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49. Rod under Suppose a rod transverse forces. AB is in equilibrium of forces perpendicular to its under the action of a number length such as P^, Pg, . . Q^, Qg etc APPLIED ELASTICITY 48 CB Since the portion in fig. 21. CB AC equilibrium the action of in is on C must be such as to balance the forces P3 and Q3. -Q, '. (S-i) which will balance the couple formed by F and the resultant of P3 and Q^. By taking moments about C for the portion and a couple CB 22) (fig. -CKXQ3 The M force is sum sum of all moment perpendicular to at C.

Then it follows that the ratio of the increase of volume to the original volume is (a-\- fi -{- y)- Call this A. 20) . ^) and n are constants, n being the modulus of E and In terms of rigidity. 22) j^ du dv dw 25. Relations We between stresses and external with a rectangular block of dimensions . '"3) ^Si + 8y + T. forces. x x dy dx dx having its centre at x, y, 2, and take the means stresses on the faces to be the stresses at the middle of the face concerned. The middle points of the two faces perpendicular to the axis of x have coordinatey shall deal APPLIED ELASTICITY 22 3'' {^'dli^-^'' ±^Sy, Y.

EIx' a sin If the we are given the component couples M,,' resultant tan~"^ clearly q). —-- couple is y M^ The with OY'. 63) has already shown, that the axis of the resultant couple does not coincide with the neutral axis except when the neutral axis coincides with one of the principal axes with OY'. of the section. 39. The form of the strained rod. The displacements of points originally on in the expressions by putting y=zo, 2 = the ji'-axis are obtained for the displacements. 65) write 2 for old' jr-axis is w strained.

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