Page 195 - Bridge and Highway Structure Rehabilitation and Repair

170 SECTION 2 STRENGTHENING AND REPAIR WORK

If %1, %2, and %3 are principal stresses, and if %1 is maximum of the three,
i.e., %1 9 %2 9 %3
Yield stress in tension 3 %yt
Yield stress in compression 3 %yc
Maximum principal stress %1 3 %yt if tension force is applied
Minimum principal stress %3 3 %yc if compression force is applied.
5. St. Venant Criteria for Maximum Principal Strain: When a point in a material is subjected
to principal strain in three directions, yield of material will occur under the maximum of
the three principal strains for applied tension or applied compression.
As an alternative, yield of material will occur under the minimum of the three principal
strains for applied compression.
If %1, %2, and %3 are principal stresses, and if %1 is maximum of the three,

i.e., %1 9 %2 9 %3
%1 3 %yt if tension force is applied
%1 3 %1/E 6 (%2 4 %3)/E 3 %yt/E, where = Poisson’s Ratio (4.30)
%1 6 (%2 4 %3) 3 %yt
Similarly, %1 = %yc if compression force is applied.
%3 6 (%1 4 %2) 3 %yc.
6. Tresca-Guest criteria for maximum shear stress: Yield will occur when maximum shear
stress T in the material reaches half the yield stress %yt.
3 (%1 6 %3)/2 3 %yt/2
(%1 6 %3) 3 %yt
For two-dimensional stress, %3 3 0
%1 3 %yt.

7. Huber-von Mises shear strain energy theory: Huber proposed that Shear Strain Energy
Theory 3 Total Strain Energy, U 6 Volumetric Strain Energy, U .
t
v
U 3 (U 6 U )
s
v
t
Based on Huber's theory, von Mises developed the following relationship between principal
stresses:
2
2
U 3 [(%1 4 %2 4 %3 ) 6 2 (%1 %2 4 %2 %3 4 %1 %3)]/2E (4.31)
2
s
2
Uv 3 (%1 4 %2 4 %3) (1 6 2 )/6E (4.32)
2
2
(%1 6 %2) 4 (%2 6 %3) 4 (%3 6 %1) 3 2 %yt 2
2
For a two dimensional system, %3 3 0; general equation can be simplifi ed as
2
2
2
(%1) 4 (%2) 6 (%1 %2) 3 %yt
8. Beltrami-Haigh theory for total strain energy: Yielding will occur when total strain energy
3 Yield stress in tension.
2
2
2
2
(%1) 4 (%2) 4 (%3) 6 2 (%1 %2 4 %2 %3 4 %1 %3)] 3 %yt
9. Recommendations based on fracture mechanics for steel:
• Ductile metals like steel exhibit yielding and subsequent plastic deformation.
• Von Mises’s shear strain theory based on principal stress difference is most accurate and