalcati a struttura mista, trasferiti alle travi principali e da queste alle colonne, fino alla fondazione. L’impalcato può essere realizzato con: pannelli in C.A. o C.A.P. C.A.P. o misti con laterizio gettati in opera (soluzione “a” “b” pannelli in C.A. o C.A.P. C.A.P. o misti con laterizio prefabbricati (sol.“c (sol.“c”, “d “d ”); ”); lamiere grecate riempite con
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Le azio azioni ni oriz orizzo zont ntal alii veng vengon ono o ripo riport rtat atee dagli elementi di facciata, i tam tampona ponam menti enti,, ai sola solai, i, e da ques questi ti alle colonne che li devono trasferire in fond fondaazion zionee. I sola solaii quin quindi di devo devono no ope operare rare com come last lastre re oltr oltree che che com come
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Lelle figure sottostanti sono riportati alcuni tipi di nodi strutturali: •
•
Nodo “a” e ”c” bullonato, e saldato, in grado di trasferire completamente il momento flettente; Nodo “b” e “d “d ” in grado di trasferire l i l il
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Le strutture di “controvento” possono essere realizzate in calcestruzzo armato, come i vani scala, o in acciaio, come come ad esempio esempio i diagonali posti tra travi e colonne. Nodi in grado di trasmettere trasmettere solo taglio e sforzo normale no rmale (cerniere).
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Esaminando nel complesso spaziale la risposta dell’edificio deve risultare che: • Ogni impalcat impalcato o deve deve operare operare come come lastra lastra vincolata vincolata ai controventi controventi verticali; verticali; • I controventi controventi devono garantire garantire almeno almeno tre tre condizioni condizioni di vincolo vincolo ad ogni piano; • L’impalca ’impalcato to deve essere essere dimensio dimensionato nato anche anche per le azioni azioni orizzonta orizzontali li provenient provenientii dal comportamento di insieme dell’edificio. Ogni Ogni elem elemen ento to di cont contro rove vent nto o vert vertic ical alee forn forniisce sce le condi ondizi zion onii di vinc vincol olo o che che
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Nel caso successivo si sono usati esclusivamente esclusivamente controventi in acciaio sia in direzione verticale che orizzontale. In presenza di “q “q” distribuita risulta:
R1
= R2 =
ql 2
R3
=0
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I difetti delle saldature sono i seguenti: - Cricche; - Soff offiature; - Manca Mancanza nza di penetr penetrazi azione one e di fusione fusione;; - Inc Incolla ollatu turra. I mezzi mezzi di indagine per scoprire scoprire eventuali imperfezioni nelle saldature sono: - Raggi X o raggi ggi γ (macchie scure sulle pellicole); - Ultras Ultrasuon uonii (onde (onde deviate deviate dai difetti difetti); ); -
Liqu Liquid idii pene penetr tran anti ti..
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Classificazione delle unioni saldate IN ORDINE DI DIFFICOLTA DIFFICOLTA’ CRESCENTE
-
In pia piano (1); 1); In vert vertic ical alee (2); (2); Front rontaale (3); (3); Sopra oprate test staa (4). (4).
VARI TIPI DI GIUNZIONI
-
Giun Giunti ti testa esta a test testaa I Sald Saldat atur uraa d’orl d’orlo o (II) (II);; Sald Saldat atur uraa d’ango d’angolo lo (III (III); ); Giun Giunto to ad “L” “L” (IV) (IV);; Gi to “T” (V); (V);
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VARI TIPI DI GIUNTI TESTA A TESTA
Nei giunti testa a testa se si vuole una completa completa penetrazione penetrazione occorre smussare i lembi in modo da creare un vano completamente completamente accessibile. 1 - A “V” (1); - A “U” (2); 2 - A “X” (3); - A “Y” (4). 3 SECONDO LA DIREZIONE DELL’AZIONE DELL’AZIONE SOLLECITANTE SOLLECITANTE
-
Laterali I Front rontaali (II) (II);; Obli Obliqu quii (III (III). ).
SECONDO LA VERIFICA DI SICUREZZA
I
4 III II I
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Resistenza della giunzione saldata GIUNTI GIUNTI TESTA A TESTA
In un giunto testa a testa privo di difetti lo stato tensionale corrisponde a quello di un mezzo continuo. La sezione resistente è quindi pari allo spessore per la lunghezza del giunto e la resistenza del materiale di apporto f d,cr viene espressa come una frazione di quella del materiale base: f d,cr = µ cr f d
con µ cr = coefficiente di efficienza. efficienza.
- σ ⊥ : tensione normale di trazione o compressione ortogonale alla gola; - τ : τ : tensione tangenziale parallela alla gola; - σ // : tensione normale di trazione o compressione parallela alla gola.
