Specific stiffness in tension The use of specific stiffness in
tension applications is straightforward. Both
stiffness in
tension and total
mass for a given length are directly proportional to
cross-sectional area. Thus performance of a beam in tension will depend on ''
Young's modulus divided by
density''.
Specific stiffness in buckling and bending Specific stiffness can be used in the design of
beams subject to
bending or Euler
buckling, since bending and buckling are stiffness-driven. However, the role that
density plays changes depending on the problem's constraints.
Beam with fixed dimensions; goal is weight reduction Examining the formulas for
buckling and
deflection, we see that the force required to achieve a given deflection or to achieve buckling depends directly on
Young's modulus. Examining the
density formula, we see that the
mass of a beam depends directly on the density. Thus if a beam's cross-sectional dimensions are constrained and weight reduction is the primary goal, performance of the beam will depend on ''
Young's modulus divided by
density''.
Beam with fixed weight; goal is increased stiffness By contrast, if a beam's weight is fixed, its cross-sectional dimensions are unconstrained, and increased stiffness is the primary goal, the performance of the beam will depend on Young's modulus divided by either density squared or cubed. This is because a beam's overall
stiffness, and thus its resistance to Euler
buckling when subjected to an axial load and to
deflection when subjected to a
bending moment, is directly proportional to both the Young's modulus of the beam's material and the
second moment of area (area moment of inertia) of the beam. Comparing the
list of area moments of inertia with formulas for
area gives the appropriate relationship for beams of various configurations.
Beam's cross-sectional area increases in two dimensions Consider a beam whose cross-sectional area increases in two dimensions, e.g. a solid round beam or a solid square beam. By combining the
area and
density formulas, we can see that the radius of this beam will vary with approximately the inverse of the square of the density for a given mass. By examining the formulas for
area moment of inertia, we can see that the stiffness of this beam will vary approximately as the fourth power of the radius. Thus the second moment of area will vary approximately as the inverse of the density squared, and performance of the beam will depend on ''
Young's modulus divided by
density squared''.
Beam's cross-sectional area increases in one dimension Consider a beam whose cross-sectional area increases in one dimension, e.g. a thin-walled round beam or a rectangular beam whose height but not width is varied. By combining the
area and
density formulas, we can see that the radius or height of this beam will vary with approximately the inverse of the density for a given mass. By examining the formulas for
area moment of inertia, we can see that the stiffness of this beam will vary approximately as the third power of the radius or height. Thus the second moment of area will vary approximately as the inverse of the cube of the density, and performance of the beam will depend on ''
Young's modulus divided by
density cubed''. However, caution must be exercised in using this metric. Thin-walled beams are ultimately limited by local buckling and
lateral-torsional buckling. These buckling modes depend on material properties other than stiffness and density, so the stiffness-over-density-cubed metric is at best a starting point for analysis. For example, most wood species score better than most metals on this metric, but many metals can be formed into useful beams with much thinner walls than could be achieved with wood, given wood's greater vulnerability to local buckling. The performance of thin-walled beams can also be greatly modified by relatively minor variations in geometry such as
flanges and stiffeners.
Stiffness versus strength in bending Note that the ultimate strength of a beam in bending depends on the ultimate strength of its material and its
section modulus, not its stiffness and second moment of area. Its deflection, however, and thus its resistance to Euler buckling, will depend on these two latter values. == Approximate specific stiffness for various materials ==