Chapter 1. Introduction. Fundamental Concepts. Introduction. Historical background. Historical background. Fundamental Concepts

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1 Chpter udmetl Cocepts Lecture Notes Dr Mohd Afed Uverst Mlys Perls N67 te lemet Alyss Itroducto A or sometmes referred s M, hs ecome powerful tool for umercl soluto of wde rge of egeerg prolems A s computtol techue used to ot ppromte solutos of oudry vlue prolems mple pplctos of A: Deformto & stress lyss of utomotve, rcrft, uldg d rdge structures eld lyss of het flu, flud flow, mgetc flu, seepge d other flow prolems Reserch out ts pplcto s stll ope to oegeerg felds such s ecoomc lyss, optmto, etc Itroducto udmetl Cocepts Reltoshp etwee CAD, CA d CAM CAD CA CAA CAM Wth the dvces computer techology d CAD systems, comple prolems c e modelled reltve ese & ccurte he cocept of A I ths method, comple rego defg cotuum s dscreted to smple geometrc shpes clled fte elemets he mterl propertes, d the goverg reltoshps re cosdered over these elemets d epressed terms of uow vlues t elemet corers A ssemly process, cosderg the lodg d costrts, results set of eutos Hstorcl cgroud Hstorcl cgroud 9, Hreoff preseted frme wor method to lye rcrft structure 9, Court used pecewse polyoml terpolto to model torso prolem 956, urer et l derved stffess mtrces for truss, em d other elemets 96, the term fte elemet ws frst coed y Clough 967, the frst oo o fte elemets y Zeewc d Cheug ws pulshed Lte 96s d erly 97s, A ws ppled to oler prolems d lrge deformtos 97, oo o oler cotu y Ode ws pulshed 5 ody, the developmet computer techology hs rought ths method wth rech of studets d egeers here re my commercl pcges vlle for M such s Pro-geer, LS- DYNA, ALGOR, LUSAS, Nstr, ANSYS, etc However, user should cover the theory developmet so tht the totl soluto d result terpretto c e doe correctly 6

2 Cosderg three-dmesol ody occupyg volume V d hvg surfce S gure he oudry s costred some rego Uder the force the ody deforms Deformto t pot (=[,y,] ) s gve y ts dsplcemet, u = [u,v,w] rcto or dstruted force per ut re, = [,y,] Dstruted force per ut volume, f = [f,f y,f ] Lod t pot, P = [P,P y,p ] 7 8 Stress compoets (depedet) ctg o pot : σ = [σ,σ y,σ,τ y,τ,τ y] gure ulrum eutos: y f y y y y f y y y f y 9 Boudry Codtos Referrg to g, there re: dsplcemet oudry codto surfce-lodg oudry codto Cosderto of eulrum log the three es drectos gves: yy y yy y y y y Boudry Codtos ulrum of elemet tetrhedro: gure

3 Str-Dsplcemet Reltos he strs vector form: [,,, ] y y,, y, Referrg to g, the strs c e wrtte s: u v w v w u w u v,,,,, y y y Str-Dsplcemet Reltos hese str reltos hold for smll deformtos gure or ler elstc mterls, the stress-str reltos come from the geerled Hooe s lw: y y v v y G y y v v G y y v v y G 5 Sher modulus (modulus of rgdty): G ( v) rom Hooe s lw reltoshps: ( v) y ( y ) 6 Reltos tht c e troduced s: Dε Where D s (66) mterl mtr gve y: v v v v v v D v v v ( v)( v) 5 v 5 v 5 v or oe dmesol cses, the reltos re smplfed s: or two dmesol cses, the prolems re modelled s ple stress or ple str 7 8

4 Ple Stress Ple Stress A th plr ody sujected to -ple lodg o ts edge surfce s sd to ple stress he stresses σ, τ d τ re set s ero gure 5 9 he Hooe s lw reltos: y v y y v y v v ( v) y y he the verse reltos re gve y: v y v y v v y y Ple Str Ple Str If log ody of uform cross secto s sujected to trsverse lodg log ts legth, smll thcess the loded re c e treted s sujected to ple str Stress σ my ot e ero ths cse Here ε, γ d γy re set to ero he verse reltos re gve y: v v y v v y ( v)( v) 5 v y y gure 6 emperture ffects emperture ffects or sotropc mterls, the temperture rse Δ results uform str It deped o the coeffcet of ler epso α of the mterl (α=δl/ Δ) emperture str s represeted s tl str: [,,,,,] Stress-str relto: σ = D(ε ε) I ple stress, we hve: [,,] I ple str, we hve: ( v)[,,] Geerlly, these eutos re for ect soluto smple prolem or comple prolems, ppromte soluto usully employ potetl eergy methods or vrtol methods to solve the prolems

5 Potetl ergy Vrtol Method otl potetl eergy of elstc ody: Str eergy ( U) + Wor potetl (WP) Str eergy: U dv V Wor potetl: WP u fdv u ds u P V dv u fdv u ds u P V V S s 5 Vrtol method s derved from Gler s Method, whch gves: ( ) dv fdv ds u P V V V where Φ s rtrry dsplcemet cosstet wth specfed oudry codto of u hs euto s lso ow s Prcple of Vrtul Wor 6 Vo Mses Stress Vo Mses Stress Vo Mses stress s used s crtero determg the oset of flure ductle mterls he flure crtero: VM Y he vo Mses stress s gve y: VM I I where I d I re the frst two vrts of the stress tesors Geerl stte of stress: I y I y y y y I term of the prcpl stresses: I I Vo Mses stress c e epressed eser wy: VM ( ) ( ) ( ) 7 8 gure elow shows system of sprgs he totl potetl eergy s gve y = -, =, where,,, d re eteso of four sprgs = -, 9 = - 5

6 6 he eulrum eutos of the system y cosderg the eulrum of ech seprte ode = = = By susttutg,, Mmg wth respect to, d K Covert to Mtr form: 5 = 6 ) ( ) (

7 Covert to Mtr form: K Prolem Determe the dsplcemet of odes of the sprg system show elow: N/mm 6 N 5 N 8 N/mm 5 N/mm 7 8 Prolem Prolem gure elow shows system of sprgs he totl potetl eergy s gve y where,, d re eteso of four sprgs N/mm 6 N 5 N 8 N/mm N/mm 5 N/mm =, =, 6 N 5 N 8 N/mm = -, 5 N/mm 9 7