Finite element analyses for limit state design of concrete structures

127 Pages·2005·2 MB·English

Finite element analyses for limit state
design of concrete structures
Posibilities and limitations with SOLVIA
Masther’s Thesis in the International Master’s programme Structural Engineering
FERNANDO LOPEZ
FEJSAL AL-HAZAM
Department of Civil and Environmental Engineering
Division of Structural Engineering
Concrete Structures
CHALMERS UNIVERSITY OF TECHNOLOGY
Göteborg, Sweden 2005
Master’s Thesis 2005:17
MASTER’S THESIS 2005:17

Finite element analyses for limit state
design of concrete structures
Posibilities and limitations with SOLVIA

Master’s Thesis in the International Master’s programme Structural Engineering
FERNANDO LOPEZ
FEJSAL AL-HAZAM

Department of Civil and Environmental Engineering
Division of Structural Engineering
Concrete Structures
CHALMERS UNIVERSITY OF TECHNOLOGY
Göteborg, Sweden 2005
Finite element analyses for limit state design of concrete structures
Posibilities and limitations with SOLVIA
Master’s Thesis in the International Master’s programme Structural Engineering
FERNANDO LOPEZ
FEJSAL AL-HAZAM

©FERNANDO LOPEZ, FEJSAL AL-HAZAM 2005

Master’s Thesis 2005:17
Department of Civil and Environmental Engineering Concrete structures
Division of Structural Engineering
Concrete Structures
Chalmers University of Technology
SE-412 96 Göteborg
Sweden
Telephone: + 46 (0)31-772 1000

Cover:
The cracking and crushing behaviour of a simply supported reinforced concrete beam,
where a half of the beam were modelled, using symmetrical boundary conditions.

Name of the printers / Department of Civil and Environmental Engineering
Göteborg, Sweden 2005

Finite element analyses for limit state design of concrete structures
FERNANDO LOPEZ
FEJSAL AL-HAZAM
Master’s Thesis in the International Master’s programme Structural Engineering
Division of Structural Engineering
Concrete Structures
Chalmers University of Technology

