Information AboutBending |
| CATEGORIES ABOUT BENDING | |
| structural system | |
| continuum mechanics | |
| statics | |
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''This article is about structural behavior. For other meanings see Bending (disambiguation) .'' In subjected to an external Load applied perpendicular to the axis of the element. A structural element subjected to bending is known as a Beam . A Closet rod Sagging under the weight of clothes on Clothes Hanger s is an example of a beam experiencing bending. Bending produces reactive parallel to the lateral loading, Compression along the top of the beam, and Tension along the bottom of the beam. These last two forces form a Couple or Moment as they are equal in magnitude and opposite in direction. This Bending Moment produces the sagging deformation characteristic of Compression Member s experiencing bending. The compressive and tensile forces shown in Figure 2 induce vary Linear ly, there therefore exists a point on the linear path between them where there is no bending stress. The Locus of these points is the neutral axis. Because of this area with no stress and the adjacent areas with low stress, using uniform cross section beams in bending is not a particularly efficient means of supporting a load as it does not use the full capacity of the beam until it is on the brink of collapse. Wide-flange beams (I-Beams) and Truss Girder s effectively address this inefficiency as they minimize the amount of material in this under-stressed region. SIMPLE OR SYMMETRICAL BENDING Beam bending is analyzed with the Euler-Bernoulli Beam Equation . The classic formula for determining the bending stress in a member is: : simplified for a beam of rectangular cross-section to: :
This equation is valid only when the stress at the extreme fiber (i.e. the portion of the beam furthest from the neutral axis) is below the Yield Stress of the material it is constructed from. At higher loadings the stress distribution becomes non-linear, and ductile materials will eventually enter a ''plastic hinge'' state where the magnitude of the stress is equal to the yield stress everywhere in the beam, with a discontinuity at the neutral axis where the stress changes from tensile to compressive. This plastic hinge state is typically used as a Limit State in the design of steel structures. COMPLEX OR UNSYMMETRICAL BENDING The equation above is, also, only valid if the cross-section is symmetrical. For unsymmetrical sections, the full form of the equation must be used (presented below): Complex Bending of Homogeneous Beams The complex bending stress equation for elastic, homogeneous beams is given as where Mx and My are the bending moments about the x and y Centroid axes, respectively. Ix and Iy are the second moments of area (also known as moments of inertia) about the x and y axes, respectively, and Ixy is the product of inertia. Using this equation it would be possible to calculate the bending stress at any point on the beam cross section regardless of moment orientation or cross-sectional shape. Note that Mx, My, Ix, Iy, and Ixy are all unique for a given section along the length of the beam. In other words, they will not change from one point to another on the cross section. However, the x and y variables shown in the equation correspond to the coordinates of a point on the cross section at which the stress is to be determined. SEE ALSO |
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