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16.20

Structural Mechanics

5-0-7 units · Undergraduate

Applies solid mechanics to analysis of high-technology structures. Structural design considerations. Review of three-dimensional elasticity theory; stress, strain, anisotropic materials, and heating effects. Two-dimensional plane stress and plane strain problems. Torsion theory for arbitrary sections. Bending of unsymmetrical section and mixed material beams. Bending, shear, and torsion of thin-wall shell beams. Buckling of columns and stability phenomena. Introduction to structural dynamics. Exercises in the design of general and aerospace structures.

Fall 2002

Topics

  • **Unit 1: Introduction and Design Overview **Why Structural Mechanics? Types of Structures; Structural Design Process; Factors in Cost.
  • Unit 2: Loads and Design Considerations Sources of Loads/Deflections; Types of Loads and Environments; Limit and Ultimate Loads; Factors and Margins of Safety; Example, the v-n Diagram; Definition of Failure; FAR’s.
  • Unit 3: Language of Stress/Strain Analysis (Review) Definition of Stress and Strain; Notation; Tensor Rules; Tensor vs. Engineering Notation; Contracted Notation; Matrix Notation.
  • Unit 4: Equations of Elasticity (Review) Equations of Elasticity (Equilibrium, Strain-Displacement, Stress-Strain); Static Determinance; Compatibility; Elasticity Tensor; Material Types and Elastic Components; Materials Axes vs. “Loading Axes”; Compliance and its Tensor; The Formal Strain Tensor; Large Strains vs. Small Strains; Linear vs. Nonlinear Srain.
  • Unit 5: Engineering Constants Engineering Constants (Longitudinal Moduli, Poisson’s Ratio, Shear Moduli, Coefficients of Mutual Influence, Chentsov Coefficients); Reciprocity Relations; Engineering Stress-strain Equations; Compliances and Engineering Constants; Purposes of Testing; Issues of Scale; Testing for Engineering Constants; Variability and Issues in Design.
  • Unit 6: Plane Stress and Plane Strain Plane Stress; Plane Strain; Applications; Approximations and Modeling Limitations.
  • Unit 7: Transformations and Other Coordinate Systems Review of Transformations: Direction Cosines; 3-D tensor form (Axis, Displacement, Stress, Strain, Elasticity Tensor); Plane Stress Case (and Mohr’s Circle); Principal Stresses/ Strains; Invariants; Extreme Shear Stresses/Strains; Reduction to 2-D; Other Coordinate Systems (Example: Cylindrical); General Curvilinear Coordinates.
  • Unit 8: Solution Procedures Exact Solution Procedures; Airy Stress Function; Biharmonic Equation; Inverse Method; Semi-Inverse Method; St. Venant’s Principle; Examples: Uniaxiallyloaded Plate, Polar Form and Stress Around a Hole; Stress Concentrations; Considerations for Orthotropic Materials.
  • Unit 9: Effects of the Environment Where Thermal Strains/“Stresses” come from; Coefficients of Thermal Expansion; Sources of Heating; Spatial Variation of Temperature; Self-equilibrating Stresses; Convection, Radiation, Conductivity (Fourier’s Equation); Solution Techniques; “Internal” Stresses; Degradation of Material Properties; Other Environmental Effects; Examples: Moisture; Piezoelectricity.
  • No Lecture
  • Unit 10: St. Venant Torsion Theory “Types” of Cross-Sections; St. Venant’s Torsion Theory; Assumptions; Considerations for Orthotropic Materials; Torsion Stress Function; Boundary Conditions; Summary of Procedure; Solution; Poisson’s Equation; Example:Circular Rod; Resultant Shear Stress; Other Cross-Sections; Warping.
  • Unit 11: Membrane Analogy Membrane Analogy; Uses; Application: Narrow Rectangular Cross-Section; Other Shapes.
  • Unit 12: Torsion of (Thin) Closed Sections Thick-walled Closed Section; Special Case – Circular Tube; Shear Flow; Bredt’s Formula; Torsion Summary.
  • Unit 13: Review of Simple Beam Theory Generic types of Loading (review); Review of Simple Beam Theory; Considerations for Orthotropic Materials.
  • Unit 14: Behavior of General Beams and Engineering Beam Theory Geometry Definitions; Assumptions; Stress Resultants; Deformation, Strain, Stress In General Shell Beams; Considerations for Orthotropic Beams; Modulus-Weighted Section Properties; “Thermal” Forces and Moments; Selective Reinforcement; Principal Axes of Cross-Section; Beams with Unsymmetric Cross-Sections; Applicability of Engineering Beam Theory; Transverse Shear Effects; Shear Center; Contribution of “Shearing” Deflection; Limitations of Engineering Beam Theory.
  • Unit 15: Behavior (Bending, Shearing, Torsion) of Shell Beams General loading of a Shell Beam; Semi-monocoque Construction; Skin/stringer Construction; Single Cell “Box Beam”; Bending Stresses; Shear Stresses; Joint Equilibrium; Pure Shear and Pure Torsion Scheme; General Solution Procedure; “No Twist” Condition; Shear Center; Torque Boundary Condition; Deflections; St. Venant Assumption; Section Properties: Bending, Shear, and Torsional Stiffness; Multicell Shell Beams; “Equal Twist” Condition; Open Section Beams; Thick Skin Shells; Effective Width.
  • Unit 16: (Review of) Bifucation Buckling Types of Buckling; Governing Equations for Bifucation Buckling; Application of Boundary Conditions; Euler Buckling Load; Coefficient of Edge Fixity; Geometrical Parameters; Considerations for Orthotropic/Composite Beams; Initial Imperfections; Primary and Secondary Moments.
  • Unit 17: The Beam-Column Beam-column Definition; Equilibrium Equations; Governing Equations; Solution for Axial Force; Buckling of Beam-Column; Primary and Secondary Moments.
  • Unit 18: Other Issues in Buckling/Structural Instability Other Issues in Buckling; Squashing; Progressive Yielding; Nonuniform Beams; Plate Buckling; Cylinders; Reinforced Plates; Postbuckling; Curvature Expression for large Deflections; Galerkin Method; Buckling and Failure.
  • Unit 19: General Dynamic Considerations (Review) System Response: The Regimes and Controlling Factors; Spring-mass System, Inertial Loads, Governing Equation; Initial Conditions; Damping; Multi-mass System, Matrix Equation Form; (Sources of) Dynamic Structural Loads; Consequences of Dynamic Structural Response.
  • Unit 20: Solutions for Single Spring-Mass System (Review) Single Degree-of-Freedom System; Free Vibration and Natural Frequency; Forced Vibration; Step Function; Unit Impulse, Dirac Delta Function; Arbitrary Force, Duhamel’s convolution) Integral; Sinusoidal Force; Dynamic Magnification Factor; Resonance.
  • Unit 21: Influence Coefficients Generalized Forces and Displacements; Flexibility Influence Coefficients; Maxwell’s Theorem of Reciprocity; Examples: Cantilevered Beam; Stiffness Influence Coefficients; Physical Interpretations.
  • Unit 22: Vibration of Multi Degree-of-Freedom Systems Governing Matrix Equation; Free Vibration; Eigenvalues and Eigenvectors–Natural Frequencies and Modes; Examples: Representation of Beam as Discrete Mass System; Physical Interpretation of Modes; Orthogonality Relations; Normal Equations of Motion; Superposition of Modal Responses; Forced Vibration.
  • Unit 23: Vibrations of Continuous Systems Generalized Beam-Column Equation with Inertia; Free Vibration; Separation of Spatial and Temporal Solutions; Example: Simply-Supported Beam; Natural Frequencies and Modes; Orthogonality Relations; Normal Equations of Motion; Forced Vibration; Superposition of Modal Responses; Resonance.