ABSTRACT
During the last decades there have been many attempts at defining robust and efficient
models that can represent the cracking and crushing behavior of concrete structures.
Among the most popular ones are the models based on the so-called “smeared crack
approach”. The advantage here is that a crack solid is modeled as a continuum and
cracked material can be described with stress-strain relationship. The crack pattern
doesn’t need to be known in advance. All these features make the smeared crack
approach particularly well suited for implementation in finite element context.
The present study considers non-linear finite element analyses performed on five
different FE-models implemented in SOLVIA FE-system, using plane stress, solid
and truss elements. The models were created for three types of simply supported
reinforced concrete beams in order to examine shear failure. The focus was on the
following concrete parameters: the shear retention factor, β and the parameter which
defines the ultimate crack strain κ.
The FE-results for each model were compared with the experimental results in order
to evaluate the accuracy of the FE-analysis. From the comparison, it was found that,
all the models gave reasonably good numerical predictions and similar results (within
a reasonable range) to the results according to experimental and theoretical
investigations. A parametric investigation was also performed. The reason for the
parametric investigation was to examine how the ultimate loads, displacements and
stresses in stirrups vary with the different combinations of β and κ.
In order to get stable analysis and a good numerical prediction of the FE-results,
certain combination of concrete parameters β and κ was selected for each model. It
was concluded that the parameter β had to be chosen between 0.85 and 0.9, while κ
could be calculated from the fracture energy of concrete and increased with respect to
the tension stiffening effect. It was found that this parameter had to be multiplied with
factors within range 1.5-4.2, depending on the FE-model, in order to get stable
analysis with good prediction of the results.
In general, it was concluded that non-linear analysis performed in SOLVIA can be
used for limit state analysis of simply supported reinforced concrete beams, analyzed
for shear type of failure.
Key words: shear retention factor, ultimate crack strain, stress, finite element method,
non-linear FE-analysis, ultimate load, displacement, crack pattern,
ultimate failure capacity, fixed orthogonal cracks, smeared crack
approach.
CHALMERS Civil and Environmental Engineering, Master’s Thesis 2005:17 I
Contents
ABSTRACT I
CONTENTS II
PREFACE IV
NOTATIONS V
1 INTRODUCTION 1
1.1 Background 1
1.2 Purpose 1
2 LIMIT STATE DESIGN OF REINFORCED CONCRETE STRUCTURES 3
2.1 Non-linear behavior of concrete structures 3
2.1.1 Tension 5
2.1.2 Compression 8
2.1.3 Bending 9
2.1.4 Shear 10
3 CRACK MODELING OF CONCRETE STRUCTURES 18
3.1 Smeared crack approach 18
3.2 The shear retention factor 19
4 THE SOLVIA FE-SYSTEM 22
4.1 General 22
4.2 The concrete material model in SOLVIA 22
4.2.1 Stress strain relationship 23
4.2.2 Failure envelopes 27
4.3 Numerical iteration methods and convergence criteria 30
4.3.1 The BFGS method with line searches 30
4.3.2 Convergence criteria for the different methods 32
5 DESCRIPTION OF TESTED BEAMS 33
5.1 Test specimen 33
5.2 Material properties 35
5.2.1 Concrete 35
5.2.2 Reinforcing steel 36
5.3 Test performance 37
5.4 Results from the experiments 37
5.4.1 Type of failures 38
5.4.2 Loads and deflections 38
II CHALMERS Civil and Environmental Engineering, Master’s Thesis 2005:17
6 FINITE ELEMENT MODELING 39
6.1 General 39
6.2 Coordinate system 39
6.3 Geometry 39
6.4 Material models 39
6.5 Boundary conditions and loads 41
6.6 Mesh for model 1 41
6.7 Mesh for model 2 42
6.8 Mesh for model 3 43
6.9 Mesh for model 4 44
6.10 Mesh for model 5 46
7 FE-RESULTS 48
7.1 Model 1 49
7.1.1 Load-displacement relationship 49
7.1.2 Crack pattern 50
7.1.3 Stresses and strains in the concrete and reinforcement 52
7.2 Model 2 53
7.2.1 Load-displacement relationship 53
7.2.2 Crack pattern 55
7.2.3 Stresses and strains in the concrete and reinforcement 57
7.3 Model 3 58
7.3.1 Load-displacement relationship 58
7.3.2 Crack pattern 60
7.3.3 Stresses and strains in the concrete and reinforcement 62
7.4 Model 4 63
7.4.1 Load-displacement relationship 63
7.4.2 Crack pattern 65
7.4.3 Stresses and strains in the concrete and reinforcement 68
7.5 Model 5 69
7.5.1 Load-displacement relationship 69
7.5.2 Crack pattern 71
7.5.3 Stresses and strains in the concrete and reinforcement 74
7.6 Evaluation of the FE results 75
8 PARAMETRIC INVESTIGATIONS 77
9 CONCLUSIONS 83
10 REFERENCES 85
APPENDIX 88
CHALMERS Civil and Environmental Engineering, Master’s Thesis 2005:17 III
Preface
This master thesis was carried out at Vägverket Konsult as a final project of the
Master program in Structural Engineering at Chalmers University of Technology,
Gothenburg, Sweden.
The examiner and supervisor was Mario Plos from the Department of Structural
Engineering at Chalmers University of Technology who supervised and examined us.
We are thankful for his guidance during the work with this project.
We would like to express our most sincere gratefulness to our supervisor at Vägverket
Konsult, Costin Pacoste for his devoted guidance, support and outstanding
encouragement throughout the development of this project.
We are also very thankful to Mikael Hermansson, for his constant contribution, co-
operation and help in the use of the FE software.
We wish to thank Mats Carlson for allowing us to work at the company and Roland
Olsson who provided the working place and all necessary equipment.
We are also very grateful to Björn Engström professor from the Department of
Structural Engineering at Chalmers University of Technology for providing us with
well documented experimental results needed partly as a base for the project.
Finally, we wish to thank our families, friends and people at the company who
supported us during the project work period.
Gothenburg, February 2005.
Fernando López.
Fejsal Al-Hazam.
IV CHALMERS Civil and Environmental Engineering, Master’s Thesis 2005:17
Notations
Roman upper case letters
A Effective tension area.
ef
A Area of longitudinal tensile reinforcing steel.
s
A ’ Area of longitudinal compressive reinforcing steel.
s
A Area of the least reinforcement in the tensile zone extending
s0
not less than l + d.
b,net
A Cross sectional area of shear reinforcement.
sv
C Compressive force in the concrete cross section.

