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Notebook containing solutions to problems is available in CSULA Library. Call No. is 3042.

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Concepts and Key Words

Lecture 1

Introduction

Review of some basic concepts.
Stress (at a point in a body). Stress is force per unit area, that is, intensity of distributed forces acting over a surface, real or imagined. A small rectangular parallelepiped (element) isolated from a loaded body. The rectangular stress components that act on this element: normal stresses and shear stresses. Subscript convention and direction convention for stress components. The stress matrix. Symmetry of the shear stresses. Six rectangular stress components define state of stress at a point in a body. Plane stress.
Strain. Element under uniaxial stress. Corresponding normal strains. Linearly elastic material: Hooke's law; Poisson's ratio. Isotropic material. Strain-stress relations for uniaxial stress, biaxial stress, triaxial stress. Inverted equations.

Lecture 2

Shear strains caused by shear stresses in a Hookean material. Shear modulus (modulus of rigidity). Relation among Young's modulus, shear modulus, and Poisson's ratio.
Strain-displacement relations. Infinitesimal deformation theory. Deformation can be decomposed into "pure deformation" characterized by strains and rigid body rotation characterized by angles of rotation.
Axial loading. Strains. Displacements. Application: bar in tension. Axial stiffness (spring constant, spring rate).
Bending of a beam with symmetrical cross-section. Shear force. Bending moment. How they act on cross-section. Sign conventions.

Lecture 3

Shear force and bending moment diagrams. Shear force-concentrated load relation. Shear force-distributed load relation. Bending moment-shear force relation.
Bending (normal) stresses in Hookean beam with symmetrical cross-section. Pure bending. Flexure formula (usually still good approximation when shear forces present).
Shear stresses in Hookean beam with narrow rectangular cross-section.
Normal and shear strains in a Hookean beam. Deflection due to bending of a Hookean beam.

Introduction to the finite-element method. One-dimensional spring element: two node and one (spring) element axial system. Displacements of nodes. External forces applied at nodes. Isolated free body diagrams of nodes and (spring) element. Internal forces that act on element. Equilibrium equations for nodes and element. Internal force-node displacement relations for element. Element stiffness matrix, relates internal forces and displacements. System stiffness matrix, relates external forces and displacements. Case when a displacement is "applied" at one node, say 1, and an external force is applied at the other node, say 2. Determination of external force at node 1 (called reaction) and displacement at node 2, with the help of an added, very stiff ground spring at node 1.

Lecture 4

Two-dimensional spring element. Example: a two-dimensional system of three noncollinear nodes connected by three springs. Application: plane truss problem.

Area moments of inertia. Product of inertia. Mohr's circle representation of inertia properties with respect to rotated axes. Principal axes. Principal moments of inertia.

Lecture 5

Pure bending of unsymmetrical Hookean beams. Out-of-plane bending moment is needed, in addition to in-plane bending moment, to produce bending in a plane not coincident with a principal centroidal axis of cross-section.
Formula for bending stress stated in terms of principal centroidal axes of cross-section. Location of neutral axis.

Transverse shear stresses revisited: the case of a beam with a circular cross section. Total shear stress over cross section at location of lateral stress-free surface acts tangent to lateral stress-free surface.
Shear flow in Hookean beams with thin, open sections. Case of cantilevered beam with end load acting through "shear center" and parallel to principal axis of cross section. Under these conditions, beam does not twist. Shear stress is proportional to first moment with respect to neutral axis of cross section between free edge and exploratory cut where shear stress computed. Product of shear stress and local thickness called shear flow.

Lecture 6

Application: symmetrical wide-flange beam. Shear stresses in web and flange. "Flow" interpretation of shear stresses.
Shear center of thin-walled open-section Hookean beams. Example: cantilevered channel beam with end load that acts through shear center; therefore, no twist. Shear center is point on line of action of load about which the moment sum of the shear stresses over cross-section vanishes.

