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Research ArticleInfluence of Opening Locations on the Structural Response ofShear Wall

G Muthukumar and Manoj Kumar

Department of Civil Engineering BITS Pilani Pilani Rajasthan 333 031 India

Correspondence should be addressed to G Muthukumar muthugpilanibits-pilaniacin

Received 30 June 2014 Revised 26 October 2014 Accepted 28 October 2014 Published 25 November 2014

Academic Editor Mesut Simsek

Copyright copy 2014 G Muthukumar and M Kumar This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Shear walls have been conferred as a major lateral load resisting element in a building in any seismic prone zone It is essential todetermine behavior of shear wall in the preelastic and postelastic stage Shear wallsmay be providedwith openings due to functionalrequirement of the buildingThe size and location of opening may play a significant role in the response of shear wallsThough it isa well known fact that size of openings affects the structural response of shear walls significantly there is no clear consensus on thebehavior of shearwalls under different opening locationsThepresent study aims to study the dynamic behavior of shearwalls undervarious opening locations using nonlinear finite element analysis using degenerated shell element with assumed strain approachOnly material nonlinearity has been considered using plasticity approach A five-parameter Willam-Warnke failure criterion isconsidered to define the yieldingcrushing of the concrete with tensile cutoff The time history responses have been plotted for allopening cases with and without ductile detailingThe analysis has been done for different damping ratios It has been observed thatthe large number of small openings resulted in better displacement response

1 Introduction

Tall reinforced concrete buildings are subjected to lateralloads due to wind and earthquake In order to resist these lat-eral loads shear walls are provided in the framed structure asa lateral load resisting element [1ndash3] Shear walls possess suffi-cient strength and stiffness under any loading conditionsTheimportance of shear wall in mitigating the damage to rein-forced concrete structures is well documented in the litera-ture [3 4] Shear walls are generally classified on the basisof aspect ratio (heightwidth ratio) Shear walls with aspectratio between 1 and 3 are generally considered to be of squattype and shear walls with aspect ratio greater than 3 are con-sidered to be of slender type In general the structural res-ponse of shear walls depends strongly on the type of loadingaspect ratio of shear wall and size and location of the open-ings in the shear walls Squat shear walls generally fail in shearmode whereas slender shear walls fail in a flexural modeThe presence of openings in shear walls makes the behaviorof shear wall slightly vulnerable under dynamic loadingconditions The structural analysis of the shear walls with

openings becomes complex due to the stress concentrationnear the openings [5] Various experimental investigationshave been performed on shear walls with and withoutopenings subjected to severe dynamic earthquake loadingconditions [6] Neuenhofer [5] in his study on shear wallwith opening has observed that for the same opening areathe reduction in stiffness for squat and slender shear walls is50 and 20 respectively [5]Thus the aspect ratio becomescritical for squat shear walls Few analytical studies have beenmade on the response of shear wall with openings [5 7ndash10]Rosman [8] developed an approximate linear elastic approachusing laminar analysis based on different assumptions toanalyze the shearwall with one row and two rows of openingsSchwaighofer used this approach to analyze the shear wallwith three rows of openings and observed that Rosmanrsquostheory predicts the behavior of shear wall with three rows ofopenings alsowith sufficient accuracy [9]Though reasonablestudies have been made on the response analysis of shearwalls with openings very few literatures exist on the influenceof opening locations on the structural response of shearwalls [11 12] Hence it is essential to study the influence of

Hindawi Publishing CorporationJournal of Nonlinear DynamicsVolume 2014 Article ID 858510 18 pageshttpdxdoiorg1011552014858510

2 Journal of Nonlinear Dynamics

opening locations on the structural response of RC shearwalls The present study analyses the response of RC shearwalls for seven different opening locations for both squatand slender shear walls with and without ductile detailing[13] for different damping ratios Moreover it was foundthat the use of conventional methods for the analysis ofshear walls with openings resulted in remarkably poor resultsespecially for the squat shear walls where the mode of failureis predominantly shear It was also shown in literature thatthe hand calculation either underestimates or overestimatesthe response in predicting the response of shear walls withopenings [5] The finite element analysis has been the mostversatile and successfully employed method of analysis inthe past to accurately predict the structural behavior ofreinforced concrete shear wall in linear as well as in non-linear range under any severe loading conditions [14] Withthe advent in computing facilities finite element methodhas gained an enormous popularity among the structuralengineering community especially in the nonlinear dynamicanalysis The nonlinearity of RC shear wall may be due togeometry or due to material Since shear wall is a hugestructure the deformation of shear wall has been assumed tobe in control and hence the geometric nonlinearity has notbeen considered The next section describes the formulationof degenerated shell element formulation

2 Degenerated Shell Element Formulation

The displacement based finite element method has beenconsidered to be the most popular choice because of itssimplicity and ease with which the computations can beperformed The use of shell element to model moderatelythick structures like shear wall is well documented in theliterature [15 16] In this section the underlying basic ideasin the formulation of the degenerated curved shell elementare described Two assumptions are made in the formulationof the curved shell element which is degenerated from three-dimensional solid First it is assumed that even for thickshells the normal to the middle surface of the elementremains straight after deformation Second the strain energycorresponding to stresses perpendicular to themiddle surfaceis disregarded that is the stress component normal to theshell mid-surface is constrained to be zero Five degrees offreedom are specified at each nodal point corresponding toits three translations and two rotations of the normal at eachnode The independent definition of the translational androtational degrees of freedom permits the transverse sheardeformation to be taken into account during the formulationof the element stiffness since rotations are not necessarilynormal to the slope of the mid-surface

Coordinate Systems The geometry of the shell can berepresented by the coordinates and normal vectors of itsmiddle surface as Figure 1 The geometry of the degeneratedshell element and kinematics of deformation are describedby using four different coordinate systems that is globalnatural local and nodal coordinate systemsThe global coor-dinate system (119909 119910 119911) is used to define the shell geometryThe shape functions are expressed in natural curvilinear

Middle surface

120578

120577

120585

k

hk

u3k

k u2k

1205732k

1205731ku1k

Figure 1 Geometry of 9-noded degenerated shell element

coordinates (120585 120578 120577) In order to easily deal with the thin shellassumption of zero normal stress in the 119911 direction the straincomponents are defined in terms of local coordinate set ofaxes (119909

1015840 1199101015840 1199111015840) At each node of shell element the nodalcoordinate set (V

1119896V2119896V3119896) with unit vectors is definedThe

four coordinate sets employed in the present formulation arenow described

Global Coordinate Set (119909 119910 119911)This is a Cartesian coordinatesystem freely chosen in relation to which the geometry ofthe structure is defined in space Nodal coordinates and dis-placements global stiffness matrix and applied force vectorare referred to this systemThe displacements correspondingto 119909 119910 and 119911 directions are 119906 V and 119908 respectively

Nodal Coordinate Set (V1119896V2119896V3119896)Anodal coordinate sys-

tem is defined at each nodal point with origin at the referencesurface (mid-surface) The vector V

3119896is constructed from

nodal coordinates at top and bottom surfaces at node 119896 andis expressed as

119881119909

3119896

119881119910

3119896

1198811199113119896

=

119909top3119896

minus 119909bot3119896

119910top3119896

minus 119910bot3119896

119911top3119896

minus 119911bot3119896

(1)

V3119896

defines the direction of the normal at any node ldquo119896rdquowhich is not necessarily perpendicular to the mid-surfaceThemajor advantage of the definition ofV

3119896with normal not

necessary to be perpendicular to mid-surface is that there areno gaps or overlaps along element boundaries

The vector V1119896

is constructed perpendicular to V3119896

andparallel to the global 119909119911 plane Hence

V1199091119896

= V1199113119896 V119910

1119896= 00 V119911

1119896= minusV1199093119896 (2)

Alternatively if the vector V3119896

is in the 119910 direction (V1199093119896

=

V1199103119896

= 00) the following expressions are assumed V1199091119896

=

minusV1199103119896 V1199101119896

= V1199111119896

= 00 (119909 direction) The superscripts referto the vector components in the global coordinate system

The vector V2119896is constructed perpendicular to the plane

defined by V1119896and V

3119896

Journal of Nonlinear Dynamics 3

HenceV2119896

= V1119896

timesV3119896 The unit vectors in the directions

of V1119896 V2119896 and V

3119896are represented by V

1119896 V2119896 and

V3119896 respectively The vectors V

1119896 V2119896

define the rotations(1205732119896

and 1205731119896 resp) of the corresponding normal

Curvilinear Coordinate Set (120576 120578 120577) In this system 120585 120578 arethe two curvilinear coordinates in the middle plane of theshell element and 120577 is a linear coordinate in the thicknessdirection It is assumed that 120585 120578 120577 vary between minus1 and +1

on the respective faces of the elementsThe relations betweencurvilinear coordinates and global coordinates are definedlater in (4) It should also be noted that 120577 direction is onlyapproximately perpendicular to the shell-surface since 120577 isdefined as a function of V

3119896

Local Coordinate Set (1199091015840 1199101015840 1199111015840)This is the Cartesian coordi-nate system defined at the sampling points wherein stressesand strains are to be calculated The direction 119911

1015840 is takenperpendicular to the surface 120585 = constant being obtained bythe cross product of the 120585 and 120578 directions The direction 119909

1015840

can be taken tangent to the 120585 direction at the sampling pointThe direction 1199101015840 is defined by the cross product of the 1199111015840and1199091015840 directions

1199111015840=

[[[[[[[

[

120597119909

120597120585120597119910

120597120585120597119911

120597120585

]]]]]]]

]

times

[[[[[[[

[

120597119909

120597120578120597119910

120597120578120597119911

120597120578

]]]]]]]

]

1199091015840=

[[[[[[[

[

120597119909

120597120585120597119910

120597120585120597119911

120597120585

]]]]]]]

]

1199101015840= 1199111015840times 1199091015840

(3)

Element Geometry The global coordinates of pairs of pointson the top and bottom surface at each node are usuallyinput to define the element geometry Alternatively the mid-surface nodal coordinates and the corresponding directionalthickness can be furnished In isoparametric formulationcoordinates of a point within an element are obtained byinterpolating the nodal coordinates through element shapefunctions and are expressed as

119909

119910

119911

=

9

sum119896=1

119873119896(120585 120578)

119909mid119896

119910mid119896

119911mid119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟mid-surface only

+

9

sum119896=1

119873119896(120585 120578)

120577ℎ119896

2

1198811199093119896

119881119910

3119896

1198811199113119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟effect of shell thickness

(4)

where 119909mid119896

119910mid119896

and 119911mid119896

are the coordinates of the shellmid-surface and ℎ

119896is the shell thickness at node 119896 In the

above expression 119873119896(120585 120578) are the element shape functions

at the point considered within the element (120585 120578) and 120577 tellsthe position of the point in the thickness direction Theunit vector in the directions of V

3119896is represented by V

3119896

The element shape functions are calculated in the naturalcoordinate system as

1198731=

1

4120585 (1 + 120585) 120578 (1 + 120578)

1198732=

1

2(1 + 120585) (1 minus 120585) 120578 (1 + 120578)

1198733= minus

1

4(1 minus 120585) 120578 (1 + 120578)

1198734= minus

1

2120585 (1 minus 120585) (1 + 120578) (1 minus 120578)

1198735=

1

4(1 minus 120585) 120578 (1 minus 120578)

1198736= minus

1

2(1 + 120585) (1 minus 120585) 120578 (1 minus 120578)

1198737= minus

1

4120585 (1 + 120585) 120578 (1 minus 120578)

1198738=

1

2120585 (1 + 120585) (1 + 120578) (1 minus 120578)

1198739= (1 + 120585) (1 minus 120585) (1 + 120578) (1 minus 120578)

(5)Based on two assumptions of the degeneration processpreviously described the element displacement field can thenbe expressed by the five degrees of freedom at each nodeTheglobal displacements are determined frommid-surface nodaldisplacements 119906

mid119896

Vmid119896

and 119908mid119896

and the relative displace-ments are caused by the two rotations of the normal as

119906

V119908

=

119899

sum119896=1

119873119896

119906mid119896

Vmid119896

119908mid119896

+

119899

sum119896=1

119873119896120577ℎ119896

2

[[[

[

V1199091119896

minusV1199092119896

V1199101119896

minusV1199102119896

V1199111119896

minusV1199112119896

]]]

]

[1205731119896

1205732119896

]

(6)where 120573

1119896and 120573

2119896are the rotations of the normals which

results in the relative displacements and 1198811119896

and 1198812119896

arethe unit vectors defined at each node and 119899 is the numberof nodes At any point on the mid-surface of the nodesan orthogonal set of local coordinates V

1119896 V2119896 and V

3119896is

constructed V1119896

and V2119896

are constructed in the followingmanner V

1119896= 119894 times V

3119896 V1119896

= V1119894

times V3119896 The vectors V

1119896

V2119896 and V

3119896are mutually perpendicular

21 StrainDisplacement Relationship TheMindlin andReiss-ner type assumptions are used to derive the strain compo-nents defined in terms of the local coordinate system of axes1199091015840 minus 1199101015840 minus 1199111015840 where 1199111015840 is perpendicular to the material surface

layer For the small deformations and neglecting the strainenergy associated with stresses perpendicular to the local1199091015840 minus 1199101015840 surface the strain components may be written as

120576 = 1205761015840119891

1205761015840119904

=

1205761199091015840

1205761199101015840

12057411990910158401199101015840

12057411990910158401199111015840

12057411991010158401199111015840

=

1205971199061015840

1205971199091015840

120597V1015840

1205971199101015840

1205971199061015840

1205971199101015840+

120597V1015840

1205971199091015840

1205971199061015840

1205971199111015840+

1205971199081015840

1205971199091015840

120597V1015840

1205971199111015840+

1205971199081015840

1205971199101015840

(7)


1205761015840119891and 1205761015840119904are the in-plane and transverse shear strains respec-

tively and 1199061015840 V1015840 and 1199081015840are the displacement components inthe local system 1199091015840 1199101015840 1199111015840 Assuming the shell mid-surfacetangential to 1199091015840 minus1199101015840 the in-plane shear strains 1205761015840

119891can further

be divided into membrane strains 1205761015840119898and bending strains 1205761015840

119887

as

1205761015840

119891= 1205761015840

119898+ 1205761015840

119887 (8)

where

1205761015840

119898=

[[[[[[[[[

[

1205971199061015840

1205971199091015840

120597V1015840

1205971199101015840

1205971199061015840

1205971199101015840+

120597V1015840

1205971199091015840

]]]]]]]]]

]

1205761015840

119887=

[[[[[[[[[[

[

11991110158401205971205791199091015840

1205971199091015840

11991110158401205971205791199101015840

1205971199101015840

1199111015840 (1205971205791199091015840

1205971199101015840

) + (1205971205791199101015840

1205971199091015840)

]]]]]]]]]]

]

(9)

The transformation matrix [T] is used to convert the strainsin local coordinate system into strains in global coordinatesystem as

[[[[[[[[

[

1205971199061015840

1205971199091015840120597V1015840

12059711990910158401205971199081015840

1205971199091015840

1205971199061015840

1205971199101015840120597V1015840

12059711991010158401205971199081015840

1205971199101015840

1205971199061015840

1205971199111015840120597V1015840

12059711991110158401205971199081015840

1205971199111015840

]]]]]]]]

]

= [T]119879

[[[[[[[[

[

120597119906

120597119909

120597V120597119909

120597119908

120597119909

120597119906

120597119910

120597V120597119910

120597119908

120597119910

120597119906

120597119911

120597V120597119911

120597119908

120597119911

]]]]]]]]

]

