Principles of Dairy Machine Design

Module 4. Beams and bending moments

Lesson 20
FLEXURAL STRESS

20.1 Introduction

Flexural strength, also known as modulus of rupture, bend strength, or fracture strength a mechanical parameter for brittle material, is defined as a material's ability to resist deformation under load. The transverse bending test is most frequently employed, in which a rod specimen having either a circular or rectangular cross-section is bent until fracture using a three point flexural test technique. The flexural strength represents the highest stress experienced within the material at its moment of rupture. It is measured in terms of stress, here given the symbol σ.

Fig. 20.1 Beam diagram

20.2 Flexural Stress

When a member is being loaded similar to that in figure 20.1 flexure stress (or bending stress) will result. Bending stress is a more specific type of normal stress. When a beam experiences load like that shown in figure 20.1 on the top fibers of the beam undergo a normal compressive stress. The stress at the horizontal plane of the neutral axis is zero. The bottom fibers of the beam undergo a normal tensile stress. It can be concluded therefore that the value of the bending stress will vary linearly with distance from the neutral axis.

Calculating the maximum bending stress is crucial for determining the adequacy of beams, rafters, joists, etc.

σ_b =MY/I

σ_b=Bending stress

M=Calculated Bending Moment, M.m

Y=Vertical distance away from neutral axis,

I=Moment of inertia about the neutral axis, m⁴

20.3 Shear Stresses in Beams Relations between Centre, Torsional and Flexural Loads

The shear stress in any section at a distance ’y’ from neutral axis is given by

σ_avg = F / b d

J_max = FAy / Ib

Fig. 20.2 Shear stress distribution in rectangular section

Where

σ_avg = Complimentary (or) Horizontal shear

stress at a layer x – x.

F = Shear Force at a section, M

A = Area above y below the section x – x, m²

y = Distance of centroid of area ‘A’ from neutral axis ,m

I = Moment of Inertia about the neutral axis, m⁴

b = Width of section x – x , m

Average shear stress is given by

σ_Avg = F / A

Where F = Shear force

A = Cross sectional Area

For Rectangular cross section

Fig. 20.3 Shear stress distribution in rectangular section

σ_avg = F/ b.d σ_max = FAy /I * b

l =1/12 b.d³

= 3/2 F/b.d

σ_max = 3/2 σ_avg

For square cross section

Same as rectangular cross section.

σ_max = 3/2 σ_avg

Fig. 20.4 Shear stress distribution in square section

For Circular cross section

Y = 4R/3л

σ_avg = F/ (л /4)D²

σ_max= 4/3 σ_avg.

Fig. 20.5 Shear stress distribution in circular section

Note:

A cross section having more area at N.A is strong in resisting shear. {except circle}.

If bending and shear are acting together in a beam, the main design criterion is the bending only. Therefore a rectangular section is commonly used as a beam.