Why Does Curved Beam Analysis Include a Negative Strain Energy Term?

[Rothlisburger]

I'm teaching a class using Shigley's Mechanical Engineering Design. There is a section on deflection of curved members. If you have this text, you'll notice it calculates the strain energy in bending, axial, and shear, as usual. Then it says that there's another "negative" strain energy component, and cites an equation for this without introduction or reference. The text explaining this term is incomprehensible.

First off, since the strain energy is the integral of the strain energy density throughout the volume, and since we've accounted for all stresses (bending, axial, and shear), why should we need another term? And since when can strain energy be negative?

Any help would be most appreciated.

-David

 

[Navin A S]

The negative strain energy component is related to the geometric stiffness of the curved member. Curved members are inherently more stiff than straight members due to their curved shape, and this increased stiffness can be accounted for in the strain energy calculations by using a negative strain energy term. The equation used to calculate this term is derived from the principle of virtual work, which states that the work done on a system can be calculated as the integral of the force multiplied by the virtual displacement. The negative strain energy term is then the integral of the moment of inertia multiplied by the virtual curvature over the entire length of the member. This additional term accounts for the increased stiffness of the curved member, and thus improves the accuracy of the strain energy calculation.