Two- and Three-Dimensional Stress Analysis

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Pauline Oliver
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1 Two- and Thee-Dimensiona Stess Anasis Stesses, w d, d d Components of Catesian stess acting at an infinitesima ome st inde = diection of pane noma nd inde = stess diection Fom otationa eqiibim abot the thee coodinate aes in tn: = = =, The state of stess at a point is compete descibed b the si components of qiibim eqations, R R R,R,R bod foce components d d d d, d d, w qiibim of an infinitesima ome R Fom eqiibim in a thee othogona diections: R R R qiibim of foces in -diection: ddd d dd d dd dd dd dd R Stains and dispacements w w w These stain dispacement eations ae aid fo sma stains i.e. sma defomations

2 Stains and dispacements in mati notation w The si components of stain ae obtained fom jst the thee dispacement components thogh the mati of diffeentia opeatos: Stess-stain eations (constittie eqations and C ae 6 6 matices, hence 36 eastic constants Since smmetic, thee ae independent eastic constants astic constants ae detemined thogh aboato testing. Fo homogeneos, isotopic mateias, we edce to constants, and homogeneos phsica popeties of continm ae the same at a points isotopic eastic popeties ae the same in a diections The si components of stess ae eated to the si components of stain thogh a set of constittie eqations: C o Stess-stain eations fo homogeneos, isotopic mateia Appication of positie noma stess in one diection cases positie stain (etension in the same diection, and negatie stain (contaction in the othe two othogona diections. The amont of contaction is popotiona to the Poisson s atio,. If a thee noma stesses ae acting the noma stain noma stess eations become: Noma stains ae independent of an sheaing behaio. The sheaing stain sheaing stess eations ae: whee Stess-stain eations fo homogeneos, isotopic mateia ( ( Mati fom of the stain-stess eations: Mati fom of the stess-stain eations: C

3 Stain compatibiit conditions The si components of stain ae epessed in tems of on thee dispacement components: Thee mst be some conditions imposed on the stain components in ode that the si stain-dispacement eqations gie a set of singeaed continos sotions fo the thee dispacement components thoghot the inteio of the bod. Ths the components of stain cannot be compete independent of one anothe. Two Dimensions Withot oss of geneait, conside the pane. Shea stesses on the othe panes wi be eo. = = = = We wi conside thee ideaiations in two dimensions.. Pane stess: =. Pane stain: = 3. Aismmetic: thee-dimensiona bod deeoped b otation of a pana section Pane stess = Thickness is sma with espect to ength and width. No oading is appied in the thickness diection. ε Pane stain = eomet and oading ae constant in ongitdina diection. ampes: stip footing, ong cinde, etaining wa, eath dam Usa conside a section of nit thickness in the diection. Incompessibe mateia: =.5 Can case tobe with nmeica anasis of pane stain conditions. Stcta eampe: wide one-wa sab ε

4 Othotopic mateias Othe tpes of mateias othe than isotopic can be deat with. We wi on discss pane stess othotopic mateias in the and ais. Define as the stain in the diection de to stain in the diection. ( In genea fo independent constants Can be edced to thee with the foowing appoimation Pobem with othotopic mateias: detemination of mateia constants Pincipa Stesses Use Moh s cice to find pincipa stesses., p tan Maimm sheaing stess: ma Stess Inaiants Same nmeica ae in an coodinate sstem Stess Intensit 3 SI on Mises stess 3 3 e (effectie, eqiaent Usef in faie theoies, which state that ieding begins when cetain stess inaiant eaches a imiting ae. Aismmet Thee-dimensiona bod that is deeoped b otation of a pana section. (,,w, Aismmet ( ( Note that this is e simia to pane stain, ecept thee is an.

5 Stess and Stain Definitions Stains: ngineeing, o nomina stain: een-lagange stain: Logaithmic, o te stain: d n Stetch: Used in age stain and age dispacement anasis. Stesses: ngineeing stess: diide b oigina aea Cach (te stess: diide b defomed aea Smma A phsica pobems ae thee-dimensiona. Using a two-dimensiona anasis impies that at east a sma amont of ideaiation has taken pace. Two-dimension simpifications Tsses and fames Pates (mats and sabs and shes Pane defomations (pane stess, pane stain, aismmetic Stains, and hence stesses, ae obtained fom deiaties (o gadients of dispacements, and hence ae ess accate. Vaios constittie modes othe than isotopic can be sed. Detemination of mateia constants is not awas eas.

= ρ. Since this equation is applied to an arbitrary point in space, we can use it to determine the charge density once we know the field.

= ρ. Since this equation is applied to an arbitrary point in space, we can use it to determine the charge density once we know the field. Gauss s Law In diffeentia fom D = ρ. ince this equation is appied to an abita point in space, we can use it to detemine the chage densit once we know the fied. (We can use this equation to ve fo the fied