Presentatie prof. Kaufmann op de IDEA Concrete infomiddag

Structural concrete design, dimensioning and detailing:
from truss models to computer-aided stress fields
Prof. Dr. Walter Kaufmann
ETH Zürich
Institute of Structural Engineering
04.06.2018 ETH Zürich | Prof. Dr. W. Kaufmann | Eurostars – DR-Design 1

Historical background of limit analysis methods
Truss models and stress fields
FE-calculations and existing computer-aided tools
DR-Design (ISD): Motivation and scope
DR-Design (ISD): Model description
DR-Design (ISD): Experimental validation
Conclusions
Structural concrete design, dimensioning and detailing:
from truss models to computer-aided stress fields

“Ancestors” of limit analysis methods – Yield line method
P. Marti et al., Aplication of yield line method (1999)A. Ingerslev «The Strength of Rectangular Slabs (1923)

“Ancestors” of limit analysis methods – Hillerborg’s strip method
H. Marcus «Die Theorie elastischer Gewebe …» (1924 / 1932) P. Marti et al., Application of Hillerborg’s Strip Method (1999)

“Ancestors” of limit analysis methods – Truss models and stress fields
E. Mörsch, «Der Eisenbetonbau» (1922)
K. W. Ritter, «Die Bauweise Hennebique» (1899)
Modern truss models and stress fields

Emil Mörsch
1872-1950
Karl Wilhelm Ritter
1847-1906

M. Ritter, «Massivbau» (ca. 1940) P. Lardy, «Massivbau» (1951)
Truss models regarded as
behavioural models
State of art: Design based on
semi-empirical models, e.g.
«admissible tensile stresses»
Situation until 1960s

M. Ritter, «Massivbau» (ca. 1940) P. Lardy, «Massivbau» (1951)
Emil Mörsch
1872-1950
Pierre Lardy
1903-1958
Max Ritter
1884-1946
Truss models regarded as
behavioural models
State of art: Design based on
semi-empirical models, e.g.
«admissible tensile stresses»
Situation until 1960s

Intrinsic problems of “admissible stress” design
Ernst Melan
1890-1963 Drawbacks of «admissible stress design»:
- Ultimate load cannot be reliably predicted (except for brittle materials)
even if stresses are accurately known → no uniform safety level
- Stresses cannot be «accurately» determined (restraint to imposed
deformations e.g. hydration, shrinkage; construction stages; …)
Ernst Melan (1938):
Since (…) typically, the sequence of loading is arbitrary, asking for the
state of stress under a certain load does not make sense.
(Translated from German: «Da (…) die Reihenfolge der Belastungen willkürlich
zu sein pflegt, hat die Frage nach einem Spannungszustand bei einer
bestimmten Belastung keinen Sinn»).

Limit analysis methods – Application to Structural Concrete
Intrinsic problems of admissible
stress design:
- initial stress state?
- safety level?
Truss models put on a consistent
mechanical basis by the Theory of
Plasticity [(Prager, Gvozdev).
Lower-bound theorem:
• Satisfy equilibrium and statical
boundary coditions
• Do not infringe yield condition
(provide required strength)
→ Safe design
→ Independent of initial stresses
( ) 0Ζ =
mΖ jσ
(S) 0=
iσ
kσ
εnΖ
z
Peter Marti
*1949
Bruno Thürlimann
1923-2008
upper bound solutions
(«failure mechanisms»)
possible range of ultimate load
lower bound solutions
(«equilibrium methods»)
P
(Among other pioneers like e.g.
D.C. Drucker, W.F. Chen,
M.P. Nielsen, M. Braestrup,
D.H. Clyde, C.T. Morley,
P. Müller, J. Witteveen ,…)

Stoffel / Marti
(1995)
Sigrist / Marti
(1992)
Kaufmann / Marti
(1995)
Bachmann / Thürlimann
(1965)
Maier / Thürlimann
(1985)
Limit analysis methods – Validation by large scale experiments
Large-scale testing indispensable
for validaton and acceptance of limit
analysis methods

J. Schlaich et al., «Toward a Consistent Design of Structural Concrete» (1987)
Jörg Schlaich
* 1934
(Among many others like e.g.
K. Schäfer, J.G. McGregor, …)
Strut-and-tie models
(«Stabwerkmodelle»):
Used for tracing the flow of
forces and form finding (often
based on elastic principal
stress trajectories, and
combined with graphic statics)
Mechanical basis?
Code compliant?
Behavioural models /
«Practitioner method»?

