Omar Hayat
ID: 108166
FOS: Structural Engineering
Table of contents
1
Introduction
2
MVLEM
3
Analytical models
4
Objectives
5
OPENSees
6
7
8
Proposed formulation for Implementation of relation in
MVLEM
Conclusions
Recommendations
INTRODUCTION
Structural wall Classification
Classification of Fiber
Models
One group refers to
procedure where
separate formulation
for shear and flexure
and then
superimposed, shear
mechanism
represented by strut
and tie model.
.
Second group
modeling techniques
focus on capturing
the mechanics of
section and using
suitable constitutive
relations for stressstrain behavior of
steel and concrete
fibers.
METHODOLOGY
REVIEW
LITERATURE
Multi Vertical Line Element Model
(MVLEM)
MVLEM Base Model
 A single 2D MVLE is a fiber modeling technique for





modeling shear walls. (L.M.Massone 2004)
Flexure response is simulated by the series of uniaxial
elements (fibers).
The external elements represents the axial stiffness of the
boundary columns.
Stiffness and force displacement relations of fibers are
defined according to the constitutive models.
A horizontal spring placed at the height ch, accounts for
shear response of the wall. ( A tri-linear force displacement
backbone curved adapted for defining it).
Shear- Flexure interaction is not considered.
METHODOLOGY
MVLEM with RC Panel Behavior
MVLEM with RC Panel
J. W. Wallace and L.M. Massone, 2006
MVLEM with RC Panel Behavior
 Panel/Membrane element actions refers to uniform
normal and shear stresses applied in the in-plane
direction.
 This wall model involves modifying the base MVLEM
by assigning a shear spring for each uniaxial element.
 Each uniaxial element is then treated as a RC panel
element.
 Therefore, the interaction between flexure and shear is
incorporated at the uniaxial element (fiber) level.
MVLEM with RC Panel Behavior
 This model involves 2D RC panel element subjected to
the membrane actions.
 The stiffness and force-deformation properties of
panel elements are derived form the material
constitutive relation. i.e the stress-strain curves for
concrete and steel.
 This constitutive panel behavior is represented by
Rotating Angle S0ftened Truss Model (RA-STM).
 RA-STM was developed by Pang for treating RC Panel
response subjected to in plane shear.
Constitutive Relations for concrete
and steel
Smeared
crack
concept
Crack reinforced is
treated as
continuous material
and the constitutive
laws for concrete
and steel are
expressed as
average stresses and
strain.
Discrete
stress
field
Allows crack
shear slip in the
formulation of
element
deformation.
Constitutive Relations
(Monotonic)
CFT
Unable to take
into account
tension
stiffening of
concrete as
concrete
tensile stress
was assumed
zero
MCFT
A
relationship
for concrete
in tension
was
proposed
RA-STM
Smeared
average stress
strain curve
embedded in
concrete was
derived but
unable to
account for
crack shear
slips , ignored
concrete
contribution
FASTM
Able to predict
the
contribution of
concrete ,
constitutive law
to relate
concrete shear
stress to
concrete strain
was
established.
SMM
Poisson effect of
cracked
reinforced
concrete was
incorporated and
is able to
successfully
predict the entire
monotonic
response curve
including post
peak.
RA-STM
 This theory is developed for RC membrane elements
subjected to in-plane shear.
 It is based on the assumption that the angle of the
cracks in the post cracking concrete coincides with the
rotating angle of the applied stresses.
 This model does not consider the contribution of
concrete as it assumes that the concrete struts are
oriented in the direction of the post cracking principal
compressive stresses and does not allow shear stress
along assume crack direction.
σι, σt = applied normal stresses in the ι and t directions
σ2c, σ1c = average normal stresses of concrete in the 2 and 1 directions
ε2,ε1 = average normal strains in the l and t directions
ρι, ρt = mild steel ratios in the l and t directions.
Cyclic Softened Membrane Model (CSMM)
Improvement
in SMM
Pinching
Mechanism
Cyclic shear
Loading
Failure
Mechanism
Damage coefficient
in concrete and
Bauschinger effect
INTRODUCTION
Typical features related with shear walls
 Reinforcement in longitudinal and vertical






