Design of Machine Foundations and Foundation Engineering Notes

Introduction to Machine Foundations

• Machine foundations differ from ordinary foundations as they must withstand two distinct categories of loading:

  • Static Loads: This consists of the combined self-weight of the machine and the structural weight of the foundation itself.

  • Dynamic Loads: These are forces that fluctuate rapidly in terms of both magnitude and direction. They typically arise from the movement of unbalanced machine components during operational cycles.

• Design Philosophy for Static Loading: The approach is virtually identical to standard foundation engineering. Engineers typically assume preliminary dimensions for the foundation and then verify that the stresses induced in the underlying soil do not exceed the allowable bearing capacity.

• Design Philosophy for Dynamic Loading: Analysis must account for the specific vibration behavior of the system. In this context, the machine and its foundation are viewed as a single, integrated vibrating system.

Fundamental Concepts of Oscillation and Vibration

• Oscillation: Defined as the repetitive movement of a rigid body or a point mass about an equilibrium position.

  • Examples: A pendulum swinging or a ship swaying on ocean waves.

  • Distinction: Oscillation involves motion without the internal deformation of the object.

• Vibration: Defined as the repeating motion or rhythmic deformation of an elastic structure.

  • Examples: Rotating industrial fans, electric motors, and machine foundations themselves.

  • Distinction: Unlike oscillation, vibration necessitates the deformation of the body as it moves.

Comparative Analysis of Static and Dynamic Loads

• Acceleration and Equilibrium: The primary differentiator between static and dynamic loading is acceleration.

• Static Analysis: Loads are assumed to be applied gradually. The system is analyzed under the assumption that it reaches a state of equilibrium where internal and external forces balance out.

• Dynamic Analysis: Acceleration is a non-negligible factor. Consequently, inertia effects must be integrated into the physical model.

• Newton's Second Law of Motion: The governing principle is formulated as:

  • $F = m \times a$

• D'Alembert's Principle: This principle states that acceleration effects can be mathematically represented as an inertia force. This force acts in the direction exactly opposite to the acceleration of the body.

  • $\text{Inertia Force} = m \times a$

  • This allow structural engineers to treat a dynamic problem as an equivalent static equilibrium problem by introducing these fictitious inertia forces.

Classifications of Vibration

• Natural Vibrations: These occur when a system is disturbed from its initial equilibrium position and then released to move freely.

  • Energy Mechanics: The vibration is sustained through the continuous exchange of kinetic energy and strain (potential) energy.

  • Frequency Determinants: The natural frequency of the vibration is a function of the system's inherent properties, specifically its mass and stiffness ($k$), and does not depend on the magnitude of the initial displacement.

• Forced Vibrations: These occur when an external, independent exciting force is applied to the system either continuously or at periodic intervals.

  • Industrial Context: Most machinery produces this type of periodic excitation through internal moving parts.

Modeling Vibrating Systems

• Single Degree of Freedom (SDOF) Systems: These systems require only one independent coordinate to fully describe their state of motion.

  • Idealized Models: SDOF systems are used extensively in vibration analysis to simplify complex equations of motion.

  • Examples: Simple mass-spring systems, a simple pendulum, or a rotating system governed by one generalized coordinate.

• Multiple Degree of Freedom (MDOF) Systems: These involve multiple discrete masses or a continuous distribution of mass.

  • Complexity: Several coordinates are required to describe the motion of the system completely.

  • Characteristics: These systems possess multiple natural frequencies and various vibration modes.

  • Example: Multi-storey buildings, where each floor is modeled as an individual mass with its own displacement coordinate.

Excitation Types and System Forces

• Classes of Excitation: Two main categories can drive SDOF and MDOF systems:

  1. Excitation Caused by a Force:

     • Sinusoidal Force: Typical of unbalanced rotating machinery.

     • Non-sinusoidal Force: Such as the rhythmic impact of a person walking.

  2. Excitation Caused by Support Displacement:

     • Sinusoidal Displacement: Controlled base movement.

     • Non-sinusoidal Displacement: Such as the erratic ground motion caused by an earthquake.

• Principal Forces in Vibrating Systems: Four forces are typically accounted for in the governing equations:

  1. Inertia Force: Defined as $m \times \ddot{x}$.

  2. Spring Force: Derived from the elasticity of the support, following Hooke's Law: $F = k \times x$.

  3. Damping Force: This force opposes the motion and is generally proportional to velocity. The most utilized model is viscous damping, defined as $c \times \dot{x}$.

  4. External Exciting Force: The primary force driving the vibration.

Industrial Machine Categories

• Reciprocating Machines: These produce periodic unbalanced forces. Steam engines are the classic example. These machines generally operate at relatively low speeds.

• Impact Machines: These generate impact loads where the force reaches a massive peak value over a very short duration. Forging hammers are the primary example.

• Rotary Machines: High-speed equipment such as turbo-generators and rotary compressors. The operating speeds for these machines are significant, often exceeding several thousand revolutions per minute (RPM).

Principles of Foundation Engineering

• Foundational Classification:

  • Shallow Foundation: A foundation where the depth ($Z$) below the ground surface is equal to or less than its least lateral dimension ($B$). Formula: $Z \le B$.

  • Deep Foundation: A foundation where the depth is greater than its least dimension. Formula: Z > B.

• Bearing Capacity Terminology:

  • Ultimate Bearing Capacity ($q_u$): The average contact pressure between the foundation and the soil that triggers a shear failure in the soil mass.

  • Safe Bearing Capacity ($q_a$): The allowable contact pressure that can be safely applied without the risk of shear failure. It involves a Factor of Safety ($F$).

  • Formula: $q_a = \frac{q_u}{F}$

Terzaghi's Bearing Capacity Formulation

• Karl Terzaghi developed the classic equation for soil bearing capacity. For a strip footing, the ultimate bearing capacity is determined by:

  • $q_u = c' N_c + q N_q + 0.5 \gamma B N_\gamma$

• Variable Definitions:

  • $c'$ = Cohesion of the soil.

  • $\gamma$ = Unit weight of the soil.

  • $q$ = Surcharge pressure applied at the foundation level ($q = \gamma \times \text{depth}$).

  • $B$ = Width of the footing.

  • $N_c, N_q, N_\gamma$ = Dimensionless bearing capacity factors that depend on the internal angle of friction ($\phi$).

• Variations: The formula is modified appropriately when applied to square or circular footing geometries.

Worked Numerical Example

• Given Parameters:

  • Load = $30,000\,kg$

  • Soil Density = $1850\,kg/m^3$

  • Internal Friction Angle ($\phi$) = $35^\circ$

  • Cohesion ($c'$) = $0$

  • Foundation Depth = $1\,m$

• Conversions and Initial Calculations:

  • Soil Unit Weight ($\gamma$): $\frac{1850 \times 9.81}{1000} \approx 18.15\,kN/m^3$

  • Total Load: $\frac{30000 \times 9.81}{1000} \approx 294.3\,kN$

• Bearing Capacity Factors (for $\phi = 35^\circ$):

  • $N_c = 57.8$

  • $N_q = 41.4$

  • $N_\gamma = 42.4$

• Design Calculation:

  • Using the modified Terzaghi equation for a square footing and a Factor of Safety ($F$) of $3$:

  • $q_a = \frac{q_u}{3}$

  • Through an iterative trial-and-error process substituting the values for width $B$:

  • Result: The required width for a square footing is approximately $0.75\,m$ .