Mechanics of Materials: Fundamentals

Examine creep as time-dependent elongation under a constant load. Describe relaxation as stress decay in a wire fixed between immovable supports.

Explore linear elasticity and Hooke's Law by examining stress–strain behavior, distinguishing elastic and linear responses, and applying Young's modulus as the parameter linking stress and strain.

Define Poisson's ratio as the ratio of lateral contraction to axial elongation under uniaxial tension in linear elastic isotropic materials, with typical values 0.25–0.35 and a theoretical limit of 0.5.

Apply linear elasticity to a steel pipe under axial compression, calculating axial shortening, lateral strain via Poisson's ratio, and diameter and thickness changes with Hooke's law.

Explore how a 2 m structural steel bar loaded axially to 6.5 mm elongation, with yield stress 250 MPa, shows elastic recovery and a 4 mm residual strain.

Analyze a circular magnesium alloy bar under tension using the stress-strain diagram to determine permanent set and the proportional limit, yielding 2.51 mm and 170 MPa.

Relate the bar's lateral strain from a 0.016 mm diameter reduction to the longitudinal strain via Poisson's ratio, then use Hooke's law to compute the axial load P.

Compute the modulus of elasticity for brass from a 20 kN tensile test on a 10 mm diameter specimen using axial stress and strain; derive Poisson's ratio from diameter contraction.

Determine the tensions in two wires supporting a lamp by applying equilibrium at point B with angles 34° and 48°, then compute normal stresses from force over area.

The lecture analyzes a long retaining wall braced by shores under triangular earth pressure, solving for shore normal stress and compressive stress via moment equilibrium.

Apply Hooke's law to compute axial stress and longitudinal strain in a 2.5 m steel bar with a square cross-section. Use Poisson's ratio to obtain lateral strain and volume change.

Explain how shear stress acts tangential to surfaces, define bearing stress at bolt contacts, and compare double versus single shear in bolted connections.

Explains the equality of shear stresses on perpendicular planes and how opposite shear on opposite faces maintains equilibrium, along with sign conventions for shear stress and shear strain.

Explore Hooke's law in shear, the shear stress–strain diagram, and how the shear modulus G relates to Young's modulus E through Poisson's ratio, with the linear initial portion.

Compute the average shear stress in the plate and the compressive stress in the punch for punching a 20 mm hole through an 8 mm steel plate at 110 kN.

Explore bearing and shear stresses in a pin connection among pins, gussets, base plate, and anchor bolts for a steel strut under a 12 k force at 40 degrees.

Explore shear stress and shear strains in elastomeric bearing pads under bridge girders, deriving average shear stress and horizontal displacement formulas.

Defines how the factor of safety sets allowable stress and allowable loads from yield or ultimate strength, and applies them to tension, shear, and bearing with net area considerations.

Learn to determine hanger allowable loads by evaluating four stress cases - main part tension, bolt-hole tension, bearing, and bolt shear - identifying bolt shear as the limiting factor.

Design for axial loads and direct shear using a two-bar truss. Compute reactions, axial force in AB, and pin diameter to meet tension and shear allowable stresses.

Determine the minimum outer diameter of a steel pipe under axial compression by applying a safety factor of 1.8 to the yield stress and using t = D/8.

Determine bolt diameter for a double shear connection under 31 kN axial load by comparing bearing and shear limits (bearing 150 MPa, shear 90 MPa) with a 15 mm thickness.

Explore how axially loaded members experience elongation or contraction, define stiffness and flexibility, and apply Hooke's law to springs and prismatic bars using K = P/Δ and F = Δ/P.

Explore thermal effects on materials, including expansion, misfits, and pre strains, and learn how the coefficient of thermal expansion alpha relates temperature change to thermal displacement.

Analyze thermal displacement and strains in a prismatic bar fixed at both ends under a uniform temperature change, deriving thermal stress via compatibility and equilibrium in a statically indeterminate system.

Explore thermal effects on welded railroad rails forming a continuous track, and calculate the compressive stress with sigma equals E alpha delta T for a 60–120 Fahrenheit rise.

