in plane bending
TRANSCRIPT
Chapter VII
IN-PLANE BENDING
1
Beam axis
Cross‐section ( to the axis)Area A
DEFINITION OF A BEAM
CROSS‐SECTION DIMENSIONS < BEAM LENGTH / 10
G
G’
Other cross‐section ( to the axis)Area A’
2
INTERNAL FORCES IN CROSS-SECTION
G
EXTERNAL FORCES AND REACTIONS CUT stresses (equilibrium )INTERNAL FORCES
3
AxisCross‐section ( to the axis)
DEXTORSUM SYSTEM OF AXES
INTERNAL FORCES IN CROSS-SECTION
4
IN-PLANE BENDING5
ShorteningCompressive stresses
ElongationTensile stresses
= 0 somewherein between(neutral axis)
BEAM DEFORMATION UNDER CONSTANT MOMENT: CIRCLE
BERNOUILLI’S PRINCIPLE6
Cross‐sections remain flat and perpendicular to the beam axis
IN-PLANE BENDING7
• Slice of unit length
IN-PLANE BENDING8
and so
IN-PLANE BENDING9
• Linear distribution of and
y
• Linear distribution of deformations
• Hooke’s law
• Linear distribution of stresses
y
E
yE E
ENA
IN-PLANE BENDING10
• Determination of the neutral axis and
• Longitudinal equilibrium (along the beam axis)
Neutral axis corresponds to the centre of gravity
0 0A A
EdA ydA
IN-PLANE BENDING11
• Determination of the neutral axis and
• Equilibrium in rotation
as
2
A A
EydA y dA M
2
A
y dA I1 M
EI
Flexural rigidity
IN-PLANE BENDING12
• Stress distribution
1 M yand EEI
: MyNavierI
yENA
IN-PLANE BENDING13
• Navier applicable to symmetrical and non‐symmetrical cross‐sections as long as the plane of bending corresponds to one of the principal axes of the cross‐section
y
z
IN-PLANE BENDING14
• Navier not directly applicable to cross‐sections subjected to bending not applied about the main axes
• Navier still assumed to be valid for beams subjected to non constant bending moments along the beam length
IN-PLANE BENDING15
Mmax = pL²/8T1 = PL/2
T2 = -PL/2
p
• Limitations of the Navier linear distribution
IN-PLANE BENDING16
b
h
Rectangularcross‐section
• Navier still assumed to be valid for beams subjected to non constant bending moments along the beam length
IN-PLANE BENDING17
Mmax = Pa(L-a)/LT1 = P(L-a)/L
T2 = -Pa/L
P
• Limitations of the Navier linear distribution
IN-PLANE BENDING18
• Limitations of the Navier linear distribution
– In practice:• Local effect (St‐Venant’s principle)
• Local yielding under the concentrated load
• Transverse stiffeners• Bearing plate plate
IN-PLANE BENDING19
• Navier still assumed to be valid for tapered beams (smooth cross‐section variation)
IN-PLANE BENDING20
• Limitations of the Navier linear distribution
IN-PLANE BENDING21
• Limitations of the Navier linear distribution
IN-PLANE BENDING22
!Stress concentration
cfr. tension
h
<y
<R =R
=(Mh/2)/I ≤R
=y
e=2y /h
IN-PLANE BENDING
y
R
• Beam cross‐section verification
h
<y
<R =R
=(Mh/2)/I ≤R
=y
e=2y /h
IN-PLANE BENDING
• Beam cross‐section verification
MyI
max 2hR for y
max
2M hR
I
2IM R WR
h
h1
IN-PLANE BENDING
• Beam cross‐section verification
11,max 1
1
22,max 2
2
Mh M RI WMh M R
I W
1 1 2 2min( ; )M W R W R
h2
G
IN-PLANE BENDING
• Adapt cross‐section shape
Similar values of A and h:• I cross‐section: W=0,32 Ah• rect. cross‐section: W=0,167Ah
I section twice more resistant
max
IN-PLANE BENDING
• Adapt cross‐section shape
Similar values of A and h:
• W=0,32Ah
• W=0,5Ah
IN-PLANE BENDING
• Adapt cross‐section shape
• Beams made of two different materials
IN-PLANE BENDING29
E0b modulus
Ea modulus Ea modulus
dy
Equal forces in the slice dy E0bbdy = Eab1dy
• Beams made of two different materials
IN-PLANE BENDING30
E0b modulus
Ea modulus Ea modulus
0
1b
a
Eb bE
• Beams made of two different materials
IN-PLANE BENDING31
b
a a=Eaa
int a,intEaintb,intE0b/Ea int
bE0b/Ea b,equ
EQUIVALENT SECTION ACTUAL SECTION
b,equEab
aEaa
intEaint
• Beams made of two different materials
IN-PLANE BENDING32
0
1 ( tan. ) ( )b b
a a
E Eb b ins loading or b after creepE E