# Bending moment and deflection relationship

### Beam Stress & Deflection | MechaniCalc

This section is to show you that there is a mathematical relationship between, Load, Shear Force, Bending Moment, Slope and Deflection. In engineering, deflection is the degree to which a structural element is displaced under a load. is the internal bending moment in the beam. If, in addition, the. Moment-Curvature Relationship m1 m2. (-ve M). (+ve v). (z) = vertical deflection at z. (z) = slope at z = v dv dz θ. ⎛ ⎞. │ │. ⎝ ⎠ z dz z dz.

Deflection of Beams Introduction: In all practical engineering applications, when we use the different components, normally we have to operate them within the certain limits i. For instance we say that the particular component is supposed to operate within this value of stress and the deflection of the component should not exceed beyond a particular value.

In some problems the maximum stress however, may not be a strict or severe condition but there may be the deflection which is the more rigid condition under operation.

### Euler–Bernoulli beam theory - Wikipedia

It is obvious therefore to study the methods by which we can predict the deflection of members under lateral loads or transverse loads, since it is this form of loading which will generally produce the greatest deflection of beams.

The following assumptions are undertaken in order to derive a differential equation of elastic curve for the loaded beam 1. Stress is proportional to strain i. Thus, the equation is valid only for beams that are not stressed beyond the elastic limit. The curvature is always small. Any deflection resulting from the shear deformation of the material or shear stresses is neglected. It can be shown that the deflections due to shear deformations are usually small and hence can be ignored.

Consider a beam AB which is initially straight and horizontal when unloaded. If under the action of loads the beam deflect to a position A'B' under load or infact we say that the axis of the beam bends to a shape A'B'.

It is customary to call A'B' the curved axis of the beam as the elastic line or deflection curve. In the case of a beam bent by transverse loads acting in a plane of symmetry, the bending moment M varies along the length of the beam and we represent the variation of bending moment in B. Uniform distributed loads result in a straight, sloped line on the shear diagram. The slope of the line is equal to the value of the distributed load.

The shear diagram is horizontal for distances along the beam with no applied load. The shear at any point along the beam is equal to the slope of the moment at that same point: The slope of the line is equal to the value of the shear.

## Euler–Bernoulli beam theory

Uniform distributed loads result in a parabolic curve on the moment diagram. The moment at any point along the beam is equal to the area under the shear diagram up to that point: The bending moment at any location along the beam can then be used to calculate the bending stress over the beam's cross section at that location.



The bending moment varies over the height of the cross section according to the flexure formula below: The negative sign indicates that a positive moment will result in a compressive stress above the neutral axis. The bending stress is zero at the beam's neutral axis, which is coincident with the centroid of the beam's cross section.

The bending stress increases linearly away from the neutral axis until the maximum values at the extreme fibers at the top and bottom of the beam. The maximum bending stress is given by: If the beam is asymmetric about the neutral axis such that the distances from the neutral axis to the top and to the bottom of the beam are not equal, the maximum stress will occur at the farthest location from the neutral axis.

In the figure below, the tensile stress at the top of the beam is larger than the compressive stress at the bottom.

The section modulus can be substituted into the flexure formula to calculate the maximum bending stress in a cross section: The shear force at any location along the beam can then be used to calculate the shear stress over the beam's cross section at that location.

The average shear stress over the cross section is given by: The shear stress is zero at the free surfaces the top and bottom of the beamand it is maximum at the centroid.

### Deflection (engineering) - Wikipedia

These terms are all constants. Shear Stresses in Rectangular Sections The distribution of shear stress along the height of a rectangular cross section is shown in the figure below: The maximum shear stress occurs at the neutral axis of the beam and is calculated by: Shear Stresses in Circular Sections A circular cross section is shown in the figure below: The equations for shear stress in a beam were derived using the assumption that the shear stress along the width of the beam is constant.

This assumption is valid at the centroid of a circular cross section, although it is not valid anywhere else.

Flexure Formula (Part 1) - Mechanics of Materials

Therefore, while the distribution of shear stress along the height of the cross section cannot be readily determined, the maximum shear stress in the section occuring at the centroid can still be calculated.