Static Equilibrium, Elasticity, and Fracture Study Notes

Overview of Static Equilibrium, Elasticity, and Fracture

• Definition of Statics: This branch of mechanics focuses on the special case where the net force and the net torque acting on an object or system are both zero.

• Key Conditions: Under these conditions, both the linear acceleration ($a$) and the angular acceleration ($\alpha$) of the object or system are zero.

• State of the Object: The object is either at rest or its center of mass moves at a constant velocity. Statics is primarily concerned with objects at rest (static).

• Importance of Force Distribution: Even when net force and torque are zero, external forces are still acting. Understanding how and where these forces act is critical for designing buildings, bridges, and machines, and for understanding the human body.

• Consequences of High Stress: If forces are excessive, objects may undergo significant deformation or fracture (break).

• Real-World Application: Architects and engineers perform these calculations to prevent structural failures, such as the 1981 Kansas City hotel elevated walkway collapse, which resulted in the loss of over 100 lives due to a failure that could have been prevented by simple physics calculations (Example 12-11).

• Medical and Athletic Context: Knowledge of forces in muscles, bones, and joints is essential for doctors, physical therapists, and athletes.

The Conditions for Equilibrium

• Newton's Second Law Context: For an object at rest, the net force ($\Sigma F$) must be zero. For example, a book on a table experiences a downward force of gravity ($mg$) and an upward normal force ($F_N$). Because it is at rest, these forces are equal in magnitude and opposite in direction.

• Distinction from Newton's Third Law: The upward normal force and downward gravitational force on the book both act on the same object to maintain equilibrium; they are not an action-reaction pair (which would act on different objects).

• Etymology: The word "equilibrium" is derived from the Latin terms for "equal forces" or "balance."

The First Condition for Equilibrium

• Mathematical Expression: The vector sum of all forces must be zero. In three dimensions:     $\Sigma F_x = 0$     $\Sigma F_y = 0$     $\Sigma F_z = 0$

• Problem-Solving Application: Most problems involve forces in a single plane ($xy$ plane), requiring only the $x$ and $y$ components. Components pointing along negative axes must be treated with a negative sign.

• Free-Body Diagrams (FBD): Solving statics problems requires a diagram showing all forces acting on a specific object and the exact points where those forces are applied.

The Second Condition for Equilibrium

• Necessity: Zero net force is necessary but not sufficient for equilibrium. An object can experience a net force of zero while still having a net torque that causes rotation.

• Definition of a Couple: A pair of equal forces acting in opposite directions but at different points on an object is called a couple.

• Mathematical Expression: The sum of all torques ($\tau$) acting on an object, calculated about any chosen axis, must be zero:     $\Sigma \tau = 0$

• Axis Selection: If an object is in equilibrium, the net torque is zero about any axis. For calculations, an axis is chosen arbitrarily, often at a point where an unknown force acts to eliminate it from the equation (lever arm = 0).

• Directional Convention: Typically, counterclockwise (CCW) torques are considered positive (\tau > 0) and clockwise (CW) torques are considered negative (\tau < 0), though any consistent convention works.

Solving Statics Problems

• General Strategy:

  1. Select one object for consideration and draw a precise Free-Body Diagram showing all forces and their points of application.

  2. Choose a coordinate system and resolve forces into components.

  3. Write the equilibrium equations for forces ($\Sigma F_x = 0$ and $\Sigma F_y = 0$).

  4. Write the equilibrium equation for torque ($\Sigma \tau = 0$) about a convenient axis (often perpendicular to the $xy$ plane).

  5. Solve for unknowns. A maximum of three unknowns can be found using the three equations ($\Sigma F_x, \Sigma F_y, \Sigma \tau$).

• Uncertain Directions: If the direction of an unknown force is uncertain, assume one direction. If the calculated result is negative, the force actually acts in the opposite direction.

