1/141/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria YIELDING AND...
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Transcript of 1/141/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria YIELDING AND...
1/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
YIELDING AND RUPTURE CRITERIA
(limit hypothesis)
2/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
The knowledge of stress and strain states and displacements in each point of a structure allows for design of its members. The dimensions of these members should assure functional and safe exploitation of a structure.
In the simplest case of uniaxial tension (compression) it can be easily accomplished as stress matrix is represented by one component 1 only, and displacement along
bar axis can be easily measured to determine axial strain 1
Measurements taken during the tensile test allow also for determination of material characteristics: elastic and plastic limits as well as ultimate strength. With these data one can easily design tensile member of a structure to assure its safety.
1
1arctanE
RH
Rm
expl<<Rm
expl<RH
expl = 1 =RH /s
expl
s-1
Safety coefficient
?Ultimate strength
Elastic limit
3/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
In the more complex states of stress (for example in combined bending and shear) the evaluation of safe dimensioning (related to elastic limit) becomes ambiguous.
zx
zx
xz
xz
xxx
z
x
1
2
2
1
1
2z
x2
1
Do we need to satisfy two independent conditions
x< RH x< RH
where RHt i RHs denote elastic limits in tension and shear, respectively?
Transformation to the principal axis of stress matrix does not help either, as we do not know whether the modulus of combined stresses is smaller then RH …
12
p |
1
2
Thus, we need to formulate a hypothesis defining which stress components should be taken as basis for safe structure design.
4/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
In general case of 3D state of stress we introduce a function in 9-dimensional space of all stress components (or 3-dimensional in the case of principal axes) which are called the exertion function:
),,()( 321 fFW ij
limW
In uniaxial sate of stress:We postulate that exertion function will take the same value in given 3D state of stress as that in uniaxial case. )()( 0 FFW ij
The solution of this equation with respect to 0: )(0 ij is called substitute stress according to the adopted hypothesis defining function F and thus – function , as well.
)( 00 FWW
limWW
0
5/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
23
22
21
2 pW
221 ,, f200 W
NR 023
22
21
10 0
N
N
R
R
Let the exertion measure be:
SUCH A HYPOTHESIS DOES NOT EXIST !A very similar one, which does exist
321 ,,max Wmis called Gallieo-Clebsh-Rankine hypothesis
Associated function appears to bo not-analytical one (derivatives on edges are indefinable)
NR
0WW
pmW
1
2
3
NRNR The ratio:
gives „the distance” from unsafe state.
stress vector in main principal axis
p
This distance can be dealt with as the exertion in a given point.
6/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
NR1
321 ,,max Wm
NR2
NR3
NN RR 1
NN RR 2
NN RR 3
NRNR
NRNR
NR
NR
1
2
3
It is seen, that materials which obey this hypothesis are isotopic with respect to their strength.
GALIEO-CLEBSH-RANKINE hypothesis (GCR)
They are also isonomic, as their strength properties are identical for tension and compression.For plane stress state it
reduces to a square.
7/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
1
1 1
1
NR1
NR2
Exertion ≤ 100%
Exertion ≤ 80%
Exertion ≤ 60%
Exertion ≤ 40%
Exertion 0%
1
1 1 NR1
NR21
8/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
scNR
1
2rNR
rNRsc
NR
Material isonomic and isotropic
Material isotropic but not isonomic
ncomrpessioN
tesionN RR
ncompressioN
tensionN RR
Material insensitive to compression. (classical Galileo hypothesis)
321 ,,max Wm
where
aa when a>0
0 when a<0
GALILEO hypothesis
9/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
COULOMB-TRESCA-GUEST hypothesis CTG
scNR
1
2rNR
rNRsc
NR
rNR1
rNR2
12
0
0TFor torsion:
Uniaxial tension
Many materials are sensitive to torsion
This hexagon represents Coulomb-Tresca hypothesis (for plane stress state); the measure of exertion is maximum shear stress:
1332210 ,,max
321 ,,max Wm
In uniaxial state of stress:
2
max0 oWm
2
,2
,2
max 133221 Wm
10/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
NR
1
2
NR
NRNR DDm fW 2
1
In uniaxial state of stress:
Small but important improvement has been made by M.T. Huber followed by von Mises and Hencky:
It is distortion energy only which decides on the material exertion:
AADDvf 2
1
2
1
For elastic materials (Hooke law obeys):
2132
322
212
6
13
4
1 GG mijijf
20
0 26
1 Gf 213
232
2210
2
1
In 3D space of principal stresses (Haigh space) this hypothesis is represented by a cylinder with open ends. In 2D plane stress state for is an ellipse shown above. 03
HUBER-MISES-HENCKY hypothesis HMH
11/14M.Chrzanowski: Strength of Materials
SM2-10: Yielding & rupture criteria
NR
1
2
NR
NRNR
NR
1
2
NR
NRNR
NR
1
2
NR
NRNR
Hypothesis
Exertion measure
3D image
2D image
GCR CTG HMHMaximum
normal stressMaximum
shear stress Deformation energy
Cube with sides equal to 2R
Hexagonal prism with uniformly inclined axis
Circular cylinder with uniformly inclined axis
Substitute stress
),max( 210 210 2122
210
Substitute stress for beams
220 42
1xyxx 22
0 4 xyx 220 3 xyx