J. Schlaich et al., «Toward a Consistent Design of Structural Concrete» (1987)
Jörg Schlaich
* 1934
(Among many others like e.g.
K. Schäfer, J.G. McGregor, …)
Strut-and-tie models
(«Stabwerkmodelle»):
Used for tracing the flow of
forces and form finding (often
based on elastic principal
stress trajectories, and
combined with graphic statics)
Mechanical background:
Limit analysis metods!

Conclusions
Computer-aided stress field analysis of
discontinuity concrete regions

Modern truss models and stress fields
consistent
mechanical
basis:
lower-bound
theorem of
limit analysis

Code provisions based on limit analysis in current EN 1992-1-1: Shear design

Code provisions based on limit analysis in current EN 1992-1-1: Horizontal shear and torsion

[Tjhin & Kuchma, 2002]

Code provisions based on limit analysis in current EN 1992-1-1: Strut and tie models

x
supF
z
infF−
supF centred
non-centred
wfO
x
z
infF−
supF centred
non-centred
wfO
supF
q q

Anchor force
NB:
= stirrup forces
+ applied load
Anchor force
[Marti & Stoffel, 1999]
 Flow of forces (transparency)
 Safe dimensioning
 Consistent detailing
 Tedious hand calculations
(iterations, many load cases)
 Even more so in assessment
 Compressive strength fc?
(depending on strain state)
 Deformation capacity?
 Serviceability checks
(deformations, crack widths)?
→ FE-calculations used in
engineering practice
→ Future of truss models?
→ Digitalisation required!
(computer-aided tools)
Design of Discontinuity: classic tools

Conclusions

FE-calculations
[Cervenka Consulting / ATENA]

FE-calculations
Linear elastic FE calculations
• Equilibrium is satisfied (application of lower bound theorem of imit analysis) → ok, but …
• Do not capture real behaviour (restraint stresses, cracking, redistribution, staged construction, …)
• Non-symmetric strength of concrete only accounted for in last step (dimensioning based on yield conditions)
• Often inefficient and / or unpractical reinforcement layouts; fc must be assumed (“safe value”)
→ Useful in design, but unable to predict realistic capacity in existing structures, nor cracking in new ones
1
sx sx x xz
sz sz z xz
a f n k n
a f n k n−
≥ +
≥ +
( )c sx sx sz sz x zhf a f a f n n≥ + − +
Direct reinforcement design
(yield regime 1):
Valid if concrete does not crush, i.e.:

FE-calculations
• Often inefficient and / or unpractical reinforcement layouts
Nonlinear FE calculations
• Capture real behaviour if correct mechanical models and material parameters are input
• Require expert users, modelling and analysis time consuming
• Input of (often many!) material parameters unknown at design stage required
• Results non-transparent and sensitive to choice of mechanical model and material parameters (often ± arbitrary in design)
→ Limited use, mainly in assessment of existing structures and research

FE-calculations
[Cervenka Consulting / ATENA]

FE-calculations
 Powerful tools with lots of
possibilities
 Account for material and
geometrical nonlinearities
 Capture “real” behaviour
if right parameters are used
 ULS and SLS results provided
 Expert users required
 Many input parameters are
unknown in design stage
 Highly sensitive to seemingly
unimportant input parameters
 fct directly contributes to
resistance in many cases
(fct > 0 for numerical stability)
→ Code compliant, safe design?
→ Useful for design stage?[Cervenka Consulting / ATENA]

FE-calculations
Nonlinear FE calculations
• Capture real behaviour if correct mechanical models and material parameters are input
• Require expert users, modelling and analysis time consuming
• Input of (often many!) material parameters unknown at design stage required
• Results non-transparent and sensitive to choice of mechanical model and material parameters (often ± arbitrary in design)
→ Limited use, mainly in assessment of existing structures and research
Alternative: Computer-aided truss models / stress fields (simplified nonlinear FE calculations)
• Not a new idea: «It is time to bring these methods from the drawing board to the computer» [Marti 1985]
• Surprising that no such tools were available until recently

Existing computer-aided tools
Design of Discontinuity Regions: Existing computer-aided tools
• AStruTie (HanGil)
[HanGil, 2017]
Idea StatiCa for specific details
(corbels, piles caps…)
AStrutTie (HanGil)
(strut-and-tie → fc=? Realistic results?)
[IDEA, 2017]
CAST (Tjhin & Kutchma, 2002)
(strut-and-tie → fc=? Realistic results?)
[Mata-Falcón & Sánchez-Sevilla, 2006]

