# Bending Moment

1.1 To examine how bending moment varies with an increasing point load. 1.2 To examine how bending moment varies at the cut position of the beam for various loading condition.

2.0 LEARNING OUTCOMES

2.1 To application the engineering knowledge in practical application 2.2 To enhance technical competency in structural engineering through laboratory application. 2.3 To communicate effectively in group.

2.4 To identify problem, solving and finding out appropriate solution through laboratory application.

3.0 THEORY

3.1 There are a number of assumptions that were made in order to develop the Elastic Theory of Bending. These are: * The beam has a constant, prismatic cross-section and is constructed of a flexible, homogenous material that has the same Modulus of Elasticity in both tension and compression (shortens or elongates equally for same stress). * The material is linearly elastic; the relationship between the stress and strain is directly proportional. * The beam material is not stressed past its proportional limit. * A plane section within the beam before bending remains a plane after bending (see AB & CD in the image below). * The neutral plane of a beam is a plane whose length is unchanged by the beam's deformation. This plane passes through the centroid of the cross-section. 3.2 In order to visualize this, think of a black rubber beam with three lines drawn on its side. The dashed lines in the diagram below represent this beam, and the neutral axis and lines AB and CD are drawn parallel to their respective sides. Lines AB and C D are separated by some distance that is determined, but is not of consequence for the following discussion. These two lines were parallel before bending. As the beam bends, these lines remain perpendicular to the neutral axis.

3.3 Thus, as the beam bends, developing a curve in the neutral plane that reflects the bending, the line CD does not remain parallel to line AD. There is a distinct shortening at the top face of the beam and elongation at the bottom face. CC' is equal to d (delta), or the shortening of the top fiber. Similarly, DD' is equal to (delta) or the elongation of the bottom fiber (tension). If these were measured, they would be found to be equal, but opposite. Knowing that the Modulus of Elasticity (E) describes a linear relationship between the strain and the amount of stress in a material and that this material is homogeneous and has a certain value for E, we can determine the stress. The magnitude of the strain at any point along C'D' can be found by using similar triangles. Therefore, the stress is also proportional to the distance from the neutral plane, as illustrated in the following diagram.

3.4 The loads and reactions acting on this beam segment cause a tendency for clockwise rotation (a clockwise moment). An internal moment counteracts this tendency so that the beam segment remains in equilibrium. Thus, the magnitude of the internal moment is exactly equal to the moment due to the external loads and reactions; but in the opposite direction. There is a state of equilibrium:

Internal Moment = External Moment

3.5 The illustration above shows the stress prisms developed due to the straining of the material. These prisms (compression and tension) can each be resolved into a single force that acts through the centroid of each prism. These two resulting forces (one is a compressive force (C) and the other a tensile force (T) ) are found to have equal but opposite magnitudes. They are separated by a distance that is approximately 2/3 of the depth of the beam (because of their triangular geometry) and create an internal couple(!) which causes the internal moment resistance of the beam. The exact location of the resultant forces of the stress prisms depends upon the location of the centroid of the entire stress prism. The location of the centroid is in turn dependent upon the...

References: * Yusof Ahmad (2001). Mekanik Bahan dan Struktur, Edisi Pertama, 2001, Malaysia: Universiti Teknologi Malaysia, pp 72 - 88.

* James M. Gere (2001). Mechanics of Materials, Fifth Edition, 2001, USA: Brooks/Cole Thomson Learning, pp 276 - 284.

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