Determine delta t for a steel rod and bolt in a double-shear setup to reach 45 MPa average bolt shear stress, using rod and bolt diameters, alpha, and E.

Design for axial loads: a hollow circular steel column on a base plate transfers 750 kN to a concrete pedestal, with t 20 mm and D 297 mm.

Analyze a two-section plastic bar fixed at both ends under a 30 degrees temperature rise to determine the compressive force, stress in the smaller-diameter AC, and displacement at point C.

Explore strain energy as the energy stored in a bar under loading, equal to the external work done, and shown by the area under the load-displacement curve.

Differentiate elastic and inelastic strain energy by analyzing loading and unloading paths, elastic limit, permanent set, and the recovered vs permanent portions of work under the load-displacement curve.

Examine linearly elastic behavior under Hooke's law, where load-displacement is linear and strain energy equals the triangular area U = P delta/2, with U = P^2 L/(2E).

strain energy is not a linear function of loads, so total energy cannot be found by summing separate loads; it is always positive and applies to both tension and compression.

Explore non uniform bars with prismatic segments under different axial forces, showing total strain energy equals the sum of each segment's energy; for linear elastic segments, use P^2 L/(2 E A).

Analyze three fixed-end bars with nonuniform cross sections to compare strain energy under the same axial load. Show how increasing cross-sectional area reduces energy, with U1, U2 equal to two-fifths of U1, and U3 equal to three-tenths of U1.

Apply strain energy to determine the vertical displacement from a single slowly applied load in a linearly elastic system, illustrated with a two-bar truss.

Apply strain energy and equilibrium to determine the vertical displacement at joint B of a two-member truss under a single vertical load P, by energy balance.

Derive the strain energy for a two-segment brass bar with diameters D and 2D under load P, and sum segment energies to get the total energy.

Analyze the strain energy of a prismatic bar hanging from the upper end under its own weight and an end load, revealing a mixed term that prevents simple energy addition.

Examine strain energy in a bar with constant axial rigidity E under single and simultaneous loads P at tip and Q at midpoint, highlighting dependence on position and interaction.

Understand stresses in beams by distinguishing pure bending from non uniform bending, linking constant bending moments to flexure and zero shear in regions of pure bending.

Relate beam curvature to its deflection and radius of curvature. Use kappa = 1/rho and d theta/ds with sign conventions to link curvature to stresses and strains.

Explore longitudinal strains in beams from bending, relating epsilon_x to curvature and distance y from neutral axis. In pure bending, epsilon_x = - y / rho, varying linearly with y.

Compute the radius of curvature and deflection for a simply supported beam under pure bending from bottom-surface strain, with rho about 200 ft and delta about 0.48 in.

Calculate the radius of curvature and end deflection of a cantilever under pure bending using the top-surface strain and the distance to the neutral surface.

Learn how normal stresses in beams arise from pure bending, and how the resultant stresses on the cross section reveal the neutral axis and the moment-curvature relationship.

Locate the neutral axis by applying the first moment of area about the centroid. Demonstrate that the axis passes through the cross-section centroid under pure bending with no axial load.

Explore the moment-curvature relationship, linking bending moment to beam curvature via EI, deriving the normal stress sigma_x equals minus E kappa y under linear elasticity.

Derives the flexural formula sigma_x = -M y / I. Explains how bending moment, inertia I, and distance y create a linear stress distribution with sign conventions.

Determine maximum bending stresses at a cross section by locating points farthest from the neutral axis; use section moduli s1 = I/C1 and s2 = I/C2 to summarize the stresses.

Explain doubly symmetric cross sections with C1 = C2 and sigma max = M/S; memorize I and S for rectangular and circular sections.

Compute the bending moment and maximum bending stress for a circular wire bent around a drum using the radius of curvature and the flexural formula. Compare with the proportional limit.

Calculate the maximum bending stress in a copper strip under pure bending using Young's modulus and the radius of curvature from its length. Show that sigma max increases with thickness.

Determine the maximum bending moment for a cantilever highway bridge girder under 11 kN/m load, then compute the maximum bending stress using sigma max = M c / I.