Lecture 7

Thick-walled Hookean cylinders subject to internal, external and end-face pressures. Implications of cylindrical symmetry on cylindrical stress components. Stress equilibrium equation in radial direction (two unknown stresses).
Implications of cylindrical symmetry on cylindrical displacement components. Strain-stress and strain-displacement relations in cylindrical coordinates. Development of a second equation involving stresses. Derivation of Cauchy's equation governing radial stress. Radial stress solution in terms of internal and external pressures. Tangential stress solution. Axial stress if cylinder ends are capped.
Stresses produced by shrink (press) fits.

Energy methods for finding deflections. Elastic strain energy. Truss example.

Lecture 8

External work. Internal work. Virtual work. Strain energy per unit volume, for Hookean material: in uniaxial stress state, in pure shear.
Strain energies for other stress states, for Hookean materials: triaxial stress, general stress state, plane stress.

Lecture 9

Strain energy in Hookean bars. Axial loading. Torsional loading. Pure bending. Transverse loading.
Strain energy and complementary energy of non-Hookean elastic materials.
Castigliano's first theorem: partial derivative of strain energy with respect to displacement equals corresponding load.

Lecture 10

Application: Two horizontal bars pinned end-to-end and pin supported at their extreme ends with a transverse load at the middle connecting pin. Bars have linear stress-strain properties. Find deflection of connecting pin. The member strain-joint displacement relations for the bars are nonlinear. Answer: transverse load varies nonlinearly with deflection of pin.
The complementary-energy theorem (Crotti-Engesser theorem): partial derivative of complementary energy with respect to load equals corresponding displacement. Application: plane truss with nonlinear stress-strain members, but with linear member strain-joint displacement relations. Find joint displacements due to joint loads by Castigliano's first theorem and the Crotti-Engesser theorem. Answer: joint deflections are nonlinear functions of applied joint loads. Castigliano's second theorem (for linear elastic structures): partial derivative of strain energy with respect to load equals corresponding displacement. Quadratic forms of strain energy in terms of displacements and loads. Example: cantilever beam with end load and couple. Find corresponding deflections.

Lecture 11

The finite-element method (continued). Element stiffness matrix for a bar - the one dimensional truss element. Assume linear displacement field. Shape (interpolation) function. Strain-nodal displacement equation. Strain energy in element. Application of Castigliano's first theorem. Element stiffness matrix.
Element stiffness matrix for a triangular plane-stress element. Assume linear displacement field. Shape functions. Strain-nodal displacement matrix equation (strain-displacement matrix). Stress-strain matrix equation (material stiffness matrix). Strain energy in element, expressed as quadratic form in nodal displacements; symmetric stiffness matrix. Application of Castigliano's first theorem. Element stiffness matrix.
Example: Rectangle consisting of two triangle elements. System (master, global, structure) stiffness matrix.

Lecture 12

Conversion of distributed loads to statically equivalent nodal forces.

Theories of failure

Stress-strain diagram for mild steel. Proportional limit, elastic limit, yield point, ultimate strength, rupture strength. Yield strength (defined by offset method). Ductile behavior. Brittle behavior. Need for failure criteria applied to stresses caused by combined loading. For brittle materials: maximum normal stress theory. For ductile materials: maximum shear stress theory (Tresca theory), and the maximum distortion energy theory (von Mises theory). Principal stresses. Principal axes. Mohr's circle for stress applied to material element aligned with principal axes. Maximum shear stress in terms of principal stresses. Maximum shear stress theory stated in terms of principal stresses. Maximum distortion energy theory stated in terms of principal stresses; von Mises stress.

Lecture 13

Limits placed on biaxial stresses based on maximum shear stress theory. Limits placed on biaxial stresses based on maximum distortion energy theory. Limits placed on biaxial stresses based on Coulomb-Mohr theory of failure for brittle materials. Failure envelope defined in Mohr's normal stress-shear stress plane.
Factors of safety. Reasons for using. Margin of safety.

Introduction to the Theory of Elasticity

The stress vector that acts on the oblique face of an isolated small right-tetrahedron. Relations between stress-vector scalar components and rectangular stress components.