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GIUNTI A CORDONE D’ANGOLO
Si considera come sezione resistente il prodotto dell’altezza della gola “a” “a” per la lunghezza del cordone. Il calcolo convenzionale delle tensioni viene eseguito ribaltando su uno dei lati del cordone la sezione di gola.
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σ ⊥2 + τ ⊥2 + τ //2
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ESEMPI
- Cordoni frontali soggetti a sforzo normale: τ ⊥
=
F 2 ⋅ l ⋅ a
- Cordoni laterali soggetti a sforzo normale:
l
F/2 F/2
F
l F/2
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- Com Combina binazi zion onee di cord cordon onii tras trasve vers rsal alii e long longit itud udin inal alii sogg sogget etti ti a tagl taglio io e moment momento o (le tensio tensioni ni norma normali li dovute dovute al flette flettente nte si valuta valutano no consi conside deran rando do attiva tutta la saldatura): a1 , l 1 σ ⊥ max (cordoni a1, a2, a3) τ //
=
F 2 l
(taglio ai cordoni
σ ' ⊥ max
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- Sollecita Sollecitazioni zioni di torsione torsione,, flessione flessione e taglio: taglio: i metodi di calcolo semplificato sono due: il metodo del momento polare e il metodo delle due forze - metodo del momento polare (cordoni laterali soggetti a torsione, taglio e momento flettente): Si calcola il baricentro G dei cordoni ribaltati sul piano della giunzione ed il loro momento polare rispetto a G ( I ); si calcola quindi la sollecitazione τ
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Completo ripristino di V e parziale di M
Cerniera
Cerniera
Cerniera
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Giunti trave - - colonna colonna
Com Complet pleto o ripr ripris isti tino no
Com Complet pleto o o parz parzia iale le ripr ripris isti tino no Com Complet pleto o o parz parzia iale le ripr ripris isti tino no
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Giunti “pendolari”
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Giunti a squadretta ed osservazioni R Nei giunti con coprigiunto a totale ripristino flessionale il R/2 R/ 2 flettente deve essere suddiviso tra ali ed anima; questa deve inoltre assorbire anche V . Nel caso di parziale ripristino flessionale si attribuisce al coprigiunto delle ali tutto il flettente ed a quelli d’anima il e1 e2 R/2 taglio. R Nei giunti flangiati il taglio viene trasmesso da d a tutti i bulloni, il b flettente sui bulloni tesi e la zona com ressa della della flangia. a H1 H2 d Nei giunti a squadretta (L) occorre tener tener conto delle eccentricità eccentricità V1 V2 H2 delle reazioni: H1
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Giunti di composizione delle sezioni Si definiscono unioni correnti quei dispositivi di collegamento fra due o più piatti o profilati la cui composizione composizione permette di ottenere un elemento elemento strutturale complesso. RICHIAMO DELL’IMPOST DELL’IMPOSTAZIONE AZIONE DEL PROBLEMA PROBLEMA ALLA JOURAWSKI JOURAWSKI Si consideri la trave composta rappresentata in Fig. 1, soggetta a M, N e T. T.
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Ipotizzando un’azione assiale costante e un momento variabile lungo un concio di trave di lunghezza dx, per l’equilibrio alla rotazione si ottiene: ovvero il taglio è legato alla variazione di momento. dM = Tdx L’equilibrio alla traslazione di un concio dell’ala superiore della trave è garantito se la saldatura tra l’ala e l’anima è in grado di assorbire gli sforzi di scorrimento (Fig. 2). Alla variazione di momento è associata una variazione di sforzi di compressione nell’ala superiore (il profilo a C in figura) figu ra) di risultante pari a:
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Unione saldata
S m τ //
= τ ⋅ b ⋅ ∆l =
S m
∆l ⋅ 2a
=
τ ⋅ b 2a
Unione bullonata
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Rotture dei giunti
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VERIFICHE DI RESISTENZA TRAZIONE
σ N =
N d
A
eff
≤ f d
(S . L L.)
Aeff è l’area del profilo o di una sua sezione netta (depurata di fori) quando il profilo è collegato in modo simmetrico rispetto al baricentro;
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FLESSIONE RETTA M
σ max = ψ w
Μ/Me ψ
ψ ≥ 1: coeff. adattamento nto 1 plastico (o di forma).
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σ D
= ψ x ⋅ f y ⋅
σ cr , D
=
M cr W
σ cr , D n σ cr , D
+ (ψ x ⋅ f y )n
n = 1,5 per travi laminate; n = 1 per travi saldate.
