Theories of Failure - cases of solids:

Let us now consider the solid shown in Figure (b).

(a)        As the principal stresses are within 260 MPa, the solid is safe according to the principal stress theory.

(b)        The maximum principal strain ε₁ = 250/E -0.3 (-150/E) = 295/E > 260/E

 ∴         The solid will fail according to principal strain theory.

(c)        Maximum shear stress, τ_max = 250-(-150)/2 = 200>130

∴          The solid is not safe as per shear stress theory.

(d)        Total strain energy density

u =  1/2E  [250²  + (- 150)²  - 2 × 0.3 × 250 (- 150)]

= 53750/E > 260²/2E

∴          The solid will fail.

(e)        Distortion energy density 1/12G ({250 - (- 150)}²  + 250² + 150²)

= 122500/6G > 260²/6G

∴          Distortion energy theory also predicts failure.