Table 1. Summary of 3-D generalization... | Emerald Publishing

Table 1.

Summary of 3-D generalization equations

Reference	Equation	∂M(θ)/∂θ at θ = ±π/6	Convexity condition^a
Mohr-Coulomb (1776)	$\frac{M (θ)}{M_{t c}} = \frac{3 \sqrt{3}}{(M_{t c} + 6) \cos (θ) - \sqrt{3} M_{t c} \sin (θ)}$	$\frac{M_{t c} (2 M_{t c} + 3)}{3 \sqrt{3}}$ $\frac{\sqrt{3} M_{t c} (M_{t c} - 3)}{{(M_{t c} + 3)}^{2}}$	unconditional
Drucker and Prager (1952)	$\frac{M (θ)}{M_{t c}} = 1$	0	unconditional
Single parameter Van Eekelen (1980)	$\frac{M (θ)}{M_{t c}} = {[\frac{1 - 0.85 \sqrt{α} \sin (3 θ)}{1 - 0.85 \sqrt{α}}]}^{- 0.229} where α = {[\frac{k^{- 4.367} - 1}{0.85 (k^{- 4.367} + 1)}]}^{2}$	0	k > 0.610
Lade and Duncan (1975)	$\frac{M (θ)}{M_{t c}} = \frac{\sin [\frac{π}{3} - \frac{1}{3} \arcsin (\sqrt{\frac{κ_{1} - 27}{κ_{1}}})]}{\sin [\frac{π}{3} + \frac{1}{3} \arcsin (\sqrt{\frac{κ_{1} - 27}{κ_{1}}} \sin (- 3 θ))]} where κ_{1} is a calibration parameter$	0	κ₁ > 36.317
Gudehus (1973)/Argyris (1974)	$\frac{M (θ)}{M_{t c}} = \frac{2 k}{(1 + k) - (1 - k) \sin (3 θ)}$	0	k > 0.777
Willam and Warnke (1974)	$\frac{M (θ)}{M_{t c}} = \frac{2 (1 - k^{2}) \cos (θ + \frac{π}{6}) + (2 k - 1) \sqrt{4 (1 - k^{2}) \cos^{2} (θ + \frac{π}{6}) + 5 k^{2} - 4 k}}{4 (1 - k^{2}) \cos^{2} (θ + \frac{π}{6}) + {(2 k - 1)}^{2}}$	0	unconditional
Jiang and Pietrusczczak (1988)	$\frac{M (θ)}{M_{t c}} = \frac{(\sqrt{1 + a} - \sqrt{1 - a}) k}{k \sqrt{1 + a} - \sqrt{1 - a} + (1 - k) \sqrt{1 - a \sin (3 θ)}} where a is a parameter which approaches 1$	0	k > 0.565
Matsuoka and Nakai (1974)	$27 - 3 M {(θ)}^{2} = A [3 - M {(θ)}^{2} + \frac{8}{9} M {(θ)}^{3} \sin (θ) (\frac{3}{4} - \sin^{2} (θ))] where A = \frac{27 - 3 M_{t c}^{2}}{3 - M_{t c}^{2} + \frac{2}{9} M_{t c}^{3}}$	0	unconditional
Jefferies and Shuttle (2011)	$\frac{M (θ)}{M_{t c}} = 1 - \frac{M_{t c}}{3 + M_{t c}} \cos (\frac{3 θ}{2} + \frac{π}{4})$	$\frac{3}{2} \frac{M_{t c}^{2}}{3 + M_{t c}} 0$	unconditional
this work	$\frac{M (θ)}{M_{t c}} = \frac{1 + k}{2} + \frac{1 - k}{2} \sin (3 a - 3 \sqrt{R^{2} - {(θ + a)}^{2}}) where R is a calibration parameterand a = \sqrt{\frac{R^{2} - \frac{π^{2}}{18}}{2}}$	0	1.508 < R < 6.258 at k = 0.75; see Figure 10

