Theory of Flexure 

Types of Beams:

Types of Beams A beam is a bar or structural member subjected to transverse loads that tend to bend it. Any structural member acts as a beam if bending is induced by external transverse forces. 

1. A simple beam is a horizontal member that rests on two supports at the ends of the beam. All parts between the supports have free movement in a vertical plane under the influence of vertical loads. 

2. A fixed beam, constrained beam, or restrained beam is rigidly fixed at both ends or rigidly fixed at one end and simply supported at the other. 

3. A continuous beam is a member resting on more than two supports. 

4. A cantilever beam is a member with one end projecting beyond the point of support, free to move in a vertical plane under the influence of vertical loads placed between the free end and the support. 

Phenomena of Flexure

Phenomena of Flexure When a simple beam bends under its own weight, the fibers on the upper or concave side are shortened, and the stress acting on them is compression; the fibers on the under or convex side are lengthened, and the stress acting on them is tension. In addition, shear exists along each cross section, the intensity of which is greatest along the sections at the two supports and zero at the middle section. 

When a cantilever beam bends under its own weight, the fibers on the upper or convex side are lengthened under tensile stresses; the fibers on the under or concave side are shortened under compressive stresses, the shear is greatest along the section at the support and zero at the free end. 

Neutral Surface: The neutral surface is that horizontal section between the concave and convex surfaces of a loaded beam, where there is no change in the length of the fibers and no tensile or compressive stresses acting upon them. 

Neutral Axis: The neutral axis is the trace of the neutral surface on any cross section of a beam. 

Elastic Curve: The elastic curve of a beam is the curve formed by the intersection of the neutral surface with the side of the beam, it being assumed that the longitudinal stresses on the fibers are within the elastic limit. 

Reaction at Support

Reactions at Supports The reactions, or upward pressures at the points of support, are computed by applying the fol- lowing conditions necessary for equilibrium of a system of vertical forces in the same plane The algebraic sum of all vertical forces must equal zero; that is, the sum of the reactions equals the sum of the downward loads. The algebraic sum of the moments of all the ver- tical forces must equal zero. Condition applies to cantilever beams and to simple beams uniformly loaded, or with equal concentrated loads placed at equal distances from the center of the beam. In the cantilever beam, the reaction is the sum of all the vertical forces acting downward, comprising the weight of the beam and the superposed loads. In the simple beam each reaction is equal to one-half the total load, consisting of the weight of the beam and the superposed loads. Condition applies to a simple beam not uniformly loaded. The reactions are computed separately, by determining the moment of the several loads about each support. The sum of the moments of the load around one support is equal to the moment of the reaction of the other support around the first support. 

Conditions of Equilibrium

Conditions of Equilibrium The fundamental laws for the stresses at any cross section of a beam in equilibrium are: Sum of horizontal tensile stresses = sum of horizontal compressive stresses. Resisting shear = vertical shear. Resisting moment = bending moment.

Vertical Shear 

At any cross section of a beam the resultant of the external vertical force sacting on one side of the section is equal and opposite to the resultant of the external vertical forces acting on the other side of the section. These forces tend to cause the beam to shear vertically along the section. The value of either resultant is known as the vertical shear at the section considered. It is computed by finding the algebraic sum of the vertical forces to the left of the section; that is, it is equal to the left reaction minus the sum of the vertical downward forces acting between the left support and the section.

Shear Diagram

A shear diagram is a graphic representation of the vertical shear at all cross sections of the beam. Thus in the uniformly loaded simple beam the ordinates to the line represent to scale the intensity of the vertical shear at the corresponding sections of the beam. The vertical shear is greatest at the supports, where it is equal to the reactions, and it is zero at the center of the span. In the cantilever beam the vertical shear is greatest at the point of support, where it is equal to the reaction, and it is zero at the free end. Graphically the vertical shear on all sections of a simple beam carrying two concentrated loads at equal distances from the supports, the weight of the beam being neglected.

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Sk Najmul (engineer)

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SK NAJMUL (munna)

Hi. I’m a Mechanical Engineer, are I’m Designer of Blog Magic. I’m Creative Website, Web Designer, Auto Cad 3D and Interaction Designer, Industrial Machine Designer, Web Developer, Business Enthusiast, CNC, Writer and Photo editing and some time research an Mechanical subjects, Inspired to make things looks better.

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