Chapter 2 Bearing capacity of foundation 承载力) In lesson of soil mechanics,we've learned Allowable bearing capacity of shallow foundations based on depth of plastic zone(从塑性区的深度计算浅基础地基 容许承载力) Design value of bearing capacity(地基承 载力的设计值)
Chapter 2 Bearing capacity of foundation (基础 承载力) In lesson of soil mechanics, we’ve learned ▪ Allowable bearing capacity of shallow foundations based on depth of plastic zone (从塑性区的深度计算浅基础地基 容许承载力) ▪ Design value of bearing capacity (地基承 载力的设计值)
2.4 Allowable bearing capacity of shallow foundations (浅基础容许承载力) The following figure shows a strip foundation of width b and infinite length carries a uniform pressure q on the surface of a homogeneous and isotropic soil.In addition,an additional surcharge pressure qo acts on the soil surface 图代表一条形基础：基础宽度为b,无限长度，均布荷载q, 地基为均质土和各性同向，基础两侧均布荷载为q。) Ground surface Ground surface Base of foundation
2.4 Allowable bearing capacity of shallow foundations (浅基础容许承载力) ▪ The following figure shows a strip foundation of width b and infinite length carries a uniform pressure q on the surface of a homogeneous and isotropic soil. In addition, an additional surcharge pressure qo acts on the soil surface (下 图代表一条形基础:基础宽度为b,无限长度,均布荷载q, 地基为均质土和各性同向,基础两侧均布荷载为qo ) Ground surface q b qo Ground surface q b d Base of foundation
2.4 Allowable bearing capacity of shallow foundations From elastic theory,the increases in principal stresses at a given point M are shown as follows(地基中任意点M的 附加大，小主应力可从弹性理论找出) -dB+sin2B) Ac3=9-Y:d(2B-sin2B) 元 元 03
2.4 Allowable bearing capacity of shallow foundations ▪ From elastic theory, the increases in principal stresses at a given point M are shown as follows (地基中任意点M的 附加大,小主应力可从弹性理论找出) b x z z 2b q 1 3 x qo= d M ( b + b) − = 2 sin 2 q d 1 ( b − b) − = 2 sin 2 q d 3
2.4 Allowable bearing capacity of shallow foundations Assume the coefficient of earth pressure at-rest(K)is 1, the increments in principal stresses can be superimposed to the principal stresses.The final principal stresses at depth z are(假设静止土压力系数K=l,这样自重应力 与附加应力可以在任意方向叠加，地基中任意点M的大 小主应力可从以下公式找出) o-9p+sn2p++d c;-9-Y-d(2B-sin2p)+y(z+d)
2.4 Allowable bearing capacity of shallow foundations ▪ Assume the coefficient of earth pressure at-rest (Ko ) is 1, the increments in principal stresses can be superimposed to the principal stresses. The final principal stresses at depth z are (假设静止土压力系数Ko =1,这样自重应力 与附加应力可以在任意方向叠加,地基中任意点M的大 小主应力可从以下公式找出 ) (2 sin 2 ) (z d) q d 1 b + b + + − = (2 sin 2 ) (z d) q d 3 b − b + + − =
2.4 Allowable bearing capacity of shallow foundations When the stresses in point M reach the limiting equilibrium,o and o3 at failure are related by the following expression from Mohr-Coulomb failure criteria (当M点达到极限平衡状态时，大小主应力必需满足以 下公式) =a5+2如m5号》 z is depth of the plastic zone and is expressed as a function of B(z为塑性区深度，是β的函数) z=9-Yd sin2B --d Y·π sinφ y.tanΦ
2.4 Allowable bearing capacity of shallow foundations ▪ When the stresses in point M reach the limiting equilibrium, 1 and 3 at failure are related by the following expression from Mohr-Coulomb failure criteria (当M点达到极限平衡状态时,大小主应力必需满足以 下公式) + + = + 2 2c tan 45 2 tan 45 2 1 3 d tan c 2 sin q d sin 2 z − − − b b − = ▪ z is depth of the plastic zone and is expressed as a function of b (z为塑性区深度,是b的函数)