direction
Lap splice/ Bond interface
Confinement Effect
Shear-flexure interaction
Pinching effect
Shear Span ratio
Failure mechanism
Objectives
Objective 1
To improve the MVLE Model for more realistic shear
behavior and its interaction with flexure by incorporating
new constitutive model
Objective 2
Inclusion of cyclic shear in established MVLEM in
OPENSees
Objective 3
To include important parameters related to shear such as
pinching effect , damage factor, stiffness degradation, cracks
opening and closure
Main
Objective
To get a
more
realistic
analytical
model
(MVLEM) for
nonlinear
shear
response of
walls
Scope of Study
1
This study is carried out considering
structural walls dominated by shear
behavior
2
Many important features related with short walls
are included through using state of art
constitutive model which are ignored in most of
the analytical models used for modeling walls
Cyclic smeared stress –strain curve for concrete (Hsu, 2006)
Tensile stress-strain envelop
 Stage T1-T2
 Stage T3 and T4
 When load is reversed from tensile direction to compressive direction
at TB, then upto point TC it represent gap closure.
 T4 represents increase in concrete stiffness before complete closing of
crack, ends at TD.
Concrete stress-strain envelop
 Stage C1-C2



Where σc = average stress of concrete
εc = average strain of concrete
ε0 = peak compressive strain at max compressive strength
 The softening coefficent is given by:

Where k= constant taken for loadings.
 Stage C3-C7
 C3 and C4 are with provided with slopes of 80 and 20 percent of Ec.
 stage C5 a straight line is assumed from point CD to TB, if loading direction is
reversed compressive reloading will follow C6.
 If compression load continues to increase, response follow stage C7.
Cyclic smeared (average) stress-strain curve for steel Mansour et al (2005)
Analytical model for steel
Dotted lines (Monotonic stress –strain)
Stage 1: Pre yield
Solid lines (cyclic stress strain)
Stage 2T: Yield in
tension
Stage 2C: Yield in
compresson
Stages 3 and 4: (Unloading
and Reloading curves)
OPENSees
 OPENSees is a software framework for simulation
applications in earthquake engineering. It is not a
code.
 An open source software which has the potential for
community code for earthquake engineering.
 OPENSees.exe is extension of Tcl interpretor for finite
element analysis which uses this framework.
Main components of OPENSees
Model Builder
Domain
Analysis
Recorder
Model Builder: For building the objects in the model and adding them
to domain
Domain: For storing the objects created by Model Builder object and
for providing analysis and recorder object the access to these objects
Analysis object: performs the analysis, e.g static analysis, transient
analysis.
Recorder: it monitors the user defined parameters in the modeling
during analysis, e.g section force deformation.
OPENSees
Overview
OPENSees
Domain
Analysis
Load Pattern
Element
Node
Constraints
Material
Uniaxial Material
Cyclic concrete
nDMaterial
Cyclic steel
Section
Elastic Fber section
Fiber Section 2D
Proposed Frame work to incorporate Cyclic
constitutive relation into MVLEM
1. Selection of plane element and coordinate system
I.
II.
III.
A 2 Dimensional element with global
and element level reference Cartesian
co ordinate system is selected.
Global co ordinate is X-Y and for
element 1-2 would be employed in
the formulations.
The uniaxial RC panel needs to be
modified to accommodate the effects
of shear Modulus of concrete, Poisson
Effect, and Damage factor.
2. Material Stiffness formulation
I.
In constructing a Stiffness matrix K for an individual
element, a material stiffness [D] is required to relate the
stresses (σ) to the strains (ε).
II. The average stress-strain relationships for concrete and
steel at element level should capture the load-deformations
characteristic for entire wall.
III. To reflect the non linear behavior of RC, [D] is modified
according to the constitutive law.
It also depends on the type of stiffness modulli used.
Equilibrium and Compatibility equations
 The equilibrium equation that relates the applied stresses in the global co
ordinate (σx , σy, τxy ) to the internal concrete stresses (σ1c , σ2c , τ12c ) and the
steel bar stresses (fsi) is given by
a