• Center of Gravity (CG): Gravity is considered to act at the center of gravity (or center of mass). For uniform, symmetrical objects, the CG is at the geometric center.

• The Cantilever Example (Fig. 12-8):

  • A uniform beam extends beyond its support like a diving board.

  • Mass of beam: $1200\,kg$ ($weight = 12,000\,N$).

  • Calculated forces: $F_B = 15,000\,N$ (upward at support B) and $F_A = -3000\,N$ (meaning support A must pull downward with $3000\,N$ using bolts or glue).

• Flexible Cables vs. Rigid Hinges:

  • Flexible cables only support force along their length ($F_{tension}$).

  • Rigid hinges can exert force in any direction (resolved into horizontal $F_{Hx}$ and vertical $F_{Hy}$ components).

Applications to Muscles and Joints

• Terminology:

  • Insertions: Points where muscles attach to bones via tendons.

  • Joints: Flexible connections between bones (e.g., elbow, knee, shoulder, hip).

  • Flexors: Muscles that bring limbs closer together (e.g., biceps).

  • Extensors: Muscles that extend limbs outward (e.g., triceps).

• Muscle Mechanics: Muscles exert pulls by contracting fibers; they cannot push.

• Bending Forward Analysis (Fig. 12-13):

  • The fifth lumbar vertebra acts as a fulcrum.

  • Force components for a person of total weight $w$:

    • Head weight ($W_H$) = $0.07w$

    • Two arms weight ($W_A$) = $0.12w$

    • Torso weight ($W_T$) = $0.46w$

  • In a scenario where a $90\,kg$ person holds a $20\,kg$ load (increasing $W_A$ to $0.34w$), the force on the spinal disk ($F_V$) can increase to nearly $3.7w$. For a $200\text{-lb}$ person, this exceeds $700\text{-lb}$.

Stability and Balance

• Three Types of Equilibrium:

  1. Stable Equilibrium: Object returns to original position after slight displacement (e.g., ball suspended from a string).

  2. Unstable Equilibrium: Object moves further away after displacement (e.g., a pencil balanced on its point).

  3. Neutral Equilibrium: Object stays in its new position (e.g., a sphere on a flat table).

• Condition for Stability: An object is stable if its CG is below the point of support or if a vertical line projected downward from the CG falls within the base of support.

• Factors Increasing Stability: A larger base of support and a lower center of gravity.

• Human Stability: Humans maintain balance by unconsciously shifting their bodies so the CG remains over the feet. Bending forward requires moving hips backward to compensate.

• Heels-to-Wall Experiment: If heels and back are against a wall, a person cannot touch their toes without falling because they cannot shift their hips back to keep the CG over the feet.

Elasticity, Stress, and Strain

• Hooke’s Law: The change in length ($\Delta l$) of an object is proportional to the applied force ($F$) for small elongations:     $F = k \Delta l$

• Deformation Regions (Fig. 12-18):

  • Proportional Limit: Extension is linear with force.

  • Elastic Limit: Beyond this, the object will not return to its original length (enters the plastic region).

  • Elastic Region: Area where the object returns to original length upon removal of force.

  • Plastic Region: Permanent deformation occurs.

  • Breaking Point: The maximum elongation before fracture.

• Young’s Modulus ($E$): A material property that determines how much it stretches or compresses:     $\Delta l = \frac{1}{E} \frac{F}{A} l_0$

  • $l_0$: Original length.

  • $A$: Cross-sectional area.

• Stress: Force per unit area ($Stress = \frac{F}{A}$) measured in $N/m^2$.

• Strain: Fractional change in length ($Strain = \frac{\Delta l}{l_0}$), which is dimensionless.

• Relationship:     $E = \frac{\text{stress}}{\text{strain}}$

Types of Stress

• Tensile Stress: Forces pull the object apart; elongates the material.

• Compressive Stress: Forces act inwardly; compresses the material. Equations for $E$ apply equally to tension and compression for most materials.

