Mechanics of Deformable Bodies

• Mechanics of Deformable Bodies covers axial stress and strain, stresses for torsion and bending, combined stresses, statically indeterminate structures, and shear and moment equations and diagrams.
• The course aims to enable understanding of fundamental concepts in the strength of materials for structural member design, explaining basic stress and strain concepts, calculating stresses due to tension, shear, bending, and torsion under various loadings, and analyzing statically determinate and indeterminate structures.

Fundamental Areas of Engineering Mechanics

• Statics deals with bodies at rest or moving with constant velocity, focusing on equilibrium.
• Dynamics is the accelerated motion of bodies.
• Strength of Materials addresses internal effects and deformations caused by applied loads on real/deformable bodies, considering internal effects as forces.
• Free Body Diagrams (FBD), equilibrium equations, and deformation diagrams are used for analysis, with a focus on determining the body's strength and rigidity.

Normal Stress

• Normal stress is a type of force that acts normal or perpendicular to cross-sectional area.
• The normal stress formula is σ = P/A, where P is the axial force perpendicular to the cross-sectional area and A is the cross-sectional area.
• SI units for normal stress (σ) include Megapascals (MPa), Kilopascals (KPa), and Pascals (Pa).
• SI units for axial force (P) include Kilonewtons (kN) and Newtons (N).
• SI units for area (A) include m² and mm².
• English units for normal stress (σ) include Kilopounds per square inch (ksi) and Pounds per square inch (psi).
• English units for axial force (P) include Kilopounds (kips) and Pounds (lbs).
• English units for area (A) include in² and ft².

Actual Stress

• Actual stresses are derived from the effects of externally applied loads, computed using equilibrium equations.

Allowable Stress

• Allowable stresses are computed from the structural properties of the member in question and indicate the member's prescribed capacity.
• To prevent member failure, the computed stress must be less than the working stress (σactual ≤ σallowable).

Shear Stress

• Shear stress acts tangent or parallel to a cross-sectional area.
• Shear Stress formula is τ = V/A, where V is the force parallel to the cross-sectional area and A is the cross-sectional area.

Bearing Stress

• Bearing stress (σb) is compressive and a special type of normal stress
• Bearing stress occurs as contact pressure between seperate bodies
• Bearing Stress formula is σb = Pb/Ab = P/(t*d)

Twisting Moment

• Twisting moment, also called torque, tends to twist or rotate the cross-sectional area of a body.

Bending Moment

• Bending moment tends to bend a member.

Thin-Walled Pressure Vessels

• Pressure vessels hold gases and liquids at pressures substantially different from ambient pressure, like tin cans and water tanks.
• A vessel is thin-walled when the ratio of its thickness to radius is small, ensuring constant internal stress throughout the material's thickness.
• Thin-walled pressure vessels have a radius to thickness ratio (r/t) greater than or equal to 10.
• Vessels with an r/t ratio less than 10 are not treated as thin-walled pressure vessels.
• For thin-walled vessels, the approximation r̄ ≈ r can be used.

Cylindrical Vessels

• Cylindrical vessels experience tangential, circumferential, girth, or hoop stress
• The formula for this stress is σc = pr/t.
• Cylindrical vessels experience longitudinal stress
• The formula for longitudinal stress is σl = pr/2t.

Spherical Vessels

• Spherical vessels experience constant stress throughout due to symmetry.
• The formula for the stress is σ = pr/2t

Deformation - Strain

• Strain is a geometric quantity that measures the deformation of a body.
• Loads cause stress, and stress causes strain.

Types of Strain

• Normal strain involves changes in dimensions, such as elongation or shortening.
• Shear strain involves changes in angles, causing distortion.

Axial Deformation

• Stress and strain exist concurrently in nature; a body under stress exhibits strain.
• Average axial strain is ε = δ/L, where δ is deformation and L is length.

Proportional Limit and Hooke's Law

• Stress is proportional to strain, represented by σ = εE.
• Hooke's Law is valid up to the proportional limit, where stress and strain vary linearly.

Elastic Limit

• A material is elastic if it returns to its original shape after the load is removed, up to the elastic limit.
• Permanent set refers to the permanent deformation remaining after load removal.
• Delta = PL/AE where P is load applied, L is length, A is cross-sectional area, E is Modulus of Elasticity

Yield Point

• The yield point on a stress-strain diagram is where the diagram becomes almost horizontal, indicating yielding.
• Beyond the yield point, there is appreciable elongation or yielding without a corresponding increase in load.

Ultimate Strength

• Ultimate strength is the highest point on the stress-strain curve.
• Ultimate strength is commonly used to define the maximum stress a material can withstand.

Rupture Strength

• Rupture strength is the stress at which failure occurs.

Deformation Formula

• δ = PL/AE is only applicable if the strain/stress in the bar is uniform, and the cross-sections, loads, and materials are constant throughout.
• Otherwise, δ = ∫(from 0 to L) (P/AE) dx should be used.
• Derived formula of the deformation of the conical frustum: δ = 4PL/πEDd

Statically Indeterminate Members

• Statically indeterminate members cannot be analyzed by equilibrium equations alone.
• Equations from equilibrium and deformation (compatibility equations) are used.
• Steps to solve: Draw the FBD, derive the compatibility equations, use Hooke's Law to express strains in terms of forces, and solve all equations simultaneously.

