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Failures Due to Static Loading

Section IV

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Static Loading? Failure Theories:

Maximum-Normal Stress Theory Maximum-Shear Stress Theory The Distortion-Energy Theory

Factor of Safety for each Failure Theory

Talking Points

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Static Load is a stationary force or moment acting on a member.

Stationary means that the force or moment does not change in magnitude.

Sometime the load is assumed to be static when it is known that some variation to be expected. Such assumptions are made to simplify the design computations when variations in load are few or minor in nature.

Static Loading?

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This theory states that failure occurs whenever the largest principal stress equals the strength.

Suppose the principal stresses are ordered as:

Then if yielding is the criteria of failure, then

failure occurs whenever:

where:

If ultimate tensile strength is used (in case of brittle materials), failure occur whenever:

where:

Failure Theoriesi. Maximum-Normal Stress Theory

Principal

Element

321

ycyt SS 31 or

stress yield ecompressiv

stress yield tensile

yc

yt

S

S

ucut SS 31 or

stress ultimate ecompressiv

stress ultimate tensile

uc

ut

S

S

Pure tensionPure torsion

Note: For pure torsion 1 = = 3 and 2 = 0. This

means that the part fails in torsion when = Sy. But

experiments show that a part in torsion will deform

at about 60% of the yield strength. This is one of

the reasons the maximum-normal stress theory is

not recommended.

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This theory is used only in the ductile materials. It states that yielding begins whenever the maximum shear stress

becomes equal to the maximum shear stress in a tension test specimen of the same material when that specimen begins to yield.

The maximum-shear stress theory predicts that failure will occur whenever:

Shear yield strength is given by:

Failure Theoriesii. Maximum-Shear Stress Theory

torsion)pure(for 2

tension)simple(for 2

31max

1max

yy SS

3121

max or 22

ysy SS 5.0

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This theory is called the shear-energy theory.

It is used also for ductile materials. Like the maximum-shear stress theory,

this theory is employed to define the beginning of yield. It predicts that yielding will occur whenever the distortion-energy in a unit volume equals the distortion energy in the same volume when uniaxially stressed to the yield strength.

The distortion energy is obtained by subtracting equation (5) from equation (2):

So yielding is predicted to occur when:

Shear yield strength is given by:

Failure Theoriesiii. The Distortion-Energy Theory

b) volume change c) distortion without volume change

a) both volume change and angular distortion

For figure (a): For unit cube, the work done in any of the

principal directions is: , where n= 1,2,3

The total strain energy is: or

The average value avg: The remaining stress, (1-avg), (2-avg), (3-avg)

will produce angular distortion, figure (c). Substituting (avg for 1, 2, 3) in equation (2) gives the amount of strain energy producing only volume change.

Substituting in equation (4) gives:

321

(1) 2nnnU

321 UUUU

(5) 26

21313221

23

22

21v

E

U

(3) 3321 avg

(4) 2123 2v EU avg

23212 3 avg

(2) 221 3132212

32

22

1 EU

2 where,

3

1

213

232

221

2d

EU

ysy SS 577.0

yS

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Factor of Safety for each Failure Theory

For maximum-normal stress theory

For maximum-shear stress theory

For the distortion-energy theory:

materials) brittle(for or

materials) ductile(for or

31

31

usus

ysys

SnSn

SnSn

31 ys Sn

2 where, 213

232

221

ys Sn

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Example A material has a yield strength of 600 MPa. Compute the fator of safety for

each of the failure theories for ductile materials. Use the following stress states:

a)- 1 = 420 MPa: 2 = 420 MPa, 3 = 0

b)- 1 = 420 MPa, 2 = 180 MPa, 3 = 0

c)- 1 = 420 MPa, 2 = 0 MPa, 3 = -180 MPa

d)- 1 = 0 MPa, 2 = -180 MPa, 3 = -420 MPa.