Theory of Bending- Bending Equation for Simple bending

Theory of Bending

Content

  1. Assumptions.
  2. Bending Equation for beams in simple bending.
  3. Beams of uniform strength.
  4. Beams of composite section (Flitched Beams).

Assumptions in the Theory of bending:-

  • The material of the beam is perfectly homogeneous (i.e. of the same kind throughout) and isotropic (i.e. of equal elastic properties in all directions).
  • The beam material is stressed within its limit and thus obeys’s Hooke’s law.
  • The transverse section which is planes before bending remains plane after bending also.
  • Each layer of the beam is free to expand or contract independently of the layer, above or below it.
  • The value of young’s modulus for the material of beam is the same in tension and compression.

Bending Equation for beams in simple bending:

The following is the Bending Equation for simple bending,…

M / I = δ / y = E / R

Where,

M= Bending moment,

I= Moment of inertia of the area of cross-section,

δ= Bending stress,

y= Distance of extreme fiber from the neutral axis,

E= Young’s modulus for the material of the beam, and

R= Radius of curvature.

Note:-

  1. The line of intersection of the neutral layer with any normal cross-section of a beam is known as the neutral axis of that section.
  2. On one side of the neutral axis, there are compressive stresses, and on another side, there are tensile stresses. At the neutral axis, there is no stress of any kind. The neutral axis of a section always passes through its centroid.
  3. Since there are compressive stresses on one side of the neutral axis and stresses on the other side, therefore these stresses form a couple whose moment must be equal to the external moment ‘M’. The moment of this couple which is the external bending moment is known as the ‘Moment of Resistance‘.
  4. From the bending equation M/ I = δ/y, we have

M = δ x I/y  ⇒  M = δ x Z

where Z is known as section modulus or modulus of section.

Beams of Uniform Strength:

A beam in which bending stress developed is constant and equal to the allowable stress, is called a beam of uniform strength. It can be achieved by…

  1. Keeping the width uniform and varying the depth,
  2. Keeping the depth uniform and varying the width,
  3. Varying the width and depth both.

The common methods of obtaining the beam of uniform strength are by keeping the width uniform and varying the depth.

 

Beams of composite section (Flitched Beams):

A beam made up of two or more different materials joined together in such a manner that they behave like a unit piece, is called a Composite beam or Fitched beam. Such beams are used when one material if used alone, requires a larger cross-section area that is not suited to the space available and also to reinforce the beam at the region of the high bending moment or to equalize the strength of the beam in tension or compression.

A beam of two materials, as shown in fig. is most common, such as wooden beams reinforced by metal strips and concrete beams reinforced with steel rods. In such cases, the total moment of resistance will be

compoosite beam
strength of material

equal to the sum of the moment of resistance of the individual section.

M = M1 + M2 = δ1 . Z1 + δ2 . + Z2       ………..(i)

We also know that at any distance, from the neutral axis, strain in both the materials will be same, i.e.

δ1 / E1 = δ2 / E2    or    δ1 = E1/E2 x δ2 = m . δ2

where,  m= E1/E2 = modular ratio.

From the above two relations, we can find out the moment of resistance of a composite beam or the stress in the two materials.

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