PGS - lezione 04 - MODELLAZIONI DISCRETE.pdf

Lezione #4:
modellazioni discrete
PONTI E GRANDI STRUTTURE - A.A. 2022/23
Prof. Ing. Franco Bontempi
Facoltà di Ingegneria Civile e Industriale
Università degli Studi di Roma La Sapienza
franco.bontempi@uniroma1.it

Index
•Part 0 – BASIS
•Part I – Trusswork & Gridwork
•Part II – Strut & Tie Models
•Part III – Stringer Panel Method
•Part IV - Applications
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Ponti e Grandi Strutture - A.A. 2022/23
lezione #4: modellazioni discrete
2

PARTE 0:
BASIS
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Modelli di analisi e modelli di progetto
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Analisi e Sintesi
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Structure Essential
• Micro-level:
local size of the sections, i.e. thickness, area, inertia, …
(Detailed Geometry)
• Meso-level:
form of the structural element or structural part (substructure), i.e. main
longitudinal axis, curvature, profile, …
(Global Geometry)
• Macro-level:
connections of the different structural parts
(Load Path)
6

Micro-level:
local size of the sections,
i.e. thickness, area,
inertia, … (Detailed
Geometry)
Meso-level:
form of the structural
element or structural part
(substructure), i.e. main
longitudinal axis, curvature,
profile, … (Global Geometry)
Macro-level:
connections of the
different structural
parts (Load Path)
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Optimization Levels
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Load Path
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Ottimizzazione topologica
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B- e D- Regions
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Principio di De Saint Venant (DSV)
• I sistemi A e B di carichi applicati sul corpo sono
staticamente equivalenti, ovvero hanno le stesse
risultanti globali.
A B
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Principio di Saint Venant: trave a sezione compatta
A
B
C
b
F
F
C = F• b
F
F
2 • F
x
x
x
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B-/D- Regions (1)
A
B
C
D E
F
G
H
1
2
3
4 5
6
7
8
22
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B-/D- Regions (2)
A
B
C
D E
F
G
H
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Suddivisione della struttura in B- e D-Regions (1)
• L'analisi di un sistema strutturale complesso difficilmente può
essere condotto in un'unica fase.
• All'interno della struttura sono infatti presenti, generalmente, due
classi di regioni, che presentano comportamenti meccanici
qualitativamente differenti.
• Si possono infatti individuare le cosiddette:
24
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• B-Regions: regioni dove lo stato di sforzo è conseguente ad un
regime deformativo semplice (con andamenti lineari); la lettera B
deriva da Bernoulli, che individuò insieme a Navier l'ipotesi sul
comportamento delle sezioni delle travi che ruotano, restando
piane;
• D-Regions: regioni dove l'assenza di una cinematica semplice,
comporta stati di sforzo comunque complessi; si hanno quindi
regioni genericamente sedi di stati di sforzo diffusivi, da cui deriva
la lettera D.
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• Le D-regions sono tutte quelle zone di singolarità per la struttura,
ove si verificano discontinuità geometriche o di materiale, o dove
sono applicate forze concentrate, sia carichi che reazioni vincolari.
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Basic stress diffusion mechanisms

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Corners

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Corbels

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Structural parts

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Beam

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Beam safety checks

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Impostazione basata su equivalenza elastica
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e

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Caratteristiche di un elemento finito (1)
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Approssimazione con tratti costanti
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Triangolo di Pascal
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Raffinamento della modellazione
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Modalità di convergenza
Convergenza monotona dall’alto
Convergenza monotona dal basso
Convergenza non monotona
e.f. completi
e.f. completi e conformi
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Impostazione basata sull’analisi limite
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p