Incremental stiffness matrix following a tensile failure.
Constant damping matrix
E Modulus of elasticity of concrete.
c
E Tangent modulus of elasticity at zero strain of concrete.
o
E Modulus of elasticity of tensile reinforcing steel.
s
E ’ Modulus of elasticity of compressive steel reinforcement.
s
E Strain hardening modulus of reinforcing steel.
sh
E Modulus of elasticity of steel plates.
sp
E Tangent modulus at which stresses normal to a crack plane
t
cease.
F Force.

Nodal point force vector equivalent to the element stresses.
F Force carried by the concrete.
c
F Force carried by the tensile reinforcing steel.
s
F ’ Force carried by the compressive reinforcing steel.
s
G Shear modulus of elasticity of concrete.
G Fracture energy.
f
K Tangent stiffness matrix based on the iteration history.

L Length.

M Constant mass matrix.

M Nominal flexural capacity.
n
M Moment due to factored loads.
u
P Unfactored concentrated load.

P* Calculated ultimate load.
P Ultimate finite element load.
a
P Diagonal tension crack load.
cr
P Calculated ultimate load governed by flexure.
f
P Ultimate test load.
u
P Calculated ultimate load governed by shear.
v
R External load vector.
RMNORM Reference moment.
RNORM Reference force.
RTOL Relative force and moment tolerance used to measure
equilibrium iteration convergence.
STOL Tolerance used to measure line search convergence.
U Vector of nodal point displacements.
U& Nodal point velocity vector.
U&& Nodal point acceleration vector.
CHALMERS Civil and Environmental Engineering, Master’s Thesis 2005:17 V
V Interface shear transfer by aggregate interlock along a diagonal
a
crack.
V Vertical component of the shear transferred across a crack.
ay
V Shear resistance of concrete without stirrups.
c
V Shear transfer in the concrete compression zone.
cz
V Shear transfer by dowel action of the longitudinal tensile
d
reinforcement.
V Shear capacity of a section with shear reinforcement.
R
V Nominal shear force carried by shear reinforcement.
s
V Shear force due to factored loads.
u
Roman lower case letters
a Shear span, distance between concentrated load and face
support.
a Level of tensile steel centroid.
tp
b Width of the cross section.
w
c Concrete cover.
d Distance from the extreme top fiber to the centroid of tension
reinforcement.
d’ Distance from the extreme top fiber to the centroid of
compression reinforcement.
d Coefficient which takes into account the concrete cover and the
ef
bar size on the effective concrete area.
d Maximum aggregate size of concrete.
max
f( ) Function of.
f Compressive strength of concrete.
c
f Ultimate compressive strength of concrete.
cu
f Yield strength of concrete.
cy
f Splitting tensile strength of concrete.
t,sp
f Tensile strength of concrete.
t
f Modulus of rupture of concrete.
t,fl
f Ultimate strength of reinforcing steel.
u
f Basic shear strength of members without shear reinforcement.
v
f Yield strength of tensile steel reinforcement.
y
f ’ Yield strength of compressive steel reinforcement.
y
f Yield strength of shear reinforcement.
yv
h Height.
i Iteration.
l Length.
l Required anchorage length.
b,net
l Clear beam span.
c
s Spacing of stirrups measured along the longitudinal
reinforcement.
s Average crack spacing
rm
t Time.
w Crack displacement.
w Ultimate crack displacement.
u
x Depth of the neutral axis.
VI CHALMERS Civil and Environmental Engineering, Master’s Thesis 2005:17