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Pure Bending : Prismatic members subjected to equal and opposite couples acting in the same longitudinal plane.. Bending stress[r]
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CHAPTER
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(3)Contents 7.1 Introduction
7.2 Bending stress
7.3 Shearing stress in bending 7.4 Strength condition
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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
(5)7.1 Introduction
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(7)7.1 Introduction
Segment BC: Mx≠0, Qy=0 => Pure Bending
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7.1 Introduction
(9)7.2 Bending stress
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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
(11)7.2 Bending stress
Neutral fiber
c d
a b
c d
d
dz 2 2 y
y a b
Due to bending moment Mx caused by the applied loading, the cross-section rotate relatively to each other by the amount of d
' ' y d d
dz c d cd y
The Normal strain of the longitudinal fiber cd that lies distance y below the neutral surface z y Compatibility
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7.2 Bending stress
Equilibrium z y E
Following Hooke’s law, we have
1 ???? 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
z z
A A
E
N dA yd A
0
x A
yd A S
0
y z
A A
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
(13)7.2 Bending stress
Mx>0: stretch top portion Mx<0: compress top portion
y z x dA x y z K Mx
x z x
A A
E E
M y dA y d A I
1 x 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
M Belong to tensile zone
(14)7.2 Bending stress
• Stress distribution
- Stresses vary linearly with the distance y from neutral axis
• Maximum stresses at a cross-section
max max x t x M y I max x c x M y I yt
max – the distance from N.A to a point farthest away from N.A in the tensile portion yc
(15)7.2 Bending stress x y min max h/2 h/2 z Mx max 2 x x x x
M h M
I W max / 2 x x I W h max max 2
t c h
y y
min
2
x x
x x
M h M
I W
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(17)7.2 Bending stress
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