Page:American Journal of Mathematics Vol. 2 (1879).pdf/25

Since $z$ is considered the only variable, such a value for $y$ may be taken that the equation will refer to that portion of a normal section of the beam which lies along the axis of symmetry of the section, for which $f\prime (y,z)=0.$

Hence

${\tfrac {dN_{1}}{ds}}=-xF_{z}^{\prime \prime }(y,z)+{\tfrac {d\Psi (y,z)}{dz}}.$........(14)

Now ${\tfrac {dN_{1}}{dz}}$ is always a positive quantity, but the function $\Psi (y,z)$ is perfectly arbitrary, and it may be given such a value and sign, if it has real existence as a function of the two variables $y$ and $z$, that the second member of equation (14) may have a sign contrary to that of its first member, whatever may be the value of $-xF_{z}^{\prime \prime }(y,z).$

In order that (14) may be a true one, therefore ${\tfrac {d\Psi (y,z)}{dz}}dz=0;$ consequently

${\tfrac {d\Psi (y,x)}{dy}}\,dy+{\tfrac {d\Psi (y,z)}{dz}}\,dz=0;{\text{ or, }}\Psi (y,z)=c\,,\,$........(15)

$c$ being a constant quanityt. In the case where the origin is taken at a section of no flexure $c=0.$ Otherwise, at the free ends of a beam, and at sections of contra flexure, there will exist normal stresses parallel to the axis, since a portion of the expression for $N_{1}$ would then be independent of $x.$

There is then established the important equation, when the origin is taken at a section of no bending,

$N_{1}=-x\left[f_{y}^{\prime }(y,z)+F_{x}^{\prime }(y,z)\right]$........(16)

It is seen by this equation that $N_{1}$ varies directly as $x.$ But in this equation there is apparently involved the condition that one external force only is acting at the distance $x$ from the section under consideration. This arises from the fact that the external forces are assumed to produce no compression at their points of application. It does not affect, however, the generality of the equation, for the last two of equations (3) show that whatever may be the bending moment, the above assumption simply means, it is so produced that the total shearing in any section is equal to that in any other, since ${\tfrac {dT_{2}}{dx}}$ and ${\tfrac {dT_{3}}{dx}}$ both equal zero.

The magnitude of the external force then, is a matter of indifference, only it must be constant for the same beam with any given system of loading.

The normal intensity $N_{1}$ is, consequently, proportional to the variable lever arm $x$ of any given constant force which may produce the bending moment to which the