Spannungsfelder
Stringer-Panel Models (Nielsen, 1971; Blaauwendraad & Hoogenboom, 1996; Marti & Heinzmann, 2012)
[Blauwendraad, 2006]

Spannungsfelder
Experimental
crack pattern
Hand-calculated
stress fields
Numerical
results EPSF
[Mata-Falcón, 2015]
[Mata-Falcón et al., 2014]
[Muttoni & Fernandez Ruiz, 2007]
EPSF elastic plastic stress fields (Fernández Ruiz & Muttoni, 2007)
 Maintains advantages of hand
calculations (transparent, safe
design with fct = 0, consistent
detailing)
 Compressive strength fc
determined automatically from
strain state
 Limited user-friendliness
 Limited use for serviceability
… no tension stiffening
… no crack width calculation
 No check of deformation
capacity (perfectly plastic
material)

Conclusions

DR-Design (Discontinuity Region Design)
Scope
• Simple method for efficient, code-compliant design and assessment of discontinuity concrete regions
• Including serviceability and deformation capacity verifications
• Direct link to conventional RC design: concrete tensile strength ONLY for stiffness, standard material properties
Inspirations
• EPSF finite-element implementation (strain compatibility, automatic determination of kc from strain state)
• Tension Chord Model TCM and Cracked Membrane Model CMM (tension stiffening, ductility and serviceability checks)
Features of DR-Design
• Maintains advantages of truss models and stress field design: Tensile strength of concrete does not contribute to strength!
• Simple uniaxial constitutive laws for reinforcement and concrete in compression
• Satisfies strain compatibility, accounting for tension stiffening
• Covers all verifications typically required in design (ULS, SLS including crack widths)
• Implemented in user-friendly FE-based software package IDEA StatiCa>Detail.
• Checks deformation capacity (explicit strain limitations of concrete and reinforcement)

DR-Design (Discontinuity Region Design)
Scope
• Simple method for efficient, code-compliant design and assessment of discontinuity concrete regions
• Including serviceability and deformation capacity verifications
• Direct link to conventional RC design: concrete tensile strength ONLY for stiffness, standard material properties
Inspirations
• EPSF finite-element implementation (strain compatibility, automatic determination of kc from strain state)
• Tension Chord Model TCM and Cracked Membrane Model CMM (tension stiffening, ductility and serviceability checks)
Development / Credits
This project has received partial funding from Eurostars-2
joint programme, with co-funding from the European Union
Horizon 2020 research and innovation programme

Conclusions

Model description
DRD verification model: main assumptions
based on [Kaufmann and Marti, 1998]
Main assumptions:
• Fictitious rotating,
stress-free cracks
(σc1,r=0) without slip
• Average strains
• Equilibrium at cracks:
i. Maximum stresses:
-σc3,r / σs,r
ii. Concrete tensile
strength neglected
except for tension-
stiffening: εm
Suitable for elements with minimum transversal reinforcement. Slender elements without shear reinforcement would
lead to conservative results.

Model description
DRD verification model: concrete
 Strain limitations of concrete specified by codes
(explicitly considers the increasing brittleness of
concrete with strength).
 Imposed to the average strain over a characteristic
crushing band length.
 kc discrete values for hand calculations

Model description
 kc (compression softening) automatically computed based
on the transversal strain state.
 Use of fib MC 2010 proposal for shear verifications
(consistent with considered max. stresses) extended for
general cases.
(standard user:
only kc currently
used, not εcu)

Model description
EN 1992-1-1, 6.5. Design with strut and tie models
6.5.2 (2): The design strength for concrete struts should
be reduced in cracked compression zones and, unless
a more rigorous approach is used, may be calculated
from Expression (6.56).
(standard user:
only kc currently
used, not εcu)

Model description
DRD verification model: bond and reinforcement
Bond model used exclusively for
verifications of gradients of
tension chord force
Tension-stiffening:
 Does not affect the
strength of the
reinforcement
 Increases the stiffness
 Reduces the ductility
(can reduce the strength
of the member)
explicit failure
criterion (rupture) *Bilinear naked steel input for design. More
realistic laws for assessment &
experimental validation.

