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I know how to translate torques and vertical forces into singularity functions, but may i also express the horizontal forces with singular function? if yes then what would be that expression? enter image description here

Here the green beam is fixed at point A, and a distributed load acts on the beam, and a horizontal force acts at the upper edge of the beam.

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Yes, it is possible to describe horizontal loading with singularity functions.

However, in order to do so you must change the loading equation from scalar to vector-form:

$$\{q\} = \left\{\matrix{q_x \\ q_y}\right\}$$

So you'll handle the horizontal and vertical load components separately.

Obviously, how you then interpret the singularity functions changes, since beams behave differently under transversal and axial loads (the fourth integral of transversal loading is the deflection, which isn't true for axial loading).

Given this, I don't know how useful singularity functions are when dealing with horizontal loading. However, your question is whether it is possible to express the horizontal loading with singularity equations and the answer to that question is "yes".

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No, it is not possible. The singularity functions are defined only for vertical forces.

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