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OkAlephZero said:dw/dx is the slope of the beam, which is assumed to be small. So dw/dx is also the angle the beam has rotated, in radians.
I understand this.AlephZero said:The top picture (Euler beam theory) assumes that cross sections of the beam stay perpendicular to the neutral axis. So the angle between a cross section and the vertical is the same as the slope of the beam.
Ok, how does the ##z\frac{dw}{dx}## come about? Is this equivalent to Z times the cos of the angle?AlephZero said:The picture is (stupidly, IMHO) drawn with a "left handed" coordinate system (z and w positive downwards not upwards) which is where the minus signs come from.
ThanksAlephZero said:In the bottom picture (Timoshenko beam theory) plane sections of the beam do not stay perpendicular to the neutral axis, so there is an extra shear strain (measured by angle gamma) involved.
bugatti79 said:Ok, how does the ##z\frac{dw}{dx}## come about? Is this equivalent to Z times the cos of the angle?
bugatti79 said:What practical examples are there where one shouldn't use Euler-Bernouilli to track beam deflection etc.
Euler-Bernoulli beam elements assume that the cross-sections of the beam remain perpendicular to the neutral axis during bending, while Timoshenko beam elements account for shearing deformation of the cross-sections. This makes Timoshenko beam elements more accurate for beams with high shear forces.
Beam elements use nodal displacements and rotations to represent boundary conditions. These are typically prescribed at the ends of the beam and are used to solve for the unknown nodal forces and moments.
Beam elements are only accurate for slender beams with small deformations. They also assume linear elastic behavior and do not account for material nonlinearity.
The stiffness and mass matrices for beam elements are calculated using numerical integration techniques, such as the Gauss-Legendre quadrature method. These matrices are then assembled into the global stiffness and mass matrices for the entire beam structure.
No, beam elements are only applicable for 1D structures. For 3D structures, other types of finite elements, such as shell or solid elements, must be used. However, beam elements can be used as part of a 3D structural model by combining them with other element types.