In structural analysis, this bending failure is called a plastic hinge, since the full load carrying ability of the structural element is not reached until the full cross-section is past the yield stress. Failure in bending will occur when the bending moment is sufficient to induce tensile/compressive stresses greater than the yield stress of the material throughout the entire cross-section. Tensile and compressive stresses increase proportionally with bending moment, but are also dependent on the second moment of area of the cross-section of a beam (that is, the shape of the cross-section, such as a circle, square or I-beam being common structural shapes). The concept of bending moment is very important in engineering (particularly in civil and mechanical engineering) and physics. Moments and torques are measured as a force multiplied by a distance so they have as unit newton-metres (N It is therefore clear that a point of zero bending moment within a beam is a point of contraflexure-that is, the point of transition from hogging to sagging or vice versa. If clockwise bending moments are taken as negative, then a negative bending moment within an element will cause " hogging", and a positive moment will cause " sagging". The forces and moments on either side of the section must be equal in order to counteract each other and maintain a state of equilibrium so the same bending moment will result from summing the moments, regardless of which side of the section is selected. The bending moment at a section through a structural element may be defined as the sum of the moments about that section of all external forces acting to one side of that section. Normal force is also termed as axial force. The resultant internal couple is called the bending moment while the resultant internal force is called the shear force (if it is transverse to the plane of element) or the normal force (if it is along the plane of the element). For equilibrium, the moment created by external forces/moments must be balanced by the couple induced by the internal loads. The internal reaction loads in a cross-section of the structural element can be resolved into a resultant force and a resultant couple. In reality, beam supports are usually neither absolutely fixed nor absolutely rotating freely. The simplest type of beam is the cantilever, which is fixed at one end and is free at the other end (neither simple or fixed). Beams can also have one end fixed and one end simply supported. Other beams can have both ends fixed (known as encastre beam) therefore each end support has both bending moments and shear reaction loads. The diagram shows a beam which is simply supported (free to rotate and therefore lacking bending moments) at both ends the ends can only react to the shear loads. The most common or simplest structural element subjected to bending moments is the beam. In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. Force tending to bend a structural element Shear and moment diagram for a simply supported beam with a concentrated load at mid-span.
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