Brinch-Hansen (1961, 1970) proposed a general equation that allows us to determine the bearing capacity of footings of any shape and size and any inclination of loading or bearing surface. The net ultimate bearing capacity from Meyerhof's equations and o Brinch-Hansen's equations are not practically significant for φ′ < 35 . Brinch-Hansen's equations are often used for footings on slopes. The general Brinch-Hansen (1970) equations are: TSA: qult = 5.14 su (1 + sc + dc - ic - bc - gc) ESA: qult = γ Df (Nq - 1) sq dq iq bq gq + 0.5 γ B Nγ sγ dγ bγ gγ Shape factors sc = 0.2 sq = 1 +
B L
; sγ = 1 - 0.4
B L
B L
(sγ > 0.6)
sin φ′ [sq was proposed by De Beer (1967, 1970) based on plate loading tests]
Depth factors Df < B
dc = 0.4
D f B
2 D f
; dγ = 1; dq = 1 + 2 tan φ′ (1 - sin φ′)
B
Df > B -1
2
-1
dc = 0.4 tan ( D f ); d ϒ = 1; dq = 1 + 2 tan φ′ (1 - sin φ′) tan ( D f ) B
B
The depth factors above are valid for failure along the long sides L of the base. For failure along the short side, B, replace B by L in the above equations.
Load Inclination factors Brinch-Hansen (1970) gave a set of equations for inclination factors. For coarse-grained o soils, he proposed equations that are valid for φ′ ≈ 45 . The author recommends the use o of Meyerhof's inclination factors since φ′ ≈ 45 are rare. Base inclination factors bc =
2 αs
π + 2
; bq = exp ( −2αs tan φ′ ) ; b γ = exp ( −2.7αs tan φ′ ) ;αs in radians
Ground inclination factors gc =
2β
π+2
; β in radians 5
g γ = g q = (1 − 0.5 tan β )
o
o
These factors apply when β <φ′ (φ′ < 45 ) and αs + β ≤ 90