Tensi ension onee crit critic icaa per per inst instaabili bilità tà fles flesso so-t -tor orsi sion onal alee in campo elastico
t
Travi laminate: ω 1 ⋅ M eq
f d (S.L.) ≤ σ (T.A.) σ = adm ψ x ⋅ W
ω 1
=
f y
⋅
h ⋅ L
0,585 ⋅ E b ⋅ t 1
ω = tabulato tabulato in funzione funzione della eometria eometria e ti o d’acciaio d’acciaio t 1 = spessore ali; l = distanza tra 2 ritegni torsionali successivi. 0,75 M max
≤ M eq ≤ M max
M eq
Travi appoggiate o continue = 1,3 M m → Travi
h
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Elementi presso- presso-inflessi inflessi Equilibrio: N ⋅ θ ⋅
⇒ per
⇒
l 2
= M = 2k (θ − θ 0 )
N = θ 0
4k l
⋅
=0
N = N cr
θ − θ 0
con θ 0 = imperfezione iniziale.
Modello cinematico:
F
θ0 θ
θ
⇒
θ − θ 0
4k Carico critico per modulo
N cr =
⇒ θ =
imperfezioni l privo di imperfezioni θ 0
1−
=
µ
−
θ 0
con
µ =
N cr
N cr
Alla plasticizzazione della molla:
k
l 2θ
v F N pl N
molla (b)
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Il comportamento dell’asta continua è descritto dalla curva (c), con una transazione più graduale dalla curva (a) alla curve (b), dovuta alla progressiva plasticizzazione della sezione critica. La norma italiana adotta una formulazione che tiene conto della forma del diagramma di momento agente sull’asta. Deve risultare: f (S.L.) β ⋅ M N con σ cr = tensione critica euleriana ω + ≤ d σ (T.A.) A N adm (anche in campo plastico) ψ ⋅ W 1 − ν
N cr
Ma
M b
β = β = 1 se M M = = costante β ⋅ M = M eq
= 0,6 M a − 0,4 M b
con
M a
≥ M b
e
M eq
> 0,4M a
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Elementi inseriti in un complesso strutturale -
Condizioni Condizioni di vincolo vincolo non perfette perfette e spesso spesso variabili variabili in funzione funzione del carico; carico; Necess Necessità ità di valut valutar aree la rigidez rigidezza za e resist resisten enza za dei vinco vincoli li e la consegu conseguent entee reale reale lunghezza libera di inflessione. P P P l1 2 π ⋅ E ⋅ I EA k=EA/l1 P cr = 2 l P cr ⋅v = k ⋅ v ⋅ l ⇒ P cr = k ⋅ l 2°
meccanismo per k <
l π
l
⋅ ⋅ l 3
In realtà l’intervento della plasticizzazione rende più complesso il problema.
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Aste vincolate agli estremi → l 0 = β l l
l
β=1
=05
Aste con vincoli intermedi
l
β = 0,7
l
=1
β=2
=2
P
P
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Aste con vincoli elastici Molto spesso i vincoli delle aste non possono esser ritenuti rigidi, allora la loro deformabilità abbassa la sicurezza nei confronti dei fenomeni d’instabilità. - La rigidezza minima efficace del vincolo elastico deve essere valutata tenendo conto delle imperfezioni iniziali;
P
l
l0=l
l0>l
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Il problema è stato inizialmente risolto da Engesser nell’ipotesi di corrente compresso rettilineo, a sezione costante, sollecitata da carico assiale costante, con estremità incernierate e supporti elastici uniformemente distribuiti. lC
k = F/v F/v
v = spost spostam amen ento to vinc vincol olo; o; F = reazione vincolo.
l P
k
lC=βl
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β =
λ y l 0
ricavando λ y in funzione di ω (da f d ⋅ A/N ⋅ A/N o o σ adm⋅ A/N A/N ). ).
i y Deve risultare:
k i
≥ k 0 (S . L.)
o
k i
≥ 1,5k 0 (T . A.) ⇐ vincoli
estremi
rigidi
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b) Sezione variabile da campo a campo, N variabile, campi di lunghezza variabile e rigidezze dei vincoli intermedi variabili: Si usa la procedura precedente assumendo: - N = N = N max - l 0 = l 0,min β = β = media aritmetica dei diversi valori di β
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Aste appartenenti a telai l 0 = distanza tra i 2 punti consecutivi di flesso della deformata critica in campo elastico di un ritto semplicemente semplicemente compresso.
