Abstract
Contributions of the second and higher modes to the overturning moment at the base of any classical linear system whose fundamental mode is given by a straight line are shown to vanish identically. A necessary and sufficient condition for a nonuniform shear beam to exhibit a linear first mode is obtained. In particular, it is shown that a shear beam with uniform cross section and mass density will have a linear fundamental mode shape if and only if the shearing stiffness is distributed parabolically. For such a beam, the higher mode shapes are given by odd degree Legendre polynomials.
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