[T] (10)

where transformation matrix [T] is given by

T =

[[[[[[[[

[

120597119909

1205971199091015840120597119909

1205971199101015840120597119909

1205971199111015840

120597119910

1205971199101015840120597119910

1205971199101015840120597119910

1205971199111015840

120597119911

1205971199111015840120597119911

1205971199101015840120597119911

1205971199111015840

]]]]]]]]

]

(11)

The derivatives of displacements with respect to Cartesiancoordinate system into derivatives of displacements withrespect to natural coordinate system are transformed as

[[[[[[[[

[

120597119906

120597119909

120597V120597119909

120597119908

120597119909

120597119906

120597119910

120597V120597119910

120597119908

120597119910

120597119906

120597119911

120597V120597119911

120597119908

120597119911

]]]]]]]]

]

= Jminus1

[[[[[[[[

[

120597119906

120597120585

120597V120597120585

120597119908

120597120585

120597119906

120597120578

120597V120597120578

120597119908

120597120578

120597119906

120597120577

120597V120597120577

120597119908

120597120577

]]]]]]]]

]

(12)

where [J] is the Jacobian matrix defined as

J =

[[[[[[[[

[

120597119909

120597120585

120597119910

120597120585

120597119911

120597120585

120597119909

120597120578

120597119910

120597120578

120597119911

120597120578

120597119909

120597

120597119910

120597120589

120597119911

120597120589

]]]]]]]]

]

(13)

The strain displacement matrix [B] relates the strain compo-nents and the nodal variables as

120576 = B120575 (14)

where

120575 = 119906 V 119908 1205731

1205732119879

(15)

The layered element formulation [17] allows the integrationthrough the element thicknesses which are divided intoseveral concrete and steel layers Each layer is assumed tohave one integration point at its mid-surface The steel layersare used to model the in-plane reinforcement onlyThe straindisplacementmatrixB and thematerial stiffnessmatrixD areevaluated at the midpoint of each layer and for all integrationpoints in the plane of the layer The element stiffness matrixK119890 is defined using numerical integration as follows

K119890 = ∭B119879DB 119889119881 (16)

where the integration ismade over the volume of the elementIn the Euclidean space the volume element is given by the

product of the differentials of the Cartesian coordinates andis expressed as 119889119881 = 119889119909119889119910119889119911 Using numerical integrationthe volume integration is converted into area integrationusing Jacobian and is expressed as

K119890 = ∬B119879DB |J| 119889120577 119889119860 (17)

Similarly the internal force vector is expressed as 119891119890 as

119891119890= ∬B119879120590 |J| 119889120577 119889119860 (18)

The element stiffness matrix relates the force vector with thedisplacement vector as

119891 = [119870] 120575 (19)

where

int119889119860=|J|∬+1

minus1

119889120585 119889120578 (Integration on layer mid-surface)

(20)

Once the displacements are determined the strains andstresses are calculated using strain displacement matrix andmaterial constitutive matrix respectively The formulation ofdegenerated shell element is completely described in Huang[18]

22 Assumed Strain Approach Nevertheless the general shelltheory based on the classical approach has been found to becomplex in the finite element formulationOn the other handthe degenerated shell element [19 20] derived from the three-dimensional element has been quite successful in modelingmoderately thick structures because of their simplicity andcircumvents the use of classical shell theoryThe degeneratedshell element is based on assumption that the normal to the


65

6 1

25

34

1radic3 1radic3

120585

120578

For 120574120585120577

(a)

1

456

23

1radic3

1radic3

120585

For 120574120578

120578

120577

(b)

Figure 2 Sampling point locations for assumed shearmembrane strains

mid-surface remain straight but not necessarily normal tothe mid-surface after deformation Also the stresses normalto the mid-surface are considered to be negligible Howeverwhen the thickness of element reduces degenerated shell ele-ment has suffered from shear locking and membrane lockingwhen subjected to full numerical integrationThe shear lock-ing andmembrane locking are the parasitic shear stresses andmembrane stresses present in the finite element solution Inorder to alleviate locking problems the reduced integrationtechnique has been suggested and adopted by many authors[21 22] However the use of reduced integration resulted inspurious mechanisms or zero energy modes in some casesThe reduced integration ignores the high ranked terms ininterpolated shear strain by numerical integration thus intro-ducing the chance of development of spurious or zero energymodes in the element The selective integration whereindifferent integration orders are used to integrate the bend-ing shear andmembrane terms of stiffnessmatrix avoids thelocking in most of the cases

The assumed strain approach has been successfullyadopted by many researchers [23 24] as an alternative toavoid locking In the assumed strain based degenerated shellelements the transverse shear strain and membrane strainsare interpolated from the assumed sampling points obtainedfrom the compatibility requirement between flexural andshear strain fields respectivelyThe assumed transverse shearstrain fields interpolated at the six appropriately locatedsampling points as shown in Figure 2 are

120574120585120577 =

3

sum119894=1

2

sum119895=1

119875119894(120578) sdot 119876

119895(120585) 120574119894119895

120585120577

120574120578120577

=

3

sum119894=1

2

sum119895=1

119875119894(120585) sdot 119876

119895(120578) 120574119894119895

120578120589

(21)

120574119894119895

120585120589and 120574

119894119895

120578120577are the shear strains obtained from Lagrangian

shape functions The interpolating functions 119875119894(119911) and 119876

119895(119911)

are

1198751(119911) =

119911

2(119911 + 1) 119875

2(119911) = 1 minus 119911

2

1198753(119911) =

119911

2(119911 minus 1)

1198761(119911) =

1

2(1 + radic3119911) 119876

2(119911) =

1

2(1 minus radic3119911)

(22)

Hence it can be observed that 120574120585120577is linear in 120585 direction and

quadratic in 120578 direction while 120574120578120577is linear in 120578 direction and

quadratic in 120585 direction The polynomial terms for curvatureof nine-node Lagrangian elements 120581

120585and 120581120578 are the same as

the assumed shear strain as given by

120581120585=

120597120579120585

120597120585(1 120585 120578 120585120578 120585

2 1205852120578 1205782 1205851205782 12058521205782)

120581120585= 120581120585(1 120585 120578 120585120578 120578

2 1205851205782)

120581120578=

120597120579120578

120597120578(1 120585 120578 120585120578 120585

2 1205852120578 1205782 1205851205782 12058521205782)

120581120578= (1 120585 120578 120585120578 120585

2 12058521205782)

120574120585120577

= 120574120585120577

(120581120585) = 120574120585120577

(1 120585 120578 120585120578 1205782 1205851205782)

120574120578120577

= 120574120578120577

(120581120578) = 120574120578120577

(1 120585 120578 120585120578 1205782 1205851205782)

(23)

The original shear strains obtained from the Lagrange shapefunctions 120574

120585120577and 120574120578120577are

120574120585120577

= 120579120585+

120597119908

120597120585= 120574120585120577

(1 120585 120578 120585120578 1205852 1205852120578 1205782 1205851205782 12058521205782)

120574120578120577

= 120579120578+

120597119908

120597120578= 120574120578120577

(1 120585 120578 120585120578 1205852 1205852120578 1205782 1205851205782 12058521205782)

(24)

The total potential energy expression has the form

120587 = 120587 + int12058213

(120574120585120577

minus 120574120585120577) 119889119881 + int120582

23(120574120578120577

minus 120574120578120577) 119889119881 (25)

12058213 and 12058223 are Lagrangian multipliers and are independentfunctions The terms 120574

120585120577and 120574

120578120577are the transverse shear

strains evaluated from the displacement field


Calculation of shape functions and its derivatives

Identify the number ofsampling points its positions

and its weights

Calculation of strain displacement matrix [BMATX] at sampling points

Calculation of substitute transverse shear and membrane strain displacement matrix

Replacement of BMATX with substitute transverse shear strain displacement matrix


Converting the strain displacement matrix into local coordinate system

Figure 3 Determination of strain displacement matrix using the assumed strain approach

The assumed shear strain fields are chosen as

120574120585120577

=

119899

sum119894=1

119877119894(120585 120578) 120574

119894

120585120577 120574

120578120577=

119899

sum119894=1

119878119894(120585 120578) 120574

119894

120578120577 (26)

The Lagrangian multipliers are taken as

12058213

=

119899

sum119894=1

12058213

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

12058223

=

119899

sum119894=1

12058223

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

(27)

By substituting the following equation can be obtained

120574120585120577

(120585119894 120578119894) = 120574120585120577

(120585119894 120578119894)

120574120578120577

(120585119894 120578119894) = 120574120578120577

(120585119894 120578119894)

(28)

120579120585and 120579

120578are the rotating normal and 119908 is the transverse

displacement of the element It can be clearly seen thatthe original shear strain and assumed shear strain are notcompatible and hence the shear locking exists for very thinshell cases The appropriately chosen polynomial terms andsampling points ensure the elimination of risk of spuriouszero energy modes The assumed strain can be considereda special case of integration scheme wherein for function120574120585120577

full integration is employed in 120578 direction and reducedintegration is employed in 120585 direction On the other hand forfunction 120574

120578120577reduced integration is employed in 120578 direction

and full integration is employed in 120585directionThemembraneand shear strains are interpolated from identical samplingpoints even though the membrane strains are expressedin orthogonal curvilinear coordinate system and transverseshear strains are expressed in natural coordinate system Theflow chart explaining the formulation of strain displacementmatrix using assumed strain approach is shown in Figure 3

3 Material Modeling

Themodeling of material may play a crucial role in achievingthe correct response The presence of nonlinearity may addanother dimension of complexity to it The nonlinearities inthe structure may accurately be estimated and incorporatedin the solution algorithm The accuracy of the solution algo-rithm depends strongly on the prediction of second-ordereffects that cause nonlinearities such as tension stiffeningcompression softening and stress transfer nonlinearitiesaround cracks

These nonlinearities are usually incorporated in the con-stitutive modeling of the reinforced concrete In order toincorporate geometric nonlinearity the second-order termsof strains are to be included In this study only material non-linearity has been considered The subsequent sections des-cribe the modeling of concrete in compression and tensionmodeling of steel

31 Concrete Modeling in Tension The presence of crack inconcrete has much influence on the response of nonlinearbehavior of reinforced concrete structures The crack inthe concrete is assumed to occur when the tensile stressexceeds the tensile strength The cracking of concrete resultsin the loss of continuity in the load transfer and hencethe stresses in both concrete and steel reinforcement differsignificantly Hence the analysis of concrete fracture hasbeen very important in order to predict the response ofstructure precisely The numerical simulation of concretefracture can be represented either by discrete crack proposedby Ngo and Scordelis [25] or by smeared crack proposedby Rashid [26] The objective of discrete crack is to simulatethe initiation and propagation of dominant cracks present inthe structure In the case of discrete crack approach nodesare disassociated due to the presence of cracks and thereforethe structure requires frequent renumbering of nodes whichmay render the huge computational cost Nevertheless when


the structurersquos behavior has been dominated by only fewdominant cracks the discretemodeling of cracking seems theonly choice On the other hand the smeared crack approachsmears out the cracks over the continuum and captures thedeterioration process through the constitutive relationshipand reduces the computational cost and time drastically

Crack modeling has gone through several stages due tothe advancement in technology and computing facilities Ear-lier researchwork indicates that the formation of crack resultsin the complete reduction in stresses in the perpendiculardirection thus neglecting the phenomenon called tensionstiffening With the rapid increase in extensive experimentalinvestigations as well as in computing facilities many finiteelement codes have been developed for the nonlinear finiteelement analysis which incorporates the tension stiffeningeffect The first tension stiffening model using degradedconcrete modulus was proposed by Scanlon and Murray[27] and subsequently many analytical models have beendeveloped such as Lin and Scordelis [28] model Vebo andGhali model Gilbert and Warner model [29] and Nayaland Rasheed model [30] The cracks are always assumed tobe formed in the direction perpendicular to the directionof the maximum principal stress These directions may notnecessarily remain the same throughout the analysis andloading and hence the modeling of orientation of crack playsa significant role in the response of structure Still due tosimplicity many investigations have been performed usingfixed crack approachwherein the direction of principal strainaxes may remain fixed throughout the analysis In this studyalso the direction of crack has been considered to be fixedthroughout the duration of the analysis However the mod-eling of aggregate interlock has not been taken very seriouslyThe constant shear retention factor or the simple function hasbeen employed to model the shear transfer across the cracksApart from the initiation of crack the propagation of crackalso plays a crucial role in the response of structure The pre-diction of crack propagation is a very difficult phenomenondue to scarcity and confliction of test results Neverthelessthe propagation of cracks plays a crucial role in the responseof nonlinear analysis of RC structures The plain concreteexhibits softening behavior and reinforced concrete exhibitsstiffening behavior due to the presence of active reinforcingsteel A gradual release of the concrete stress is adopted inthis present study as shown in Figure 4 [31] The reduction inthe stress is given by the following expression

119864119894= 1205721198911015840

119905(1 minus

120576119894

120576119898

)1

120576119894

120576119905le 120576119894le 120576119898 (29)

120572 and 120576119898

are the tension stiffening parameters 120576119898

is themaximum value reached by the tensile strain at the pointconsidered 120576

119894is the current tensile strain in material

direction 119894 The coefficient depends on the percentage ofsteel in the section In the present study values of 120572 and 120576

119898

are taken as 05 and 00020 respectively It has also beenreported that the influence of the tension stiffening constantson the response of the structures is generally small and hencethe constant value is justified in the analysis [31] Generallythe cracked concrete can transfer shear forces through dowel

Compression

Tension

Stre

ss

Strain

ft

120572ft

120590i

E

120576t 120576i 120576m

Figure 4 Tension stiffening effect of cracked concrete

action and aggregate interlockThemagnitude of shear mod-uli has been considerably affected because of extensive crack-ing in different directions

Thus the reduced shear moduli can be put to incorporatethe aggregate interlock and dowel action In the plain con-crete aggregate interlock is the major shear transfer mech-anism and for reinforced concrete dowel action is the majorshear transfermechanism with reinforcement ratio being thecritical variable In order to incorporate the aggregate inter-lock and dowel action the appropriate value of cracked shearmodulus [32] has been considered in the material modelingof concrete

Cracked in One Direction The stress-strain relationship forcracked concrete where cracking is assumed to take place inonly one direction is given as

[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[

[

0 0 0 0 0

0 119864 0 0 0

0 0 119866119888

120 0

0 0 0 11986611988813

0

0 0 0 0 11986623

]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

12= 025 times 119866(1 minus

1

0004) if 120576

1lt 0004

119866119888

12= 0 if 120576

1ge 0004

119866119888

13= 119866119888

12 119866

23=

5119866

6

(30)

Cracked in Two Directions The stress-strain relationship forcracked concrete where cracking is assumed to take place inboth directions is given as

[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0119866119888

12

20 0

0 0 0 119866119888

130

0 0 0 0 119866119888

23

]]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

13= 025 times 119866(1 minus

1205761

0004) if 120576

1lt 0004

119866119888

13= 0 if 120576

1ge 0004


120591oct120579 = 60∘

120579 = 0∘

120590oct

(a) Deviatoric plane in octahedral stress

Smooth 1205901

1205902 1205903

(b) Meridian plane in principal stress

Figure 5 Willam-Warnke failure model

119866119888

23= 025 times 119866(1 minus

1205762

0004)

119866119888

12= 05 times 119866

119888

13if 119866119888

23lt 119866119888

13

(31)

It has also beenmentioned by Hinton and Owen [33] that thetensile strength of concrete is a relatively small and unreliablequantity which is not highly influential to the response ofstructures In the above stress-strain relationship the crackedshearmodulus (119866119888) is assumed to be a function of the currenttensile strain 119866 is the uncracked concrete shear modulus Ifthe crack closes the uncracked shear modulus 119866 is assumedin the corresponding direction Even after the formation ofinitial cracks the structure can often deform further withoutfurther collapse In addition to the formation of new cracksthere may be a possibility of crack closing and opening of theexisting cracks If the normal strain across the existing crackbecomes greater than that just prior to crack formation thecrack is said to have opened again otherwise it is assumedto be closed Nevertheless if all cracks are closed then thematerial is assumed to have gained the status equivalent tothat of noncracked concrete with linear elastic behavior

