Linear Strain Triangle and other tpes o D elements B S. Ziaei Rad
Linear Strain Triangle (LST or T6 This element is also called qadratic trianglar element. Qadratic Trianglar Element
Linear Strain Triangle There are si nodes on this element: three corner nodes and three midside nodes. Each node has two degrees o reedom (DOF as beore. The displacements (, are assmed to be qadratic nctions o (,, where bi (i,,..., are constants. From these, the strains are ond to be,
Linear Strain Triangle which are linear nctions. Ths, we hae the linear strain triangle (LST, which proides better reslts than the CST. In the natral coordinate sstem we deined earlier, the si shape nctions or the LST element are, in which. Each o these si shape nctions represents a qadratic orm on the element as shown in the igre.
Linear Strain Triangle Shape Fnction or LST Displacements can be written as, The element stiness matri is still gien b bt here is qadratic in and. In general, the integral has to be compted nmericall.
Linear Qadrilateral Element (Q Linear Qadrilateral Element
Linear Qadrilateral Element (Q There are or nodes at the corners o the qadrilateral shape. In the natral coordinate sstem, the or shape nctions are, ote that at an point inside the element, as epected. The displacement ield is gien b which are bilinear nctions oer the element.
Isoparametric Element I we se the same parameters (shape nctions to epress Geometr, we are sing an isoparametric ormlation.
Isoparametric Element or [ ] [ ] ] [ ε det } { A
Isoparametric Element ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( A } ]{ [ } ]{ ][ [ } { d B d A A ε ]det ][ [ ] [ ] [ dd B E B t K T e
Qadratic Qadrilateral Element (Q8 This is the most widel sed element or -D problems de to its high accrac in analsis and leibilit in modeling. Qadratic Qadrilateral Element
Qadratic Qadrilateral Element (Q8 F (, G (, Fi G i i i (, (, i Gie a ale o zero along the sides o the element That the gien node does not contact Select sch that when mltipl b Fi, it will prodce A ale o nit at node i and a ale o zero at other neighboring nodes. Eample: Consider F (, ( ( G (, c c c G (, ; G(, ; (, F (, G(,
Qadratic Qadrilateral Element (Q8 ( (, (, (, c c c c c G c c G / / / c c c ( ( (
Qadratic Qadrilateral Element (Q8 There are eight nodes or this element, or corners nodes and or midside nodes. In the natral coordinate sstem, the eight shape nctions are,
which are qadratic nctions oer the element. Strains and stresses oer a qadratic qadrilateral element are linear nctions, which are better representations. otes: Q and T are sall sed together in a mesh with linear elements. Q8 and T6 are sall applied in a mesh composed o qadratic elements. Qadratic elements are preerred or stress analsis, becase o their high accrac and the leibilit in modeling comple geometr, sch as cred bondaries. Qadratic Qadrilateral Element (Q8 Again, we hae The displacement ield is gien b at an point inside the element.
Eample. A sqare plate with a hole at the center and nder pressre in one direction. The dimension o the plate is in. in., thickness is. in. and radis o the hole is in. Assme E E6 psi,. and p psi. Find the maimm stress in the plate.
Eample. FE Analsis: From the knowledge o stress concentrations, we shold epect the maimm stresses occr at points A and B on the edge o the hole. Vale o this stress shold be arond p ( psi which is the eact soltion or an ininitel large plate with a hole. We se the ASYS FEA sotware to do the modeling (meshing and analsis, sing qadratic trianglar (T6 or LST linear qadrilateral (Q and qadratic qadrilateral (Q8 elements. Linear triangles (CST or T is OT aailable in ASYS. The stress calclations are listed in the ollowing table, along with the nmber o elements and DOF sed, or comparison.
Eample. Discssions: Check the deormed shape o the plate Check conergence (se a iner mesh, i possible Less elements (~ shold be enogh to achiee the same accrac with a better or smarter mesh We ll redo this eample in net chapter emploing the smmetr conditions.
Eample. FEA Mesh (Q8, 9 elements FEA Stress Plot (Q8, 9 elements
Transormation o Loads Concentrated load (point orces, srace traction (pressre loads and bod orce (weight are the main tpes o loads applied to a strctre. Both traction and bod orces need to be conerted to nodal orces in the FEA, since the cannot be applied to the FE model directl. The conersions o these loads are based on the same idea (the eqialent-work concept which we hae sed or the cases o bar and beam elements. Traction on a Q element
Transormation o Loads Sppose, or eample, we hae a linearl aring traction q on a Q element edge, as shown in the igre. The traction is normal to the bondar. Using the local (tangential coordinate s, we can write the work done b the traction q as, where t is the thickness, L the side length and the component o displacement normal to the edge AB. For the Q element (linear displacement ield, we hae The traction q(s, which is also linear, is gien in a similar wa,
Transormation o Loads Ths, we hae, and the eqialent nodal orce ector is,
Transormation o Loads ote, or constant q, we hae, For qadratic elements (either trianglar or qadrilateral, the traction is conerted to orces at three nodes along the edge, instead o two nodes. Traction tangent to the bondar, as well as bod orces, are conerted to nodal orces in a similar wa.
Transormation o Loads @ @ L s s d L ds ( L s L q q s ( ( ( ( q q q
Transormation o Loads [ ] d L q q t W q ( ( ( ( 6 q q tl L q q s [ ] d L q q t W q ( ( tlq
Transormation o Loads V - Point Load considered in a sal manner b haing a Strctral node at the point. - Traction Force As it was seen in the preios eample First the orce and the delection along the side epress b Use o shape nctions and then nmerical integration will Be sed to calclate the eqialent nodal orces. - Bod Force A Bod orce which is a distribted orce Per nit olme, contribte to the global orce ector F. Assme { F } [ ] T as constant within each element. T FdV e e d T t e Where the 8* element bod orce is gien b e T det dd
Stress Calclation The stress in an element is determined b the ollowing relation, where B is the strain-nodal displacement matri and d is the nodal displacement ector which is known or each element once the global FE eqation has been soled. Stresses can be ealated at an point inside the element (sch as the center or at the nodes. Contor plots are sall sed in FEA sotware packages (dring post-process or sers to isall inspect the stress reslts.
The on Mises Stress The on Mises stress is the eectie or eqialent stress or -D and -D stress analsis. For a dctile material, the stress leel is considered to be sae, i where is the on Mises stress and the ield stress o the material. This is a generalization o the -D (eperimental reslt to -D and -D sitations. The on Mises stress is deined b in which and are the three principle stresses at the considered point in a strctre.
Aeraged Stresses: Stresses are sall aeraged at nodes in FEA sotware packages to proide more accrate stress ales. This option shold be trned o at nodes between two materials or other geometr discontinit locations where stress discontinit does eist. The on Mises Stress For -D problems, the two principle stresses in the plane are determined b Ths, we can also epress the on Mises stress in terms o the stress components in the coordinate sstem. For plane stress conditions, we hae,
Discssions Know the behaiors o each tpe o elements: T and Q: linear displacement, constant strain and stress; T6 and Q8: qadratic displacement, linear strain and stress. Choose the right tpe o elements or a gien problem: When in dobt, se higher order elements or a iner mesh. Aoid elements with large aspect ratios and corner angles: where Lma and Lmin are the largest and smallest characteristic lengths o an element, respectiel. Elements with Bad Shapes Elements with ice Shapes
Discssions Connect the elements properl: Don t leae nintended gaps or ree elements in FE models. Improper connections (gaps along AB and CD