β = 0,7
β = 0,5
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INFLUENZA DELLA DEFORMABILITA’ A TAGLIO SUL CARICO DI PUNTA Il tagl taglio io dete etermin rminaa una una rota otazione one aggiuntiva del concio yT ' = χ ⋅
T
= χ ⋅
G ⋅ A con : χ = fattore
P ⋅ y ' G ⋅ A di taglio Y
che non gener generaa flett flettent entii e a detr detratt atto o dall’eq. della linea elastica: II E ⋅ J ( y IV − y IV T ) + Py
= 0 ⇒ y IV +
P E J *
Pϕ
⋅ y II = 0
P
+d
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Nel caso di sezione piena in acciaio (l = altezza asta incernierata, b = base minore sez.) si trova: χ ⋅ P E G ⋅ A
π 2 ⋅ E ⋅ J 2 2 l b = 1,2 = 2,6 E ⋅ A l 2(1 + ν )
T ⋅ Lt
m Cos α ⋅ E ⋅ Ad s 3
Lt T
α T ⋅ Lt
Valore generalmente trascurabile. Invece tale riduzione L0 assume importanza nel caso di aste calastrellate e tralicciate.
γ
ESEMPIO: TRALICCIO
m = Lt
T E ⋅ A
Cos α ⋅ E ⋅ Ad 2
T
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ESEMPIO: TRALICCIO (asta calastrellata) s1 s 2
= =
T ⋅ L ⋅ Lt 2 0
24 ⋅ E ⋅ J t T ⋅ L30 48 ⋅ E ⋅ J 0
: Momento in
calastrell i
: Momento in
corrente
Lt ⋅ L0 L20 χ γ γ ' = + = = G ⋅ A T 12 ⋅ E ⋅ J 24 ⋅ E ⋅ J 0 t
T/2
T/2
L0
In entrambi i casi il problema si può impostare nella forma seguente:
σ
=
π 2 ⋅ E
dove :
T/2
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INSTABILITA’ LATERALE L’instabilità laterale interessa tipicamente tipicamente le travi inflesse con anima molto sottile. In In alcuni casi, un leggero sbandamento laterale o una torsione, possono portare alla instabilità. Esaminiam Esaminiamo o il caso di una trave rettang rettangolar olaree soggetta soggetta a momenti momenti costanti costanti applicati applicati sulle estremità. Si osse osserv rvaa che che la conf config igur uraz azio ione ne vari variat ataa dete determ rmin inaa dei dei mom momenti enti aggi aggiun unti tivi vi che che dovranno rispettare le seguenti relazioni:
EJ '
=
=−
⋅
GJ '
=
⋅
Derivando la seconda relazione in z e sostituendo so stituendo si ottiene: G J t ϕ '' z
ϕ = M x 0 ⋅ ϕ ' y = M x 0 ⋅ −M x 0 ⋅ z EJ
da cui:
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La soluzione è: ϕz = A cos λ z + Bsenλ z
Se imponiamo troviamo: A = 0
e
e
ϕ z ( z = 0) = 0
ϕ z ( z
= l ) = 0
(rotazioni impedite sugli appoggi),
B ⋅ s e nλ z = 0
Oltre alla soluzione banale ( B = 0) , si trova λ l = K π
M x 0
ossia
(GJt ) ( EJ y )
⋅ l = K π
(con K = K = 1,2…)
Il alore del carico carico critico critico è il il più più piccolo piccolo fra quelli associati associati ai diversi diversi alori di K :
( M x 0 )critico = π ⋅
(GJt ) ( EJ y ) l
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INSTABILITA’ E COLLASSO IN REGIME ELASTO -PLASTICO ELASTO- PLASTICO DELLE TRAVI E ⋅ J
d 2 v dx 2
= P (e + f − v )
f
e
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M P P P
=
σ 0 ⋅ b ⋅ h 2 4
= σ 0 ⋅ b ⋅ h
M P = 1 − M P P P
2
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ANALISI DI STRUTTURE IN ACCIAIO I procedimenti dell’analisi limite trovano alcuni limiti nei seguenti problemi: p roblemi: - Necessità Necessità di tenere tenere in conto di un un nume numero ro cospicuo cospicuo di condizioni condizioni di carico; carico; - Presenza Presenza di imperfe imperfezion zionii geomet geometriche riche (aste non rettil rettilinee inee), ), costru costruttive ttive (vertical (verticalità ità delle colonne) colonne) , Autotensioni Autotensioni..
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TRAVI
-
Profilati Profilati di classe classe III e IV : calcol calcolo o elastic elastico o lineare lineare,, il caric carico o ultimo ultimo corrispon corrisponde de al raggiungimento del momento momento ultimo nella n ella prima sezione; Profilati Profilati classe classe I e II : E’ possibile possibile effettua effettuare re una analisi analisi limite. limite.
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