32 Concrete Modeling in Compression The theory of plas-ticity has been used in the compression modeling of theconcrete The failure surface or bounding surface has beendefined to demarcate plastic behavior from the elastic behav-ior Failure surface is the important component in the con-crete plasticity Sometimes the failure surface can be referredto as yield surface or loading surface The material behavesin the elastic fashion as long as the stress lies below thefailure surface Several failure models have been developedand reported in the literature [34] Nevertheless the five-parameter failure model proposed by Willam and Warnke[35] seems to possess all inherent properties of the failure sur-face The failure surface is constructed using two meridiansnamely compression meridian and tension meridian Thetwo meridians are pictorially depicted in a meridian planeand cross section of the failure surface is represented in thedeviatoric plane

The variations of the average shear stresses 120591119898119905

and 120591119898119888

along tensile (120579 = 0∘) and compressive (120579 = 60∘) meridians

as shown in Figure 5 are approximated by second-order para-bolic expressions in terms of the average normal stresses 120590

119898

as follows

120591119898119905

1198911015840119888

=120588119905

radic51198911015840119888

= 1198860+ 1198861(

120590119898

1198911015840119888

) + 1198862(

120590119898

1198911015840119888

)

2

120579 = 0∘

120591119898119888

1198911015840119888

=120588119888

radic51198911015840119888

= 1198870+ 1198871(

120590119898

1198911015840119888

) + 1198872(

120590119898

1198911015840119888

)

2

120579 = 60∘

(32)

These twomeridiansmust intersect the hydrostatic axis atthe same point120590

1198981198911015840119888= 1205850(corresponding to hydrostatic ten-

sion) the number of parameters that need to be determined isreduced to five The five parameters (119886

0or 1198870 1198861 1198862 1198871 1198872) are

to be determined from a set of experimental data with whichthe failure surface can be constructed using second-orderparabolic expressions The failure surface is expressed as

119891 (120590119898 120591119898 120579) = radic5

120591119898

120588 (120590119898 120579)

minus 1 = 0

120588 (120579) = (2120588119888(1205882

119888minus 1205882

119905) cos 120579 + 120588

119888(2120588119905minus 120588119888)

times [4 (1205882

119888minus 1205882

119905) cos2120579 + 5120588

2

119905minus 4120588119905120588119888]12

)

times (4 (1205882

119888minus 1205882

119905) cos2120579 + (120588

119888minus 2120588119905)2

)minus1

(33)

The formulation of Willam-Warnke five-parameter materialmodel is described in Chen [34] Once the yield surface isreached any further increase in the loading results in theplastic flowThemagnitude and direction of the plastic strainincrement are defined using flow rule which is described inthe next section

321 Flow Rule In this method associated flow rule isemployed because of the lack of experimental evidencein nonassociated flow rule The plastic strain incrementexpressed in terms of current stress increment is given as

119889120576119901

119894119895= 119889120582

120597119891 (120590)

120597120590119894119895

(34)


Subsequent yield surfaceInitial yield surface

Yield limitStrain hardening

Failure surface

Failure

1205901

1205902 120590

120576

Figure 6 Isotropic hardening with expanding yield surfaces and the corresponding uniaxial stress-strain curve

119889120582 determines the magnitude of the plastic strain incrementThe gradient 120597119891(120590)120597120590

119894119895defines the direction of plastic strain

increment to be perpendicular to the yield surface119891(120590) is theloading condition or the loading surfaces

322 Hardening Rule The relationship between loading sur-faces (or effective stress) and the plastic work (accumulatedplastic strain) is represented by a hardening ruleThe ldquoMadridparabolardquo is used to define the hardening rule The isotropichardening is adopted in the present study as shown in Figure6

120590 = 1198640120576 minus

1

2

1198640

1205760

1205762 (35)

1198640is the initial elasticity modulus 120576 is the total strain and

1205760is the total strain at peak stress 1198911015840

119888 The total strain can be

divided into elastic and plastic components as120576 = 120576119890+ 120576119901

= [119863119890] ( 120576 minus 120576

119901)

120576119901 = 119886 119886 =

120597119865

120597120590

(36)

119865 is the yield function and is the consistency parameterwhich defines the magnitude of the plastic flow

The loading unloading conditions (Kuhn-Tucker condi-tions) can be stated as

ge 0 119865 le 0 119865 = 0 (37)The first of these Khun-Tucker conditions indicates that theconsistency parameter is nonnegative the second conditionimplies that the stress states must lie on or within the yieldsurface The third condition ensures that the stresses lie onthe yield surface during the plastic loading

119886119879

= 0 119886119879[119863119890] ( 120576 minus 120576

119901) = 0

119886119879[119863119890] ( 120576 minus 119886) = 0

=119886119879[119863119890] 120576

119886119879[119863119890] 119886

(38)

The elastoplastic constitutive matrix is given by the followingexpression

[119863119890119901] = [119863] minus

[119863] 119886 119886119879[119863]

119867 + 119886119879[119863] 119886

(39)

119886 = flow vector defined by the stress gradient of the yieldfunction119863 = constitutive matrix in elastic rangeThe secondterm in (39) represents the effect of degradation of materialduring the plastic loading

33 Modeling of Reinforcement in Tension and CompressionReinforcing bars in structural concrete are generally assumedone-dimensional elements without transverse shear stiffnessor flexural rigidity The reinforcing bar can generally betreated as either discrete or smeared The major advantage ofdiscrete representation of reinforcing bar is existence of one-to-one correspondence between the real structure andmodelIn the smeared reinforcement the average stress-strain rela-tionship is calculated for an element area and incorporateddirectly as part of the overall concrete element stiffnessmatrix In the present investigation the smeared layeredapproach is adoptedThe bilinear stress-strain curve with lin-ear elastic and strain hardening region is adopted in this studyas shown inFigure 7 Sometimes the trilinear idealization hasalso been adopted as the stress-strain curve in tension Typi-cally the hardening strainmodulus is assumed to be 1of ini-tial elasticity modulus The position and thickness of steellayers are to be defined as input parameters along with theelasticity modulus and hardening modulus The direction ofsteel (horizontal or vertical) can be set up by defining theangle with respect to local 119909-axisThere can only be two statesof stress for the reinforcing bar namely elastic and linearstrain hardening

4 Dynamic Analysis of RC Shear Wall

The dynamic analysis of structure can be performed by threeways namely (i) equivalent lateral forcemethod (ii) responsespectrummethod and (iii) time historymethodThe equiva-lent lateral force method determines the equivalent dynamiceffect in the static manner The response spectrum methodaims at determining the maximum response quantity of thestructure For tall and irregular buildings dynamic analysisby response spectrum method seems to be a popular choiceamong designers The time history analysis of the structurehas been successfully used to analyze the structure especiallyof huge importance Even though time history analysis con-sumes time it is the only method capable of giving resultscloser to the actual one especially in the nonlinear regime Inthe dynamic analysis the loads are applied over a period oftime and the response is obtained at different time intervals

10 Journal of Nonlinear DynamicsSt

ress

Strain

Strain hardening region

fy Esh

Es

120576y

Figure 7 Stress-strain curve of steel

The equation of dynamic equilibrium at any time ldquo119905rdquo is givenby (1)

[119872] [119905] + [119862] [

119905] + [119870] [119880

119905] = [119877

119905] (40)

119872 119862 and 119870 are the mass damping and stiffness matricesrespectively The mass matrix can be formulated either byusing consistent mass approach or by using lumped massapproach Since damping cannot be precisely determinedanalytically the damping can be considered proportional tomass or stiffness or both depending on the type of the prob-lem The direct time integration [24] of the equation ofmotion can be performed using explicit (central differencescheme) and implicit (Houbolt method Newmark Betamethod and Wilson Theta method) time integration In theexplicit time integration the formation of complete stiffnessmatrix of the structure is not required and hence saves a lot ofcomputer time andmoney in storing and saving of those dataMoreover in the case of all explicit time integration schemesthe iterations are not required as the equilibrium at time119905 + Δ119905 depends on the equilibrium at time 119905 Nevertheless themajor drawback of explicit time integration is that the timestep (Δ119905) used for calculation of response has to be smallerthan the critical time step (Δ119905cr) to ensure the stable solution

Δ119905 le Δ119905cr =119879119899

120587=

2

120596 (41)

On the other hand implicit time integration requires theiterations to be carried out within the time step as the solutionat time 119905 + Δ119905 involves the equilibrium equation at 119905 + Δ119905The Newmark 120573 method converges to various implicit andexplicit schemes for different values of Beta called the stabi-lity parameter In this study for 120573 = 025 the Newmark120573 method converges to the constant acceleration implicitmethod known as trapezoidal rule The trapezoidal rule isunconditionally stable and hence allows larger time step tobe used in the calculation of response Nevertheless the timestep can be made smaller from the accuracy point of viewThe formulation of implicit Newmark Beta method (trape-zoidal rule) is mentioned in the literature [24]

41 Formulation of Mass Matrix In a dynamic analysis acorrect estimate of mass matrix is very important in predict-ing the dynamic response of RC structures There are twodifferent ways namely (i) consistent approach and (ii)lumped approach by which the element mass matrix can bedeveloped In the case of consistent approach the masses areassumed to be distributed over the entire finite elementmeshIn this approach the shape functions (N

119894) used for the com-

putation of mass matrix are the same shape functions usedfor the development of stiffness matrix and hence the nameldquoconsistentrdquo approach The mass matrix developed using theconsistent approach is known as consistent mass matrix Theconsistent element mass matrix (M

119890) is given by

M119890= ∭

119881

120588119898119873119894119873119894119889119881 (42)

where 120588119898is the mass density ldquo119894rdquo is the node number and

the integration is performed over the entire volume of theelement The consistent mass matrix contains off-diagonalterms and hence is computationally expensive

On the other hand the lumped mass matrix is purelydiagonal and hence computationally cheaper than the con-sistent mass matrix Nevertheless the diagonalization of themass matrix from the full mass matrix results in the lossof information and accuracy [18] Nodal quadrature rowsum and special lumping are the three lumping proceduresavailable to generate the lumped mass matrices All the threemethods of lumping lead to the same mass matrix for nine-node rectangular elements Nevertheless one of the mostefficient means of lumping is to distribute the element massin proportion to the diagonal terms of consistent massmatrix[36] and also discarding the off-diagonal elements This wayof lumping has been successfully used in many finite elementcodes in practice

The advantage of this special lumping scheme is the assu-rance of positive definiteness of mass matrix The use oflumped mass matrix is mostly employed in lower order ele-ments For higher order elements the use of lumped massmatrix may not be an appropriate option and hence the pre-sent study uses only consistent mass matrix Moreover thelumped mass matrix may be an ideal option in the case ofRC framed structures in which the masses can be lumped atfloor level As RC shear wall is the concrete structure it maybe appropriate to use consistent mass matrix The elementmass matrices for consistent and lumped mass matrices areas mentioned below

[119872119888

119890] =

[[[[[

[

1198681

0 0 0 1198682

0 1198681

0 minus1198682

0

0 0 1198681

0 0

0 minus1198682

0 1198683

0

1198682

0 0 0 1198683

]]]]]

]

(43)

[119872119871

119890] =

[[[[[

[

1198681198711

0 0 0 0

0 1198681198711

0 0 0

0 0 1198681198711

0 0

0 0 0 1198681198712

0

0 0 0 0 1198681198712

]]]]]

]

(44)


Equation (43) is very general and can be used to developthe consistent mass matrix for any displacement based finiteelement In (43) ldquo119868

1rdquo represents the contribution of mass to

resisting the linear or translational motion and is defined as1198681

= int120588119898119889119911 Hence masses corresponding to translational

degrees of freedom are represented by ldquo1198681rdquo On the other

hand ldquo1198683rdquo represents the contribution of mass to resisting the

change in the rotatory motion and is expressed mathemati-cally as 119868

3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass

matrix 119872 is the sum of the element mass matrices M119890 119911 is

the position of layermiddle surface from shellmiddle surfaceEquation (44) represents the lumped mass matrix which isnot used in the present analytical study

42 Formulation of Damping Matrix Mass and stiffnessmatrices can be represented systematically by overall geome-try and material characteristics However damping can onlybe represented in a phenomenological manner thus makingthe dynamic analysis of structures in a state of uncertaintyThe quantification and representation of damping is certainlycomplicated by the relationship between its mathematicalrepresentation and the physical sources The damping maybe assumed to be contributed through friction hysteretic andviscous characteristicsThere is no single universally acceptedmethodology for representing damping because of the natureof the state variables which control damping Neverthelessseveral investigations have been done in making the repre-sentation of damping in a simplistic yet logical manner [37]Only for the mathematical convenience the damping hasbeen modeled as equivalent viscous damping represented asthe percentage of critical damping The governing equationof motion second-order differential equation with constantcoefficients is rewritten as

119872 + 119862 + 119870119906 = 119877 (119905) (45)

The trial solution is given by

119906 = 119888119890119904119905 (46)

On substituting the trial solution and simplifying theroots of quadratic equation are as

119904 =minus119888

2119898plusmn radic(

119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0

997904rArr 2 real roots (over damped)

997888rarr 1199041= minus120585120596 + 120596radic1205852 minus 1 119904

2= minus120585120596 minus 120596radic1205852 minus 1

(119888

2119898)2

minus 1205962= 0

997904rArr 1 real root (critically damped)

997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0

997904rArr 2 complex roots (under damped)



(47a)

In the above equation the damping ratio 120585 is given by

120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)

Over damped system does not vibrate it all The classicalexample is automatic door closing Critical damping has thelinear exploding function and hence the amplitude is higherthan the over damped system followed by exponentiallyexploding function resulting in the fast movement over thetimeThe bottom line is that both over damped and criticallydamped systems do not vibrate at all Nevertheless in a build-ing structure critically damped and over damped situationmay not arise Damping matrix can be formulated analogousto mass and stiffness matrices [38 39] It is also importantto note that the damping matrix should be formulated fromdamping ratio and not from the member sizes Rayleighdissipation function assumes that the dissipation of energytakes place and can be idealized as the function of velocityWhen Rayleigh damping is used the resultant dampingmatrix is of the same size as stiffness matrix Rayleigh damp-ing is being used conveniently because of its versatility in seg-regating each mode independently The damping can bedefined as the linear combination of mass and stiffness mat-rices as

[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)

It is to be noted that the damping is controlled by only twoparameters (Figure 8) From (49) it is observed that if 120573 iszero the higher modes of the structure will be assigned verylittle dampingWhen120572 alpha is zero the highermodes will beheavily damped as the damping ratio is directly proportionalto circular frequency (120596) [40]Thus the choice of damping isproblem dependent Hence it is inevitable to performmodalanalysis to determine the different frequencies for differentmodes However in the present study the fundamentalnatural period (119879) and fundamental natural frequency (119891)have been calculated using the formulas asmentioned inGoeland Chopra [41] The coupling of the modes usually can beavoided easily in the case of undamped free vibration Thesame is not true for damped vibration Hence in order torepresent the equation ofmotion in uncoupled form it is sug-gested to have a damping matrix proportional to uncoupledmass and stiffness matrices Thus Rayleighrsquos proportionaldamping has the specific advantage that the equation ofmotion can be uncoupled when it is proportional tomass andstiffness matrices Thus it is proposed to use Rayleigh damp-ing in this study

12 Journal of Nonlinear DynamicsD

ampi

ng ra

tio (120585

)

Total damping

120573 damping

Circular frequency (120596)

