Structural Mechanics: Key Concepts and Definitions
Isotropic: An isotropic material behaves the same in all directions.
Anisotropic: Direction-dependent materials are anisotropic – not isotropic.
Bearing Stress: Internal stress caused by compressive forces (contact pressure between separate bodies).
Center of Mass: The point at which there is equal mass in all directions.
Centroid: The point at which there is equal volume on all sides, first moment area (ydA).
Characteristics of a Force: Location/point of application (dimensions), magnitude (units), and direction (arrowhead).
Free Body Diagram: A graphical illustration of all forces on a body, must include the three characteristics of each force.
Translation: Movement in one direction.
Degree of Freedom: The number of variables that can be changed arbitrarily.
Degrees of Freedom in 1-D: Two degrees of freedom: x-translation and y-translation.
Degrees of Freedom in 2-D: Three degrees of freedom: x-translation, y-translation, and rotation about the z-axis.
Degrees of Freedom in 3-D: Six degrees of freedom: x-translation, y-translation, z-translation, rotation about the x-axis, rotation about the y-axis, rotation about the z-axis.
Determinant System: If the number of unknown variables is less than or equal to the number of equations in the system, the system is said to be determinant.
Ductility: Ductility refers to the long, flat segment of strain, where energy is absorbed. A more ductile material will absorb more energy, while a more brittle material will absorb less energy before the rupture point. Ductility = plastic length / elastic length.
Dynamic: Changing, not constant, dependent on time.
Static: Not changing.
Engineering Strain/Strain: = change in length / original length, normalized deflection not considering time.
Natural/True Strain: = ln(1 + (change in length / original length)), must be used for cable structures due to large deflections.
Force Couple: Two equal and opposite forces separated by a distance.
Newton’s Equation of Motion: F = ma + cv + kx
Elasticity/Elastic Behavior: Elastic behavior is repeatable.
Plasticity: Non-repeatable behavior, permanent deformation.
Homogeneous: One material. If an object is homogeneous, the center of gravity and the centroid coincide.
Hooke’s Law: stress = E * strain, constitutive law relating stress and strain.
Indeterminate System: A system where the number of unknown variables is greater than the number of equations is indeterminate.
Internal Hinge/Pin: A point of zero moment inside a structural member.
Moment: A force times a distance; moment creates rotation.
Moment of Inertia: A member’s moment of inertia expresses its tendency to resist rotation/angular acceleration or curvature, how much of the member’s mass is located towards the outside of the member (more mass towards the outside will mean a larger moment of inertia).
Normal Stress: normal stress = Force / Area, when a force acts normal to a surface, it exerts a normal stress.
Normal Strain: strain = change in length / original length, normal strain occurs when the elongation of an object is in response to a normal stress.
Poisson’s Ratio: = -lateral strain / longitudinal strain, only for linear/elastic, isotropic materials, approximately 0.3 for steel and 0.5 for rubber.
Prismatic: A prismatic member has a constant cross section.
Rigid: Rigid bodies undergo no deformation.
Shear Strain: strain = change in length / original length, strain due to shear stress.
Shear Deformation: deformation = change in length.
Shear Stress/Average Shear Stress: stress = Force / Area, normalized force parallel to surface.
Slenderness Ratio: A member is slender if the ratio of span to depth is less than 5. Slenderness = span/depth = L/r where r = radius of gyration.
Flexural Member in Terms of Span to Depth Ratio: Members in which moment is the governing factor for design. A slenderness ratio greater than five in theory or 10 in practice corresponds to a slender member.
Shear Member in Terms of Span to Depth Ratio: Shear stress governs member design for members with a slenderness ratio of less than three.
Static Equation of Motion: F = ma. Note that time is not a parameter in static analysis; we consider only initial and final states.
Stress: Stress is the measure of what the material feels from externally applied forces, ratio of the external forces to the cross sectional area of the material.
Transmissibility of a Force: The principle of transmissibility applies to rigid bodies in particular. It states that the external effects on a body remain unchanged when a force F1 acting at point A is replaced by a force F2 of equal magnitude at point B provided that both forces have the same sense and line of action.
Two Force Member: A member with only two axial forces and no other loading.
Yield Stress: The stress at which plastic behavior begins.
Biaxial/Oblique Bending: A cross-section is subject to biaxial bending if the normal stresses from bending reduce to two moments Mz and My. Oblique bending is generally accompanied by oblique shearing. If the shear forces Vz = Vy = 0, then we have pure oblique bending.
Elastic Neutral Axis: Centroid.
Small Deflection Theory: Original shape looks like final shape.
Kip: 1000 pounds.
Shape Factor: Mp/My
Bernoulli Assumptions of Beam Bending: Linear strain distribution over the cross section, plane cross sections remain plane (no warping), homogeneous, isentropic, prismatic cross section (in general terms, a cross section that may be described by a differential equation; for now we are analyzing beams with constant cross-section (the simplest differential equation)), small deformations and deflections (small strain, small deflections cannot be seen with the naked eye, large deflections can; for practical purposes, small strain may be defined by 0.01 > strain > 0.0001).
Axial Stress Equation: stress = Force / Area
Axial Deformation Equation: deformation = (Force * Length) / (Area * Elastic Modulus), derived from Hooke’s Law, valid only for small deflections.
Constitutive Law: Relates force to deflection.
Isentropic: Properties are the same in all directions.
Anisotropic: Properties are different in different directions.
Shear and Moment Diagrams Relationship: The integral of the shear curve is equal to the change in moment over that same distance. Graphical integration may be used to find the change in moment by calculating the area under the shear curve.
Small Deformation: Cannot be seen with the naked eye, while large deflections can. For practical purposes, small strain may be defined by 0.01 > strain > 0.0001.
Elastic Shear Stress Distribution Over a Cross-Section in Bending: shear stress = (V * Q) / (I * t) (shear stress due to bending), only valid for slender members.
Elastic Normal Stress Distribution Over a Cross-Section in Bending: Varies linearly with the distance from the neutral surface.
Parallel Axis Theorem (Equation): I + Ad2, used to find moment of inertia about an axis other than a central axis.
Section Modulus: S = I/c
Flexural Deformation: Deflection caused by bending/flexing of the member.
Flexural Stress: Normal stress caused by bending/flexing of the member is referred to as flexural stress.
Free Surface: Shear is zero at free surfaces.
Plastic Modulus: Zx, used instead of Ix for designing in the plastic range, also known as plastic inertia.
Thin Wall Traverse Stress Distribution: Unit shear, the shear stress is assumed to be distributed evenly along the length of the wall.
Hooke’s Law for Shear Stress: shear stress = G * shear strain, where G = shear modulus.
Plastic Neutral Axis: Does not coincide with the centroid of the cross section.
Compact Section: A section is considered compact if the ratios of its cross-sectional dimensions are sufficiently small such that the member may reach yield stress and its fully plastic moment without experiencing any local buckling.
Mohr’s Circle: Geometric, graphical representation of transformation of stresses in two dimensions, used to determine normal and shear stress components for different orientations of the stress element.