MEMBRANE ELEMENT WITH NORMAL ROTATIONS

Transcription

1 9. MEMBRANE ELEMENT WITH NORMAL ROTATIONS Rotatons Mst Be Compatble Between Beam, Membrane and Shell Elements 9. INTRODUCTION { XE "Membrane Element" }The comple natre of most bldngs and other cvl engneerng strctres reqres that frame, plate bendng and membrane elements est n the same compter model. The threedmensonal beam element normall has s degrees-of-freedom per node three dsplacements and three rotatons per node. The plate bendng element, presented n the prevos chapter, has two rotatons n the plane of the element and one dsplacement normal to the element at each node. The standard plane stress element, sed to model the membrane behavor n shell elements, has onl two n-plane dsplacements at each node and cannot carr moments appled normal to the plane of the element. { XE "Normal Rotatons" }A frame element embedded normal to a shear wall or slab s ver common n the modelng of bldngs and man other tpes of strctral sstems. It s possble to se a constrant to transfer the frame element moment to a force-cople appled n the plane of the element. However, for shells connected to edge beams and man other common tpes of strctral sstems, there s a need for a membrane element that has a normal rotaton as a basc DOF at each node.

2 9- STATIC AND DYNAMIC ANALYSIS { XE "Qadrlateral Element" }The search for a membrane element wth normal rotatons was a frtless endeavor for the frst 30 ears of the development of fnte element technolog. Wthn the last ears, however, a practcal qadrlateral element has evolved. Rather than refer to the man research papers (smmarzed n reference []) that led to the development of the element crrentl sed n the general strctral analss program SAP000, the fndamental eqatons wll be developed n ths chapter. In addton, nmercal eamples wll be presented to llstrate the accrac of the element. 9. BASIC ASSUMPTIONS The development of the membrane element s ver smlar to the plate bendng element presented n the prevos chapter. The qadrlateral element s shown n Fgre (a) 7 6 (b) s 3 r 3 ABSOLUTE ROTATIONS RELATIVE ROTATIONS (d) (c) Fgre 9. Qadrlateral Membrane Element wth Normal Rotatons Development of the element can be dvded nto the followng for steps:. The startng pont s the nne-node qadrlateral element, 6 DOF, shown n Fgre 9.a.

3 MEMBRANE ELEMENT 9-3. The net step s to rotate the md-sde relatve dsplacements to be normal and tangental to each sde and to set the relatve tangental dsplacement to zero, redcng the element to the DOF shown n Fgre 9.b. 3. The thrd step s to ntrodce parabolc normal dsplacement constrants to elmnate the for md-sde normal dsplacements and to ntrodce for relatve normal rotatons at the nodes shown n Fgre 9.c.. The fnal step s to convert the relatve normal rotatons to absolte vales and to modf the shape fnctons to pass the patch test. Ths reslts n the b element stffness wth respect to the DOF shown n Fgre 9.d. 9.3 DISPLACEMENT APPROXIMATION The basc assmpton s that n-plane and dsplacements are defned b the followng eqatons: ( r, ( r, The eght shape fnctons are gven b: N ( r)( / N ( r)( / N ( r)( )/ N ( r)( )/ N 3 s ( r )( s 7 ( r )( s s 6 ( r)( s ( r)( s (9.) )/ N )/ (9.) N )/ N )/ The frst for shape fnctons are the natral blnear shape fnctons for a fornode qadrlateral and are not zero at nodes to. The last for shape fnctons for the md-sde nodes and center node are an addton to the blnear fnctons and are referred to as herarchcal fnctons. 9. INTRODUCTION OF NODE ROTATION A tpcal element sde j s shown n Fgre 9..

4 9- STATIC AND DYNAMIC ANALYSIS j θ L L j α j j θ j,,3, j,3,, j m,6,7, ( j ) ( j ) Fgre 9. Tpcal Sde of Qadrlateral Element If t s assmed that the relatve normal dsplacement of the sde s parabolc, the followng eqaton mst be satsfed: Lj j ( θ j θ ) (9.3) Becase the tangental md-sde dsplacement s zero, the global relatve md-sde dsplacements are gven b: Lj cosα j j cosα j ( θ j θ ) L (9.) j sn α j j sn α j ( θ j θ ) Eqaton (9.) can be appled to all for sdes and the global dsplacements, Eqaton (9.), can be wrtten as: ( r, ( r, M M ( r, θ ( r, θ Therefore, the sstem has been redced to DOF. (9.)