120572 damping

120596i 120596j

Figure 8 Variation of damping with circular frequency

43 Nonlinear Solution The numerical procedure for non-linear analysis employs the iterative procedure to satisfy theequilibrium at the end of the load step Once the convergenceof the solution is achieved the algorithm proceeds to the nextstep It is always desirable to keep the load step very smallespecially after the onset of nonlinear behavior The stiffnessmatrix is updated at the beginning of each load step Theconvergence is said to be achieved if the out of balance forcescalculated as follows are less than the specified tolerance

120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)

5 Development of Computer Program

In the present study an analysis module NLDAS was devel-oped using Fortran 77 and used to perform the nonlineardynamic finite element analysis of RC shear walls Thecomputer codes developed by Huang [23] and Owen andHinton [31] for the static and dynamic elastoplastic analysis ofRC structures based on simple Owen-Figurious yieldfailurecriterion have been taken as the base programs in thisstudy These programs have been merged and modified toinclude state-of-the-art five-parameter yieldfailure modeland concrete crackingAmodular approachhas been adoptedfor the program development To this end several new sub-routines have been incorporated and few subroutines havebeen enhanced

The program consists of 19 subroutines which are devel-oped to perform various operations In the programNLDASthough the input files for static and dynamic analysis aredifferent all the data sets have to be read in at the beginning ofthe program The program mainly consisted of the followingsubroutines input loading incremental loading stiffnessmass and damping matrices development and assemblysolution of equations residual force calculations and conver-gence check and output results in addition to modules forstorage of global arrays such as nodal coordinates elementconnectivity material properties and boundary conditionsFigure 9 shows the program layout explaining the process ofthe program

Start

Input data

Finite element discretization

Strain displacement matrix

Incremental loading

Mass matrix damping matrix assembly

Constitutive material matrix

Assemblage of stiffness matrix

Iterative solver

Residual force calculation

Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop

Figure 9 Program layout NLDAS

Table 1 Material property of RC shear wall

Material property MagnitudeYoungrsquos modulus of concrete 27 times 1010 kNm2

Youngrsquos modulus of steel 20 times 1011 kNm2

Poissonrsquos ratio of concrete 0100Uniaxial compressive strength of concrete 30 times 106 kNm2

Tensile strength of concrete 3 times 106 kNm2

Ultimate crushing strain of concrete 00035Yield stress of steel 50 times 107 kNm2

Tension stiffening coefficient (120572ts) 06Tension stiffening constant (120576ts) 00020

6 Displacement Time HistoryResponse of Shear Wall withDifferent Opening Locations

In order to identify the safe regions where the openings canbe provided in a shear wall two representative problems often storeys (35m high 8m wide and 03 thick) and fivestoreys (175m high 8mwide and 03 thick) as shown in Fig-ures 10(a) and 10(b) behaviorally slender and squat typeare chosen and analyzed for dynamic loading condition sub-jected to Figure 10(c) using the finite element analysis The


Table 2 Displacement time history responses of shear walls with different door window opening locations

A typical storey Slender shear wall (10 storeys) Squat shear wall (5 storeys)

Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Not strengthenedStrengthened

minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15

degenerated shell element of size 1m times 05m has been usedto discretize the shear wall geometry The opening locationis varied keeping all practical positions In total there areseven cases considered as the possible opening locations pre-vailed in practice namely (i) door cum window (ii) twodoors (iii) two windows (iv) three windows (v) four win-dows (vi) staggered two windows and (vii) staggered fourwindows The size of the opening in all cases is kept at 14The openings are provided uniformly in all the storeys Forsimplicity only a typical storey is represented in Figure 10The material properties used for the material modeling areas mentioned in Table 1

The analysis has been done for all practical damping casesnamely (i) no damping (iii) 2 damping (iii) 5 damping

and (iv) 10 damping The reinforcement ratio of 025 isadopted for both vertical and horizontal reinforcement Thestrengthening of shear walls around openings was consideredas per IS 13920 requirements The simulated EL-Centroearthquake shown in Figure 8 is applied as the input groundaccelerationwithmaximumamplitude of 105 gThe responseof the structure is traced for 15 seconds of duration Severalinvestigators have also adopted this way of response calcu-lation by predicting the response only for the most intenseearthquake period in order to simplify the computation Thetotal duration of the ground motion has been taken as 15seconds The analysis has been carried out with the timestep of 0015 seconds The undamped top displacement timehistories of shear walls with different opening locations have


Table 3 Influence of opening locations on the maximum displacement of slender shear walls

Case

Maximum displacement of shear wall at0 damping 2 damping 5 damping 10 damping

Door cum window 135 8004 377 376 255 255 160 160Two doors 114 80 304 303 212 212 137 137Two windows 1678 8323 390 390 258 259 163 163Three windows 1851 8744 378 378 273 273 177 177Four windows 9946 6168 302 302 213 213 139 139Staggered two windows 1613 167 386 387 248 248 155 155Staggered four windows 1265 5683 347 346 237 237 150 150

Table 4 Influence of opening locations on the maximum displacement of squat shear walls

Case



Not strengthened strengthened

(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)

Figure 10 (a) Slender shear wall with openings (b) Squat shear wall with openings (c) Dynamic ground acceleration applied at the base ofthe structure


Table 5 Maximum displacement response on RC shear walls with different opening locations for different damping ratios

Case Slender shear wall (10 storeys) Squat shear wall (5 storeys)

No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case

been presented in Table 2 Though the response analyses ofshear walls have been conducted for different damping ratiosonly undamped responses have been plotted for brevity

Tables 3 and 4 show the influence of opening locationson themaximum displacement response of slender and squatshear walls respectively

As evidenced through Tables 3 and 4 the displacementhas been found to be higher in the case of slender shear wallsthan squat shear walls It was inferred from Tables 2 and 3that the maximum displacement responses of slender shearwalls have been found to be in the case of three windowopenings followed by two window openings Nevertheless


the strengthening around openings reduces the displacementresponse to 53 The influence of strengthening has beenconsidered massive in the case of staggered openings (twowindows and four windows) as seen in Table 3 In generalthe strengthening has resulted in better behavior of the shearwall

On the other hand in the case of squat shear walls (Tables2 and 4) the displacement response has been found to be leastin the case of shear wall with four windows regular and stag-gered Also from Table 4 it is further observed that the squatshear walls undergo high displacement in the presence of twowindow openings Hence it is better to provide large num-ber of small openings to have better structural performance

Table 5 shows the maximum displacement response onthe RC shear wall with different opening locations for differ-ent damping ratios It is observed that the influence ofstrengthening has been found to be significant in case of shearwall with no damping On the other hand the influence ofstrengthening has been offset by the presence of damping asseen in Table 2 for 2 5 and 10

7 Conclusions

On the basis of displacement time history responses ofshear walls with different opening locations and with variousdamping ratios it has been concluded that shear wallsare penetrated by large number of small openings thansmall number of large openings Moreover the influence ofstrengthening has been considered essential especially forundamped shear wall The shear wall with four windows hasbeen considered best for both slender and squat shear wallsThe strengthening (ductile detailing) has been consideredsignificant in the case of slender shear wall with staggeredopenings However for higher damping the strengtheninghas not been considered essential

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Rahimian ldquoLateral stiffness of concrete shear walls for tallbuildingsrdquo ACI Structural Journal vol 108 no 6 pp 755ndash7652011

[2] H-S Kim and D-G Lee ldquoAnalysis of shear wall with openingsusing super elementsrdquo Engineering Structures vol 25 no 8 pp981ndash991 2003

[3] J S Kuang andY BHo ldquoSeismic behavior and ductility of squatreinforced concrete shear walls with nonseismic detailingrdquo ACIStructural Journal vol 105 no 2 pp 225ndash231 2008

[4] M Fintel ldquoPerformance of buildings with shear walls in earth-quakes of the last thirty yearsrdquo PCI Journal vol 40 no 3 pp62ndash80 1995

[5] A Neuenhofer ldquoLateral stiffness of shear walls with openingsrdquoJournal of Structural Engineering vol 132 no 11 pp 1846ndash18512006

[6] Y LMo ldquoAnalysis and design of low-rise structural walls underdynamically applied shear forcesrdquo ACI Structural Journal vol85 no 2 pp 180ndash189 1988

[7] I A MacLeod Shear Wall-Frame Interaction Portland CementAssociation 1970

[8] R Rosman ldquoApproximate analysis of shear walls subject tolateral loadsrdquo Proceedings of ACI Journal vol 61 no 6 pp 717ndash732 1964

[9] J Schwaighofer andH FMicroys ldquoAnalysis of shear walls usingstandard computer programmesrdquo ACI Journal vol 66 no 12pp 1005ndash1007 1969

[10] C P Taylor P A Cote and J W Wallace ldquoDesign of slenderreinforced concrete walls with openingsrdquo ACI Structural Jour-nal vol 95 no 4 pp 420ndash433 1998

[11] M Tomii and S Miyata ldquoStudy on shearing resistance of quakeresisting walls having various openingsrdquo Transactions of theArchitectural Institute of Japan vol 67 1961

[12] D J Lee Experimental and theoretical study of normal andhigh strength concrete wall panels with openings [PhD thesis]Griffith University Queensland Australia 2008

[13] ldquoDuctile detailing of reinforced concrete structures subjected toseismic forcesrdquo Tech Rep IS-13920 Bureau of Indian StandardsNew Delhi India 1993

[14] C Damoni B Belletti and R Esposito ldquoNumerical predictionof the response of a squat shear wall subjected to monotonicloadingrdquoEuropean Journal of Environmental andCivil Engineer-ing vol 18 no 7 pp 754ndash769 2014

[15] Y Liu and S Teng ldquoNonlinear analysis of reinforced concreteslabs using nonlayered shell elementrdquo Journal of Structural Engi-neering vol 134 no 7 pp 1092ndash1100 2008

[16] B Belletti C Damoni and A Gasperi ldquoModeling approachessuitable for pushover analyses of RC structural wall buildingsrdquoEngineering Structures vol 57 pp 327ndash338 2013

[17] S Teng Y Liu and C K Soh ldquoFlexural analysis of reinforcedconcrete slabs using degenerated shell element with assumedstrain and 3-D concrete modelrdquoACI Structural Journal vol 102no 4 pp 515ndash525 2005

[18] H C Huang ldquoImplementation of assumed strain degeneratedshell elementsrdquoComputers and Structures vol 25 no 1 pp 147ndash155 1987

[19] S Ahmad B M Irons and O C Zienkiewicz ldquoAnalysis ofthick and thin shell structures by curved finite elementsrdquo Inter-national Journal for Numerical Methods in Engineering vol 2no 3 pp 419ndash451 1970

[20] T Kant S Kumar and U P Singh ldquoShell dynamics with three-dimensional degenerate finite elementsrdquo Computers and Struc-tures vol 50 no 1 pp 135ndash146 1994

[21] O C Zienkiewicz R L Taylor and JM Too ldquoReduced integra-tion technique in general analysis of plates and shellsrdquo Interna-tional Journal for Numerical Methods in Engineering vol 3 no2 pp 275ndash290 1971

[22] S F Paswey andRWClough ldquoImprovednumerical integrationof thick shell finite elementsrdquo International Journal for Numeri-cal Methods in Engineering vol 3 no 4 pp 575ndash586 1971

[23] H-C Huang Static and Dynamic Analyses of Plates and ShellsTheory Software and Applications Springer Bedford UK 1989

[24] K J Bathe Finite Element Procedures Prentice Hall NewDelhiIndia 2006

[25] D Ngo and A C Scordelis ldquo Finite element analysis of rein-forced concrete beamsrdquo ACI Journal vol 64 no 3 pp 152ndash1631967


[26] Y R Rashid ldquoUltimate strength analysis of prestressed concretepressure vesselsrdquo Nuclear Engineering and Design vol 7 no 4pp 334ndash344 1968

[27] A Scanlon andDWMurray ldquoTime dependent reinforced con-crete slab deflectionsrdquo Journal of the StructuralDivision vol 100no 9 pp 1911ndash1924 1974

[28] C-S Lin and A C Scordelis ldquoNonlinear analysis of RC shellsof general formrdquo ASCE Journal of Structural Division vol 101no 3 pp 523ndash538 1975

[29] R I Gilbert and R F Warner ldquoTension stiffening in reinforcedconcrete slabsrdquoASCE Journal of Structural Division vol 104 no12 pp 1885ndash1900 1978

[30] R Nayal and H A Rasheed ldquoTension stiffening model forconcrete beams reinforced with steel and FRP barsrdquo Journal ofMaterials in Civil Engineering vol 18 no 6 pp 831ndash841 2006

[31] D R G Owen and E Hinton Finite Elements in PlasticityTheory and Practice Pineridge Press 1980

[32] Z P Bazant and L Cedolin ldquoFinite element modeling of crackband propagationrdquo Journal of Structural Engineering vol 109no 1 pp 69ndash92 1983

[33] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press London UK 1984

[34] W F ChenPlasticity in Reinforced ConcreteMcGraw-Hill NewYork NY USA 1982

[35] K Willam and E Warnke ldquoConstitutive model for triaxialbehavior of concreterdquo in Proceedings of the International Asso-ciation for Bridge and Structural Engineering vol 19 pp 1ndash30Zurich Switzerland 1975

[36] G C Archer and T M Whalen ldquoDevelopment of rotationallyconsistent diagonalmassmatrices for plate and beam elementsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 675ndash689 2005

[37] F A Charney ldquoUnintended consequences of modelling damp-ing in structuresrdquo ASCE Journal of Structural Engineering vol134 no 4 pp 581ndash592 2008

[38] A Chopra Dynamics of Structures Theory and Application toEarthquake Engineering Prentice-Hall Englewood Cliffs NJUSA 3rd edition 2006

[39] S K Duggal Earthquake Resistant Design of Structures OxfordHigher Education Oxford University Press New Delhi India2007

[40] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2003

[41] R K Goel and A K Chopra ldquoPeriod formulas for concreteshear wall buildingsrdquo Journal of Structural Engineering vol 124no 4 pp 426ndash433 1998

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Submit your manuscripts athttpwwwhindawicom

VLSI Design



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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



opening locations on the structural response of RC shearwalls The present study analyses the response of RC shearwalls for seven different opening locations for both squatand slender shear walls with and without ductile detailing[13] for different damping ratios Moreover it was foundthat the use of conventional methods for the analysis ofshear walls with openings resulted in remarkably poor resultsespecially for the squat shear walls where the mode of failureis predominantly shear It was also shown in literature thatthe hand calculation either underestimates or overestimatesthe response in predicting the response of shear walls withopenings [5] The finite element analysis has been the mostversatile and successfully employed method of analysis inthe past to accurately predict the structural behavior ofreinforced concrete shear wall in linear as well as in non-linear range under any severe loading conditions [14] Withthe advent in computing facilities finite element methodhas gained an enormous popularity among the structuralengineering community especially in the nonlinear dynamicanalysis The nonlinearity of RC shear wall may be due togeometry or due to material Since shear wall is a hugestructure the deformation of shear wall has been assumed tobe in control and hence the geometric nonlinearity has notbeen considered The next section describes the formulationof degenerated shell element formulation

2 Degenerated Shell Element Formulation

The displacement based finite element method has beenconsidered to be the most popular choice because of itssimplicity and ease with which the computations can beperformed The use of shell element to model moderatelythick structures like shear wall is well documented in theliterature [15 16] In this section the underlying basic ideasin the formulation of the degenerated curved shell elementare described Two assumptions are made in the formulationof the curved shell element which is degenerated from three-dimensional solid First it is assumed that even for thickshells the normal to the middle surface of the elementremains straight after deformation Second the strain energycorresponding to stresses perpendicular to themiddle surfaceis disregarded that is the stress component normal to theshell mid-surface is constrained to be zero Five degrees offreedom are specified at each nodal point corresponding toits three translations and two rotations of the normal at eachnode The independent definition of the translational androtational degrees of freedom permits the transverse sheardeformation to be taken into account during the formulationof the element stiffness since rotations are not necessarilynormal to the slope of the mid-surface