Readings

  • Rivello. Theory and Analysis of Flight Structures . McGraw-Hill, 1969.
    Listed reading
  • Timoshenko, and Goodier. Theory of Elasticity . McGraw-Hill, 1970.
    Listed reading
  • Megson. Aircraft Structures for Engineering Students . Halsted Press, 1990.
    Listed reading
  • Bisplinghoff, Mar, and Pian. Statics of Deformable Solids . Addison-Wesley, 1965. (also: Dover, 1990)
    Listed reading
  • Haisler. An Introduction to Aerospace Structural Analysis . Wiley, 1985.
    Listed reading
  • Perry. Aircraft Structures . McGraw-Hill, 1950.
    Listed reading
  • Jones. Mechanics of Composite Materials . McGraw-Hill. (also: Hemisphere, 1988)
    Listed reading
  • Cutler. Understanding Aircraft Structures . Granada, 1981.
    Listed reading
  • Timoshenko, and Gere. Theory of Elastic Stability . McGraw-Hill, 1961.
    Listed reading
  • Gere, and Timoshenko. Mechanics of Materials . 4th ed. PWS, 1997.
    Listed reading
  • Meirovitch. Elements of Vibration Analysis . 2nd ed. McGraw-Hill, 1986.
    Listed reading

Prerequisites & course information

Prereq: 16.001 U (Spring) 5-0-7 units

Sources