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The load is distributed to a part of the cantilever length, having linearly varying magnitude from w_1 to w_2 , while the remaining length is unloaded. The dimensions of w_1 and w_2 are force per length. The total amount of force applied to the beam is W={L-a-b\over2}(w_1+w_2) , where L the beam length and a , b the unloaded lengths at the left and right side of the beam respectively.

The following table contains the formulas describing the static response of the cantilever beam under a varying distributed load, of trapezoidal shape.

In this case, a moment is imposed in a single point of the beam, anywhere across the span. In practical terms, it could be a force couple, or a member in torsion, connected out of plane and perpendicular to the beam.

This is the most generic case. The formulas for partially distributed uniform and triangular loads can be derived by appropriately setting the values of w_1 and w_2 . Furthermore, the respective cases for fully loaded span, can be derived by setting a and b to zero.

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The last two assumptions satisfy the kinematic requirements for the Euler Bernoulli beam theory that is adopted here too.

The following table contains the formulas describing the static response of the cantilever beam under a partially distributed trapezoidal load.

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Removing the singe support or inserting an internal hinge, would render the cantilever beam into a mechanism: a body the moves without restriction in one or more directions. This is unwanted situation for a load carrying structure. As a result, the cantilever beam offers no redundancy in terms of supports. If a local failure occurs the whole structure would collapse. These type of structures, that offer no redundancy, are called critical or determinant structures. To the contrary, a structure that features more supports than required to restrict its free movements is called redundant or indeterminate structure. The cantilever beam is a determinant structure.

The following table contains the formulas describing the static response of the cantilever beam under a trapezoidal load distribution, due to a slab, as depicted in the schematic above.

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These rules, though not mandatory, are rather universal. A different set of rules, if followed consistently would also produce the same physical results.

This load distribution is typical for cantilever beams supporting a slab. The distribution looks like a right trapezoid, with an increasing part close to the fixed support and a constant part, with magnitude equal to w , at the remaining length, up to the tip. The dimensions of w are force per length. The total amount of force applied to the beam is W=w (L-a/2) , where, L , is the cantilever length and, a , is the length close to the fixed support, where the load distribution is varying (triangular).

The following table contains the formulas describing the static response of the cantilever beam under a concentrated point force P , imposed at a random distance a from the fixed support.

For the calculation of the internal forces and moments, at any section cut of the beam, a sign convention is necessary. The following are adopted here:

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This tool calculates the static response of cantilever beams under various loading scenarios. The tool calculates and plots diagrams for these quantities:

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The force is concentrated in a single point, anywhere across the cantilever length. In practice however, the force may be spread over a small area. In order to consider the force as concentrated, though, the dimensions of the application area should be substantially smaller than the beam length. In the close vicinity of the force, stress concentrations are expected and as result the response predicted by the classical beam theory maybe inaccurate. This is only a local phenomenon however, and as we move away from the force location, the discrepancy of the results becomes negligible.

The following table contains the formulas describing the static response of the cantilever beam under a uniform distributed load w .

The load w is distributed throughout the cantilever span, having constant magnitude and direction. Its dimensions are force per length. The total amount of force applied to the cantilever beam is W=w L , where L the beam length. Either the total force W or the distributed force per length w may be given, depending on the circumstances.

The cantilever beam is one of the most simple structures. It features only one support, at one of its ends. The support is a, so called, fixed support that inhibits all movement, including vertical or horizontal displacements as well as any rotations. The other end is unsupported, and therefore it is free to move or rotate. This free end is often called the tip of the cantilever.

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At any case, the moment application area should spread to a small length of the cantilever, so that it can be successfully idealized as a concentrated moment to a point. Although in the close vicinity the application area, the predicted results through the classical beam theory are expected to be inaccurate (due to stress concentrations and other localized effects), the predicted results become perfectly valid, when we move away, as stated by the Saint-Venant principle.

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The following table contains the formulas describing the static response of the cantilever beam under a partially distributed uniform load.

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The values of w_1 and w_2 can be freely assigned. It is not mandatory for the former to be smaller than the latter. They may take even negative values (one or both of them).

If w_1=0 , the formulas in the following table correspond to a triangular distributed load, with increasing magnitude (peak at the tip).

The following table contains the formulas describing the static response of the cantilever beam under a concentrated point force P , imposed at the tip.

The static analysis of any load carrying structure involves the estimation of its internal forces and moments, as well as its deflections. Typically, for a plane structure, with in plane loading, the internal actions of interest are the axial force N , the transverse shear force V and the bending moment M . For a cantilever beam that carries only transverse loads, the axial force is always zero, provided the deflections are small. Therefore it is rather common to neglect axial forces.

The following table contains the formulas describing the static response of the cantilever beam under a concentrated point moment M , imposed at a distance a from the fixed support.

If w_2=0 , the formulas in the following table correspond to a triangular distributed load, with decreasing magnitude (peak at the fixed support).

The load is distributed to a part of the cantilever length, with constant magnitude w , while the remaining length is unloaded. The dimensions of w are force per length. The total amount of force applied to the beam is W=w\left(L-a-b\right) , where L the cantilever length and a , b the unloaded lengths at the left and right side of the beam, respectively.

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The load is distributed throughout the cantilever length, having linearly varying magnitude, starting from w_1 at the fixed support, to w_2 at the free end. The dimensions of w_1 and w_2 are force per length. The total amount of force applied to the beam is W={L\over2}(w_1+w_2) , where L the cantilever length.

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The lens first devised by Augustin Fresnel (pronounced fraynel) (1788-1827) is a beehive shape of concentric annular sections that cast parallel beams of light.

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The force is concentrated in a single point, located at the free end of the beam. In practice however, the force may be spread over a small area, although the dimensions of this area should be substantially smaller than the cantilever length. In the close vicinity of the force application, stress concentrations are expected and as result the response predicted by the classical beam theory is maybe inaccurate. This is only a local phenomenon however. As we move away from the force location, the results become valid, by virtue of the Saint-Venant principle.

Please take in mind that the assumptions of Euler-Bernoulli beam theory are adopted, the material is elastic and the cross section is constant over the entire beam span (prismatic beam).

The values of w_1 and w_2 can be freely assigned. It is not mandatory for the former to be smaller than the latter. They may take even negative values (one or both of them).