Coordinate Systems The geometry of the shell can berepresented by the coordinates and normal vectors of itsmiddle surface as Figure 1 The geometry of the degeneratedshell element and kinematics of deformation are describedby using four different coordinate systems that is globalnatural local and nodal coordinate systemsThe global coor-dinate system (119909 119910 119911) is used to define the shell geometryThe shape functions are expressed in natural curvilinear

Middle surface

120578

120577

120585

k

hk

u3k

k u2k

1205732k

1205731ku1k

Figure 1 Geometry of 9-noded degenerated shell element

coordinates (120585 120578 120577) In order to easily deal with the thin shellassumption of zero normal stress in the 119911 direction the straincomponents are defined in terms of local coordinate set ofaxes (119909

1015840 1199101015840 1199111015840) At each node of shell element the nodalcoordinate set (V

1119896V2119896V3119896) with unit vectors is definedThe

four coordinate sets employed in the present formulation arenow described

Global Coordinate Set (119909 119910 119911)This is a Cartesian coordinatesystem freely chosen in relation to which the geometry ofthe structure is defined in space Nodal coordinates and dis-placements global stiffness matrix and applied force vectorare referred to this systemThe displacements correspondingto 119909 119910 and 119911 directions are 119906 V and 119908 respectively

Nodal Coordinate Set (V1119896V2119896V3119896)Anodal coordinate sys-

tem is defined at each nodal point with origin at the referencesurface (mid-surface) The vector V

3119896is constructed from

nodal coordinates at top and bottom surfaces at node 119896 andis expressed as

119881119909

3119896

119881119910

3119896

1198811199113119896

=

119909top3119896

minus 119909bot3119896

119910top3119896

minus 119910bot3119896

119911top3119896

minus 119911bot3119896

(1)

V3119896

defines the direction of the normal at any node ldquo119896rdquowhich is not necessarily perpendicular to the mid-surfaceThemajor advantage of the definition ofV

3119896with normal not

necessary to be perpendicular to mid-surface is that there areno gaps or overlaps along element boundaries

The vector V1119896

is constructed perpendicular to V3119896

andparallel to the global 119909119911 plane Hence

V1199091119896

= V1199113119896 V119910

1119896= 00 V119911

1119896= minusV1199093119896 (2)

Alternatively if the vector V3119896

is in the 119910 direction (V1199093119896

=

V1199103119896

= 00) the following expressions are assumed V1199091119896

=

minusV1199103119896 V1199101119896

= V1199111119896

= 00 (119909 direction) The superscripts referto the vector components in the global coordinate system

The vector V2119896is constructed perpendicular to the plane

defined by V1119896and V

3119896


HenceV2119896

= V1119896


of V1119896 V2119896 and V


1119896 V2119896 and


1119896 V2119896





3119896



1015840


1199111015840=

[[[[[[[

[

120597119909

120597120585120597119910

120597120585120597119911

120597120585

]]]]]]]

]

times

[[[[[[[

[

120597119909

120597120578120597119910

120597120578120597119911

120597120578

]]]]]]]

]

1199091015840=

[[[[[[[

[

120597119909

120597120585120597119910

120597120585120597119911

120597120585

]]]]]]]

]

1199101015840= 1199111015840times 1199091015840

(3)


119909

119910

119911

=

9

sum119896=1

119873119896(120585 120578)

119909mid119896

119910mid119896

119911mid119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟mid-surface only

+

9

sum119896=1

119873119896(120585 120578)

120577ℎ119896

2

1198811199093119896

119881119910

3119896

1198811199113119896


(4)

where 119909mid119896

119910mid119896

and 119911mid119896






3119896


1198731=

1

4120585 (1 + 120585) 120578 (1 + 120578)

1198732=

1

2(1 + 120585) (1 minus 120585) 120578 (1 + 120578)

1198733= minus

1

4(1 minus 120585) 120578 (1 + 120578)

1198734= minus

1

2120585 (1 minus 120585) (1 + 120578) (1 minus 120578)

1198735=

1

4(1 minus 120585) 120578 (1 minus 120578)

1198736= minus

1

2(1 + 120585) (1 minus 120585) 120578 (1 minus 120578)

1198737= minus

1

4120585 (1 + 120585) 120578 (1 minus 120578)

1198738=

1

2120585 (1 + 120585) (1 + 120578) (1 minus 120578)

1198739= (1 + 120585) (1 minus 120585) (1 + 120578) (1 minus 120578)


mid119896

Vmid119896

and 119908mid119896


119906

V119908

=

119899

sum119896=1

119873119896

119906mid119896

Vmid119896

119908mid119896

+

119899

sum119896=1

119873119896120577ℎ119896

2

[[[

[

V1199091119896

minusV1199092119896

V1199101119896

minusV1199102119896

V1199111119896

minusV1199112119896

]]]

]

[1205731119896

1205732119896

]

(6)where 120573

1119896and 120573



and 1198812119896


1119896 V2119896 and V

3119896is


and V2119896


1119896= 119894 times V

3119896 V1119896

= V1119894


1119896

V2119896 and V




120576 = 1205761015840119891

1205761015840119904

=

1205761199091015840

1205761199101015840

12057411990910158401199101015840

12057411990910158401199111015840

12057411991010158401199111015840

=

1205971199061015840

1205971199091015840

120597V1015840

1205971199101015840

1205971199061015840

1205971199101015840+

120597V1015840

1205971199091015840

1205971199061015840

1205971199111015840+

1205971199081015840

1205971199091015840

120597V1015840

1205971199111015840+

1205971199081015840

1205971199101015840

(7)




119891can further


119887

as

1205761015840

119891= 1205761015840

119898+ 1205761015840

119887 (8)

where

1205761015840

119898=

[[[[[[[[[

[

1205971199061015840

1205971199091015840

120597V1015840

1205971199101015840

1205971199061015840

1205971199101015840+

120597V1015840

1205971199091015840

]]]]]]]]]

]

1205761015840

119887=

[[[[[[[[[[

[

11991110158401205971205791199091015840

1205971199091015840

11991110158401205971205791199101015840

1205971199101015840

1199111015840 (1205971205791199091015840

1205971199101015840

) + (1205971205791199101015840

1205971199091015840)

]]]]]]]]]]

]

(9)


[[[[[[[[

[

1205971199061015840

1205971199091015840120597V1015840

12059711990910158401205971199081015840

1205971199091015840

1205971199061015840

1205971199101015840120597V1015840

12059711991010158401205971199081015840

1205971199101015840

1205971199061015840

1205971199111015840120597V1015840

12059711991110158401205971199081015840

1205971199111015840

]]]]]]]]

]

= [T]119879

[[[[[[[[

[

120597119906

120597119909

120597V120597119909

120597119908

120597119909

120597119906

120597119910

120597V120597119910

120597119908

120597119910

120597119906

120597119911

120597V120597119911

120597119908

120597119911

]]]]]]]]

]

[T] (10)


T =

[[[[[[[[

[

120597119909

1205971199091015840120597119909

1205971199101015840120597119909

1205971199111015840

120597119910

1205971199101015840120597119910

1205971199101015840120597119910

1205971199111015840

120597119911

1205971199111015840120597119911

1205971199101015840120597119911

1205971199111015840

]]]]]]]]

]

(11)


[[[[[[[[

[

120597119906

120597119909

120597V120597119909

120597119908

120597119909

120597119906

120597119910

120597V120597119910

120597119908

120597119910

120597119906

120597119911

120597V120597119911

120597119908

120597119911

]]]]]]]]

]

= Jminus1

[[[[[[[[

[

120597119906

120597120585

120597V120597120585

120597119908

120597120585

120597119906

120597120578

120597V120597120578

120597119908

120597120578

120597119906

120597120577

120597V120597120577

120597119908

120597120577

]]]]]]]]

]

(12)


J =

[[[[[[[[

[

120597119909

120597120585

120597119910

120597120585

120597119911

120597120585

120597119909

120597120578

120597119910

120597120578

120597119911

120597120578

120597119909

120597

120597119910

120597120589

120597119911

120597120589

]]]]]]]]

]

(13)


120576 = B120575 (14)

where

120575 = 119906 V 119908 1205731

1205732119879

(15)


K119890 = ∭B119879DB 119889119881 (16)



K119890 = ∬B119879DB |J| 119889120577 119889119860 (17)


119891119890= ∬B119879120590 |J| 119889120577 119889119860 (18)


119891 = [119870] 120575 (19)

where

int119889119860=|J|∬+1

minus1


(20)




65

6 1

25

34

1radic3 1radic3

120585

120578

For 120574120585120577

(a)

1

456

23

1radic3

1radic3

120585

For 120574120578

120578

120577

(b)




120574120585120577 =

3

sum119894=1

2

sum119895=1

119875119894(120578) sdot 119876

119895(120585) 120574119894119895

120585120577

120574120578120577

=

3

sum119894=1

2

sum119895=1

119875119894(120585) sdot 119876

119895(120578) 120574119894119895

120578120589

(21)

120574119894119895

120585120589and 120574

119894119895



119895(119911)

are

1198751(119911) =

119911

2(119911 + 1) 119875

2(119911) = 1 minus 119911

2

1198753(119911) =

119911

2(119911 minus 1)

1198761(119911) =

1

2(1 + radic3119911) 119876

2(119911) =

1


(22)






120581120585=

120597120579120585

120597120585(1 120585 120578 120585120578 120585

2 1205852120578 1205782 1205851205782 12058521205782)

120581120585= 120581120585(1 120585 120578 120585120578 120578

2 1205851205782)

120581120578=

120597120579120578

120597120578(1 120585 120578 120585120578 120585

2 1205852120578 1205782 1205851205782 12058521205782)

120581120578= (1 120585 120578 120585120578 120585

2 12058521205782)

120574120585120577

= 120574120585120577

(120581120585) = 120574120585120577

(1 120585 120578 120585120578 1205782 1205851205782)

120574120578120577

= 120574120578120577

(120581120578) = 120574120578120577

(1 120585 120578 120585120578 1205782 1205851205782)

(23)


120585120577and 120574120578120577are

120574120585120577

= 120579120585+

120597119908

120597120585= 120574120585120577

(1 120585 120578 120585120578 1205852 1205852120578 1205782 1205851205782 12058521205782)

120574120578120577

= 120579120578+

120597119908

120597120578= 120574120578120577

(1 120585 120578 120585120578 1205852 1205852120578 1205782 1205851205782 12058521205782)

(24)


120587 = 120587 + int12058213

(120574120585120577

minus 120574120585120577) 119889119881 + int120582

23(120574120578120577

minus 120574120578120577) 119889119881 (25)


120585120577and 120574






and its weights








120574120585120577

=

119899

sum119894=1

119877119894(120585 120578) 120574

119894

120585120577 120574

120578120577=

119899

sum119894=1

119878119894(120585 120578) 120574

119894

120578120577 (26)


12058213

=

119899

sum119894=1

12058213

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

12058223

=

119899

sum119894=1

12058223

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

(27)


120574120585120577

(120585119894 120578119894) = 120574120585120577

(120585119894 120578119894)

120574120578120577

(120585119894 120578119894) = 120574120578120577

(120585119894 120578119894)

(28)

120579120585and 120579






3 Material Modeling







119864119894= 1205721198911015840

119905(1 minus

120576119894

120576119898

)1

120576119894

120576119905le 120576119894le 120576119898 (29)

120572 and 120576119898





119898


Compression

Tension

Stre

ss

Strain

ft

120572ft

120590i

E

120576t 120576i 120576m





[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[

[

0 0 0 0 0

0 119864 0 0 0

0 0 119866119888

120 0

0 0 0 11986611988813

0

0 0 0 0 11986623

]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

12= 025 times 119866(1 minus

1

0004) if 120576

1lt 0004

119866119888

12= 0 if 120576

1ge 0004

119866119888

13= 119866119888

12 119866

23=

5119866

6

(30)


[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0119866119888

12

20 0

0 0 0 119866119888

130

0 0 0 0 119866119888

23

]]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

13= 025 times 119866(1 minus

1205761

0004) if 120576

1lt 0004

119866119888

13= 0 if 120576

1ge 0004


120591oct120579 = 60∘

120579 = 0∘

120590oct


Smooth 1205901

1205902 1205903



119866119888

23= 025 times 119866(1 minus

1205762

0004)

119866119888

12= 05 times 119866

119888

13if 119866119888

23lt 119866119888

13

(31)




and 120591119898119888



119898

as follows

120591119898119905

1198911015840119888

=120588119905

radic51198911015840119888

= 1198860+ 1198861(

120590119898

1198911015840119888

) + 1198862(

120590119898

1198911015840119888

)

2

120579 = 0∘

120591119898119888

1198911015840119888

=120588119888

radic51198911015840119888

= 1198870+ 1198871(

120590119898

1198911015840119888

) + 1198872(

120590119898

1198911015840119888

)

2

120579 = 60∘

(32)




0or 1198870 1198861 1198862 1198871 1198872) are


119891 (120590119898 120591119898 120579) = radic5

120591119898

120588 (120590119898 120579)

minus 1 = 0

120588 (120579) = (2120588119888(1205882

119888minus 1205882

119905) cos 120579 + 120588

119888(2120588119905minus 120588119888)

times [4 (1205882

119888minus 1205882

119905) cos2120579 + 5120588

2

119905minus 4120588119905120588119888]12

)

times (4 (1205882

119888minus 1205882

119905) cos2120579 + (120588

119888minus 2120588119905)2

)minus1

(33)



119889120576119901

119894119895= 119889120582

120597119891 (120590)

120597120590119894119895

(34)




Failure surface

Failure

1205901

1205902 120590

120576






120590 = 1198640120576 minus

1

2

1198640

1205760

1205762 (35)





= [119863119890] ( 120576 minus 120576

119901)

120576119901 = 119886 119886 =

120597119865

120597120590

(36)




119886119879

= 0 119886119879[119863119890] ( 120576 minus 120576

119901) = 0

119886119879[119863119890] ( 120576 minus 119886) = 0

=119886119879[119863119890] 120576

119886119879[119863119890] 119886

(38)


[119863119890119901] = [119863] minus

[119863] 119886 119886119879[119863]

119867 + 119886119879[119863] 119886

(39)






ress

Strain


fy Esh

Es

120576y



[119872] [119905] + [119862] [

119905] + [119870] [119880

119905] = [119877

119905] (40)


Δ119905 le Δ119905cr =119879119899

120587=

2

120596 (41)





119890) is given by

M119890= ∭

119881

120588119898119873119894119873119894119889119881 (42)





[119872119888

119890] =

[[[[[

[

1198681

0 0 0 1198682

0 1198681

0 minus1198682

0

0 0 1198681

0 0

0 minus1198682

0 1198683

0

1198682

0 0 0 1198683

]]]]]

]

(43)

[119872119871

119890] =

[[[[[

[

1198681198711

0 0 0 0

0 1198681198711

0 0 0

0 0 1198681198711

0 0

0 0 0 1198681198712

0

0 0 0 0 1198681198712

]]]]]

]

(44)









3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass




119872 + 119862 + 119870119906 = 119877 (119905) (45)


119906 = 119888119890119904119905 (46)


119904 =minus119888


119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0




(119888

2119898)2

minus 1205962= 0


997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0




(47a)


120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)


[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)



ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










HenceV2119896

= V1119896


of V1119896 V2119896 and V


1119896 V2119896 and


1119896 V2119896





3119896



1015840


1199111015840=

[[[[[[[

[

120597119909

120597120585120597119910

120597120585120597119911

120597120585

]]]]]]]