• Shear Stress: Equal and opposite forces applied across opposite faces of an object (e.g., a book parallel to a tabletop).

  • Shear Modulus ($G$):     $\Delta l = \frac{1}{G} \frac{F}{A} l_0$

  • Note: Area ($A$) for shear is the surface parallel to the force.

• Bulk Modulus ($B$): Applies when an object is subjected to pressure from all sides (e.g., submerged in a fluid), causing volume change ($\Delta V$):     $\Delta V = -\frac{1}{B} V_0 \Delta P$

  • $\Delta P$: Change in pressure.

  • The minus sign indicates volume decreases as pressure increases.

Fracture and Strength

• Ultimate Strength: The maximum stress a material can withstand before breaking.

• Safety Factors: Due to variations in material specimens, actual stresses are usually kept between one-tenth to one-third of ultimate strength (safety factor of 3 to 10+).

• Representative Material Strengths (Table 12-2):

  • Iron (Cast): Tensile $170 \times 10^6\,N/m^2$, Compressive $550 \times 10^6\,N/m^2$.

  • Concrete: Very weak in tension ($2 \times 10^6\,N/m^2$) but reasonably strong in compression ($20 \times 10^6\,N/m^2$).

• Reinforced Concrete: Uses iron rods to provide tensile strength.

• Prestressed Concrete: Iron rods are put under tension while concrete is poured and released after it dries. This keeps the concrete under permanent compression, preventing it from entering tension even when loaded.

Trusses and Bridges

• Truss Definition: A framework of struts joined at ends (joints) by pins or rivets, always arranged in triangles for stability.

• Triangles vs. Rectangles: Rectangles easily collapse into parallelograms under sideways forces; triangles are rigid.

• Assumptions in Truss Analysis: Struts are typically assumed to be massless and subject only to pure tension or compression along their length.

• Reality of Struts: Real struts have mass, meaning forces at joints do not act precisely along the strut due to gravity ($mg$).

• Example 12-12: Truss Bridge Analysis:

  • Bridge length: $64\,m$.

  • Total mass supported by one truss: $7.0 \times 10^5\,kg$.

  • Method of Joints: Analyzing forces on each pin (joint) where $\Sigma F = 0$.

  • Results for symmetrical equilateral triangle truss:         $F_{AB} = \frac{1}{\sqrt{3}} Mg$         $F_{AC} = \frac{1}{2\sqrt{3}} Mg$

• Suspension Bridges: For very large spans (e.g., Golden Gate Bridge), truss structures are replaced by light suspension cables under tension.

Arches and Domes

• Arches:

  • Semicircular (Round) Arch: Developed by Romans ($2000$ years ago). Wedge-shaped stones are kept under compression.

  • Pointed Arch: Developed around $1100\,A.D.$ (Gothic cathedrals). Higher arches reduce horizontal thrust at the base.

  • Mechanical Advantage: Calculation shows that for an $8.0\text{-m}$ span, a pointed arch ($8.0\,m$ high) requires only half the horizontal buttressing force ($F_H = 1.5 \times 10^4\,N$) compared to a round arch ($4.0\,m$ high, $F_H = 3.0 \times 10^4\,N$).

• Domes:

  • Pantheon (Rome): Hemispherical stone dome, $2000$ years old.

  • Florence Cathedral: Filippo Brunelleschi designed a pointed dome ($43\,m$ diameter) that could be built in horizontal layers without massive wooden framework.

  • Superdome (New Orleans): Modern dome ($200\,m$ diameter) made of steel trusses and concrete.

Questions & Discussion

• How does a person bending over stay balanced? By moving the hips backward, the person's center of gravity remains directly over the base of support (the feet).

• Why does FA in Figure 12-8 come out negative? In the calculation for the cantilever, the weight of the overhanging board creates a clockwise torque. To balance this, the left-hand support must pull downward, which is indicated by the negative sign in the calculation results ($F_A = -3000\,N$).