Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros

General info M, W, F 8:00-8:50 A.M. at Room G-83 ESB Office: Room G-19 ESB Tel: ext.2310 Course notes: USER NAME: cairns PASSWORD: materials Facebook : Konstantinos Sierros (using courses: Mechanics of Materials) Office hours: M, W 9:00-10:30 A.M. or by appointment

Course textbook Mechanics of Materials, 6 th edition, James M. Gere, Thomson, Brooks/Cole, 2006

1.1: Introduction to Mechanics of Materials Definition: Mechanics of materials is a branch of applied mechanics that deals with the behaviour of solid bodies subjected to various types of loading Compression Tension (stretched) Bending Torsion (twisted) Shearing

1.1: Introduction to Mechanics of Materials Fundamental concepts stress and strain deformation and displacement elasticity and inelasticity load-carrying capacity Design and analysis of mechanical and structural systems

1.2: Normal stress and strain Most fundamental concepts in Mechanics of Materials are stress and strain Prismatic bar: Straight structural member with the same cross- section throughout its length Axial force: Load directed along the axis of the member Axial force can be tensile or compressive Type of loading for landing gear strut and for tow bar?

Examples A truss bridge is a type of beam bridge with a skeletal structure. The forces of tension, or pulling, are represented by red lines and the forces of compression, or squeezing, are represented by green lines. The Howe Truss was originally designed to combine diagonal timber compression members and vertical iron rod tension members

Normal stress Continuously distributed stresses acting over the entire cross-section. Axial force P is the resultant of those stresses Stress (σ) has units of force per unit area If stresses acting on cross-section are uniformly distributed then: Units of stress in USCS: pounds per square inch (psi) or kilopounds per square inch (ksi) SI units: newtons per square meter (N/m 2 ) which is equal to Pa

Limitations The loads P are transmitted to the bar by pins that pass through the holes High localized stresses are produced around the holes !! Stress concentrations

Normal strain A prismatic bar will change in length when under a uniaxial tensile force…and obviously it will become longer… Definition of elongation per unit length or strain (ε) If bar is in tension, strain is tensile and if in compression the strain is compressive Strain is a dimensionless quantity (i.e. no units!!)

Line of action of the axial forces for a uniform stress distribution It can be demonstrated that in order to have uniform tension or compression in a prismatic bar, the axial force must act through the centroid of the cross-sectional area. *In geometry, the centroid or barycenter of an object X in n-dimensional space is the intersection of all hyperplanes that divide X into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of X. *The geometric centroid of a physical object coincides with its center of mass if the object has uniform density, or if the object's shape and density have a symmetry which fully determines the centroid. These conditions are sufficient but not necessary.

Problem A circular aluminum tube of length L = 400 mm is loaded in compression by forces P (see figure). The outside and inside diameters are 60 mm and 50 mm, respectively. A strain gage is placed on the outside of the bar to measure normal strains in the longitudinal direction. (a) If the measured strain is 550 x 10 -6, what is the shortening of the bar? (b) If the compressive stress in the bar is intended to be 40 MPa, what should be the load P?

Problem Two steel wires, AB and BC, support a lamp weighing 18 lb (see figure). Wire AB is at an angle α = 34° to the horizontal and wire BC is at an angle β = 48°. Both wires have diameter 30 mils. (Wire diameters are often expressed in mils; one mil equals in.) Determine the tensile stresses AB and BC in the two wires.

Problem A reinforced concrete slab 8.0 ft square and 9.0 in. thick is lifted by four cables attached to the corners, as shown in the figure. The cables are attached to a hook at a point 5.0 ft above the top of the slab. Each cable has an effective cross-sectional area A 0.12 in2. Determine the tensile stress σ t in the cables due to the weight of the concrete slab. (See Table H-1, Appendix H, for the weight density of reinforced concrete.)