Read e-book online A First Course in Finite Elements [With CDROM] PDF

By Jacob Fish, Ted Belytschko

ISBN-10: 0470035803

ISBN-13: 9780470035801

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Read Online or Download A First Course in Finite Elements [With CDROM] PDF

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F1e ; F2e de ¼ ! 4 Elongation of an element and free-body diagrams, showing the positive sense of pe and FIe . 14 DIRECT APPROACH FOR DISCRETE SYSTEMS The element stiffness matrix Ke that relates these matrices will now be developed. 6) follows. Note that when ue1 ¼ ue2 , which is rigid body translation, the elongation vanishes. 9) can be written in the matrix form as ! e! u1 F1e ke Àke ¼ : F2e ue2 Àke ke |fflffl{zfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflffl{zffl ffl} Ke Fe de ð2:9Þ ð2:10Þ Using the underscored definitions, we can write the relation between the nodal forces and nodal displacements as e e e F ¼K d ; where ke K ¼ Àke e Àke ke !

The elongation of the segment is given by uðx þ ÁxÞ À uðxÞ and the original length is Áx; therefore, the strain is given by eðxÞ ¼ elongation uðx þ ÁxÞ À uðxÞ ¼ : original length Áx Taking the limit of the above as Áx ! 0, we recognize that the right right-hand side is the derivative of uðxÞ. Therefore, the strain–displacement equation is eðxÞ ¼ du : dx ð3:3Þ The stress–strain law for a linear elastic material is Hooke’s law, which we already saw in Chapter 2: sðxÞ ¼ EðxÞeðxÞ; where E is Young’s modulus.

1). 4). Compatibility is satisfied by requiring the displacement to be continuous. More will be said later about the degree of smoothness, or continuity, which is required. To solve the above differential equation, we need to prescribe boundary conditions at the two ends of the bar. For the purpose of illustration, we will consider the following specific boundary conditions: at x ¼ l, the displacement, uðx ¼ lÞ, is prescribed; at x ¼ 0, the force per unit area, or traction, denoted by "t, is prescribed.

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A First Course in Finite Elements [With CDROM] by Jacob Fish, Ted Belytschko

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