 Where ρsi is the steel ratio in ith direction and T[- θ] are the transformation
matrices.
 The compatibility equations which defines the relationship between steel
strains (εsi) and concrete strains is represented as:
b
It should be noted that the steel strain and concrete strains are in equation b
are bi axial strains, which will then take into account the Poisson ratio effect
using Hsu/Zhu ratios for cracked concrete.
To obtain set of uniaxial strains needed to compute the uniaxial material tangent
moduli of concrete and steel , the matrix [V] is given below which obtains set of
uniaxial strain from set of biaxial strain (Hsu/ Zhu)
Where ν1 2 represents compression strain in 2 direction on
tenisle strain in 1 direction. Under cyclic loading this ratio is
defined as a linear function of tensile strain.
Once uniaxial strains are found using the above relations, equation
a can be then easily evaluated using the Uniaxial constitutive cyclic
constitutive relation.
3. Evaluation of Material stiffness matrix
I.
For forming the element stiffness matrix [K]e
constitutive matrix [D] is evaluated in the tangent
stiffness form is given as:
Model non linear material response
 To model nonlinear material response, the constitutive relations (that
are the stress-strain curves for concrete and steel for cyclic loadings)
which were discussed earlier are used.
 The material stiffness matrix D for the element is to be defined wrt
global axes, which is done by first defining a stiffness matrix for
concrete component and for each reinforced component.
 The total stiffness is then determined by combining the contribution
from each of components, using appropriate transformation to take
into account the anisotropy of the materials
4. Element Stiffness Matrix
I.
The element stiffness matrix [K]e is evaluated using basic finite
element procedure
II.
[B] is a matrix which depends on assumed element displacement
functions. In this case, the displacement field is arranged to be
consistent with finite element formulation, in terms of axial
displacement (u), total lateral displacement (w) section rotation (θ).
5. Global Stiffness Matrix
Global stiffness matrix [K] is formed by summing corresponding
matrices at element level and finally global equilibrium is
checked for overall wall model by comparing the applied and
resisting forces.
6.Check for equilibrium and convergence.
The element resisting forces are determined which is then
assembled to form global resisting force increment vector and an
iteration is formed till convergence is achieved and is checked for
equilibrium.
Through the iteration, the material stiffness [D] and element
stiffness matrices [K] are progressively refined until the
convergence is satisfied.
Selection of the
Element, forces [P]
and displacements at
nodes
Using Hsu/Zhu
Matrix to obtain
set of uniaxial
strain
Obtain tangential
uniaxial constitutive
matrix for rebar
and concrete
Selection of
coordinate system
to define local
and global axis
Solution procedure for
determining concrete
and steel stress vector
[σ] and strain vector [ε]
Equilibrium
satisfaction at
element level
Evaluation
of the
constitutive
matrix
Form the
element
stiffness
matrix [K]e
Iterative solution
Convergence /
Equilibrium Check
Form the
element
resisting
forces
Assemble global
stiffness matrix
and global
resisting forces
Results and discussion
 Validation of selected constitutive laws for RC Panel Behavior
 Three panels are selected of same dimension with only varying reinforcement.
 Dimensions: 1.4m x 1.4m x 178mm
 Panel subjected to the bi axial membrane stresses i.e. principal tensile and compressive
stresses of equal magnitude are applied in the vertical and horizontal directions.
 WALL 01 Properties ( Experimental work performed by Pang)
Concrete
Panel
Steel
fc' (ksi)
εo (in/in)
fy, (ksi)
l-directions(%)
t-direction (%)
l/t
ratio
A2
6
0.0021
65
1.2
1.2
1
A3
6
0.00194
65
1.8
1.8
1
A4
6
0.0022
65
3
3
1
Loading
Pattern
Pure Shear
Shear Stress (MPa)
Panel A4
8
Analysis
Test
4
0
0
0.01
0.02
12
Panel A3
8
Analysis
Test
4
0
0.03
0
Shear Strain
Shear Stress (MPa)
Shear Stress (MPa)
12
0.01
0.02
Shear Strain
12
Panel A2
8
Test
Analysis
4
0
0
0.01
0.02
Shear Strain
0.03
0.03
Comparison of Panels
12
Shear Stress (Mpa)
Panels
8
A4
A3
4
A2
0
0
0.01
0.02
Shear Strain
0.03