Presupposti
1) Il materiale ha comportamento perfettamente e indefinitamente plastico:
si è in presenza di duttilità infinita;
2) Non esistono effetti geometrici ovvero problemi di instabilità: non ci sono
ripercussione degli spostamenti (e delle deformazioni) sull’equilibrio.
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Teorema del limite inferiore(teorema statico)
• Se
i. esiste un sistema di carichi esterni in equilibrio con una distribuzione di
sforzi interni che non ecceda in alcun punto la resistenza del materiale,
segue che
1) il collasso non si verifica
2) il sistema di carichi esterni è minore o uguale al vero carico di collasso.
• Lo stato di sforzo interno non deve essere necessariamente quello che
effettivamente si verifica al collasso, ma solo essere equilibrato e compatibile
con la resistenza del materiale non appaiono le condizioni di congruenza.
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Teorema del limite superiore (teorema cinematico)
• Se esistono:
i. un sistema di carichi esterni e
ii. un meccanismo di collasso plastico tale che il lavoro dei carichi esterni
per un incremento di spostamento sia uguale al lavoro delle forze
interne,
segue che:
1) si verifica il collasso;
2) il sistema di carichi esterni è maggiore o uguale al vero carico di
collasso.
• Il meccanismo di collasso deve essere congruente, ma non necessariamente
quello reale; non appaiono le condizioni di equilibrio.
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Primo principio dell’analisi limite
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Secondo principio dell’analisi limite
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PARTE I:
TRUSSWORK & GRIDWORK
Elasticità
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Trusswork (Traliccio)
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Equivalenze
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Trusswork (1)
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Trusswork (2)
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Gridwork (Graticcio)
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Continuo vs. Discreto
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PARTE II:
STRUT AND TIE MODEL
(S&T)
Plasticità
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Traliccio di Morsch
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Travature
reticolari
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Equilibrio
sezionale
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Equivalenza
Equilibrio
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Principal Stresses
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Individuazione
diagonali
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Individuazione
correnti
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Cls
Acc
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Kupfer - Gerstle
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Kupfer - Gerstle
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Conseguenze
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Es.
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Es.
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Torsione
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S&T catalogue
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Example
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Example
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Elastic stress trajectories and the corresponding S&T
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Application of the load path method
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Example
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Risk of unacceptable width of cracks at discontinuities
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Centred concentrated load
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Eccentric concentrated load
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Centered opposite concentrated loads
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Eccentric opposite concentrated loads
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Gerber saddle
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Beam with recess
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Walls under (a) compression and (b) tension.
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Deep Beams
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Deep beam with hole
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Holes
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3D Strut & Tie
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Example of a 3D STM for a pile plinth
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S&T at whole structure scale
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Failure Criteria
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Strut, Tie, Nodal zone
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Compression stress field: (a) prism, (b) bottle, (c) fan

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Geometric shapes of struts

Kupfer - Gerstle
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Bottle-shaped strut

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Types of struts in a beam web

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Classification of nodes

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Width of tie to determine dimensions of nodal zone

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Anchor forces perpendicular to the plane of beam

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Lap joint of tension reinforcement

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Lap joint of compression reinforcement

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Curved reinforcement with deviation forces

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Modeling of continuous deep beam showing the possible locations of
openingswithout disturbing the flow of forces

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Behavior of corner joint due opening moment

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Stresses and cracking in a closing corner joint

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Exterior beam column connection

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Suggested anchorage of bars

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PARTE III:
STRINGER PANEL METHOD
(SPM)
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Precursori elastici

Precursori plastici
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Airframe Strinegr & Panel
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Reinforced Concrete Plane Element
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PARTE IV:
APPLICATIONS
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Design of details
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Testate
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End Anchorage Zones (1)
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End Anchorage Zones (2)
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Testata morta
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Ancoraggi intermedi (Blister)
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External anchor pocket
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Anchor block
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Spinte a vuoto
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Diaphragm & Piers
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Bridge Diaphragm (1)

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Bridge Diaphragm (2)

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EVOLUTION OF THE DESIGN
OF A CABLE-STAYED BRACKET

CONCEPTUAL DESIGN
Definition and optimization
of the structural configuration

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Ponti e Grandi Strutture - A.A. 2022/23 lezione
#4: modellazioni discrete
240
STRUCTURAL SCHEME
Versione iniziale
Versione finale
beam SX beam DX
column
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241
LOAD SCHEMES
Reinforcement
Bars
Vsd
Reinforcement
Bars
Vsd
Reinforcement
Bars
Vsd
Reinforcement
Bars
Vsd
SYM ASYM
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243
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STRUCTURAL PARTS
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246
FIRST ANALYSIS (A):
two dimensional geometry
column
a
Vsd
Vsd
a/2
Vsd*=Vsd/2
Vsd* =Vsd/2
column
a
Vsd
Vsd
column
a
Vsd
Vsd
a/2
Vsd*=Vsd/2
Vsd* =Vsd/2
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247
• the steel parts, the longitudinal bars and the stirrups are represented by
bars working both in tension and in compression, while concrete parts
are lumped into bars with no tension behavior;
• one model a segment of concrete column sufficient to extinguish the
diffusive effects connected with this D-region, i.e. until a B-region is
reached, governed by the so-called Bernoulli stress regime;
Mechanical modeling by S&T
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248
S & T Model Definition
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249
Strut & Tie Models
Reinforcement
Bars
Vsd
Reinforcement
Bars
Vsd
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250
Reinforcement
Bars
Vsd
Reinforcement
Bars
Vsd
Strut & Tie Results
stirrups longitudinal bars
concrete
steel bracket
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251
Hybrid models
Reinforcement
Bars
Vsd
Reinforcement
Bars
Vsd
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252
Global response
End of external bracket displacement
-8,00
-7,00
-6,00
-5,00
-4,00
-3,00
-2,00
-1,00
0,00
0 500 1000 1500 2000
Load [KN]
Uy
[mm]
Vsd=600 KN - th=8mm
Vsd=850 KN - th=10mm
Y
X
End of external bracket displacement
-8,00
-7,00
-6,00
-5,00
-4,00
-3,00
-2,00
-1,00
0,00
0 500 1000 1500 2000
Load [KN]
Uy
[mm]
Vsd=600 KN - th=8mm
Y
X
Y
X
Y
X
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Local response
>290
<-290
>290
<-290
>290
<-290
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Micro-level:
local size of the sections,
i.e. thickness, area,
inertia, … (Detailed
Geometry)
Meso-level:
form of the structural
element or structural part
(substructure), i.e. main
longitudinal axis, curvature,
profile, … (Global Geometry)
Macro-level:
connections of the
different structural
parts (Load Path)
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Optimization Levels
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255
EVOLUTION OF THE FORM (1)
600.0
250.0
15.0
60.2
70.0
145.0
56°
66°
50°
378.5
188.0
320.1
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600.0
369.4
55°
66°
50°
224.4
15.0
60.0
70.0
145.0
280.0
399.4
126.0
100.8
195.0
230.7
188.0
69.7
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Versione finale
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CONSTRUCTABILITY (1)
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260
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STRUCTURAL MODELING (model #1)
Strut & Tie modeling of the stayed bracket
STEP #1 STEP #2
STEP #3 STEP #4

