Fiber Modelling of Structural Walls dominated by Shear

Fiber Modeling of Shear Walls using OPENSees Omar HayatID: 108166FOS: Structural EngineeringIntroduction1MVLEM2Analytical models3Objectives4OPENSees56Conclusions7Recommendations8Table of contentsProposed formulation for Implementation of relation in MVLEM2Chapter 1

Structural wall Classification

INTRODUCTIONOne group refers to procedure where separate formulation for shear and flexure and then superimposed, shear mechanism represented by strut and tie model.Classification of Fiber ModelsSecond group modeling techniques focus on capturing the mechanics of section and using suitable constitutive relations for stress-strain behavior of steel and concrete fibers.

.Multi Vertical Line Element Model (MVLEM)

METHODOLOGYLITERATURE REVIEWMVLEM Base ModelA 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.

MVLEM with RC Panel BehaviorMETHODOLOGY

MVLEM with RC Panel

J. W. Wallace and L.M. Massone, 2006MVLEM with RC Panel BehaviorPanel/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 BehaviorThis 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.Allows crack shear slip in the formulation of element deformation.Crack reinforced is treated as continuous material and the constitutive laws for concrete and steel are expressed as average stresses and strain.Smeared crack conceptDiscrete stress field Constitutive Relations for concrete and steelCFTMCFTRA-STMFASTMUnable to take into account tension stiffening of concrete as concrete tensile stress was assumed zero A relationship for concrete in tension was proposed Able to predict the contribution of concrete , constitutive law to relate concrete shear stress to concrete strain was established.Smeared average stress strain curve embedded in concrete was derived but unable to account for crack shear slips , ignored concrete contribution SMMPoisson effect of cracked reinforced concrete was incorporated and is able to successfully predict the entire monotonic response curve including post peak.Constitutive Relations (Monotonic) RA-STMThis 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.

, t = applied normal stresses in the and t directions2c, 1c = average normal stresses of concrete in the 2 and 1 directions2,1 = average normal strains in the l and t directions, t = mild steel ratios in the l and t directions.

Cyclic Softened Membrane Model (CSMM)Pinching MechanismCyclic shear LoadingDamage coefficient in concrete and Bauschinger effect Failure MechanismImprovement in SMM

Typical features related with shear walls

Reinforcement in longitudinal and vertical directionLap splice/ Bond interfaceConfinement EffectShear-flexure interactionPinching effectShear Span ratioFailure mechanismINTRODUCTIONObjectivesMain ObjectiveTo get a more realistic analytical model (MVLEM) for nonlinear shear response of walls Objective 1To improve the MVLE Model for more realistic shear behavior and its interaction with flexure by incorporating new constitutive model

Objective 2Inclusion of cyclic shear in established MVLEM in OPENSees

Objective 3To include important parameters related to shear such as pinching effect , damage factor, stiffness degradation, cracks opening and closure Scope of Study1This study is carried out considering structural walls dominated by shear behavior2Many 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 envelopStage T1-T2

Stage T3 and T4When 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 envelopStage C1-C2

Where c = average stress of concrete c = average strain of concrete0 = peak compressive strain at max compressive strengthThe softening coefficent is given by:

Where k= constant taken for loadings.Stage C3-C7C3 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 steelSolid lines (cyclic stress strain)Dotted lines (Monotonic stress strain)Stage 1: Pre yieldStage 2T: Yield in tensionStage 2C: Yield in compressonStages 3 and 4: (Unloading and Reloading curves)

OPENSeesOPENSees 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 OPENSeesDomainModel BuilderAnalysisRecorderModel Builder: For building the objects in the model and adding them to domainDomain: For storing the objects created by Model Builder object and for providing analysis and recorder object the access to these objectsAnalysis 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

OverviewProposed Frame work to incorporate Cyclic constitutive relation into MVLEMA 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.

1. Selection of plane element and coordinate system

I. In constructing a Stiffness matrix K for an individual element, a material stiffness [D] is required to relate the stresses () to the strains ().

