Page:American Journal of Mathematics Vol. 2 (1879).pdf/22

These are the only equations of condition resulting from the consideration of the principles of statics alone, and are, in general, insufficient to determine the six unknown intensities which enter them.

In the following discussion the piece or beam subjected to bending will be supposed to occupy a horizontal position; the bending forces (including the reäction) will be supposed to act in a direction normal to the axis of the beam; the beam will be supposed straight and uniform in normal section; the axis of $x$ will be taken to be parallel to the axis of the beam; the axis of $z$ will be vertical and the axis of $y$ horizontal and perpendicular to that of $x$. The axes of $z$ and $y$ will thus be parallel to axes of symmetry of the section, if that section be symmetrical and the beam be properly placed. No other kind of section or position will be considered. In the generality of cases the coefficients of elasticity for tension and compression will be considered equal. In the one or two cases where they are not supposed to be equal, the axis of $x$ will still be taken parallel to the axis of the beam, and not coincident with it.

Now in the case of flexure, generally considered, on account of the distortion of the material subjected to stress, the six stresses $N_{1},$ $N_{2},$ $N_{3},$ $T_{1},$ $T_{2},$ $T_{3}$ actually exist, some of them are so small that they may be considered differential quantities, i. e., they owe their existence to the indefinitely small difference of the intensities of stresses on two small portions of the material indefinitely close together. The omission of these quantities will evidently produce no essential error in the results, though it is true that it takes from the mathematical exactness of the equations.

Beams whose sections, i. e. normal sections, are symmetrical in respect to the axis of $y$ and $z$ will first be considered, and it will be assumed that $N_{2}=0,$ $N_{3}=0,$ and $T_{1}=0$. It should be stated that the sections considered will not only be symmetrical ones but such that they will not have re-entrant contours.

The case of rectangular sections when $N_{3}$ is not equal to zero will be taken up afterwards. It might be treated as existing in all beams if the external forces were so applied that $T_{1}$ is still zero, but that is an exceptional case and will not be taken up. $T_{1}$ may in reality exist as a very small quantity, in some cases, on account of the variable value of $T_{2}$ at the neutral surface.