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Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) [1] is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral ...
The starting point is the relation from Euler-Bernoulli beam theory = Where is the deflection and is the bending moment. This equation [7] is simpler than the fourth-order beam equation and can be integrated twice to find if the value of as a function of is known.
The Euler–Bernoulli beam equation defines the behaviour of a beam element (see below). It is based on five assumptions: Continuum mechanics is valid for a bending beam. The stress at a cross section varies linearly in the direction of bending, and is zero at the centroid of every cross section.
The conjugate-beam methods is an engineering method to derive the slope and displacement of a beam. A conjugate beam is defined as an imaginary beam with the same dimensions (length) as that of the original beam but load at any point on the conjugate beam is equal to the bending moment at that point divided by EI. [1]
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.
[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-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of fourth order but, unlike Euler ...
The following procedure provides a method that may be used to determine the displacement and slope at a point on the elastic curve of a beam using the moment-area theorem. Determine the reaction forces of a structure and draw the M/EI diagram of the structure.
Releasing the vertical reaction for A allows the beam to rotate to Δ. Likewise for part (c). Δ is typically taken as positive upwards. Part (d) of the figure shows the influence line for shear at point B. Using the beam sign convention and cutting the beam at B, we can deduce the figure shown.