]

times

[[[[[[[

[

120597119909

120597120578120597119910

120597120578120597119911

120597120578

]]]]]]]

]

1199091015840=

[[[[[[[

[

120597119909

120597120585120597119910

120597120585120597119911

120597120585

]]]]]]]

]

1199101015840= 1199111015840times 1199091015840

(3)


119909

119910

119911

=

9

sum119896=1

119873119896(120585 120578)

119909mid119896

119910mid119896

119911mid119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟mid-surface only

+

9

sum119896=1

119873119896(120585 120578)

120577ℎ119896

2

1198811199093119896

119881119910

3119896

1198811199113119896


(4)

where 119909mid119896

119910mid119896

and 119911mid119896






3119896


1198731=

1

4120585 (1 + 120585) 120578 (1 + 120578)

1198732=

1

2(1 + 120585) (1 minus 120585) 120578 (1 + 120578)

1198733= minus

1

4(1 minus 120585) 120578 (1 + 120578)

1198734= minus

1

2120585 (1 minus 120585) (1 + 120578) (1 minus 120578)

1198735=

1

4(1 minus 120585) 120578 (1 minus 120578)

1198736= minus

1

2(1 + 120585) (1 minus 120585) 120578 (1 minus 120578)

1198737= minus

1

4120585 (1 + 120585) 120578 (1 minus 120578)

1198738=

1

2120585 (1 + 120585) (1 + 120578) (1 minus 120578)

1198739= (1 + 120585) (1 minus 120585) (1 + 120578) (1 minus 120578)


mid119896

Vmid119896

and 119908mid119896


119906

V119908

=

119899

sum119896=1

119873119896

119906mid119896

Vmid119896

119908mid119896

+

119899

sum119896=1

119873119896120577ℎ119896

2

[[[

[

V1199091119896

minusV1199092119896

V1199101119896

minusV1199102119896

V1199111119896

minusV1199112119896

]]]

]

[1205731119896

1205732119896

]

(6)where 120573

1119896and 120573



and 1198812119896


1119896 V2119896 and V

3119896is


and V2119896


1119896= 119894 times V

3119896 V1119896

= V1119894


1119896

V2119896 and V




120576 = 1205761015840119891

1205761015840119904

=

1205761199091015840

1205761199101015840

12057411990910158401199101015840

12057411990910158401199111015840

12057411991010158401199111015840

=

1205971199061015840

1205971199091015840

120597V1015840

1205971199101015840

1205971199061015840

1205971199101015840+

120597V1015840

1205971199091015840

1205971199061015840

1205971199111015840+

1205971199081015840

1205971199091015840

120597V1015840

1205971199111015840+

1205971199081015840

1205971199101015840

(7)




119891can further


119887

as

1205761015840

119891= 1205761015840

119898+ 1205761015840

119887 (8)

where

1205761015840

119898=

[[[[[[[[[

[

1205971199061015840

1205971199091015840

120597V1015840

1205971199101015840

1205971199061015840

1205971199101015840+

120597V1015840

1205971199091015840

]]]]]]]]]

]

1205761015840

119887=

[[[[[[[[[[

[

11991110158401205971205791199091015840

1205971199091015840

11991110158401205971205791199101015840

1205971199101015840

1199111015840 (1205971205791199091015840

1205971199101015840

) + (1205971205791199101015840

1205971199091015840)

]]]]]]]]]]

]

(9)


[[[[[[[[

[

1205971199061015840

1205971199091015840120597V1015840

12059711990910158401205971199081015840

1205971199091015840

1205971199061015840

1205971199101015840120597V1015840

12059711991010158401205971199081015840

1205971199101015840

1205971199061015840

1205971199111015840120597V1015840

12059711991110158401205971199081015840

1205971199111015840

]]]]]]]]

]

= [T]119879

[[[[[[[[

[

120597119906

120597119909

120597V120597119909

120597119908

120597119909

120597119906

120597119910

120597V120597119910

120597119908

120597119910

120597119906

120597119911

120597V120597119911

120597119908

120597119911

]]]]]]]]

]

[T] (10)


T =

[[[[[[[[

[

120597119909

1205971199091015840120597119909

1205971199101015840120597119909

1205971199111015840

120597119910

1205971199101015840120597119910

1205971199101015840120597119910

1205971199111015840

120597119911

1205971199111015840120597119911

1205971199101015840120597119911

1205971199111015840

]]]]]]]]

]

(11)


[[[[[[[[

[

120597119906

120597119909

120597V120597119909

120597119908

120597119909

120597119906

120597119910

120597V120597119910

120597119908

120597119910

120597119906

120597119911

120597V120597119911

120597119908

120597119911

]]]]]]]]

]

= Jminus1

[[[[[[[[

[

120597119906

120597120585

120597V120597120585

120597119908

120597120585

120597119906

120597120578

120597V120597120578

120597119908

120597120578

120597119906

120597120577

120597V120597120577

120597119908

120597120577

]]]]]]]]

]

(12)


J =

[[[[[[[[

[

120597119909

120597120585

120597119910

120597120585

120597119911

120597120585

120597119909

120597120578

120597119910

120597120578

120597119911

120597120578

120597119909

120597

120597119910

120597120589

120597119911

120597120589

]]]]]]]]

]

(13)


120576 = B120575 (14)

where

120575 = 119906 V 119908 1205731

1205732119879

(15)


K119890 = ∭B119879DB 119889119881 (16)



K119890 = ∬B119879DB |J| 119889120577 119889119860 (17)


119891119890= ∬B119879120590 |J| 119889120577 119889119860 (18)


119891 = [119870] 120575 (19)

where

int119889119860=|J|∬+1

minus1


(20)




65

6 1

25

34

1radic3 1radic3

120585

120578

For 120574120585120577

(a)

1

456

23

1radic3

1radic3

120585

For 120574120578

120578

120577

(b)




120574120585120577 =

3

sum119894=1

2

sum119895=1

119875119894(120578) sdot 119876

119895(120585) 120574119894119895

120585120577

120574120578120577

=

3

sum119894=1

2

sum119895=1

119875119894(120585) sdot 119876

119895(120578) 120574119894119895

120578120589

(21)

120574119894119895

120585120589and 120574

119894119895



119895(119911)

are

1198751(119911) =

119911

2(119911 + 1) 119875

2(119911) = 1 minus 119911

2

1198753(119911) =

119911

2(119911 minus 1)

1198761(119911) =

1

2(1 + radic3119911) 119876

2(119911) =

1


(22)






120581120585=

120597120579120585

120597120585(1 120585 120578 120585120578 120585

2 1205852120578 1205782 1205851205782 12058521205782)

120581120585= 120581120585(1 120585 120578 120585120578 120578

2 1205851205782)

120581120578=

120597120579120578

120597120578(1 120585 120578 120585120578 120585

2 1205852120578 1205782 1205851205782 12058521205782)

120581120578= (1 120585 120578 120585120578 120585

2 12058521205782)

120574120585120577

= 120574120585120577

(120581120585) = 120574120585120577

(1 120585 120578 120585120578 1205782 1205851205782)

120574120578120577

= 120574120578120577

(120581120578) = 120574120578120577

(1 120585 120578 120585120578 1205782 1205851205782)

(23)


120585120577and 120574120578120577are

120574120585120577

= 120579120585+

120597119908

120597120585= 120574120585120577

(1 120585 120578 120585120578 1205852 1205852120578 1205782 1205851205782 12058521205782)

120574120578120577

= 120579120578+

120597119908

120597120578= 120574120578120577

(1 120585 120578 120585120578 1205852 1205852120578 1205782 1205851205782 12058521205782)

(24)


120587 = 120587 + int12058213

(120574120585120577

minus 120574120585120577) 119889119881 + int120582

23(120574120578120577

minus 120574120578120577) 119889119881 (25)


120585120577and 120574






and its weights








120574120585120577

=

119899

sum119894=1

119877119894(120585 120578) 120574

119894

120585120577 120574

120578120577=

119899

sum119894=1

119878119894(120585 120578) 120574

119894

120578120577 (26)


12058213

=

119899

sum119894=1

12058213

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

12058223

=

119899

sum119894=1

12058223

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

(27)


120574120585120577

(120585119894 120578119894) = 120574120585120577

(120585119894 120578119894)

120574120578120577

(120585119894 120578119894) = 120574120578120577

(120585119894 120578119894)

(28)

120579120585and 120579






3 Material Modeling







119864119894= 1205721198911015840

119905(1 minus

120576119894

120576119898

)1

120576119894

120576119905le 120576119894le 120576119898 (29)

120572 and 120576119898





119898


Compression

Tension

Stre

ss

Strain

ft

120572ft

120590i

E

120576t 120576i 120576m





[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[

[

0 0 0 0 0

0 119864 0 0 0

0 0 119866119888

120 0

0 0 0 11986611988813

0

0 0 0 0 11986623

]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

12= 025 times 119866(1 minus

1

0004) if 120576

1lt 0004

119866119888

12= 0 if 120576

1ge 0004

119866119888

13= 119866119888

12 119866

23=

5119866

6

(30)


[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0119866119888

12

20 0

0 0 0 119866119888

130

0 0 0 0 119866119888

23

]]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

13= 025 times 119866(1 minus

1205761

0004) if 120576

1lt 0004

119866119888

13= 0 if 120576

1ge 0004


120591oct120579 = 60∘

120579 = 0∘

120590oct


Smooth 1205901

1205902 1205903



119866119888

23= 025 times 119866(1 minus

1205762

0004)

119866119888

12= 05 times 119866

119888

13if 119866119888

23lt 119866119888

13

(31)




and 120591119898119888



119898

as follows

120591119898119905

1198911015840119888

=120588119905

radic51198911015840119888

= 1198860+ 1198861(

120590119898

1198911015840119888

) + 1198862(

120590119898

1198911015840119888

)

2

120579 = 0∘

120591119898119888

1198911015840119888

=120588119888

radic51198911015840119888

= 1198870+ 1198871(

120590119898

1198911015840119888

) + 1198872(

120590119898

1198911015840119888

)

2

120579 = 60∘

(32)




0or 1198870 1198861 1198862 1198871 1198872) are


119891 (120590119898 120591119898 120579) = radic5

120591119898

120588 (120590119898 120579)

minus 1 = 0

120588 (120579) = (2120588119888(1205882

119888minus 1205882

119905) cos 120579 + 120588

119888(2120588119905minus 120588119888)

times [4 (1205882

119888minus 1205882

119905) cos2120579 + 5120588

2

119905minus 4120588119905120588119888]12

)

times (4 (1205882

119888minus 1205882

119905) cos2120579 + (120588

119888minus 2120588119905)2

)minus1

(33)



119889120576119901

119894119895= 119889120582

120597119891 (120590)

120597120590119894119895

(34)




Failure surface

Failure

1205901

1205902 120590

120576






120590 = 1198640120576 minus

1

2

1198640

1205760

1205762 (35)





= [119863119890] ( 120576 minus 120576

119901)

120576119901 = 119886 119886 =

120597119865

120597120590

(36)




119886119879

= 0 119886119879[119863119890] ( 120576 minus 120576

119901) = 0

119886119879[119863119890] ( 120576 minus 119886) = 0

=119886119879[119863119890] 120576

119886119879[119863119890] 119886

(38)


[119863119890119901] = [119863] minus

[119863] 119886 119886119879[119863]

119867 + 119886119879[119863] 119886

(39)






ress

Strain


fy Esh

Es

120576y



[119872] [119905] + [119862] [

119905] + [119870] [119880

119905] = [119877

119905] (40)


Δ119905 le Δ119905cr =119879119899

120587=

2

120596 (41)





119890) is given by

M119890= ∭

119881

120588119898119873119894119873119894119889119881 (42)





[119872119888

119890] =

[[[[[

[

1198681

0 0 0 1198682

0 1198681

0 minus1198682

0

0 0 1198681

0 0

0 minus1198682

0 1198683

0

1198682

0 0 0 1198683

]]]]]

]

(43)

[119872119871

119890] =

[[[[[

[

1198681198711

0 0 0 0

0 1198681198711

0 0 0

0 0 1198681198711

0 0

0 0 0 1198681198712

0

0 0 0 0 1198681198712

]]]]]

]

(44)









3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass




119872 + 119862 + 119870119906 = 119877 (119905) (45)


119906 = 119888119890119904119905 (46)


119904 =minus119888


119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0




(119888

2119898)2

minus 1205962= 0


997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0




(47a)


120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)


[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)



ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119891can further


119887

as

1205761015840

119891= 1205761015840

119898+ 1205761015840

119887 (8)

where

1205761015840

119898=

[[[[[[[[[

[

1205971199061015840

1205971199091015840

120597V1015840

1205971199101015840

1205971199061015840

1205971199101015840+

120597V1015840

1205971199091015840

]]]]]]]]]

]

1205761015840

119887=

[[[[[[[[[[

[

11991110158401205971205791199091015840

1205971199091015840

11991110158401205971205791199101015840

1205971199101015840

1199111015840 (1205971205791199091015840

1205971199101015840

) + (1205971205791199101015840

1205971199091015840)

]]]]]]]]]]

]

(9)


[[[[[[[[

[

1205971199061015840

1205971199091015840120597V1015840

12059711990910158401205971199081015840

1205971199091015840

1205971199061015840

1205971199101015840120597V1015840

12059711991010158401205971199081015840

1205971199101015840

1205971199061015840

1205971199111015840120597V1015840

12059711991110158401205971199081015840

1205971199111015840

]]]]]]]]

]

= [T]119879

[[[[[[[[

[

120597119906

120597119909

120597V120597119909

120597119908

120597119909

120597119906

120597119910

120597V120597119910

120597119908

120597119910

120597119906

120597119911

120597V120597119911

120597119908

120597119911

]]]]]]]]

]

[T] (10)


T =

[[[[[[[[

[

120597119909

1205971199091015840120597119909

1205971199101015840120597119909

1205971199111015840

120597119910

1205971199101015840120597119910

1205971199101015840120597119910

1205971199111015840

120597119911

1205971199111015840120597119911

1205971199101015840120597119911

1205971199111015840

]]]]]]]]

]

(11)


[[[[[[[[

[

120597119906

120597119909

120597V120597119909

120597119908

120597119909

120597119906

120597119910

120597V120597119910

120597119908

120597119910

120597119906

120597119911

120597V120597119911

120597119908

120597119911

]]]]]]]]

]

= Jminus1

[[[[[[[[

[

120597119906

120597120585

120597V120597120585

120597119908

120597120585

120597119906

120597120578

120597V120597120578

120597119908

120597120578

120597119906

120597120577

120597V120597120577

120597119908

120597120577

]]]]]]]]

]

(12)


J =

[[[[[[[[

[

120597119909

120597120585

120597119910

120597120585

120597119911

120597120585

120597119909

120597120578

120597119910

120597120578

120597119911

120597120578

120597119909

120597

120597119910

120597120589

120597119911

120597120589

]]]]]]]]

]

(13)


120576 = B120575 (14)

where

120575 = 119906 V 119908 1205731

1205732119879

(15)


K119890 = ∭B119879DB 119889119881 (16)



K119890 = ∬B119879DB |J| 119889120577 119889119860 (17)


119891119890= ∬B119879120590 |J| 119889120577 119889119860 (18)


119891 = [119870] 120575 (19)

where

int119889119860=|J|∬+1

minus1


(20)




65

6 1

25

34

1radic3 1radic3

120585

120578

For 120574120585120577

(a)

1

456

23

1radic3

1radic3

120585

For 120574120578

120578

120577

(b)




120574120585120577 =

3

sum119894=1

2

sum119895=1

119875119894(120578) sdot 119876

119895(120585) 120574119894119895

120585120577

120574120578120577

=

3

sum119894=1

2

sum119895=1

119875119894(120585) sdot 119876

119895(120578) 120574119894119895

120578120589

(21)

