Chapter 8 Bearing capacity In this lecture well

第八章 地基承载力 Chapter 8 Bearing capacity In this lecture, we’ll learn § § Types of foundation failures (基础破坏的类型) Ultimate bearing capacity of shallow Foundations (浅基础 地基极限承载力) § Prandtl’s method: smooth foundation and weightless soil (普朗特公式: 基底光滑与土无重量) § Terzaghi’s method: rough foundation (太沙基公式: 基 底粗糙)

8. 1 Introduction (前言) § Ultimate bearing capacity (qu) is defined as the pressure which would cause shear failure of the supporting soil immediately below and adjacent to a foundation. (当基底 压力增大到极限承载力时, 地基出现剪切破坏) § A foundation must satisfy two fundamental requirements: ( 设计基础要满足两个要求 ) § the factor of safety Fs against shear failure of the supporting soil must be adequate, a value between 2 and 3 normally being specified (地基达到剪切破坏的 安全糸数 Fs 要适当, 一般在 2至 3之间 )

§ § the settlement of the foundation should be tolerable and, in particular, differential settlement should not cause any unacceptable damage of the structure (基础的沉降和沉 降差必须在该建筑物所允许的范围之内) Allowable bearing capacity (qa) is defined as the maximum pressure which may be applied to the soil such that the above two requirements are satisfied. From first requirement, qa is defined as: (地基的容许承载力 qa 定义 为当上述两个条件满足时的基底最大压力. 当条件一满 足时qa 定义如下)

8. 2 Types of foundation failures (基础破坏的类型) (i) General shear failure (整体剪切破坏 ) § § State of plastic equilibrium is fully developed throughout the soil above the failure surfaces (地基形成连续滑动面而破坏) Heaving of the ground surface occurs (基础两侧地面向上隆起) Typical failure mode for soils of low compressibility, i. e. dense or stiff soils (紧砂或硬粘土的典型破坏模式) Pressure Settlement § Ground surface (地面) Sliding surface ( 滑动面)

(ii) Local shear failure (局部剪切破坏) § § Partial development of the state of plastic equilibrium (剪切 破坏区限制在地基内部某一区域) Slight heaving occurs (基础两侧地面只是微微隆起) It is characterised by the occurrence of relatively large settlements (这种破坏型式的特征是出现相对大的沉降 ) Pressure Ground surface (地面) Settlement § Sliding surface ( 滑动面)

(iii) Punching shear failure (冲剪破坏) § There is relatively high compression of the soil under the foundation accompanied by shearing in the vertical direction around the edges of the foundation (基础边缘下地基产生垂 直剪切破坏 ) No heaving of the ground surface (基础两侧地面不出现隆 起) Pressure Settlement § Ground surface (地面) Sliding surface ( 滑动面)

8. 3 Allowable bearing capacity determined from plastic zone 1. Ultimate bearing capacity of shallow foundations (浅基础地基极限承载力) § Foundations can be classified as shallow and deep foundations. In general, if the depth of a foundation (D) is smaller than or equal to its breadth (B), i. e. D/B 1, the foundation is classified as shallow foundation. (一般认为, 当基础的埋深D 小于或等于基础宽度B时, 称为浅基础)

2. Allowable bearing capacity of shallow foundations (浅基础地基容许承载力) § 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) qo b q Ground surface b q d Ground surface Base of foundation

§ 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 q qo= d x z M 1 3 2 x z

§ 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 (假设静止土压力糸数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 )

§ 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) § z is depth of the plastic zone and is expressed as a function of (z为塑性区深度, 是 的函数)

§ Differentiate z with respect to , the maximum value of z can be determined by equating dz/d to 0 (z最大值从设 导数dz/d 为零找出 ) § Rearrange the above expression into the following form:

§ The bearing capacity equation is better to express into the following form: or where is unit weight of soil below the base of the foundation, o is unit weight of soil above the base of the foundation

§ If zmax is chosen as b/3 § If zmax is chosen as b/4 § If zmax is chosen as 0 § q 1/3, q 1/4 and qc are allowable bearing capacities of the foundation (q 1/3, q 1/4 and qc 为地基容许承载力)

8. 4. Prandtl’s Theory (普朗特公式) § Assumption: the base of the foundation is smooth and the soil is weightless. (假设基底光滑与土体无重) § The failure mechanism for a strip foundation is shown in following figure. (下图代表条形基础的假设滑动面) b qo qu qo

§ Plastic equilibrium exists above the assumed failure surface EDCGF such that: (在滑动面EDCGF以上土体达到朔性平衡状态) § ABC is active Rankine zone (ABC为朗肯主动区) § ACD and BCG are radial shear zones (ACD与BCG为径向剪切区 ) § ADE and BGF are passive Rankine zones ADE and BGF (ADE与 BGF为朗肯被动区 ) b qo qu qo

§ ABC and BAC are 45 + /2 § DAE, DEA, GBF and GFB are 45 - /2 § The surfaces CD and CG are logarithmic spirals to which BC and ED, or AC and FG are tangential (ED与FG为直线) b qo qu qo

§ Using plastic theory, the following exact solution is obtained for the ultimate bearing capacity qu of a strip footing on the surface of a weightless soil. (根据塑性理 论, 条形基础在无重量地基上的极限承载力为以下公 式)

§ The ultimate bearing capacity equation is simplified to the following expression: (极限承载力公式可以简化为以下公式) where d is depth of foundation, o is unit weight of soil above the base of the foundation, Nc and Nq are bearing capacity factors (d为 基础的埋深, o基底以上两侧土体的重度, Nq与Nc为承载力因素)

8. 5. Terzaghi’s Theory 太沙基理论 (i) Smooth foundation (基底光滑) § Assumption: the principle of superposition can be used in deriving the ultimate bearing capacity equation. The ultimate bearing capacity can be expressed in the following form: (当考虑地基重量时, 可应用叠加 原埋, 以下公式中的首项代表地基自重的贡献, Nq与Nc分别从普朗 特公式找出 ) and

(ii) Rough foundation – Terzaghi’s method ( 基底粗糙) § If the foundation is rough, so that no slip takes place on AB, ABC is not considered to be an active Rankine zone (ABC不是朗肯主动区) § Assumption: ABC and BAC equal to b qo qu qo

§ § The ultimate bearing capacity equation is shown as follows: The ultimate bearing factors are shown in the next graph.

8. 5 Design values of bearing capacity ( 地基承载力的设计值) (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) § 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)