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Alternative S&T modeling of the stayed bracket
STEP #1
STEP #3 STEP #4
STEP #2

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Alternative S&T modeling of the stayed bracket
STEP #1
STEP #3 STEP #4
STEP #2

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STRUCTURAL MODELING
OF CONCRETE PART (I):
trusswork discretization
ab
l
sl
A
A
a
b
sa
A
A
b
a
sb
A
A
b
a
b
a
b
a
d
d
y
y
x
x
2
2
2
2
2
8
3
2
3
8
3
2
3
8
3
2
,
,
,
=
=
=


































=
−
=
−
=

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4
3
2
1
,
,
, u
u
u
u
VI
V
IV
III
II
I
N
N
N
N
N
N ,
,
,
,
,
a
x
y
u
x

 


=
b
y
y
v
y

 


=
a
b
x
v
y
u y
x


 +



+


=
b
l
a
N
N
N
N
N
VI
V
II
I
x

+
+
+
=












a
l
b
N
N
N
N
N
VI
V
IV
III
y

+
+
+
=












l
N
N
N VI
V
xy
−
=
xy
y
x N
N
N ,
,
STRUCTURAL MODELING
OF CONCRETE PART (II):
stress representation

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LOADING SYSTEMS: SYM.
Reinforcment
Bars
Vsd
C + Steel
C
Steel
Vsd
Vsd

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LOADING SYSTEMS: ASYM.
Reinforcment
Bars
C + Steel
C
Steel
Vsd Vsd

Model S&T #1
Results for SYM
loading system

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Vsd = 1050 kN – cap element stress
• max tension = 389,7 MPa
• min compression = -232,5 MPa
• tension = 582,7 MPa

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Vsd = 1050 kN – reinforcement bar stress
stirrups longitudinal

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Vsd = 1050 kN – concrete stress
• max tension = 0 MPa

Model S&T #1
Results for ASYM
loading system

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Model S&T #2
Results for SYM
loading system

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Model S&T #2
Results for ASYM
loading system

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Model S&T #3
Results for SYM
loading system

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FINITE ELEMENT
ANALYSIS BY
SUBSTRUCTING
TECHNIQUE AND S&T

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STRUCTURAL MODELING
Reinforcement

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STRUCTURAL MODELING: CAP

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RIGID
LINKS
BEAM ELEMENTS
STRUCTURAL MODELING: LINKS

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Reinforcement
Vsd Vsd
C + Steel
C
Steel
Vsd
SYMMETRIC CONFIGURATION

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Vsd = 1050 kN – cap element stress:
elastic analysis (stress X)
>290
<-290

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elastic analysis (stress Y)
>290
<-290

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>290
<-290
elastic analysis (Von Mises) (I)

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elastic analysis (Von Mises) (II)
>580
<-580

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concrete
Vsd = 1050 kN – ties and concrete stress

