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The Timoshenko–Ehrenfest beam theory was developed by Stephen Timoshenko and Paul Ehrenfest [1] [2] [3] early in the 20th century. [ 4 ] [ 5 ] The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich composite beams , or beams subject to high ...
Castigliano's method for calculating displacements is an application of his second theorem, which states: If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Q i then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement q i in the direction of Q i.
The strain energy in the form of elastic deformation is mostly recoverable in the form of mechanical work. For example, the heat of combustion of cyclopropane (696 kJ/mol) is higher than that of propane (657 kJ/mol) for each additional CH 2 unit. Compounds with unusually large strain energy include tetrahedranes, propellanes, cubane-type ...
The total elastic energy due to strain can be divided into two parts: one part causes change in volume, and the other part causes a change in shape. Distortion energy is the amount of energy that is needed to change the shape. Fracture mechanics was established by Alan Arnold Griffith and George Rankine Irwin. This important theory is also ...
In other words, the summation of the work done on the system by the set of external forces is equal to the work stored as strain energy in the elements that make up the system. The virtual internal work in the right-hand-side of the above equation may be found by summing the virtual work done on the individual elements.
Thus it is referred to as Timoshenko-Ehrenfest beam theory. This fact was testified by Timoshenko. [21] The interrelation between Timoshenko-Ehrenfest beam and Euler-Bernoulli beam theory was investigated in the book by Wang, Reddy and Lee. [22] He died in 1972 and his ashes are buried in Alta Mesa Memorial Park, Palo Alto, California.
For an isotropic hyperelastic material, the function relates the energy stored in an elastic material, and thus the stress–strain relationship, only to the three strain (elongation) components, thus disregarding the deformation history, heat dissipation, stress relaxation etc.
Energy principles in structural mechanics express the relationships between stresses, strains or deformations, displacements, material properties, and external effects in the form of energy or work done by internal and external forces.