Applied Element Method (AEM) in Dynamic and Seismic ... · PDF fileDynamic and Seismic Analysis Earthquake Engineering and Engineering Seismology ... in Dynamic and Seismic Analysis

IntroductionOverview of the AEM

Dynamic analysis with the AEM

Applied Element Method (AEM) inDynamic and Seismic Analysis

Earthquake Engineering and Engineering Seismology

Prof dr Stanko Brčić, Mr Mladen Ćosić

Faculty of Civil EngineeringUniversity of Belgrade

Borsko Jezero, May 19-22, 2014.

S.Brčić, M.Ćosić Applied Element Method



Numerical methods - FEM/AEMComparrison: FEM - AEMApplicability of the AEM

Contents

1 IntroductionNumerical methods - FEM/AEMComparrison: FEM - AEMApplicability of the AEM

2 Overview of the AEMSprings in the AEMDiscretization in the AEMLarge deformation static analysis with AEM

3 Dynamic analysis with the AEMDifferential equations of motionSeismic analysisProgresive collapse analysisExtreme Loadingr for Structures





Applied Element Method

Numerical methodsApplied Element Method (AEM) - relatively new numericalmethod (15-20 years)Two general approaches in numerical methods:

- Continuum based numerical methods (FEM, BEM, . . . )- Discrete formulations (Rigid Body and Spring Models, Discrete

Element Method, EDEM, . . . )

AEM evolved from Extended DEM and RBSM as the newdiscrete numerical method






Numerical methods

1st comprehensive formulation:H.Tagel-Din: “A new efficient method for nonlinear, largedeformation and collapse analysis of structures”, PhD thesis,Civ.Eng.Dpt., Univ. of Tokyo, 1998 (mentor Prof. KimiroMeguro)1st use of the name AEM:Meguro K. Tagel-Din H.: “Applied Element Method forStructural Analysis: Theory and Applications for LinearMaterials”, Struct.Eng./Earthquake Eng., JSCE Vol 17, No 1,pp 21-35, 2000






Numerical methodsFinite Element Method (FEM) - still the most widely usednumerical methodFEM simulations in all engineering fields (solid, fluid, acoustic,thermal, electro-magnetic, chemical engineering, . . . )FEM applied in non-engineering fields too (medicine,. . . )FEM applied in various interdisciplinary areas (fluid-structure,soil-structure, etc.)





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Connectivity between elementsIn the FEM elements are connected at nodesAll deformation is happening inside the element itselfIn the AEM elements are connected through the whole surfaceusing springsEach spring is actually 3 springs: 1 spring for normal and 2springs for shear deformationsIn the AEM deformation is outside elements (in surfacesprings)Elements in AEM are rigid bodies





Various Finite Elements





Applied Elements





FEM and AEM - Comparison






Transition from large elements to small elements

Due to various reasons (stress concentration,. . . ), areas offiner mesh are necessaryCoarse and fine mesh in FEM: special transition elementsIn the AEM transition elements are not necessaryConnection between elements in AEM is realized through thesurface springs, not nodal contact






FEM AEM






Partial element connectivityFEM - connectivity and continuity at nodes: C0, C1

In the FEM partial connectivity may be achieved only withadditional elements and new dofsIn the AEM partial connectivity is easily achieved with surfacespringsIn the AEM discontinuity between elements may be easilyachieved (zero spring stiffness)











Numerical methodsProblems with FEM in fields with discontinuities in materialSimulations of crack initiation, development, propagationSpecial techniques in FEM for crack analysis (smeared crackapproach, discrete crack approach,. . . )FEM simulation of conditions that define initiation ofstructural collapse - YESHowever, progressive structural collapse with FEM - NO





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Numerical methodsAEM is the only numerical method which can acuratelyanalyze, track and vizualize structural behavior through all 3stages of the loading:

- Small displacements (linear elastic, non-linear)- Large displacements (geometric & material change, Element

separation)- Collision and collapse (including progresive collapse)

The only software related to AEM: Extreme Loadingr forStructures (ELS)





Linear elastic analysis





Nonlinear analysis





Separation of elements





Collision and progressive collapse














Springs in the AEMDiscretization in the AEMLarge deformation static analysis with AEM


Overview of the AEMDiscretization of a structure into nodes, surrounded by a smallvolume elements (rectangular in 2D, parallelopiped in 3D)Each node in AEM has 3 or 6 dofs (2D or 3D)Elements are connected by normal and shear springsSprings are responsible for internal deformations and stressfield in the structureDiscrete elements around the nodal points move as the rigidbodies, but the element assembly is deformableStructural deformation is represented by the spring deformation





Degrees of freedom in the AEM





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Three types of springs in the AEMThere are 3 types of springs in the AEM:

- Matrix (material) springs- Reinforcement springs- Contact springs

Linear and/or non-linear (normal and shear) springs arerepresenting the material behaviorFor the RC structures each reinforcement bar is represented bythe corresponding normal and shear springsContact springs are generated when elements collide with eachother or with ground





Material springs in the AEM





Reinforcement springs in the AEM






Automatic element contact and collision in AEMContact or collision in AEM is detected automaticallyElements in AEM are able to contact and separate, re-contactagain without user interventionThere are 3 types of contacts in the AEM:

- Corner-to-Face- Edge-to-Edge- Corner-to-Ground





Contact Corner-to-Face in the AEM





Contact Edge-to-Edge in the AEM





Contact Corner-to-Ground in the AEM





Collision between elements in the AEM






Automatic element separation in the AEMWhen the average strain at the element face reaches theseparation strain, all springs at that face are removedSeparation strain is the average strain along the element facecorresponding to material propertiesAfter removal of springs adjacent elements are no longerconnected, until collision occursIf the collision occurs, elements collide as the rigid bodies