120574119894119895

120585120589and 120574

119894119895



119895(119911)

are

1198751(119911) =

119911

2(119911 + 1) 119875

2(119911) = 1 minus 119911

2

1198753(119911) =

119911

2(119911 minus 1)

1198761(119911) =

1

2(1 + radic3119911) 119876

2(119911) =

1


(22)






120581120585=

120597120579120585

120597120585(1 120585 120578 120585120578 120585

2 1205852120578 1205782 1205851205782 12058521205782)

120581120585= 120581120585(1 120585 120578 120585120578 120578

2 1205851205782)

120581120578=

120597120579120578

120597120578(1 120585 120578 120585120578 120585

2 1205852120578 1205782 1205851205782 12058521205782)

120581120578= (1 120585 120578 120585120578 120585

2 12058521205782)

120574120585120577

= 120574120585120577

(120581120585) = 120574120585120577

(1 120585 120578 120585120578 1205782 1205851205782)

120574120578120577

= 120574120578120577

(120581120578) = 120574120578120577

(1 120585 120578 120585120578 1205782 1205851205782)

(23)


120585120577and 120574120578120577are

120574120585120577

= 120579120585+

120597119908

120597120585= 120574120585120577

(1 120585 120578 120585120578 1205852 1205852120578 1205782 1205851205782 12058521205782)

120574120578120577

= 120579120578+

120597119908

120597120578= 120574120578120577

(1 120585 120578 120585120578 1205852 1205852120578 1205782 1205851205782 12058521205782)

(24)


120587 = 120587 + int12058213

(120574120585120577

minus 120574120585120577) 119889119881 + int120582

23(120574120578120577

minus 120574120578120577) 119889119881 (25)


120585120577and 120574






and its weights








120574120585120577

=

119899

sum119894=1

119877119894(120585 120578) 120574

119894

120585120577 120574

120578120577=

119899

sum119894=1

119878119894(120585 120578) 120574

119894

120578120577 (26)


12058213

=

119899

sum119894=1

12058213

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

12058223

=

119899

sum119894=1

12058223

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

(27)


120574120585120577

(120585119894 120578119894) = 120574120585120577

(120585119894 120578119894)

120574120578120577

(120585119894 120578119894) = 120574120578120577

(120585119894 120578119894)

(28)

120579120585and 120579






3 Material Modeling







119864119894= 1205721198911015840

119905(1 minus

120576119894

120576119898

)1

120576119894

120576119905le 120576119894le 120576119898 (29)

120572 and 120576119898





119898


Compression

Tension

Stre

ss

Strain

ft

120572ft

120590i

E

120576t 120576i 120576m





[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[

[

0 0 0 0 0

0 119864 0 0 0

0 0 119866119888

120 0

0 0 0 11986611988813

0

0 0 0 0 11986623

]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

12= 025 times 119866(1 minus

1

0004) if 120576

1lt 0004

119866119888

12= 0 if 120576

1ge 0004

119866119888

13= 119866119888

12 119866

23=

5119866

6

(30)


[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0119866119888

12

20 0

0 0 0 119866119888

130

0 0 0 0 119866119888

23

]]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

13= 025 times 119866(1 minus

1205761

0004) if 120576

1lt 0004

119866119888

13= 0 if 120576

1ge 0004


120591oct120579 = 60∘

120579 = 0∘

120590oct


Smooth 1205901

1205902 1205903



119866119888

23= 025 times 119866(1 minus

1205762

0004)

119866119888

12= 05 times 119866

119888

13if 119866119888

23lt 119866119888

13

(31)




and 120591119898119888



119898

as follows

120591119898119905

1198911015840119888

=120588119905

radic51198911015840119888

= 1198860+ 1198861(

120590119898

1198911015840119888

) + 1198862(

120590119898

1198911015840119888

)

2

120579 = 0∘

120591119898119888

1198911015840119888

=120588119888

radic51198911015840119888

= 1198870+ 1198871(

120590119898

1198911015840119888

) + 1198872(

120590119898

1198911015840119888

)

2

120579 = 60∘

(32)




0or 1198870 1198861 1198862 1198871 1198872) are


119891 (120590119898 120591119898 120579) = radic5

120591119898

120588 (120590119898 120579)

minus 1 = 0

120588 (120579) = (2120588119888(1205882

119888minus 1205882

119905) cos 120579 + 120588

119888(2120588119905minus 120588119888)

times [4 (1205882

119888minus 1205882

119905) cos2120579 + 5120588

2

119905minus 4120588119905120588119888]12

)

times (4 (1205882

119888minus 1205882

119905) cos2120579 + (120588

119888minus 2120588119905)2

)minus1

(33)



119889120576119901

119894119895= 119889120582

120597119891 (120590)

120597120590119894119895

(34)




Failure surface

Failure

1205901

1205902 120590

120576






120590 = 1198640120576 minus

1

2

1198640

1205760

1205762 (35)





= [119863119890] ( 120576 minus 120576

119901)

120576119901 = 119886 119886 =

120597119865

120597120590

(36)




119886119879

= 0 119886119879[119863119890] ( 120576 minus 120576

119901) = 0

119886119879[119863119890] ( 120576 minus 119886) = 0

=119886119879[119863119890] 120576

119886119879[119863119890] 119886

(38)


[119863119890119901] = [119863] minus

[119863] 119886 119886119879[119863]

119867 + 119886119879[119863] 119886

(39)






ress

Strain


fy Esh

Es

120576y



[119872] [119905] + [119862] [

119905] + [119870] [119880

119905] = [119877

119905] (40)


Δ119905 le Δ119905cr =119879119899

120587=

2

120596 (41)





119890) is given by

M119890= ∭

119881

120588119898119873119894119873119894119889119881 (42)





[119872119888

119890] =

[[[[[

[

1198681

0 0 0 1198682

0 1198681

0 minus1198682

0

0 0 1198681

0 0

0 minus1198682

0 1198683

0

1198682

0 0 0 1198683

]]]]]

]

(43)

[119872119871

119890] =

[[[[[

[

1198681198711

0 0 0 0

0 1198681198711

0 0 0

0 0 1198681198711

0 0

0 0 0 1198681198712

0

0 0 0 0 1198681198712

]]]]]

]

(44)









3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass




119872 + 119862 + 119870119906 = 119877 (119905) (45)


119906 = 119888119890119904119905 (46)


119904 =minus119888


119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0




(119888

2119898)2

minus 1205962= 0


997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0




(47a)


120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)


[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)



ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










65

6 1

25

34

1radic3 1radic3

120585

120578

For 120574120585120577

(a)

1

456

23

1radic3

1radic3

120585

For 120574120578

120578

120577

(b)




120574120585120577 =

3

sum119894=1

2

sum119895=1

119875119894(120578) sdot 119876

119895(120585) 120574119894119895

120585120577

120574120578120577

=

3

sum119894=1

2

sum119895=1

119875119894(120585) sdot 119876

119895(120578) 120574119894119895

120578120589

(21)

120574119894119895

120585120589and 120574

119894119895



119895(119911)

are

1198751(119911) =

119911

2(119911 + 1) 119875

2(119911) = 1 minus 119911

2

1198753(119911) =

119911

2(119911 minus 1)

1198761(119911) =

1

2(1 + radic3119911) 119876

2(119911) =

1


(22)






120581120585=

120597120579120585

120597120585(1 120585 120578 120585120578 120585

2 1205852120578 1205782 1205851205782 12058521205782)

120581120585= 120581120585(1 120585 120578 120585120578 120578

2 1205851205782)

120581120578=

120597120579120578

120597120578(1 120585 120578 120585120578 120585

2 1205852120578 1205782 1205851205782 12058521205782)

120581120578= (1 120585 120578 120585120578 120585

2 12058521205782)

120574120585120577

= 120574120585120577

(120581120585) = 120574120585120577

(1 120585 120578 120585120578 1205782 1205851205782)

120574120578120577

= 120574120578120577

(120581120578) = 120574120578120577

(1 120585 120578 120585120578 1205782 1205851205782)

(23)


120585120577and 120574120578120577are

120574120585120577

= 120579120585+

120597119908

120597120585= 120574120585120577

(1 120585 120578 120585120578 1205852 1205852120578 1205782 1205851205782 12058521205782)

120574120578120577

= 120579120578+

120597119908

120597120578= 120574120578120577

(1 120585 120578 120585120578 1205852 1205852120578 1205782 1205851205782 12058521205782)

(24)


120587 = 120587 + int12058213

(120574120585120577

minus 120574120585120577) 119889119881 + int120582

23(120574120578120577

minus 120574120578120577) 119889119881 (25)


120585120577and 120574






and its weights








120574120585120577

=

119899

sum119894=1

119877119894(120585 120578) 120574

119894

120585120577 120574

120578120577=

119899

sum119894=1

119878119894(120585 120578) 120574

119894

120578120577 (26)


12058213

=

119899

sum119894=1

12058213

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

12058223

=

119899

sum119894=1

12058223

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

(27)


120574120585120577

(120585119894 120578119894) = 120574120585120577

(120585119894 120578119894)

120574120578120577

(120585119894 120578119894) = 120574120578120577

(120585119894 120578119894)

(28)

120579120585and 120579






3 Material Modeling







119864119894= 1205721198911015840

119905(1 minus

120576119894

120576119898

)1

120576119894

120576119905le 120576119894le 120576119898 (29)

120572 and 120576119898





119898


Compression

Tension

Stre

ss

Strain

ft

120572ft

120590i

E

120576t 120576i 120576m





[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[

[

0 0 0 0 0

0 119864 0 0 0

0 0 119866119888

120 0

0 0 0 11986611988813

0

0 0 0 0 11986623

]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

12= 025 times 119866(1 minus

1

0004) if 120576

1lt 0004

119866119888

12= 0 if 120576

1ge 0004

119866119888

13= 119866119888

12 119866

23=

5119866

6

(30)


[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0119866119888

12

20 0

0 0 0 119866119888

130

0 0 0 0 119866119888

23

]]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

13= 025 times 119866(1 minus

1205761

0004) if 120576

1lt 0004

119866119888

13= 0 if 120576

1ge 0004


120591oct120579 = 60∘

120579 = 0∘

120590oct


Smooth 1205901

1205902 1205903



119866119888

23= 025 times 119866(1 minus

1205762

0004)

119866119888

12= 05 times 119866

119888

13if 119866119888

23lt 119866119888

13

(31)




and 120591119898119888



119898

as follows

120591119898119905

1198911015840119888

=120588119905

radic51198911015840119888

= 1198860+ 1198861(

120590119898

1198911015840119888

) + 1198862(

120590119898

1198911015840119888

)

2

120579 = 0∘

120591119898119888

1198911015840119888

=120588119888

radic51198911015840119888

= 1198870+ 1198871(

120590119898

1198911015840119888

) + 1198872(

120590119898

1198911015840119888

)

2

120579 = 60∘

(32)




0or 1198870 1198861 1198862 1198871 1198872) are


119891 (120590119898 120591119898 120579) = radic5

120591119898

120588 (120590119898 120579)

minus 1 = 0

120588 (120579) = (2120588119888(1205882

119888minus 1205882

119905) cos 120579 + 120588

119888(2120588119905minus 120588119888)

times [4 (1205882

119888minus 1205882

119905) cos2120579 + 5120588

2

119905minus 4120588119905120588119888]12

)

times (4 (1205882

119888minus 1205882

119905) cos2120579 + (120588

119888minus 2120588119905)2

)minus1

(33)



119889120576119901

119894119895= 119889120582

120597119891 (120590)

120597120590119894119895

(34)




Failure surface

Failure

1205901

1205902 120590

120576






120590 = 1198640120576 minus

1

2

1198640

1205760

1205762 (35)





= [119863119890] ( 120576 minus 120576

119901)

120576119901 = 119886 119886 =

120597119865

120597120590

(36)




119886119879

= 0 119886119879[119863119890] ( 120576 minus 120576

119901) = 0

119886119879[119863119890] ( 120576 minus 119886) = 0

=119886119879[119863119890] 120576

119886119879[119863119890] 119886

(38)


[119863119890119901] = [119863] minus

[119863] 119886 119886119879[119863]

119867 + 119886119879[119863] 119886

(39)






ress

Strain


fy Esh

Es

120576y



[119872] [119905] + [119862] [

119905] + [119870] [119880

119905] = [119877

119905] (40)


Δ119905 le Δ119905cr =119879119899

120587=

2

120596 (41)





119890) is given by

M119890= ∭

119881

120588119898119873119894119873119894119889119881 (42)





[119872119888

119890] =

[[[[[

[

1198681

0 0 0 1198682

0 1198681

0 minus1198682

0

0 0 1198681

0 0

0 minus1198682

0 1198683

0

1198682

0 0 0 1198683

]]]]]

]

(43)

[119872119871

119890] =

[[[[[

[

1198681198711

0 0 0 0

0 1198681198711

0 0 0

0 0 1198681198711

0 0

0 0 0 1198681198712

0

0 0 0 0 1198681198712

]]]]]

]

(44)









3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass




119872 + 119862 + 119870119906 = 119877 (119905) (45)


119906 = 119888119890119904119905 (46)


119904 =minus119888


119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0




(119888

2119898)2

minus 1205962= 0


997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0




(47a)


120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)


[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)



ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












and its weights








120574120585120577

=

119899

sum119894=1

119877119894(120585 120578) 120574

119894

120585120577 120574

120578120577=

119899

sum119894=1

119878119894(120585 120578) 120574

119894

120578120577 (26)


12058213

=

119899

sum119894=1

12058213

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

12058223

=

119899

sum119894=1

12058223

119894120575 (120585119894minus 120585) 120575 (120578

119894minus 120578)

(27)


120574120585120577

(120585119894 120578119894) = 120574120585120577

(120585119894 120578119894)

120574120578120577

(120585119894 120578119894) = 120574120578120577

(120585119894 120578119894)

(28)

120579120585and 120579






3 Material Modeling







119864119894= 1205721198911015840

119905(1 minus

120576119894

120576119898

)1

120576119894

120576119905le 120576119894le 120576119898 (29)

120572 and 120576119898





119898


Compression

Tension

Stre

ss

Strain

ft

120572ft

120590i

E

120576t 120576i 120576m





[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[

[

0 0 0 0 0

0 119864 0 0 0

0 0 119866119888

120 0

0 0 0 11986611988813

0

0 0 0 0 11986623

]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

12= 025 times 119866(1 minus

1

0004) if 120576

1lt 0004

119866119888

12= 0 if 120576

1ge 0004

119866119888

13= 119866119888

12 119866

23=

5119866

6

(30)


[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0119866119888

12

20 0

0 0 0 119866119888

130

0 0 0 0 119866119888

23

]]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

13= 025 times 119866(1 minus

1205761

0004) if 120576

1lt 0004

119866119888

13= 0 if 120576

1ge 0004


120591oct120579 = 60∘

120579 = 0∘

120590oct


Smooth 1205901

1205902 1205903



119866119888

23= 025 times 119866(1 minus

1205762

0004)

119866119888

12= 05 times 119866

119888

13if 119866119888

23lt 119866119888

13

(31)




and 120591119898119888



119898

as follows

120591119898119905

1198911015840119888

=120588119905

radic51198911015840119888

= 1198860+ 1198861(

120590119898

1198911015840119888

) + 1198862(

120590119898

1198911015840119888

)

2

120579 = 0∘

120591119898119888

1198911015840119888

=120588119888

radic51198911015840119888

= 1198870+ 1198871(

120590119898

1198911015840119888

) + 1198872(

120590119898

1198911015840119888

)

2

120579 = 60∘

(32)