Thermal Stresses

• Changes in temperature cause dimensional changes: increased temp results in expansion, decreased temp produces contraction.
• Deformation is isotropic and proportional to the temperature change.
• Thermal strain is δt = αΔTL, where α is the coefficient of thermal expansion, ΔT is the temperature change (Final - Initial), and L is the length.
• If the temperature change is uniform, thermal strain is also uniform. δT = εTL = α(ΔT)L

Procedures for deriving compatibility equations

• Remove the constraints that prevent thermal deformation.
• Apply forces to restore constraint conditions.
• Relate thermal deformations to constraint force deformations.

Torsion

• Structural members must carry torsional loads in many engineering applications.

Assumptions of Deformation

• Circular sections remain circular and plane sections remain plane without warping, retaining perpendicularity to the shaft's axis
• The cross-section does not deform
• The distance between cross-section does not change meaning there is no axial strain
• The projection upon a transverse section of straight radial lines remains straight.
• The shaft is loaded by twisting couples in planes perpendicular to the shaft's axis.
• Stresses do not exceed the proportional limit.

Torsion Formulas

• θ = TL/JG, where θ is the angle of twist in radians, L is the shaft length, J is the polar moment of inertia, and G is the modulus of rigidity.

Shear Stress

• Shear Stress at a section is τ = Tρ/J or τmax = Tr/J where T is torque, ρ is radial distance, J is the polar moment of inertia and r is the radius.
• For a solid shaft: J = (πr^4)/2
• For a hollow shaft: J = (π/2) * (R^4 - r^4)

Maximum Shear Stress

• Solid shaft: τmax = 2T/(πr^3)
• Hollow shaft: τmax = 2TR/(π(R^4 − r^4))

Power Transmission

• In practical applications, shafts transmit power.
• Power (P) transmitted by torque (T) rotating at angular speed (ω) is P = Tω.
• If the shaft rotates at f revolutions per unit time, ω = 2πf, so P = T(2πf).
• Torque can therefore be expressed as T = P/(2πf).
• Angular speed is measured in radians per unit time.
• P = ωT, ω = 2πf, P = 2πfT

Torsion of Thin-Walled Tubes

• Torsional Shearing Stress: τ = T/2At, where A is the area bounded by the centerline and t is the wall thickness at the point of consideration.

Shear and Bending Diagram - Bending Stress

• Beams are structural members subject to bending moment.
• Types of beams include: Non-prismatic beams which have a variable cross-section along its length and Prismatic beams which have a constant cross-section throughout its length.

Types of Beams

• Determinate beams include: Simply Supported, Cantilever, and Overhanging.
• Indeterminate beams include: Propped Cantilever, Fixed/Restrained, and Continuous.

Internal Supports

• Internal Roller Support
• Internal Hinge Support allows increased beam flexibility, because when cut at an internal hinge the bending moment at that section is zero, reducing bending stresses.

Types of Loads

• Point Load
• Uniformly Distributed Load
• Triangular Distributed Load
• Trapezoidally Distributed Load
• Point Moment

Methods to Sketch Shear and Moment Diagram

• Equation Method
• Area Method

Steps for Equation Method

• Divide the beam into segments where loading is continuous, marking endpoints at discontinuities.
• Sketch the FBD to compute support reactions.
• For each segment, define shear and moment equations by isolating each segment and introducing an imaginary cutting plane at distance x.
• Isolate the left or right part of the beam, showing exposed forces at the cut section.
• Sum forces to get V, and moments to get M.
• Summarize equations and plot expressions of V and M for each segment.

Steps for Area Method

• Horizontal line set where all values are zero plotted on the shear diagram.
• Upward reactions/concentrated loads define a rise; downward define a fall of their own value.
• Shear force at a section equals the slope of the bending moment diagram at that section.
• Upward distributed loads define a rise with a slope
• Downward distributed loads define a fall with a slope
• Area Above Shear = Moment Goes Up
• Area Below Shear = Moment Goes Down
• The value at a point increases/decreases by the area of the load.
• Follow Ellipse Rule for second-degree curves and above
• The Shear One diagram is one degree higher than the Load diagram
• The Moment diagram is always one degree higher than the Shear diagram

Bending Stress

• Bending Stress formula is σ = My/I, where M is the bending moment at the section considered, y is the distance from the neutral axis to the fiber, and I is the moment of inertia of the cross-section about the neutral axis.
• Flexure Formula: σMAX = Mc/I
• In a moment diagram, areas above zero are positive bending, and areas below are negative bending.
• In Positive Bending, fibers at the top of the Neutral Axis experience compression, while fibers at the bottom experience tension.
• In Negative Bending, fibers at the top of the Neutral Axis experience tension, while fibers at the bottom experience compression.
• Flexure formula: σMAX = Mc/I = M/S where S = I/c is the section modulus.

Shear Stresses

• Shear stress formula is τ = VQ/Ib, where V is the shear force at section being considered, Q is the statical moment of area or 1st moment of area, I is the moment of inertia of the cross section about the neutral axis, and b is the width of the fiber where shearing stress is desired.

Design of Fasteners in Built-Up Beams - Spacing of Rivets

• Spacing of rivets is s = RI/VQ, where R is the force capacity of one connector, I is the moment of inertia of the cross-section about the neutral axis, V is the maximum shear force, and Q is the statical moment of area of the attached flange.