Chapter 5

1
Chapter 5 Part ? Stresses in Beams ????

• 1 Normal stresses in beams?????2 Moment of
  Inertia???3 Shear stresses in beams?????4
  Strength analysis of beams??????5 Rational
  design of beams?????? 6 Combined bending and
  axial load
• ??(?)??

2
1 Normal stresses in beams?????

• ? Introduction
• ? Experiment Assumptions
• ? Normal stresses in beams

3
? Introduction
? ????
????
4
?????
? ??????,???? ??B??.?????????????BC???,?????BA???
?????,?BA??????????????????????
5
? ??????(1680)
? ?????????????
? ???????????B??,???????? ???????B??????????
? ????????,??????
A
C
B
P
A
C
B
P
6
? ??????????
?????
?????
Stresses in beams
? Normal stresses -s
? Shear stresses -t
7
? Symmetric bending ????
?????????????????T???

• ???????????(????????????),????????????
  ??,??????????????,??????????????? The applied
  loads act in the plane of symmetry and the
  bending of the beam occurs in that plane

Pure bending ??? ???????? The condition of
constant bending moment (no shear force)
8
? ?????????????
?????????????
? ??????????????
Experiment
(Pure and symmetric bending)
? Transverse line??????,?????? ? Longitudinal
line??????,???,??? ? Thickness?????, ??????
9
? Experiment Assumptions
Assumptions
? Plane Assumption(????) The entire transverse
section of the beam , originally plane , remains
plane and normal to the longitudinal fibers of
the beam after bending???????????,??????

• Assumption on the fibers under uni-axial tension
  and compression????? ,????????
• no interaction between longitudinal
  fibers

10
????,????
??????,??????
? The top surface is in compression and the
bottom in tension. Obviously, there must exist a
surface parallel to the upper and lower faces of
the member where no elongation and no stress
exist. This surface is called the neutral surface
(???).
? Neutral axis(???) the intersection of neutral
surface with any cross section, ?symmetric axis
????????????
? ??????????????????
? ?????????????????,?????
?????????????????????
11
? 1700??????.????????????????
?????????,?????????????? ???????????????
? 1713???????????????????,???
????????????????????????? ???????????
? 1819?,????????????????????
????????????????????????
? 1826?,?????????????,?????? ???
12
? Normal stresses in symmetric bending beams
Compatibility equation
Constitutive equation
?? Where is the neutral axis?
13
Statics equation
?
(a)?(b)
The neutral axis passes through the centroid of
cross section?????????
parallel force system in space
(a)?(c)
?
Moment of inertia of cross section with respect
to z
EIz flexural rigidity
?
(d)?(a)
14
?
section modulus in bending
??
? ?????
????????
? ?????
? ?????
, ??????
? ????
,????
15
????????
????symmetric bending????pure bending
????-?????,?????????
???? ???????? ? ?
?-???????????????
??????
???neutral axis????Centroidal axis
???-????????????? ???-???????????
??????flexural rigidity???????section modulus
????EI-??????????????
??????Wz-????????????
?????
16
2 Moment of Inertia???

• ? Moment of first order and moment of inertia
• ? Moment of inertia of simple section
• ? Parallel axis theorem
• ? Examples

17
Moment of first order and moment of
inertia??????
First moment of area (??,???)
-the first moment of the area of cross section
with respect to z(???z????)
Second moment of area (moment of inertia???)
-the moment of inertia of cross section with
respect to z(??? z ?????)
18
? Moment of inertia of simple section(???????)
Rectangular section
Circular section
Polar second moment of area
Moment of inertia
19
? Parallel axis theorem
? ???????????????????
? ???????????? ??????
Parallel axis theorem (?????)
C-centroid
Similarly
20
? ????????????
21
? ????????z?????????
1???????
2??????????
22
? Examples
A beam section with the dimensions shown is
subjected to a bending moment of 1800 N-m about
its centroidal axis, as shown. Determine the
normal stress due to bending.
T beam
The centroid is determined using the table below
The moment of inertia is determined from
23
The stress at A is tensile. Point A is 63 - 24.75
38.25 mm (0.0382 m) from the neutral bending
axis
The stress at B is compressive. Point B is 24.15-
18 6.75 mm (0.00675 m) from the neutral bending
axis
24
? Examples
Beam AB is subjected to the two 6 kN forces
shown. The cross section of the beam has the
dimensions shown(channel beam). Determine the
maximum tensile and compressive stresses in the
beam.
The centroid is determined using the table below.
Note that section 2 is subtracted from section
25
The moment of inertia is determined from
26
The normal stress due to bending on the bottom
(0.1155 m from the neutral bending axis) is
compressive and is defined by
The normal stress due to bending on the top of
the beam (0.0645 m from the neutral bending axis)
is tensile and is defined by
27
? Examples
?????18 ???( I beam)??, Me20 kNm, E200
GPa??????????smax, ?????? r
?1. ???
???????????????????? (GB 706-88)
?18 ???
28
Me20 kNm,E200 Gpa,? smax ? r
2. ????
3. ????
29
?????d2mm, ?b6mm, D1400mm, E200GPa??????
smax ? M
?1. ????
??????,????????
? ??????
? ??????
30
?? d2 mm, ? b 6mm, D 1400mm, E 200GPa,?
smax ? M
2. ????
3. ????
31
3 Shear stresses in beams?????

• ? Shear stresses in rectangular beam
• ??????????
• ? Shear stresses in a thin-walled beam
• ????????
• ? Comparison of normal and shear stresses in a
  beam ???????????
• ? Examples

32
? ??????,???????????????, ??,???????????????????
?
??
????
D.J.Jourawski(???????1821-1891) ?????
???????
33
? Shear stresses in rectangular beam
Problem
slender rectangular beam (hgtb)
Assumptions
? The shear stresses are parallel to the shear
force Fs and the vertical sides of the cross
section t (y) // ???? (Shear stress in pair
)
? The distribution of the shear stresses is
uniform across the width of the beam t
(y)?????????
34
Shear stresses in a beam of narrow rectangular
cross-section??????????
Sz(w)-the first area moment with respect to the
neutral axis z of that part of the cross section
w located under the line m-n. ?? w???? z ???
35
?
? h/b??,?????(h/bgt2?,????)
36

• Warping of cross sections of a beam due to shear
• ????????

Non-uniform shear stress?Non-uniform shear strain
?Warping of cross sections?????
The deformation of longitudinal fibers is
unaffected by Fs???????????????
? When FS? Const,
?l h?,??????????????
If l gt 5h,the formula of s derived for pure
bending has enough precision in the case of
non-uniform bending
37
? Shear stresses in a thin-walled beam
Thin-walled I-beam ??????
Assumptions ?//??(??)?? ? // sides of the web(or
flange) and????????uniformly distributed across
the thickness?

• - y ?????????? z ???(first area moment with
  respect to the neutral axis z of that part of the
  cross section located under y. )

Variation of shear stresses in an I-beam
38
??
??
?????????
???????,???????
39
?????,A?B??????0
A box beam
A
B
??
?????????
???????,???????
40
Comparison of normal and shear stresses in a
beam???????????
When l gtgt h ,smax gtgt tmax
41
? Examples
Cantilevered beam BC has a 14kN end load applied
as shown. At section D, determine the shear
stress at point A (42 mm below the top of the
beam), knowing
For this cantilevered beam the reactions at the
fixed end are
FB 14kN , M 16.8 kN-m
42
Since the shear force is constant, the Fs and M
diagrams are as shown
To compute the shear stress at A, we will
determine Sz for the flange and then the web.
Sz,flange (200)(28)(61) 341,600 mm3 Sz,web
(100)(14)(40) 56,000 mm3
43
At A the first area moment is
Sz,flange Sz,web
tA 1.128 MPa
44
? Examples
?
45
?
46
(No Transcript)
47
Two beam sections are fastened by 16mm-diameter
bolts spaced 25mm apart as shown. A 3.2kN shear
force is applied in the direction indicated.
Determine the shear stress in each bolt.
Using the beam sections shown, we must determine
the area moment of inertia for the composite beam
and q at the interface of the two beams, where
the bolts are located.
48
The moment of inertia with respect to the neutral
axis of the entire beam is
The first area moment of inertia at the bolts is
The shear flow is defined by
The shear flow is related to the bolt spacing and
force carried by each bolt through qF/s
49
Therefore, the force along the top row of bolts
is F qs (6.165 kN/m)(0.025 m) 0.154 kN
The area of each bolt over which the force is
distributed is
The shear stress in each bolt is
50
???

• ?????????,???-????????,?              ???

B?????????????????????????????????????????(???????
?)?
?????????????????????
??????????????ey/r???????
??????????????????????????
???????????????????????????
51
4 Strength analysis of beams??????

• ? State of stress in a beam
• ??????
• ? Strength analysis of beams
• ??????
• ? Examples ??

52
? State of stress in a beam
Beam with solid section ?????
Element a, c -uni-axial stress????
Element b -pure shearing ???
53
Thin-walled beam ?????
a ??- pure shearing stress???
c , d - uni-axial stress????
b -complicated state of stress??????
54
? Strength analysis of beams ??????
? uni-axial stress
smax Maximum stress in bending ???????
s - allowable stress under uni-axial load
??????????
? pure shearing stress
tmax - Maximum shear stress in bending ???????
t - allowable stress under pure shearing
stress ?????????
(Chapter 8)
? complicated state of stress
55
? Strength analysis of beams?????
? Long beam except thin-walled one ??????
If the allowable stresses are different for
tension and compression , it may be desirable to
use an unsymmetrical cross section

• Short, or thin-walled beam
• ????????? M ? FS???

56
? Examples
A steel bar of rectangular cross-section, 10 cm
deep and 5 cm wide, is bent in the planes of the
longer sides. Estimate the greatest allowable
bending moment if the bending stresses are not to
exceed 150 MN/m2 in tension and compression.
Solution
The bending moment is applied about Cx.
The second moment of area about this axis is Ix
1/12 (0.05) (0.10)3 4.16 x 10-6 m2
The bending stress
If the greatest stresses are not to exceed 150
MN/m2, we must have
The greatest bending stresses occur in the
extreme fibers where y 5 cm. Then
M 6250 Nm
57
? Examples
?1. ????
58
2. ??? s ?????
? ?22a, Wz3.0910-4 m4
3. ????????
59
?????????????,???????????,?????????P???P8kN,L6m,
d0.8m,?????No20a?????, ?? 160MPa,????????
?(1)???Mmax??????
???RA
M(x)0,????????? x1L/2-d/4
??????,? 
2)??????
???,No20a?????? Wz237cm3
?????
60
?
?????????,??????????? Mmax  ??
????
-?? D, B
MD-?????
MB-?????
61
a, b, c
???-
62
Un-symmetric Bending of Beams with Doubly
Symmetric Cross Section????????????

• ???????????????--???????????
• ???????????? ????????????????????????????,???????

(1)???????P?y??a?,??y,z????? PyPcos a
PzPsin a (2)???Py???xy?????????,Pz???xz????????
?,?????x??mm??????
MzPyxPcos a x   MyPzxPsin a x 
?????????,?????My?Mz??????,???????????????????A???
????????????????
????mm?y?z????????Iy?Iz
63
??????
????????????????,?????????? 
???????,???n-n??????????C????????n-n?z?????,???P?y
?????????M?z?????a,?
?????, ??a,????????????????????(?8.1-6),?????,???
? 00,900?(???Mz???My???),???IyIz?(??????,??????)
,???????????????????????
64
Beams having two axes of symmetry in the
cross-section
?1. ????
? ????x-y ?x-z ???????????? ? ??????
65
Fy Fz F
2. ????
????
-??A
66
3. ????


???
????st,max?sc,max, ?????
-d, f
4. ????
???????????
67
?????,????F1?F2??,??F1800N,F21.6kN,l1m,????s
160MPa??????????????? (1)?????,h2b(2) ??????
(1)
(2)
68
5 Rational design of beams????????

• ? Selection of a beam cross section
• ??????????
• ? Non prismatic beam ????
• ? Rational load distribution of beam
• ??????

69
? Selection of a beam cross section
? Most of the material of beam should be
located as far as possible from neutral
axis.????????????????,
The distances to the extreme fibers in tension
and compression are in the same ratio as the
respective allowable stresses.
Symmetric
70
? Mainly pay attention to normal strength, with
consideration of shear strength and stability of
the web??????,?????????????
The web can not be too thin ??????
71
? Non prismatic beam
Beam of constant strength (fully stressed
beam) ????-???????????
72
(No Transcript)
73
Rational load distribution of beam??????
Arrangement of restrictions ??????
a ? F is maximum
74
Arrangement of loads ????????
75
(No Transcript)
76
6 Combined bending and axial load ??(?)??

• ? Combined bending and axial load ??(?)??
• ? Eccentric axial load ????
• ? Core of a cross section ????
• ? Examples ??

77
? Combined bending and axial load??(?)??
Combined bending and axial load ????
Eccentric axial load ????
(????????)
(?????????)
78
Stresses caused by combined bending and axial load
Internal force-FN,M
Critical points???-axial load ????
79
? Eccentric axial load ????
????(???????)
Eccentric compression
?
Combined bending and compressing????
80
? Core of a cross section ????
Neutral axis ???
Function of neutral axis ?????
?????,????????
81
Core of a cross section ????
? There is a small region around the centroid
such that a compressive load acting within that
region will produce no tension at any point of
cross section. This region is called the core of
the section.??????????????????,??????

• When a compressive load acts on a material that
  is very weak in tension, such as ceramic material
  or concrete, the load should act within the core
  of the section
• ??????????,???????????????

82
? ??
?1. ????
83
2. ????
3. ??????
?????????
? ?12.6, Wz7.7510-4 m4 , A1.8110-3 m2
4. ???????
?12.6 ??????,??????
84
?1. ? 1 ??????????
?1??????????-?? 1-2
85
2. ????????????
-??2-3
????? 3-4 ? 4-1
86
Thanks
87
(No Transcript)
88
????

• ???????

????????????
????????? ????????????????
????????? ??-?, ??-?,??-?
????,?????( ???????)???(???,???)???
?????????????????????????????? ????
89
???????
????
???????????(???????,??????,?????????)?????(???)(?
???)?????(????)?????(??)??(??????,??????)
90
(No Transcript)
91
???
??1.FN ???? 2.T??????????
3.Fs???????? M???????? ??
???M ?????,??????
????????(???????????????)

• ?????????(????)
• Fs ?????,?????q??
• M ??????,?????Fs??

????????
????
??????????????????????????????????,??????????????
?????????????????,??
92
??????t,s??????????
??????????????,?????????????
A, IP, WP, Iz, Wz??????
Iz??????
1???????????
??????????
2??????????
3??????????????
????,????
????,?????????
93
?????????
??????? ???????,?????
???????(????????)

• ??????
• (??????)

????
? ??????
??????????
94
 
???????
????