Lecture 14

Transformation (rotation) of coordinate axes. Transformation of rectangular stress components.
Tensor transformation law. Orthogonal tensor. Symmetric tensor. The eigenvalue and eigenvector problem associated with the stress matrix. Characteristic equation. Stress matrix has three real eigenvalues and three orthonormal eigenvectors.

Lecture 15

If the faces of a cubical element are normal to the orthonormal eigenvectors of the stress matrix then only normal stresses act on these faces. The normal stresses are called principal stresses.
Stress invariants: first, second, third.
Total shear stress and direction on oblique surface of right-tetrahedron. Tensor transformation of strain matrix: "mathematical strain". "Engineering" strain. Equations of equilibrium. Generalized Hooke's law. Elastically homogeneous and isotropic material. Generalization of Hooke's law involves 36 constants in a 6 x 6 matrix [C]. Symmetry of [C] can be deduced from application of Castigliano's first theorem to strain energy per unit volume expressed as quadratic form in strains. Symmetry of [C] leaves 21 independent constants. Consequences of material symmetries on constants in [C] matrix.

Lecture 16

(a) Elastic symmetry with respect to one plane. Monoclinic material: 13 independent constants in [C] matrix. (b) Elastic symmetry with respect to two (thereby three) planes. Orthotropic material: Nine independent constants. (c) [C] matrix invariant for all rotated coordinate axes. Isotropic material. Two independent constants. Lamé constants. Compatibility equations (for strains). Sufficient to ensure the existence of a single-valued, continuous displacement field, if body is simply-connected.

Plane Elastostatic Problems

Prismatical cylinder subject to axially uniform, transverse loads only, and zero axial displacement. Body is in state of plane strain. Stress-strain equations. Equilibrium equations.

Lecture 17

Assume stresses are given by taking certain partial derivatives of a scalar function of the lateral coordinates. Stresses satisfy equilibrium equations. Substitute stresses in the one governing stress compatibility equation to obtain biharmonic equation in the scalar function, now identified as Airy stress function.
Another case, thin body in nearly plane stress state. Assume stresses and strains are nearly two-dimemsional. Assume again stresses given by Airy stress function. Substitute stresses in stress compatibility equation in lateral coordinates to obtain biharmonic equation in the Airy stress function. Other compatibility equations are not necessarily satisfied. Conclusion: plane strain and plane stress problems are essentially the same; can convert one solution into the other. Plane stress example: cantilever beam with uniform load. Boundary conditons. Trial polynomial solution to biharmonic equation; certain terms can be eliminated from consideration of boundary conditions. The determination of the unknown constant coefficients in polynomial from substitution into biharmonic equation and boundary conditions. Some stress boundary conditions cannot be satisfied pointwise but only as stress resultants. Comparison of calculated stress field with that from strength of materials.
Lecture 18

Torsion of Prismatical Cylinders

Cylinder of uniform cross-section. Assume one end fixed against rotation. Each cross-section rotates through angle (angle of twist) that is proportional to distance from fixed end.

Assume angle of twist is small. Develop lateral components of displacement field in terms of angle of twist and position of particle relative to reference longitudinal axis. Assume longitudinal component of displacement field is function of lateral position coordinates only. This function is proportional to what is called the warping function. Resulting strain and stress fields. Enforcement of equilibrium equations leads to Laplace equation for the warping function. Enforcement of the lateral stress-free boundary conditons together with the Laplace equation forms a Neumann boundary value problem. Application: bar with elliptical cross section. Develop formula for torsional constant. Formula for maximum shear stress. The rectangular cross section case. Formulas for maximum shear stress and torsional constant in terms of ratio of two sides of rectangular cross section. The case of an elongated rectangular cross section. Approximate stress distribution. Extension to torsion of thin-walled open sections. Maximum shear stresses and torsional constants.

The final exam will be open textbook. You will also be allowed to refer to three 8 1/2 x 11 in. sheets of notes during the exam.

THE END

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