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Discretization in the AEMThe structure is virtually divided into small elements(rectangular, 2D, or cubic, 3D)Elements have one nodal point in their centersSprings on element faces are connection between elementsEach reinforcement is presented by the corresponding spring(each spring is, actually, 2 or 3 springs: normal and shear)






Discretization in the AEMAt the location of reinforcement two pairs of springs are used:for concrete and for reinforcement barReinforcement spring and concrete spring have the same strainDue to stress conditions, separation between elements occursbecause of failure of a concrete-spring before thereinforcement-springTherefore, relative displacement between reinforcement andconcrete may be taken into account automatically





Discretization in the AEM






Spring stiffness in the AEMWith reference to previous figure, spring stiffness are:

- Normal spring

Kn =E T d

a

- Shear spring

Ks =GT d

a

E and G are the modulus of elasticity and the shear modulus(of the material)






Spring stiffness in the AEMT is the thickness of the element, a the distance betweenelement centroids (nodal points), i.e. the length of the spring,while d is the distance between adjacent springsTherefore, T d is the affected area of the spring, while a is thelength of the springFor the reinforcement spring, T d is replaced by the area of thereinforcement bar T a→ Aa





Two elements in an arbitrary position






Stiffness matrix in the AEMStiffness matrix components corresponding to each dof aredetermined by assuming unit displacement in considered dofand by determining the corresponding force in the centroid ofeach elementFor two elements in 2D, and dofs u1, u2, . . . , u6, the stifnessmatrix is 6× 6:

K =

[K11 K12

K21 K22

]where submatrices K12 and K21 are symmetric: K12 = K21






Stiffness matrix in the AEMElements of the submatrix K11 are given by kij = kji,(i, j = 1, 2, 3)

Diagonal elements

k11 = sin2(Θ + α)Kn + cos2(Θ + α)Ks

k22 = sin2(Θ + α)Ks + cos2(Θ + α)Kn

k33 = L2 cos2 αKn + L2 sin2 αKs






Stiffness matrix in the AEMOff-diagonal elements

k12 = −L sin(Θ + α) sin(Θ + α)Kn

+ L sin(Θ + α) cos(Θ + α)Ks

k13 = L cos(Θ + α) sinαKs − L sin(Θ + α) cosαKn

k23 = L cos(Θ + α) cosαKn + L sin(Θ + α) sinαKs

where L,Θ, α are given in the previous figure, while Kn andKs are the normal and shear spring stiffness






Stiffness matrix in the AEMPresented stiffness coefficients are due to only one pair ofsprings (normal and shear)The stiffness coefficients depend on the spring stiffness andthe spring locationThe global stiffness matrix is obtained by summing up thestiffness matrices of all pairs of springs around each elementThe order of the global stiffness matrix is 3N or 6N , where Nis the number of elements (i.e. the nodal points)






Linear static analysis in the AEMThe governing equation in the linear elastic static analysis is

Ku = f

Matrix K is the global stifness matrix, u is the unknowndisplacement vector and f is the applied load vectorSmall displacement theoryObtained results (for sufficient number of elements andsprings) are equal to the theoretical results






Number of elements and springs in the AEMVarious parametric analyses in order to determine the numberof elements and the number of springs over the face of eachelementFor framed structures the optimum is

- ≈ 10 elements along the cross-sectional height- ≈ 10 springs over each element face

More elements and more springs give better accuracy, but alsomuch greater CPU time





Parametric analysis of the cantilever beam





Parametric analysis of the cantilever beam





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Large deformation static analysis in the AEMLarge deformation static analysis: geometric and materialnon-linearityThe governing equation for large deformation analysis:

K ∆u = ∆f + Rm + Rg

- K is the nonlinear stiffness matrix, ∆u incrementaldisplacement vector, ∆f incremental load vector

- Rm is the residual force vector due to cracking orincompatibitity between spring strains and stresses

- Rg is the residual force vector due to geometric changes in thestructure due to loading





Postbuckling behavior of the cantilever beam





Load-displacement relations of the cantileverbeam





Postbuckling behavior of the sway frame




Differential equations of motionSeismic analysisProgresive collapse analysisExtreme Loadingr for Structures

Contents









Dynamic linear/nonlinear analysis in the AEM

The governing equation for large deformation dynamic analysis:

M ∆u + C ∆u + K ∆u = ∆f(t) + Rm + Rg

- M are C are the mass and damping matrices and K is thenonlinear stiffness matrix

- ∆u, ∆u and ∆u are incremental acceleration, velocity anddisplacement vectors

- ∆f(t) is the incremental load vector






Dynamic linear/nonlinear analysis in the AEM

- Rm is the residual force vector due to cracking orincompatibitity between spring strains and stresses

- Rg is the residual force vector due to geometric changes in thestructure due to loading





Eigenvalue analysis of a building





Contents








Seismic analysis of a building





Seismic analysis of a building





Seismic analysis of a building: model

Model on the shaking table and numerical model





Seismic analysis of a building: model





Seismic analysis of a building: results





Contents








Progresive collapse of a building





Progresive collapse of a building





AEM - Demolition of a building





AEM - Demolition of a building





Contents








Reinforcement presentation in ELS





Reinforcement yield in ELS





Automatic crack propagation in ELS





Collapse of a building in ELS


Documents

Applied Element Method (AEM) in Dynamic and Seismic ... · PDF fileDynamic and Seismic Analysis Earthquake Engineering and Engineering Seismology ... in Dynamic and Seismic Analysis