Model description
DRD verification model: tension stiffening
Stabilised crack pattern
 Implementation of
Tension Chord Model
(TCM) [Alvarez, 1998;
Marti et al., 1998]
 Average crack spacing:
assumed λ=0.67
for ρ>ρcr≈0.6%  Reinforcement is able to
carry the cracking load without yielding 0
1
1sr y ctm
cr
f f n
 
σ = = + − 
ρ 

Model description
Non-stabilised crack pattern
for ρ<ρcr≈0.6%  Reinforcement is NOT able to carry the cracking load without
yielding. Cracks are controlled by other reinforcement.
 Independent cracks are
assumed + bond model of
Tension Chord Model.
 Crack localization (size
effect): stiffness of the
whole rebar embedded in
concrete > local stiffness
near the crack
→ considered strain:
average over lavg = length at
which rebar is fully anchored
(for ft )
Will be
released for
stirrups in
ISD 9.1

Model description
Resulting tension chord behaviour
 Fully cracked behaviour
considered for design.
 Uncracked initial stiffness
can be considered for
refined verification
models.

Model description
DRD verification model: effective area of concrete in tension
→ suitable for numerical implementation and valid for automatic definition of ρc,eff in any region
Maximum concrete area each
rebar can activate (concrete at fct)
(illustrated for rebars 3 and 4) Areas used in calculation

Model description
DRD verification model: crack width – stabilised crack pattern
WT4
[Walther, 1967]

Model description
DRD verification model: crack width – non-stabilised crack pattern
[Zhu et al., 2003]
Assumed independent cracks at SLS Considered for:
a) Regions with ρ < 0.6% (ρmin)
b) Cracks triggered by geometric
discontinuities at low loads
T6Will be
released in
ISD 9.1

Model description
DRD verification model: crack width – crack kinematics
s

Conclusions

Experimental validation
DRD experimental validation
• Direct tension experiments – Alvarez and Marti (1996)
 Ultimate limit state
 Load deformation behaviour
 Crack width
• Pure bending experiments – Frantz and Breen (1978)
 Crack width distribution
• Cantilever shear walls – Bimschas, Hannewald and Dazio (2010, 2013)
 Load deformation behaviour under combined loading
 Bearing capacity under combined loading
• Beams with low amount of transversal reinforcement – Huber, Huber and Kolleger (2016)
 Bearing capacity in shear (failures due to insufficient ductility of the transversal reinforcement)

Alvarez and Marti (1996) - experimental setup/specimens
[Avarez and Marti, 1996]
Z1 Z1
Specimen Z1 Z2 Z4 Z8
Long.
reinforcement
14xØ14
(ρ = 1%)
14xØ14
(ρ = 1%)
14xØ14
(ρ = 1%)
10xØ14
(ρ = 0.7%)
Steel quality
(ductility class)
High High Normal High
fck_cube (MPa) 50 90 50 50
Loading: pure tension

Alvarez and Marti (1996) - ultimate state
[Avarez and Marti, 1996]
Specimen Z1 Z2 Z4 Z8
Experiment
Vexp (kN)
εm,exp (%)
1294
6.7
1295
6.8
1275
0.6
924
6.4
DR-Design
Vcalc (kN)
εm,calc (%)
1275
7.0
1282
4.6
1242
0.4
918
6.5
Safety factor
Strength: Vexp/Vcalc
Deform. capacity: εm,exp/εm,calc
1.01
0.96
1.01
1.48
1.03
1.50
1.01
0.98
V: Peak load
εm: Average tensile strain

Alvarez and Marti (1996)
Load deformation behaviour
Neglecting tension-stiffening
overestimates the deformation
capacity up to 5 times
(depending on ρ, the ductility of
the reinforcement…)

Alvarez and Marti (1996) -
crack width

Frantz and Breen (1980) - experimental setup/specimen
Specimen RS-3
Main
reinforcement
2xØ15.88
6xØ12.7
Web
reinforcement
6xØ6
Loading: pure bending
[Frantz and Breen, 1980]
d (mm)
885 mm

Frantz and Breen (1980) – crack width

Bimschas et al. (2010, 2013) – experimental setup/specimens
VK1: first yielding of
reinforcement [Bimschas, 2010]
1370 kN
±V
Specimen VK1 VK3 VK6
Effective height
(m)
3.30 3.30 4.50
Section depth (m) 1.50 1.50 1.50
Section width (m) 0.35 0.35 0.35
ρsl (%) 0.82 1.23 1.23
ρst (%) 0.08 0.08 0.08
Loading: constant normal force N = -1370kN; quasi-static cyclic
loading with increasing amplitudes in horizontal direction.
Note: DR-Design aims to describe the backbone of the cyclic
response using a monotonic model. Strain penetration into the
foundation is not considered.
εu=8.4%