2.4 Allowable bearing capacity of shallow foundations Differentiate z with respect to B,the maximum value ofz can be determined by equating dz/dβtoO(z最大值从设 导数dz/dB为零找出) -d YπV ytanΦ Rearrange the above expression into the following form: +yd1+ πcotΦ cot-7+
2.4 Allowable bearing capacity of shallow foundations ▪ Differentiate z with respect to b, the maximum value of z can be determined by equating dz/db to 0 (z最大值从设 导数dz/db为零找出 ) ▪ Rearrange the above expression into the following form: d tan c 2 cot q d z max − − + − − = + − + + − + + + − = 2 cot cot c 2 cot d 1 2 cot z q max
2.4 Allowable bearing capacity of shallow foundations The bearing capacity equation is better to express into the following form: q=y.Zmx N'y+y.d.Ng+c.Ne or q=y.ZmxN'y+Yo.d.Ng+c.Ne where y is unit weight of soil below the base of the foundation,Yo is unit weight of soil above the base of the foundation 元 N。=π：cotφ 元 N'y= Ng=1+ cot中-元+4 cotφ- +中 cot- 二+ 2
2.4 Allowable bearing capacity of shallow foundations ▪ The bearing capacity equation is better to express into the following form: max q Nc q = z N' + dN +c or max o q Nc q = z N' + dN +c where is unit weight of soil below the base of the foundation, o is unit weight of soil above the base of the foundation + − = 2 cot N' + − = 2 cot cot Nc + − = + 2 cot Nq 1
2.4 Allowable bearing capacity of shallow foundations Ifmax is chosen as b/3 9写-b,。dN,+eN。=Y Y.b.NI +Yo.d.Na +c.Ne Ifzmax is chosen as b/4 q1-iY-b-N,+Yo'd-Ng+c-N.-Y.b-N1+Yo-d.Ng+c-Ne 4 4 ·Ifmax is chosen as0 Hey =Yo'd.Ng +c.Ne qi3,qi4 and dey are allowable bearing capacities of the foundation(q13,q14 and der为地基容许承载力)
2.4 Allowable bearing capacity of shallow foundations ▪ If zmax is chosen as b/3 o q c 3 o q c 1 3 1 b N d N c N 2 1 b N' d N c N 3 1 q = + + = + + ▪ If zmax is chosen as b/4 o q c 4 o q c 1 4 1 b N d N c N 2 1 b N' d N c N 4 1 q = + + = + + ▪ If zmax is chosen as 0 c o q Nc q = dN +c ▪ q1/3 , q1/4 and qc are allowable bearing capacities of the foundation (q1/3 , q1/4 and qc 为地基容许承载力)
2.5 Design values of bearing capacity 基承载力的设计值) (i)Theoretical Methods(理论公式) Determine the ultimate bearing capacity qu from theory likes plastic theory(从理论公式如塑性理论计算极限承载力qu) Calculate allowable bearing capacity qa from dividing qu by a factor of safety Fs(极限承载力qu除以一个安全糸数作为 容许承载力qa) a Ju If qa also satisfies the settlement criteria,qa will be the design value of bearing capacity(当基底压力为qa时，沉降 小于所允许的范围之内，q。为承载力的设计值)
2.5 Design values of bearing capacity (地 基承载力的设计值) ▪ Determine the ultimate bearing capacity qu from theory likes plastic theory (从理论公式如塑性理论计算极限承载力qu ) ▪ Calculate allowable bearing capacity qa from dividing qu by a factor of safety Fs (极限承载力qu除以一个安全糸数作为 容许承载力qa ) s u a F q q = ▪ If qa also satisfies the settlement criteria, qa will be the design value of bearing capacity (当基底压力为qa时, 沉降 小于所允许的范围之内, qa 为承载力的设计值) (i) Theoretical Methods (理论公式)
2.5 Design value of bearing capacity 基承载力的设计值) (i)Codes of practice(规范查表方法) 从承载力表，可依据原位标准贯入击数(SPT N values)查 取承载力标准值6 从承载力表，可依据室内试验指标查取承载力基本值￡，然 后将f,乘以回归修正系数φ得到 ·当基础宽度小于或等于3m,基础埋深小于或等于0.5m,f 为承载力的设计值(design value of bearing capacity) 当基础宽度与埋深超出以上范围时，须经深宽修正得到承 载力的设计值
2.5 Design value of bearing capacity (地 基承载力的设计值) ▪ 从承载力表,可依据原位标准贯入击数 (SPT N values) 查 取承载力标准值 fk ▪ 从承载力表,可依据室内试验指标查取承载力基本值fo ,然 后将fo乘以回归修正糸数f得到fk ▪ 当基础宽度小于或等于3m,基础埋深小于或等于0.5m, fk 为承载力的设计值 (design value of bearing capacity) ▪ 当基础宽度与埋深超出以上范围时,须经深宽修正得到承 载力的设计值 (ii) Codes of practice (规范查表方法)
