Equations:
σ x1 = | σ---------------------x + σ y | + | σ---------------------x – σ y | ⋅ COS(2 ⋅ θ ) +τxy ⋅ SIN(2 ⋅ θ ) |
| 2 | | 2 | |
σx1 +σ y 1 = σ x +σ y
σ x –σ y
τx1y 1 = –---------------------⋅ SIN(2 ⋅ θ ) + τxy ⋅ σ y 2
Example:
Given: σx=15000_kPa, σy=4755_kPa, τ xy=7500_kPa, θ=30_°.
Solution: σx1=18933.9405_kPa, σy1=821.0595_kPa, τ x1y1= -686.2151_kPa.
Mohr’s Circle (14, 4)
Equations:
σ 1 = | σ-----x-----+-----σ----y-- | + | σ-----x-----–----σ-----y- | + ρxy2 |
| 2 | | 2 | |
σ1 + σ 2 = σ x + σ y
SIN(2 ⋅ θ p1) = ------- | τxy | | | | |
σ-----x-----–-----σ----y------ | 2------------------- |
| + τxy | 2 | |
| ----------2----- | ----- | | | |
| | | |
| | | | | |
θ p2 = θ p1 + 90 | | | τmax = | σ-----1-----–----σ-----2- |
| | | | | | 2 |
θ s = θ p1 – 45 | | | σ avg = | σ-----x-----+-----σ----y-- |
| | | | | | 2 |
Example:
Given: σx=-5600_psi, σy=-18400_psi, τ xy=4800_psi.
Solution: σ1= -4000_psi, σ2= -20000_psi, θp1=18.4349_°, θp2=108.4349_°, τ max=8000_psi, θs= -26.5651_°, σavg= -12000_psi.
5-58 Equation Reference
