An example of double forces taken from structural analysis

The classic Bernoulli–Euler beam is revisited in relation to the Timoshenko beam, using a T-type cross-section. A practical example of double forces is worked out, where the double forces appear due to the shearing of the horizontal flange of the T-section. It is shown that a non-local estimate of the mean value of the vertical deflection of the beam leads to a shear gradient that is energetically consistent with the double force. Therefore, an additional energy term that includes the derivative of the beam’s curvature leads to a consistent non-local beam model, simple enough to be used in structural analysis. Such beam models are useful in micro- and nano-technology. Moreover, this simple example opens the discussion for the importance of non-local differentiation in the averaging theories of Mechanics.