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com a sec¸a\u2dco transversal e´ a linha neutra (LN). M > 0 { Fibras superiores a` LN sa\u2dco comprimidas / encurtadas Fibras inferiores a` LN sa\u2dco tracionadas / alongadas 69 Se Sd xd Se Sd Tracao~ ~ ~Compressao \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd 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\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd LN y = E.S. M > 0 \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd 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\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd Superficie neutra M N d o \u3b8 Superficie neutra ds = dx~ \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd Figura 3.47: Elemento de volume sob flexa\u2dco Seja o elemento de volume gene´rico, limitado pelas sec¸o\u2dces Se e Sd, de comprimento elementar dx. Na configurac¸a\u2dco deformada, d\u3b8 e´ o a\u2c6ngulo entre Se e Sd, o ponto O e´ o centro de curvatura e OM = ON = \u3c1 e´ o raio de curvatura da linha ela´stica na superf´\u131cie neutra. A curvatura e´: \u3ba = 1 \u3c1 = d\u3b8 ds ' d\u3b8 dx Considerando ds ' dx para vigas horizontais ou de pequena inclinac¸a\u2dco e para pequenas deformac¸o\u2dces. Uma paralela a \u201dSe\u201d pelo ponto N mostra (sombreado) os encurtamento das fibras superiores e os alongamentos das fibras inferiores a` superf´\u131cie neutra. Estas de- formac¸o\u2dces longitudinais du sa\u2dco mostradas na fig(3.48b) . As figs3.48(c) e 3.48(d) mostram as correspondentes deformac¸o\u2dces espec´\u131ficas ²x e tenso\u2dces normais \u3c3x. Seja uma camada de fibras gene´rica, paralela a` superf´\u131cie neutra, de ordenada y em relac¸a\u2dco a` LN (\u2212ds \u2264 y \u2264 di). \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd 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\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd 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\ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd y ds di o x \u3c3 \u3c3 \u3c3s i a) b) d)c) d\u3b8 \u3b5x Figura 3.48: Diagramas de deformac¸a\u2dco longitudinal, especif´\u131ca e tenso\u2dces 70 du = d\u3b8 y ²x = du dx = d\u3b8 dx y \u3c3x = E²x = E d\u3b8 dx y Func¸o\u2dces Diretamente proporcionais a y (variac¸a\u2dco linear), sendo \u3c3x = K y, e K = E d\u3b8 dx = K . E para calcular a constante K e determinar a posic¸a\u2dco da LN, lembramos da sec¸a\u2dco 3: Esforc¸o normal N = \u222b A \u3c3xdA = \u222b A KydA = K \u222b A ydA = 0 para valores arbitra´rios de K, temos \u222b A ydA = 0 A ordenada do baricentro em relac¸a\u2dco a` LN: y = \u222b A ydA A = 0 Conclu´\u131mos que a LN passa pelo baricentro da sec¸a\u2dco. Momento fletor Mz = \u222b A y \u3c3x dA = \u222b A y K y dA = K \u222b A y 2 dA =M , onde: \u222b A y 2 dA = I (momento de ine´rcia da sec¸a\u2dco em relac¸a\u2dco a` LN) enta\u2dco: K I =M \u2192 K =M/I \u2192 \u3c3x = M I y (3.77) (I = Iz = J = Jz \u2192 dimensional L4, unidade mm4 ou cm4) 71 Observac¸a\u2dco: \u2022 O diagrama de tenso\u2dces da fig3.48(d) e´ a vista longitudinal do so´lido de tenso\u2dces (fig3.49 para um sec¸a\u2dco retangular). Nas aplicac¸o\u2dces, o diagrama de tenso\u2dces e´ suficiente para representar a variac¸a\u2dco das tenso\u2dces normais na sec¸a\u2dco transversal. LN C\u2019 C B\u2019 BA\u2019 A\u2019 D D\u2019 o Figura 3.49: So´lido de tenso\u2dces \u2022 Ca´lculo das Tenso\u2dces Extremas (Ma´ximas) y = \u2212ds\u2192 \u3c3s = M I (\u2212ds) = \u2212 M I/ds y = di\u2192 \u3c3i = M I (di) = M I/di Fazendo I/ds = Ws, I/di = Wi - Mo´dulos de resiste\u2c6ncia a` flexa\u2dco (dimensional L3), Obtemos \u3c3s = \u2212M/Ws e \u3c3 =M/Wi\u2192 \u3c3max =M/W em valor absoluto. M > 0 { \u3c3s = Max. Tensa\u2dco de compressa\u2dco \u3c3i = Max. Tensa\u2dco de trac¸a\u2dco M < 0 { \u3c3s = Max. Tensa\u2dco de trac¸a\u2dco \u3c3i = Max. Tensa\u2dco de compressa\u2dco 2. Tenso\u2dces Normais na Flexa\u2dco Simples e Reta Sa\u2dco va´lidas as mesmas propriedades da flexa\u2dco pura e reta. Como o momento fletor e´ varia´vel, nas aplicac¸o\u2dces e´ necessa´rio analisar 2 sec¸o\u2dces cr´\u131ticas: momentos fletor maximo positivo(+) e negativo(-). Caso particular: sec¸a\u2dco sime´trica em relac¸a\u2dco a` LN \u2192basta analisar uma sec¸a\u2dco cr´\u131tica (momento fletor ma´ximo absoluto). 72 3.3.3 Exerc´\u131cios 1. A viga representada na fig3.50 tem sec¸a\u2dco constante, circular com dia\u2c6metro 0,25 m. Dados L = 1,5