2. Material Stiffness formulationIII. 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.

II. The average stress-strain relationships for concrete and steel at element level should capture the load-deformations characteristic for entire wall.Equilibrium and Compatibility equationsThe equilibrium equation that relates the applied stresses in the global co ordinate (x , y, xy ) to the internal concrete stresses (1c , 2c , 12c ) and the steel bar stresses (fsi) is given by

Where si is the steel ratio in ith direction and T[- ] are the transformation matrices.

The compatibility equations which defines the relationship between steel strains (si) and concrete strains is represented as:

abTo 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)

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.Where 1 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.For forming the element stiffness matrix [K]e constitutive matrix [D] is evaluated in the tangent stiffness form is given as:

3. Evaluation of Material stiffness matrix

Model non linear material responseTo 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 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 ().

I.The element stiffness matrix [K]e is evaluated using basic finite element procedure4. Element Stiffness MatrixThe 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.

5. Global Stiffness MatrixGlobal 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.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 nodesSelection of coordinate system to define local and global axisSolution procedure for determining concrete and steel stress vector [] and strain vector []Equilibrium satisfaction at element levelForm the element resisting forces Form the element stiffness matrix [K]eAssemble global stiffness matrix and global resisting forcesIterative solutionConvergence / Equilibrium CheckUsing Hsu/Zhu Matrix to obtain set of uniaxial strainYESNONOObtain tangential uniaxial constitutive matrix for rebar and concrete

Evaluation of the constitutive matrix

Results and discussionValidation of selected constitutive laws for RC Panel BehaviorThree panels are selected of same dimension with only varying reinforcement.Dimensions: 1.4m x 1.4m x 178mmPanel 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)PanelConcreteSteel Loading Patternfc' (ksi)o (in/in)fy, (ksi)l-directions(%)t-direction (%)l/t ratioA260.0021651.21.21Pure ShearA360.00194651.81.81A460.002265331Affect 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.

PanelConcreteSteel Loading Patternfc' (ksi)o (in/in)fy, (ksi)l/t ratioA260.002165451Cyclic shearA360.0019465701A460.002265901WALL 01 with varying steel orientationsShear strain history for all the panels

Panel A2 ( = 45) Panel A4 ( = 90) Panel A3 ( = 70) Comparison of shear stress-strain hysteritic for cyclic loadings. (Dotted line are from model and solid for experiment) Shear stress(MPA)Shear strainDiscussion on WALL 01In 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)Specimen ConcreteReinforcementfc' (Mpa)fy, GradeLongitudnal (at boundaries)Web ReinforcementSpacingWall 0227.660 (414 MPa)8 - # 3 , Ab = 71 mm2# 2 bar, Ab = 32 mm2189 mm center

Cyclic load pattern ( Drift ration on right side and top displacement in mm on left side)

Drifts (%)0.10%0.25%0.50%0.75%1%1.50%2%2.50%SpecimenDirectionTop displacement(mm)Average Stiffness (kN/m)Top displacementAverage StiffnessTop displacementAverage StiffnessTop displacementAverage StiffnessTop displacementAverage StiffnessTop displacementAverage StiffnessTop displacementAverage StiffnessTop displacementAverage StiffnessWall 02Positive2.933007.2311516.1289524.5250033.1207650.6135667.989086.2450Negative3.27.615.924.232.849.866.583.8

Modeling Wall 02 using 8 fiber elementsLateral Load applications and corresponding displacements and shear deformationsChapter 145

Lateral load versus lateral top displacement for WALL 02

Comparison of back bone curvesDiscussion on WALL 02The 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. SpecimenDimensionsSteel Ratio (%)Span RatioAxial Load(cm)web horizontal web vertical(kN)A 130 x 1800.250.250.690B170 x 1200.250.250.350C170 x 1700.570.611533

WALL 03 properties

ab

cLateral 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 ConclusionsThe 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,.

RecommendationsThis 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.

THANK YOU Analysis

OPENSees Domain

Node

Constraints

Material

Uniaxial Material

Load Pattern

Element

nDMaterial

Section

Cyclic concrete

Cyclic steel

Elastic Fber sectionFiber Section 2D