Lecture Strength of Materials I: Chapter 7 - PhD. Tran Minh Tu

Pure Bending : Prismatic members subjected to equal and opposite couples acting in the same longitudinal plane.. Bending stress[r]

STRENGTH OF MATERIALS

CHAPTER

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BENDING

Contents 7.1 Introduction

7.2 Bending stress

7.3 Shearing stress in bending

7.4 Strength condition

7.5 Sample Problems

7.6 Deflections of beam

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7.1 Introduction

In previous charters, we considered the stresses in the bars caused

by axial loading and torsion Here we introduce the third fundamental loading: bending When deriving the relationship between the bending moment and the stresses causes, we find it again necessary to make certain simplifying assumptions

We use the same steps in the analysis of bending that we used for torsion in chapter 6

7.1 Introduction

Classification of Beam Supports

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7.1 Introduction

 Limitation

7.1 Introduction

 Segment BC: Mx≠0, Qy=0

=> Pure Bending

 Segments AB,CD: Mx≠0, Qy≠0

=> Nonuniform Bending

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7.1 Introduction

Pure Bending: Prismatic members subjected to equal and opposite couples acting in the same longitudinal plane

7.2 Bending stress

 Simplifying assumptions

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7.2 Bending stress

The positive bending moment causes the

material within the bottom portion of the beam

to stretch and the material within the top portion

to compress Consequently, between these two

regions there must be a surface, called the

neutral surface, in which longitudinal fibers of

the material will not undergo a change in

length

Neutral axis

7.2 Bending stress

Neutral fiber

dz

Due to bending moment Mx caused

by the applied loading, the

cross-section rotate relatively to each other

by the amount of d

dz c d cd     y

        

The Normal strain of the longitudinal

fiber cd that lies distance y below the

neutral surface

y





 Compatibility

Consider a segment of the beam

bounded by two cross-sections that

are separated by the infinitesimal

distance dz

12

7.2 Bending stress

 Equilibrium

z

y E









y

z x

dA 

x

y

z K

Mx

Because of the loads applied in the

plane yOz, thus: Nz=My=0 and Mx≠0

0

E

N    dA   yd A 



0

x A

yd A  S 



0

E

M   x  dA   xyd A 



0

xy A

xyd A  I 



x – neutral axis (the neutral axis passes through the centroid C of the cross-section)

y - axis – the axis of symmetry of the cross-section

7.2 Bending stress

Mx>0: stretch top portion

Mx<0: compress top portion

y

z x

dA 

x

y

z K

Mx

2

M   y  dA   y d A  I

x

M EI





EIx – stiffness of beam

Mx – internal bending moment

 – radius of neutral longitudinal fiber

x z

x

M

y I





y – coordinate of point

Belong to tensile zone

 Flexure formula – section modulus

7.2 Bending stress

• Stress distribution

- Stresses vary linearly with

the distance y from neutral axis

• Maximum stresses at a cross-section

x t x

M

y I

 



x

M

y I

 



y t

max – the distance from N.A to a point farthest away from N.A in the tensile portion

y c

max – the distance from N.A to a point farthest away from N.A in the compressive portion

7.2 Bending stress

x y

min

max

h/2

h/2

z

Mx

max

2

/ 2

x x

I W

h



max max

2

min

2

   



with called the section modulus of the beam

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7.2 Bending stress

7.2 Bending stress

Properties of American Standard Shapes

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7.2 Bending stress

7.2 Bending stress

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7.2 Bending stress