In the following table, the formulas describing the static response of the simple beam under a uniform distributed load w are presented.

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

The simply supported beam is one of the most simple structures. It features only two supports, one at each end. One pinned support and a roller support. Both of them inhibit any vertical movement, allowing on the other hand, free rotations around them. The roller support also permits the beam to expand or contract axially, though free horizontal movement is prevented by the other support.

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

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

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The orientation of the triangular load is important! The formulas presented in this section have been prepared for the case of an ascending load (left-to-right), as shown in the schematic. For a descending load you may mirror the beam, so that its left end (point A) is the least loaded one. The x axis and all results will be mirrored too.

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Kohärentsein

The load is distributed to a part of the beam span, having linearly varying magnitude from w_1 to w_2 , while the remaining span 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 span length and a , b the unloaded lengths at the left and right side of the beam respectively.

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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 simply supported beam that carries only transverse loads, the axial force is always zero, therefore it is often neglected. The calculated results in the page are based on the following assumptions:

KohärentSynonym

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

Die Einhaltung dieser Leitfragen erzeugt den roten Faden Ihrer Arbeit. Der rote Faden ist eine Metapher für die Kohärenz auf Textebene und den Zusammenhalt bzw. die innere und inhaltliche Geschlossenheit Ihrer Arbeit und aller Teile Ihrer Arbeit. Der rote Faden hat einige metastrukturelle (übergeordnete) Ankerpunkte:

Prüfen Sie Schritt für Schritt ausgehend von der Gesamtebene, ob die Aneinanderreihung der Textelemente logisch ist. Achten Sie darauf, dass ein argumentativer Pfad nachvollziehbar ist und Leser*innen Ihren Gedanken folgen können. Nutzen Sie Ihren Freundeskreis und lassen Sie Ihren Text gegenlesen. Das hilft Ihnen, unlogische oder unklare Gedanken im Text herauszufiltern. Auch die Schreibmentor*innen und -assistent*innen des CTL können Ihnen Feedback auf Auszüge aus Ihrem Text oder Ihre Gliederung geben.

In the following table, the formulas describing the static response of the simple beam, under a partially distributed uniform load, are presented.

In the following table, the formulas describing the static response of the simple beam under a concentrated point force P , imposed in the middle, are presented.

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).

Jeder Absatz, jedes Kapitel und im Endeffekt die ganze Arbeit haben nur einen Zweck – die Forschungsfrage zu beantworten und Ihren Leser*innen diese verständlich zu vermitteln. Wenn Sie sich nicht sicher sind, ob sie den roten Faden erfolgreich entwickelt haben, können Sie Schrittweise den Zusammenhang der jeweiligen Absätze und Kapitel mit der Forschungsfrage vergleichen.

Kohärentes Verhalten

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In the following table, the formulas describing the static response of the simple beam under a varying distributed load, of trapezoidal form, are presented.

In the following table, the formulas describing the static response of the simple beam, under a partially distributed trapezoidal load, are presented.

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

Kohärenz

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In the following table, the formulas describing the static response of the simple beam under a trapezoidal load distribution, as depicted in the schematic above, are presented.

Removing any of the supports or inserting an internal hinge, would render the simply supported beam to a mechanism, that is body the moves without restriction in one or more directions. Obviously this is unwanted for a load carrying structure. Therefore, the simply supported 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.

This load distribution is typical for the beams in the perimeter of a slab. The distribution is of trapezoidal shape, with maximum magnitude w at the interior of the beam, while at its two ends it becomes zero. The dimensions of (\w\) are force per length. The total amount of force applied to the beam is W=w (L-a/2-b/2) , where L the span length and a , b the lengths at the left and right side of the beam respectively, where the load distribution is varying (triangular).

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KohärentBeispiel

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.

The load is distributed throughout the beam span, however, its magnitude is not constant but is varying linearly, starting from zero at the left end to its peak value w_1 at the right end. The dimensions of w_1 are force per length. The total amount of force applied to the beam is W={1\over2}w L , where L the span length.

KohärentBedeutung

The force is concentrated in a single point, anywhere across the beam span. 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 span 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.

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).

Kohärenz bezeichnet den Zusammenhang zwischen unterschiedlichen Texteilen. Dieser sollte sich durch die Struktur der Arbeit und die logische Aneinanderreihung von Inhalten in Form von Absätzen ergeben. Man unterscheidet drei Ebenen der Kohärenz:

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KohärentEnglisch

At any case, the moment application area should spread to a small length of the beam, 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), as we move away, the predicted results are perfectly valid, as stated by the Saint-Venant principle.

In the following table, the formulas describing the static response of the simple beam under a linearly varying (triangular) distributed load, ascending from the left to the right, are presented.

In the following table, the formulas describing the static response of the simple beam under a concentrated point moment M , imposed at a distance a from the left end, are presented.

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

In the following table, the formulas describing the static response of the simple beam under a concentrated point force P , imposed at a random distance a from the left end, are presented.

The force is concentrated in a single point, located in the middle 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 beam span 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.

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Jeder Absatz besteht wie jedes Kapitel aus Einleitung, Hauptgedanke und Zusammenfassung des Geschriebenen. Sie können sich jede der Ebenen wie eine Sanduhr vorstellen: Die Information, die am Anfang eingeleitet wird, bildet in der Mitte in Form der Kernaussage des Absatzes einen Höhepunkt und wird mit einem Fazit abgeschlossen, von dem aus Sie auf den nächsten Gedanken / das nächste Argument überleiten können. So wie der Sand, der am Anfang im oberen Teil der Uhr gesammelt ist, durch die Mitte fließt und sich im unteren Teil der Uhr sammelt.

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).

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The load is distributed throughout the beam span, having linearly varying magnitude, starting from w_1 at the left end, to w_2 at the right 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 span length.

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Achten Sie bei der Lektüre von wissenschaftlichen Texten auf Absatzstrukturen der Autor*innen. Betrachten Sie die Absatzgestaltung und identifizieren Sie Einleitung-Kernaussage-Zusammenfassung/Überleitung.

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: