bending stress strain of bar exposed to bending momentfast10.vsb.cz/lausova/lesson05-18.pdfcentre of...
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Bending stress –
strain of bar exposed
to bending moment
• Basic principles and conditions of solution
• Calculation of bending (direct) stress
• Design of bar exposed to bending moment
• Combined stress of bar
Department of Structural Mechanics
Faculty of Civil Engineering, VSB - Technical University Ostrava
Elasticity and Plasticity
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Bars under bending
Basic principles and conditions of solution
The bending moments and shear forces become in the bar in the course of
bending.
ba
M
l
M
MM
M
V
Simple bending
0, yz MV0 zxy MMVNIn xz plane hold true:
Plane bending: inner and external forces are situated in xy plane or xz plane
– principal plains.
0, zy MV0 yxz MMVNIn xy plane hold true:
x
a
Raz
b
Rbzl
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Simple bending
Laboratory test
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Simple bending
Testing of structures
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Basic conditions
b) axial fibres are not mutually in compression
0 zy
a) deformated cross-sections stay on plane
figure and perpendicular to deformated axis
(Bernoulli hypothesis)
Character of condition is deformation-geometrical.
a
M
b
M
z
x
z
y
y
z
Daniel Bernoulli(1700 - 1782)
Basic principles and conditions of solution
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Relations between inner forces and stress in cross-section
+z
+x
+y
N
zV
yV
Cross-section
Centre of gravity
Central line
y
z
AN x d.d ANA
xd
xxy
xz AzzNMA
xy d..
AyyNMA
xz d..
likewise
Calculation of bending (direct) stress
Placement of
inner forces
resultant
The resultant force can be
replaced by the normal force
on the normal and the two
shear forces in y, z
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Normal stress in bending
z
x
max
max
Neutral axes is the same as the
central line only at simple
loadind by the bending moment.zAB
C ED
d
xdxdxd
r
x = n
y
y
xI
z.M
e
IW
y
y
Extrem of stress is on outer fibres where z = e.
y
y
y
y
W
Me
I
Mmax
e
Distribution of normal stress x in
bending is linear over the hight of
beam and extreme values are in
outer fibres.
Zerro value of x is on neutral axes.
Wy - section modulus
for outer fibres [m3]
Iy - moment of inertia
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Normal ( bending) stress at simple bending
ANA
xd
Simple bending: suma N = 0
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Extrem of normal stress in bending - symmetrical cross section
z
y
x
lowery
y
lowerxW
M
,
,
Upper fibres:
uppery
y
upperW
M
,
Lower fibres:
y
y
lowerxupperxW
M max,,max,,
upperx,
lowerx,
Minus stress
Positive stress
We can determine signe of stress according to bending moment, after
deformation in bending there are clear tensile or compressed fibres.
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x1e
z
y 2e
Neutral axes in centre of gravity of section
fibrestensileW
Me
I
M
ey
y
y
y
ex
1,
11, .
fibrescompressedW
Me
I
M
ey
y
y
y
ex
2,
22, .
1
1e
IW
y
e,y 2
2e
IW
y
e,y
… Section modullus for outer fibres [m3]
0x
upper
upperW
M
lower
lowerW
M
lowerxupperx max,,max,,
Distance of outer fibres from axes of center of
gravity e1,2 (or c1,2)
In farther fibres from neutral axes there are with higher
stress ( je Wy,min )
Extrem of normal stress in bending - asymmetrical cross section
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Combination of stresses
a
Raz
b
Rbzl
Rax
F
σx
A
N xN
zI
M
y
yM
N
M
V
N MM
0N
c
In section c stress is
calculated by
superposition and it is
possible to gain:
x = n
Movement of neutral axes
-
+
- -
+
- - -
+ +
-
-
+
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Limited validitation of derived relation
Limited validation
y
y
xI
zM .
• Relation is valid for case of simple bending, constant cross-section and
the height of beam h << l (span).
xz
x
max
ha
Raz
b
Rbzl
(compression)
(tension)
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Limited validation of derived relation
x
a
Raz
b
Rbzl
Relation is not valid in abrupt changes of cross-section.
hy
y
xI
zM .
Limited validation
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Limited validation of derived relation
Limited validation
x
a
Raz
b
Rbzl
Relation is not valid in
case of bearing walls,
where l < 3h .
h
z (tension)
(compression)
y
y
xI
zM .
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Cross sectional characteristics
Cross sectional characteristics
x1c
2,cx
z
y
1,cx
2c
Neutral axis in center of gravity
1,
11, .cy
y
y
y
cxW
Mc
I
M
2,
22, .cy
y
y
y
cxW
Mc
I
M
Cross-section modulus calculation in case of simple shapes
b
h
1
1,c
IW
y
cy 2
2,c
IW
y
cy
3..12
1hbI y
hbI z ..12
1 3
2..
6
1
2
hbh
IW
y
y
hbb
IW
y
z ..6
1
2
2
d64
. 4dI
32
.
2
3d
d
IW
… Cross-section modulus to outer fibres [m3]
0x
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Design and reliability assessment of bar exposed to bending moments
Design of bar exposed to bending moment
Reliability assessment of design
Limit state of carrying capacity
Design of carrying
structure
Realization
Dimensioning
dEd fWM ,, min
dRdEd fWMM .min
RdMAdjusted design
d
Ed
f
MW min
M
kd
ff
dEd MM max
Assumption in design:
The same strength of
material in case of tension
and compression (steel), no
shear stresses influence
1Rd
Ed
M
M
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Vertical, horizontal and unsymmetrical bending
z
y
y
y
xI
zM .
z
zx
I
yM .
a b
Vertical bending Horizontal bending
z
z
y
y
xI
yM
I
zM .. My and Mz – combined stress of bar
(unsymmetrical bending)
Combined stress of bar
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Eccentric tension and compression
+z
+x
+y
N
Centre of gravity
Central line of beam
ze
ye
Neutral axis
nz
ny
Tension
zy eNM . yz eNM .
- acting N +My + Mz or another expression is when the N force whose position is placed
against the centrer of gravity on eccentrities ey and ey. Positive N force on positive
eccentrities causes moments:
Bending stress is the sume of individual stresses:
yI
Mz
I
M
A
N
z
z
y
y
x ..
(the expres. in brackets =0, y intersection is
obtained by substituting zero for z-coordinate)
A
Ii
y
y 2
A
Ii zz 2
22
..1.
z
y
y
zx
i
ye
i
ze
A
N
Segments of neutral axis:
0z 0.
1.2
z
y
i
ye
A
N y
zn
e
iy
2
0yz
y
ne
iz
2
Combined stress of bar
into:
is possible to modify by substitution:
0x