0or 1198870 1198861 1198862 1198871 1198872) are


119891 (120590119898 120591119898 120579) = radic5

120591119898

120588 (120590119898 120579)

minus 1 = 0

120588 (120579) = (2120588119888(1205882

119888minus 1205882

119905) cos 120579 + 120588

119888(2120588119905minus 120588119888)

times [4 (1205882

119888minus 1205882

119905) cos2120579 + 5120588

2

119905minus 4120588119905120588119888]12

)

times (4 (1205882

119888minus 1205882

119905) cos2120579 + (120588

119888minus 2120588119905)2

)minus1

(33)



119889120576119901

119894119895= 119889120582

120597119891 (120590)

120597120590119894119895

(34)




Failure surface

Failure

1205901

1205902 120590

120576






120590 = 1198640120576 minus

1

2

1198640

1205760

1205762 (35)





= [119863119890] ( 120576 minus 120576

119901)

120576119901 = 119886 119886 =

120597119865

120597120590

(36)




119886119879

= 0 119886119879[119863119890] ( 120576 minus 120576

119901) = 0

119886119879[119863119890] ( 120576 minus 119886) = 0

=119886119879[119863119890] 120576

119886119879[119863119890] 119886

(38)


[119863119890119901] = [119863] minus

[119863] 119886 119886119879[119863]

119867 + 119886119879[119863] 119886

(39)






ress

Strain


fy Esh

Es

120576y



[119872] [119905] + [119862] [

119905] + [119870] [119880

119905] = [119877

119905] (40)


Δ119905 le Δ119905cr =119879119899

120587=

2

120596 (41)





119890) is given by

M119890= ∭

119881

120588119898119873119894119873119894119889119881 (42)





[119872119888

119890] =

[[[[[

[

1198681

0 0 0 1198682

0 1198681

0 minus1198682

0

0 0 1198681

0 0

0 minus1198682

0 1198683

0

1198682

0 0 0 1198683

]]]]]

]

(43)

[119872119871

119890] =

[[[[[

[

1198681198711

0 0 0 0

0 1198681198711

0 0 0

0 0 1198681198711

0 0

0 0 0 1198681198712

0

0 0 0 0 1198681198712

]]]]]

]

(44)









3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass




119872 + 119862 + 119870119906 = 119877 (119905) (45)


119906 = 119888119890119904119905 (46)


119904 =minus119888


119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0




(119888

2119898)2

minus 1205962= 0


997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0




(47a)


120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)


[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)



ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119864119894= 1205721198911015840

119905(1 minus

120576119894

120576119898

)1

120576119894

120576119905le 120576119894le 120576119898 (29)

120572 and 120576119898





119898


Compression

Tension

Stre

ss

Strain

ft

120572ft

120590i

E

120576t 120576i 120576m





[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[

[

0 0 0 0 0

0 119864 0 0 0

0 0 119866119888

120 0

0 0 0 11986611988813

0

0 0 0 0 11986623

]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

12= 025 times 119866(1 minus

1

0004) if 120576

1lt 0004

119866119888

12= 0 if 120576

1ge 0004

119866119888

13= 119866119888

12 119866

23=

5119866

6

(30)


[[[[[

[

1205901

1205902

12059112

12059113

12059123

]]]]]

]

=

[[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0119866119888

12

20 0

0 0 0 119866119888

130

0 0 0 0 119866119888

23

]]]]]]

]

[[[[[

[

1205761

1205762

12057412

12057413

12057423

]]]]]

]

119866119888

13= 025 times 119866(1 minus

1205761

0004) if 120576

1lt 0004

119866119888

13= 0 if 120576

1ge 0004


120591oct120579 = 60∘

120579 = 0∘

120590oct


Smooth 1205901

1205902 1205903



119866119888

23= 025 times 119866(1 minus

1205762

0004)

119866119888

12= 05 times 119866

119888

13if 119866119888

23lt 119866119888

13

(31)




and 120591119898119888



119898

as follows

120591119898119905

1198911015840119888

=120588119905

radic51198911015840119888

= 1198860+ 1198861(

120590119898

1198911015840119888

) + 1198862(

120590119898

1198911015840119888

)

2

120579 = 0∘

120591119898119888

1198911015840119888

=120588119888

radic51198911015840119888

= 1198870+ 1198871(

120590119898

1198911015840119888

) + 1198872(

120590119898

1198911015840119888

)

2

120579 = 60∘

(32)




0or 1198870 1198861 1198862 1198871 1198872) are


119891 (120590119898 120591119898 120579) = radic5

120591119898

120588 (120590119898 120579)

minus 1 = 0

120588 (120579) = (2120588119888(1205882

119888minus 1205882

119905) cos 120579 + 120588

119888(2120588119905minus 120588119888)

times [4 (1205882

119888minus 1205882

119905) cos2120579 + 5120588

2

119905minus 4120588119905120588119888]12

)

times (4 (1205882

119888minus 1205882

119905) cos2120579 + (120588

119888minus 2120588119905)2

)minus1

(33)



119889120576119901

119894119895= 119889120582

120597119891 (120590)

120597120590119894119895

(34)




Failure surface

Failure

1205901

1205902 120590

120576






120590 = 1198640120576 minus

1

2

1198640

1205760

1205762 (35)





= [119863119890] ( 120576 minus 120576

119901)

120576119901 = 119886 119886 =

120597119865

120597120590

(36)




119886119879

= 0 119886119879[119863119890] ( 120576 minus 120576

119901) = 0

119886119879[119863119890] ( 120576 minus 119886) = 0

=119886119879[119863119890] 120576

119886119879[119863119890] 119886

(38)


[119863119890119901] = [119863] minus

[119863] 119886 119886119879[119863]

119867 + 119886119879[119863] 119886

(39)






ress

Strain


fy Esh

Es

120576y



[119872] [119905] + [119862] [

119905] + [119870] [119880

119905] = [119877

119905] (40)


Δ119905 le Δ119905cr =119879119899

120587=

2

120596 (41)





119890) is given by

M119890= ∭

119881

120588119898119873119894119873119894119889119881 (42)





[119872119888

119890] =

[[[[[

[

1198681

0 0 0 1198682

0 1198681

0 minus1198682

0

0 0 1198681

0 0

0 minus1198682

0 1198683

0

1198682

0 0 0 1198683

]]]]]

]

(43)

[119872119871

119890] =

[[[[[

[

1198681198711

0 0 0 0

0 1198681198711

0 0 0

0 0 1198681198711

0 0

0 0 0 1198681198712

0

0 0 0 0 1198681198712

]]]]]

]

(44)









3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass




119872 + 119862 + 119870119906 = 119877 (119905) (45)


119906 = 119888119890119904119905 (46)


119904 =minus119888


119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0




(119888

2119898)2

minus 1205962= 0


997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0




(47a)


120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)


[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)



ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120591oct120579 = 60∘

120579 = 0∘

120590oct


Smooth 1205901

1205902 1205903



119866119888

23= 025 times 119866(1 minus

1205762

0004)

119866119888

12= 05 times 119866

119888

13if 119866119888

23lt 119866119888

13

(31)




and 120591119898119888



119898

as follows

120591119898119905

1198911015840119888

=120588119905

radic51198911015840119888

= 1198860+ 1198861(

120590119898

1198911015840119888

) + 1198862(

120590119898

1198911015840119888

)

2

120579 = 0∘

120591119898119888

1198911015840119888

=120588119888

radic51198911015840119888

= 1198870+ 1198871(

120590119898

1198911015840119888

) + 1198872(

120590119898

1198911015840119888

)

2

120579 = 60∘

(32)




0or 1198870 1198861 1198862 1198871 1198872) are


119891 (120590119898 120591119898 120579) = radic5

120591119898

120588 (120590119898 120579)

minus 1 = 0

120588 (120579) = (2120588119888(1205882

119888minus 1205882

119905) cos 120579 + 120588

119888(2120588119905minus 120588119888)

times [4 (1205882

119888minus 1205882

119905) cos2120579 + 5120588

2

119905minus 4120588119905120588119888]12

)

times (4 (1205882

119888minus 1205882

119905) cos2120579 + (120588

119888minus 2120588119905)2

)minus1

(33)



119889120576119901

119894119895= 119889120582

120597119891 (120590)

120597120590119894119895

(34)




Failure surface

Failure

1205901

1205902 120590

120576






120590 = 1198640120576 minus

1

2

1198640

1205760

1205762 (35)





= [119863119890] ( 120576 minus 120576

119901)

120576119901 = 119886 119886 =

120597119865

120597120590

(36)




119886119879

= 0 119886119879[119863119890] ( 120576 minus 120576

119901) = 0

119886119879[119863119890] ( 120576 minus 119886) = 0

=119886119879[119863119890] 120576

119886119879[119863119890] 119886

(38)


[119863119890119901] = [119863] minus

[119863] 119886 119886119879[119863]

119867 + 119886119879[119863] 119886

(39)






ress

Strain


fy Esh

Es

120576y



[119872] [119905] + [119862] [

119905] + [119870] [119880

119905] = [119877

119905] (40)


Δ119905 le Δ119905cr =119879119899

120587=

2

120596 (41)





119890) is given by

M119890= ∭

119881

120588119898119873119894119873119894119889119881 (42)





[119872119888

119890] =

[[[[[

[

1198681

0 0 0 1198682

0 1198681

0 minus1198682

0

0 0 1198681

0 0

0 minus1198682

0 1198683

0

1198682

0 0 0 1198683

]]]]]

]

(43)

[119872119871

119890] =

[[[[[

[

1198681198711

0 0 0 0

0 1198681198711

0 0 0

0 0 1198681198711

0 0

0 0 0 1198681198712

0

0 0 0 0 1198681198712

]]]]]

]

(44)









3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass




119872 + 119862 + 119870119906 = 119877 (119905) (45)


119906 = 119888119890119904119905 (46)


119904 =minus119888


119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0




(119888

2119898)2

minus 1205962= 0


997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0




(47a)


120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)


[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)



ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Failure surface

Failure

1205901

1205902 120590

120576






120590 = 1198640120576 minus

1

2

1198640

1205760

1205762 (35)





= [119863119890] ( 120576 minus 120576

119901)

120576119901 = 119886 119886 =

120597119865

120597120590

(36)




119886119879

= 0 119886119879[119863119890] ( 120576 minus 120576

119901) = 0

119886119879[119863119890] ( 120576 minus 119886) = 0

=119886119879[119863119890] 120576

119886119879[119863119890] 119886

(38)


[119863119890119901] = [119863] minus

[119863] 119886 119886119879[119863]

119867 + 119886119879[119863] 119886

(39)






ress

Strain


fy Esh

Es

120576y



[119872] [119905] + [119862] [

119905] + [119870] [119880

119905] = [119877

119905] (40)


Δ119905 le Δ119905cr =119879119899

120587=

2

120596 (41)





119890) is given by

M119890= ∭

119881

120588119898119873119894119873119894119889119881 (42)





[119872119888

119890] =

[[[[[

[

1198681

0 0 0 1198682

0 1198681

0 minus1198682

0

0 0 1198681

0 0

0 minus1198682

0 1198683

0

1198682

0 0 0 1198683

]]]]]

]

(43)

[119872119871

119890] =

[[[[[

[

1198681198711

0 0 0 0

0 1198681198711

0 0 0

0 0 1198681198711

0 0

0 0 0 1198681198712

0

0 0 0 0 1198681198712

]]]]]

]

(44)









3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass




119872 + 119862 + 119870119906 = 119877 (119905) (45)


119906 = 119888119890119904119905 (46)


119904 =minus119888


119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0




(119888

2119898)2

minus 1205962= 0


997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0




(47a)


120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)


[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)



ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ress

Strain


fy Esh

Es

120576y



[119872] [119905] + [119862] [

119905] + [119870] [119880

119905] = [119877

119905] (40)


Δ119905 le Δ119905cr =119879119899

120587=

2

120596 (41)





119890) is given by

M119890= ∭

119881

120588119898119873119894119873119894119889119881 (42)





[119872119888

119890] =

[[[[[

[

1198681

0 0 0 1198682

0 1198681

0 minus1198682

0

0 0 1198681

0 0

0 minus1198682

0 1198683

0

1198682

0 0 0 1198683

]]]]]

]

(43)

[119872119871

119890] =

[[[[[

[

1198681198711

0 0 0 0

0 1198681198711

0 0 0

0 0 1198681198711

0 0

0 0 0 1198681198712

0

0 0 0 0 1198681198712

]]]]]

]

(44)









3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass




119872 + 119862 + 119870119906 = 119877 (119905) (45)


119906 = 119888119890119904119905 (46)


119904 =minus119888


119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0




(119888

2119898)2

minus 1205962= 0


997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0




(47a)


120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)


[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)



ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















3= int120588

119898119911 times 119911119889119911 119868

2= int120588

119898119911119889119911 The total mass




119872 + 119862 + 119870119906 = 119877 (119905) (45)


119906 = 119888119890119904119905 (46)


119904 =minus119888


119888

2119898)2

minus 1205962

(119888

2119898)2

minus 1205962gt 0




(119888

2119898)2

minus 1205962= 0


997888rarr 1199041= minus120585120596

(119888

2119898)2

minus 1205962lt 0




(47a)


120585 =119888

2119898120596997904rArr

119888

2119898= 120585120596 120585 =

119888

119888cr (47b)


[119862] = 120572 [119872] + 120573 [119870] (48)

120589119894=

120572

2120596119894

+120573120596119894

2 (49)



ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ampi

ng ra

tio (120585

)

Total damping

120573 damping


120572 damping

120596i 120596j



120595119899

119894= 119891119899minus 119901119899

119894= 119891119899minus int119881

119861119879120590119899

119894119889119881 lt Tolerance (00025)

(50)




Start

Input data



Incremental loading




Iterative solver


Convergence check

Output module

Stop

Incr

emen

tal l

oop

Itera

tive l

oop














Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Door cum window

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus200

minus100


minus20

minus10

0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Two doors Disp

lace

men

t (m

m)


0

100

00 03 06 09 12 15Time (s)

minus200

minus100


0

10

20

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus20

minus10

Two windows


0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus200

minus100


0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

minus40

minus20

Three windows


minus200

minus100

0

100

200

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


minus40

minus20

0

20

40

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)


Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table 2 Continued


Fourwindows


minus150

minus100

minus50

0

50

100

00 03 06 09 12 15D

ispla

cem

ent (

mm

)Time (s)


minus10

minus5

0

5

10

00 03 06 09 12 15

Disp

lace

men

t (m

m)

Time (s)

Staggered two

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


minus50

0

50

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)

Staggered four

windows


minus200

minus100

0

100

200

Disp

lace

men

t (m

m)

00 03 06 09 12 15Time (s)


00 03 06 09 12 15Time (s)

Disp

lace

men

t (m

m)

minus15

minus10

minus5

0

5

10

15






Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Case




Case




(a) (b)

minus15

minus10

minus5

0

5

10

15

00 03 06 09 12 15Time (s)

Acce

lera

tion

(ms2)

(c)





No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












No damping

0

50

100

150

200

1 2 3 4 5 6 7Max

imum

disp

lace

men

t (m

m)

Case

Max

imum

disp

lace

men

t (m

m)

0

10

20

30

40

1 2 3 4 5 6 7Case

2 damping

Max

imum

disp

lace

men

t (m

m)

00

10

20

30

40

50

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

05

1

15

2

1 2 3 4 5 6 7Case

5 damping

Max

imum

disp

lace

men

t (m

m)

0

1

2

3

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

02

04

06

08

1 2 3 4 5 6 7Case

10 damping

Max

imum

disp

lace

men

t (m

m)

00

05

10

15

20

1 2 3 4 5 6 7Case

Max

imum

disp

lace

men

t (m

m)

0

01

02

03

04

05

1 2 3 4 5 6 7Case








7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













7 Conclusions




References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Influence of Opening Locations on the ...downloads.hindawi.com/archive/2014/858510.pdf · Research Article Influence of Opening Locations on the Structural Response