Affect of varying steel angle on cyclic response of wall 01
 The hysteresis of shear stress-strain curves to the experimental curves are compared.

It is noticed that the cyclic MVLEM can successfully predict the cyclic shear response of RC Panel elements
when the steel orientation to the principal stresses is varied from 45 to 90 degress.
WALL 01 with varying steel orientations
Concrete
Steel
Panel
fc' (ksi)
εo (in/in)
fy, (ksi)
α
A2
6
0.0021
65
45
1
A3
6
0.00194
65
70
1
A4
6
0.0022
65
90
1
l/t ratio Loading Pattern
Shear strain history for all the panels
Cyclic shear
Shear stress(MPA)
Shear strain
Panel A2 ( α = 45)
Panel A3 ( α = 70)
Panel A4 ( α = 90)
Comparison of shear stress-strain hysteritic for cyclic loadings.
(Dotted line are from model and solid for experiment)
Discussion on WALL 01
 In the first part, (subjected to pure shear), the response from the cyclic
MVLEM are agreeable with the experimental results.
 It can clearly show three three distinct stages that are elastic range, post
cracking and plastic stage.
 Panel A4 shows over reinforced state, while A2 under reinforced state.
 In the second part, (subjected to cyclic shear) the hysteresis obtained from
the model excellently shows the pinching mechanism which is the foremost
important feature in the shear dominated walls.
 It emphasizes the importance of the orientation of steel bar to the applied
stresses, as such the pinching mechanism is absent when the steel grid lines
are in 90 degrees with the applied principal stresses.
Slender Intermediate Wall (Wall 02)
Cyclic load pattern ( Drift ration on right side and
top displacement in mm on left side)
Concrete
Specimen
Wall 02
fc' (Mpa)
fy, Grade
27.6
60 (414 MPa)
Reinforcement
Longitudnal (at
Web
boundaries)
Reinforcement
8 - # 3 , Ab = 71
mm2
# 2 bar, Ab =
32 mm2
Spacing
189 mm
center
Lateral Load applications and corresponding displacements and shear
deformations
Drifts (%)
Speci
men
Wall
02
Direction
Positive
Negative
0.10%
0.25%
0.50%
0.75%
1%
1.50%
2%
2.50%
Top
displacemen
t(mm)
Averag
e
Stiffne
ss
(kN/m)
Top
displace
ment
Avera
ge
Stiffn
ess
Top
displace
ment
Avera
ge
Stiffn
ess
Top
displace
ment
Avera
ge
Stiffn
ess
Top
displace
ment
Avera
ge
Stiffn
ess
Top
displace
ment
Averag
e
Stiffnes
s
Top
displac
ement
Averag
e
Stiffnes
s
Top
displac
ement
Average
Stiffness
2.9
3.2
3300
7.2
7.6
311
5
16.1
15.9
289
5
24.5
24.2
250
0
33.1
32.8
207
6
50.6
49.8
1356
67.9
66.5
890
86.2
83.8
450
Modeling Wall 02 using 8 fiber elements
Lateral load versus lateral top displacement for WALL 02
Comparison of back bone curves
Discussion on WALL 02
 The cyclic MVLEM takes into account the lateral stiffness
of the wall, where the original MVLEM considering
monotonic response curve for cyclic loading results in over
estimating the actual response.
 The model is an excellent agreement with the experimental
results for lateral displacement versus lateral load plot.
 For higher lateral drifts, the model slightly underestimates
the lateral stiffness.
 The backbone curve obtained is well matching with the
experimental one.
Wall 03 (Squat walls)
 3 short walls are selected and their properties are shown in the table,
the experimental work for specimens is by Hidalgo et al (2002).
 Lateral load is applied at the mid height of the specimens A and B to
avoid the rotation that might occur.
 Lateral load is applied at the top of the wall for specimen C, and a
constant axial load is also applied.
 These wall are provided are meant to fail in shear by providing a
relatively large longitudnal reinforcement at wall boundaries.
WALL 03 properties
Dimensions
Specimen
Steel Ratio (%)
Axial Load
Span Ratio
(cm)
web
horizontal
web vertical
A
130 x 180
0.25
0.25
0.69
0
B
170 x 120
0.25
0.25
0.35
0
C
170 x 170
0.57
0.61
1
533
(kN)
a
b
c
Lateral load versus displacement response for wall span a.) 1.0, b.) 0.69 c.) 0.35
Comparison of affect of variation of reducing the span ratios
Conclusions
The model developed in the study provides an accurate prediction of
the membrane/panels behavior which is subjected to in plane shear
loadings, thus it verifies for predicting the RC panel response
subjected to reverse cyclic loading and justifies replacing RA-STM
used in original MVLEM for predicting panel response.
 The wall model predicts pinching mechanism which is a key feature in
shear dominant structures.
 In comparing the response of intermediate wall subjected to reverse
cyclic loading, it is clearly observed that the developed model
succeeds significantly in capturing the characteristics of the cyclic
wall responses.
 The ability of the developed model in tracking short span walls
response is noticeable, as it matches well with the experimental
results even for quite short span ratio of walls where the previous
model shows very poor results,.

Recommendations

This study should be extended to verify for different shapes of walls like
T-shape and tested under different loading conditions.

The formulation can also be extended for analysis of rectangular shape 3D
walls. Bond slip and buckling behavior of reinforced bars can be
incorporated in these cyclic constitutive laws.

More research should be carried out to follow the failure mechanism and
post yield behavior of walls using the mentioned constitutive relations.