Nonlinear Composite Beam Theory by Hodges, Dewey H.

By Hodges, Dewey H.

From an authoritative professional whose paintings on glossy helicopter rotor blade research has spanned over 3 many years, comes the 1st constant and rigorous presentation of beam concept. starting with an summary of the idea constructed over the past 60 years, Dr. Hodges addresses the kinematics of beam deformation, offers an easy approach to symbolize pressure in an in the beginning curved and twisted beam, and gives cross-sectional research for beams with arbitrary move sections and composed of arbitrary fabrics. He is going directly to current how to properly get better all parts of cross-sectional pressure and rigidity sooner than supplying a usual one-dimensional (1D) thought of beams. pattern effects for either cross-sectional and 1D research are offered as is a parallel remedy for thin-walled beams.
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We will be more precise in defining frame B later, but this level of detail is sufficient for our present purposes. In order to represent the deformed state mathematically, we must first deal with the rotation. 6) where ∆ is the identity tensor, which can be represented by a dyadic bi bi with bi representing any orthonormal set of base vectors. Now the displacement field can be specified. 7) where w = wi bi is a vector that represents the (small) warping displacement field such that wi = wi (x1 , x2 , x3 ).

The force strain is so named because its components are conjugate to the section forces. It is related to the unit tangent vectors of the deformed and undeformed beam. Recall that the unit tangent vector of the undeformed beam is r = b1 , and the unit tangent vector of the deformed beam is ∂R/∂s or R /s , where s is equal to unity plus a small stretching strain of the reference line. Now, γ is the difference of the unit tangent of the deformed beam, “pulled back” to the frame b orientation by premultiplication with C bB , and the unit tangent of the undeformed beam.

84)     2γ13 B3 0 1 T3 where 2γ12 and 2γ13 are the small angles caused by the transverse shear deformation. The distinction between these two frames is important for the development of different levels of approximation. KINEMATICS OF BEAMS 53 2 Fig. 3 Planar view of coordinate systems used for transverse shear formulation with 2γ13 exaggerated in magnitude. 85) where R is the position vector to a point on the reference line of the deformed beam, and wi are the components of warping. Note in this specialized formulation that we choose T 1 to be tangent to the deformed beam reference line, which means we classify the transverse shear deformation as part of the warping field.

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