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SUMMARY OF RESULTS (SYM) Vsd= 1050 kN
SIMM Vsd= 1050 kN Limit
Model 1 substruct Design
SMAXBIEL [N/mm^2] 582,71 582,72 580
TENSION [kN] 696,1 696,1
SMAXTEL
(SMTEL_x)
[N/mm^2] 389,75 653,2 290
TENSION [kN] 423,2 388,07
only “substructured” SMTEL_y [N/mm^2] 291,5 290
only “model 1” SMINTEL [N/mm^2] -232,46 -290
only “substructured” SmTEL_x [N/mm^2] -530,4 -290
only “substructured” SmTEL_y [N/mm^2] -641,62 -290
SMAXSTAF [N/mm^2] 96,3 90,32 374
SMINSTAF [N/mm^2] -0,02 -6,93 - 374
SMAXLONG [N/mm^2] -52,93 -55,52 374
SMINLONG [N/mm^2] -59,16 -61,29 - 374
SMAXCLS [N/mm^2] 0 0 1,5
SMINCLS [N/mm^2] -17,72 -18,21 -28
Linear elastic Steel

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304
ELASTIC- PLASTIC MATERIAL LAW
WITH VON MISES CRITERION
62519
.
4
]
N/mm
[
10000
max
2
max
=
=


00138
.
0
]
N/mm
[
290 2
=
=
y
y


]
[N/mm
210000 2
0 =
E
*
100
/
1 0
1 E
E =


x10^(-3)
E0
E1
y

y

max


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>290
<-290
e-plastic analysis (stress X)

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306
>290
<-290
e-plastic analysis (stress Y)

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307
>290
<-290
e-plastic analysis (Von Mises) (I)

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>580
<-580
e-plastic analysis (Von Mises) (II)

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Vsd = 1050 kN – cap element strain:
e-plastic analysis (Von Mises strain)

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310

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concrete

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SUMMARY OF RESULTS (SYM) Vsd= 1050 kN
SIMM Vsd= 1050 kN Limit
Model elastic e-plastic Design
SMAXBIEL [N/mm^2] 582,72 582,72 580
TENSION [kN] 696,1 696,1
SMTEL_x [N/mm^2] 653,2 560 290
TENSION [kN] 388,07 371,09
SMTEL_y [N/mm^2] 291,5 324,26 290
SmTEL_x [N/mm^2] -530,4 -515,65 -290
SmTEL_y [N/mm^2] -641,62 -632,07 -290
SMAXSTAF [N/mm^2] 90,32 122,93 374
SMINSTAF [N/mm^2] -6,93 15,55 - 374
SMAXLONG [N/mm^2] -55,52 -42,81 374
SMINLONG [N/mm^2] -61,29 -54,89 - 374
SMAXCLS [N/mm^2] 0 0 1,5
SMINCLS [N/mm^2] -18,21 -19,77 -28
elastic steel
e-plastic steel

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-25
-20
-15
-10
-5
0
0 200 400 600 800 1000 1200
Load
Uy
Load application
Structural response (1)

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314
0,00
2,00
4,00
6,00
8,00
10,00
12,00
14,00
0 200 400 600 800 1000 1200
Load
Ux
Spigolo alto

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0,000
0,001
0,001
0,002
0,002
0,003
0 200 400 600 800 1000 1200
Load
Elastic
Strain_x
Centre of Diaphram
-0,010
0,000
0,010
0,020
0,030
0,040
0,050
0,060
0 200 400 600 800 1000 1200
Load
Plastic
Strain_x
Centre of Diaphram
0,000
0,010
0,020
0,030
0,040
0,050
0,060
0 200 400 600 800 1000 1200
Load
Total
Strain_x
Centre of Diaphram

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0
50
100
150
200
250
300
350
400
450
0,0000 0,0100 0,0200 0,0300 0,0400 0,0500 0,0600
Total Strain_x
Stress_x
Centre of Diaphram
0
50
100
150
200
250
300
350
400
450
0 200 400 600 800 1000 1200
Load
Stress_x
Centre of Diaphram
0,000
0,010
0,020
0,030
0,040
0,050
0,060
0 200 400 600 800 1000 1200
Load
Total
Strain_x
Centre of Diaphram

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C + Steel
C
Steel
Vsd
ASYMMETRIC CONFIGURATION
Reinforcement Bars
Vsd
e-plastic steel

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e-plastic analysis (stress X)
>290
<-290

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319
e-plastic analysis (stress Y)
>290
<-290

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>290
<-290
e-plastic analysis (Von Mises) (I)

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e-plastic analysis (Von Mises) (II)
>580
<-580

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322

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concrete

CONCLUSIONI
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Index
•Part 0 – BASIS
•Part I – Trusswork & Gridwork
•Part II – Strut & Tie Models
•Part III – Stringer Panel Method
•Part IV - Applications
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325

https://sites.google.com/a/uniroma1.it/francobontempi/
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326

https://www.youtube.com/channel/UCW3IyXTBJVIiS6OZeSdIN7g
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327

https://fr.linkedin.com/in/francobontempi
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