Bimschas et al. (2010, 2013) – peak load
[Bimschas, 2010]
VK1: peak strength VK1: failure
Concrete
crushing in
compression
Specimen VK1 VK3 VK6
Experiment*
Vexp (kN)
728 876 647
DR-Design
Vcalc(kN)
730 860 650
Vexp/Vcalc 1.00 1.02 1.00
Note: DR-D aims to describe the behaviour
of the backbone until concrete peak
horizontal strength is reached, (≠ to loss of
vertical bearing capacity).
*mean peak horizontal load of North and
South directions.

Bimschas et al. (2010, 2013) – load deformation behaviour
Failure mode: concrete crushing in compression. Failure is considered when the strain limit criteria specified in codes for sectional
analysis is reached on average over the crushing band length.

Bimschas et al. (2010, 2013) – stress fields specimen VK1
Note: Refined analysis considers the initial uncracked stiffness, as well as the actual stress-strain relationship of the
reinforcement. Moreover, no concrete strain limitation is considered.

DRD experimental validation: Bimschas et al. (2010, 2013) – load deformation behaviour
[%]
σsr/ft
σc3r/(fc·kc)
σsr>fy
1370 kN
250 kN
84º
1370 kN
500 kN
80º
1370 kN
750 kN
79º
σsr<0

Huber et al. (2016) – experimental setup/specimens
Øw
(mm)
fy
(MPa)
ft
(MPa)
εu
(%)
4 653 710 4.9
6 569 658 3.1
12 552 654 3.4
Specimen R1000m35 R1000m60 R500m352 R500m351
Section depth 1.00 m 1.00 m 0.50 m 0.50 m
Section width 0.30 m 0.30 m 0.15 m 0.15 m
ρw 0.094 % 0.094 % 0.084 % 0.094 %
Øw Ø6 Ø12 Ø4 Ø6
fc 29.6 MPa 60.9 MPa 35.9 MPa 37.9 MPa

Huber et al. (2016) – ultimate load
• Neglecting tension
stiffening leads to
unsafe load predictions
and does not capture
the real failure mode
(stirrup rupture).
• Higher impact of strain
localization in real size
elements  use of
existing experimental
databases could
underestimate the
impact of these failures.
Cold-formed steel with same ft & fy  less ductile & less
predicted load (≈10%) than standard bilinear steel law.

Huber et al. (2016) – stress fields specimen R1000m35
776 kN
θ=40.5º
937 kNStirrups
yielding
θ=36.5º
εz=20‰
σsrz=600 MPa<ft
ε1=23‰  kc=0.41
σc3r=12 MPa
σc3r/(fc·kc)=1.00
εz=5.4‰
σsrz=638 MPa=ft
ε1=6.4‰  kc=0.64
σc3r=7.7 MPa
σc3r/(fc·kc)=0.42
*Results at the most restrictive
concrete and steel finite elements
(minimum kc & maximum σsrz)
DRD (No tens.-stiff.)
DRD
[%]
σsr/ft
σc3r/(fc·kc)
σsr>fy

Huber et al. (2016) – shear concrete crushing verifications
 Is there a clear link to kc prescribed for hand
calculations? Impact of strain localization?
[SIA 262:213;
fib MC 2010]
Concrete crushing
 Tension-stiffening required to capture the failure
mode.

Conclusions

Conclusions
Why use truss models and stress fields (and limit analysis methods in general)
• Powerful tools for the design of concrete structures.
• They are transparent, allow to trace the of flow of forces and give the engineer full control over the design.
Future of truss models and stress fields
• Due to their drawbacks (time-consuming, not useful for SLS) these methods will not survive as hand calculations.
• They need to be implemented in user-friendly computer programs, but maintaining their advantages.
• The DRD-method, implemented in the program Idea Statica Detail, developed jointly by ETH Zürich and Idea-RS, will
hopefully contribute to the survival of these methods in structural concrete design.

Acknowledgements
DR-DESIGN Project Team
Funding:

Thank you for the attention

Presentatie prof. Kaufmann op de IDEA Concrete infomiddag

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