5 MEMBRANE ELEMENT 9-9. STRAIN-DISPLACEMENT EQUATIONS { XE "Stran Dsplacement Eqatons:D Plane Elements" }The strandsplacement eqatons can now be constrcted from the followng fndamental eqatons: ε, ε and γ (9.6) Alternatvel, the 3 b stran-dsplacement eqatons wrtten n sb matr form are the followng: ε ε [ B B ] θ (9.7) γ { XE "Correcton Matr" }In order that the element satsfes the constant stress patch test, the followng modfcaton to the 3 b B matr mst be made: B B B da A (9.) The development of ths eqaton s presented n the chapter on ncompatble elements, Eqaton (6.). 9.6 STRESS-STRAIN RELATIONSHIP { XE "Stress-Stran Relatonshp" }The stress-stran relatonshp for orthotropc plane stress materals can be wrtten as: σ D D D3 ε σ D D D3 ε (9.9) τ D3 D3 D33 γ The onl restrcton on the stress-stran matr s that t mst be smmetrc and postve defnte. 9.7 TRANSFORM RELATIVE TO ABSOLUTE ROTATIONS The element b stffness matr for a qadrlateral element wth normal rotatons s obtaned sng for-pont nmercal ntegraton. Or:

6 9-6 STATIC AND DYNAMIC ANALYSIS T K B DBdV (9.0) { XE "Relatve Rotatons" }The stffness matr for the membrane element, as calclated from Eqaton (9.9), has for nknown relatve rotatons at the nodes. An eamnaton of the propertes of the stffness matr ndcates that t has a zero energ mode n addton to the three rgd bod modes. Ths spros deformaton mode, relatve to the rgdbod rotaton of the element, s shown n Fgre 9.3. Fgre 9.3 Zero Energ Dsplacement Mode { XE "Zero Energ Mode" }The zero energ dsplacement mode has eqal rotatons at all nodes and zero md-sde dsplacements. To elmnate ths mode, t s onl necessar to add a rank one matr to the element stffness matr that has stffness assocated wth the mode. From the elastct defnton of rotaton, the absolte rotaton at the center of the element, or an estmaton of the rgd-bod rotaton of the element, can be calclated from: θ 0 b 0 (9.) { XE "Relatve Rotatons" }where b 0 s a b matr. The dfference between the absolte rotaton and the average relatve rotaton at the center of the element s: 0 N 0,0) d θ ( θ b 0 (9.)

7 MEMBRANE ELEMENT 9-7 A stffness k 0 (or a penalt term) can now be assgned to ths deformaton to create, sng one pont ntegraton, the followng rank one stffness matr: K T T 0 b0 k 0 b dv k0 Vol b0 b0 0 (9.3) Eperence wth the solton of a large nmber of problems ndcates that the followng vale for rotatonal stffness s effectve: k D33 (9.) where D 33 s the shear modls for sotropc materals. When ths rank one matr s added to the b stffness matr, the zero energ mode s removed and the node rotaton s converted to an absolte rotaton. 9. TRIANGULAR MEMBRANE ELEMENT { XE "Tranglar Elements" }The same appromatons sed to develop the qadrlateral element are appled to the tranglar element wth three md-sde nodes. The resltng stffness matr s 9 b 9. Appromatel 90 percent of the compter program for the qadrlateral element s the same as for the tranglar element. Onl dfferent shape fnctons are sed and the constrant assocated wth the forth sde s skpped. However, the trangle s sgnfcantl more stff than the qadrlateral. In fact, the accrac of the membrane behavor of the trangle wth the drllng degrees of freedom s nearl the same as the constant stran trangle. 9.9 NUMERICAL EXAMPLE { XE "Dstorted Elements" }The beam shown n Fgre 9. s modeled wth two membrane elements wth drllng degrees-of-freedom. a E,00 υ 0. d M V L a L Fgre 9. Beam Modeled wth Dstorted Elements

8 9- STATIC AND DYNAMIC ANALYSIS Reslts for both dsplacements and stresses are smmarzed n Table 9.. Table 9.. Reslts of Analss of Cantlever Beam Mesh Dstorton Factor a TIP MOMENT LOADING Normalzed Tp Dsplacement Normalzed Mamm Stress At Spport TIP SHEAR LOADING Normalzed Tp Dsplacement Normalzed Mamm Stress At Spport Eact { XE "Shear Lockng" }For rectanglar elements sbjected to end moment, the eact reslts are obtaned and shear lockng does not est. For a tp shear loadng, the dsplacements are n error b onl percent; however, the bendng stresses are n error b percent. Ths behavor s almost dentcal to the behavor of plane elements wth ncompatble modes. As the element s dstorted, the dsplacements and stresses deterorate. All reslts were obtaned sng for-pont ntegraton. The end moment can be appled as two eqal and opposte horzontal forces at the end of the beam. Or, one half of the end moment can be appled drectl as two concentrated moments at the two end nodes. The reslts for the two dfferent methods of loadng are almost dentcal. Therefore, standard beam elements can be attached drectl to the nodes of the membrane elements wth normal rotatonal DOF. 9.0 SUMMARY The membrane plane stress element presented n ths chapter can be sed to accratel model man comple strctral sstems where frame, membrane and plate elements nterconnect. The qadrlateral element prodces ecellent reslts. However, the performance of the tranglar membrane element s ver poor.

9 MEMBRANE ELEMENT REFERENCES. { XE "Ibrahmbegovc, Adnan" }Ibrahmbegovc, Adnan, R. Talor, and E. Wlson "A Robst Membrane Qadrlateral Element wth Drllng Degrees of Freedom," Int. J. of Nm. Meth.