Show that the deflection at the center of a straight member with respect to its ends is given by…

Show that the deflection at the center of a straight member
with respect to its ends is given by

where _1 and _3 are the curvatures at the two ends;
l is the distance between the two ends; and _2 is the curvature at the
middle. The variation of _ is assumed to be a second-degree parabola.

This is a geometric relation which can be derived by
integration of the equation _ = _d2y/dx2, where y is the deflection
and x is the distance along the member, measured from the left end. The
equation can be derived more easily by the method of elastic weights, using
equivalent concentrated loading (

Show that the deflection at the center of a straight member
with respect to its ends is given by

where _1 and _3 are the curvatures at the two ends;
l is the distance between the two ends; and _2 is the curvature at the
middle. The variation of _ is assumed to be a second-degree parabola.

This is a geometric relation which can be derived by
integration of the equation _ = _d2y/dx2, where y is the deflection
and x is the distance along the member, measured from the left end. The
equation can be derived more easily by the method of elastic weights, using
equivalent concentrated loading (Figure 10.11).

In practice, the expression derived can be used for
continuous or simple beams having a constant or a variable cross section when
the parabolic variation of _ is acceptable.

Figure 10.11