Reference	Equation	∂M(θ)/∂θ at θ = ±π/6	Convexity condition^a
Mohr-Coulomb (1776)	$\frac{M (θ)}{M_{t c}} = \frac{3 \sqrt{3}}{(M_{t c} + 6) \cos (θ) - \sqrt{3} M_{t c} \sin (θ)}$	$\frac{M_{t c} (2 M_{t c} + 3)}{3 \sqrt{3}}$ $\frac{\sqrt{3} M_{t c} (M_{t c} - 3)}{{(M_{t c} + 3)}^{2}}$	unconditional
Drucker and Prager (1952)	$\frac{M (θ)}{M_{t c}} = 1$	0	unconditional
Single parameter Van Eekelen (1980)	$\frac{M (θ)}{M_{t c}} = {[\frac{1 - 0.85 \sqrt{α} \sin (3 θ)}{1 - 0.85 \sqrt{α}}]}^{- 0.229} where α = {[\frac{k^{- 4.367} - 1}{0.85 (k^{- 4.367} + 1)}]}^{2}$	0	k > 0.610
Lade and Duncan (1975)	$\frac{M (θ)}{M_{t c}} = \frac{\sin [\frac{π}{3} - \frac{1}{3} \arcsin (\sqrt{\frac{κ_{1} - 27}{κ_{1}}})]}{\sin [\frac{π}{3} + \frac{1}{3} \arcsin (\sqrt{\frac{κ_{1} - 27}{κ_{1}}} \sin (- 3 θ))]} where κ_{1} is a calibration parameter$	0	κ₁ > 36.317
Gudehus (1973)/Argyris (1974)	$\frac{M (θ)}{M_{t c}} = \frac{2 k}{(1 + k) - (1 - k) \sin (3 θ)}$	0	k > 0.777
Willam and Warnke (1974)	$\frac{M (θ)}{M_{t c}} = \frac{2 (1 - k^{2}) \cos (θ + \frac{π}{6}) + (2 k - 1) \sqrt{4 (1 - k^{2}) \cos^{2} (θ + \frac{π}{6}) + 5 k^{2} - 4 k}}{4 (1 - k^{2}) \cos^{2} (θ + \frac{π}{6}) + {(2 k - 1)}^{2}}$	0	unconditional
Jiang and Pietrusczczak (1988)	$\frac{M (θ)}{M_{t c}} = \frac{(\sqrt{1 + a} - \sqrt{1 - a}) k}{k \sqrt{1 + a} - \sqrt{1 - a} + (1 - k) \sqrt{1 - a \sin (3 θ)}} where a is a parameter which approaches 1$	0	k > 0.565
Matsuoka and Nakai (1974)	$27 - 3 M {(θ)}^{2} = A [3 - M {(θ)}^{2} + \frac{8}{9} M {(θ)}^{3} \sin (θ) (\frac{3}{4} - \sin^{2} (θ))] where A = \frac{27 - 3 M_{t c}^{2}}{3 - M_{t c}^{2} + \frac{2}{9} M_{t c}^{3}}$	0	unconditional
Jefferies and Shuttle (2011)	$\frac{M (θ)}{M_{t c}} = 1 - \frac{M_{t c}}{3 + M_{t c}} \cos (\frac{3 θ}{2} + \frac{π}{4})$	$\frac{3}{2} \frac{M_{t c}^{2}}{3 + M_{t c}} 0$	unconditional
this work	$\frac{M (θ)}{M_{t c}} = \frac{1 + k}{2} + \frac{1 - k}{2} \sin (3 a - 3 \sqrt{R^{2} - {(θ + a)}^{2}}) where R is a calibration parameterand a = \sqrt{\frac{R^{2} - \frac{π^{2}}{18}}{2}}$	0	1.508 < R < 6.258 at k = 0.75; see Figure 10

^a

Note: In all cases it is assumed that M_tc/2 ≤ M_te ≤ M_tc or 0.5 ≤ k ≤ 1 which bounds the shapes between an equilateral triangle and a circle.