Mechanics of Materials Part 1

Explore deformation and strain, including normal strain and shear strain, and learn how forces cause length changes and angle shifts under tensile and compressive loading.

Compute strain in cable AB using AB' minus AB over AB, with AB' found via law of cosines after AB increases from 1000 to 1004.18 mm when CB rotates 0.5°.

Analyze shear strain at corners A and B of a deformed square plate by calculating angle changes with phi and theta, converting to radians yields gamma A and gamma B.

Apply epsilon equals (L' - L)/L to compute normal strains for AB, AC, and BD, using original and deformed lengths; AB and AC shorten, BD lengthens.

Explains stress–strain diagrams from tension and compression tests, defining engineering stress and strain, Hooke's law and Young's modulus, and elastic to plastic transitions including yielding and necking.

Compare ductile and brittle materials; define strain energy and energy density, and derive modulus of resilience and modulus of toughness from elastic behavior and the proportional limit.

Apply Hooke's law to determine AB's stretch in an elastic A36 steel cable. Use E = 29x10^3 ksi to compute epsilon, then multiply by the original length.

Compute the modulus of elasticity from the elastic slope, then determine the yield and ultimate loads from the stress-strain diagram for an aluminum alloy with a 0.5 in diameter.

From the stress–strain diagram, calculate elastic recovery and permanent set for an original 2 inch gauge length loaded to 60 ksi, using Hooke’s law and elastic strain.

Poisson's ratio links axial and radial strains in the elastic range, relating longitudinal strain to lateral contraction via nu equals negative ratio of epsilon_lat to epsilon_long, with nu in [0,0.5].

Apply Hooke’s law to relate longitudinal and lateral strains for a 60 kN load on a 12.7 mm aluminum specimen with Poisson’s ratio 0.35, yielding final length and diameter.

Explore the shear stress–strain diagram, define tau and gamma, and apply tau equals G gamma to torsion tests, linking the shear modulus to the modulus of elasticity and nu.

Determine E from G using G = E/(2(1+ν)) for ν = 0.3, and compute P from τ_y = 50 ksi and A = πd^2/4 with d = 0.25 in.

Study axial loads and their deformation effects on bars, covering tension and compression, Saint-Venant's principle, and deriving delta from stress and strain via Hooke's law.

Calculate the end-to-end displacement of a three-section aluminum copper steel shaft and the normal stresses in each section using section areas and the E values for each material.

Apply axial deformation theory to a three-pin steel assembly, compute internal loads and elongations with E and area, then use law of sines to get B's horizontal displacement.

An in-depth calculation of a four-wire stainless steel system carrying a 500 lb load, solving internal forces and displacements via elasticity to determine vertical movement.

Apply the principle of superposition to combine stresses or displacements from multiple loads, and learn to handle statically indeterminate axially loaded members using a compatibility condition and the four-force method.

Determine the maximum elastic load on a steel bolt inside a bronze sleeve by applying yield stresses, cross-sectional areas, and a compatibility condition to relate steel and bronze deflections.

Compute the average normal stress in the copper AB/CD rods and the steel F rod attached to a rigid cap, solving for internal forces via symmetry and a compatibility condition.

Expand with rising temperature, contract with falling, delta L = alpha delta T L. Vary temperature along the length, integrate; confinement yields thermal stresses.

Analyze a992 steel rod heated from 40°F to 160°F, causing thermal expansion that compresses springs; compute total compression and spring force using alpha, delta t, and kx.

Compute the normal stress in a hollow A36 steel pipe confined between rigid supports as temperature varies along its length, using an integral delta t and resulting internal force p.

Explore how cross-section changes create stress concentrations with maximum stress at a hole, and learn to compute the stress concentration factor K as sigma max over sigma average.

Determine the maximum axial force on a bar with a hole and a fillet by using stress concentration graphs and geometry to compare two failure locations.

Explore torsion in circular shafts by deriving angle of twist and shear stresses, applying tau max = t c / j and tau = t rho / j.

Calculate the largest torque and maximum shear stresses in regions CD and DE for a 50 mm aluminum shaft under multiple torsional loads, using tau = Tc/J.

Determine the minimum wall thickness of a hollow tubular shaft under torsion using the allowable shear stress, outer diameter 160 mm, polar moment of inertia, and internal torques across sections.

Explore how power transmits through a tractor shaft, deriving power as torque times angular velocity and linking omega to frequency via omega = 2πf.

Calculate the smallest solid shaft diameter by converting 0.1 hp at 80 rpm to torque and applying the shear stress relation with four ksi.

Explore the angle of twist in shafts with varying cross-sections, deriving phi from the torsion formula under linear elastic behavior and applying the right-hand rule sign convention.

Calculate the angle of twist at point A for a 50 mm a992 steel shaft under multiple torques by summing twists with phi = Tl/(JG) and the right-hand rule.

Compute the shaft diameter (about 25 mm) to satisfy 0.05 rad twist between gears B and D and a 75 MPa shear limit, using p = t ω.

analyze angle of twist for a one‑inch a992 steel shaft under two opposing torques by evaluating BC and CD and summing twists via phi = TL over g.

Explore statically indeterminate shafts with fixed ends under torque, derive equilibrium and compatibility conditions using angle of twist, and equate phi1 and phi2 to solve for unknowns.

This lecture solves a torsion problem on a hollow bronze shaft, using a free body diagram, torque compatibility, and sign conventions to determine internal torques and the maximum shear stress.

Determine external reactions at fixed supports A and B for two A36 steel shafts connected by gears under a 500 N·m torque, using free-body diagrams and a gear-based compatibility condition.

Examine stress concentrations in shafts with cross-section changes (shoulder fillet) under torsion, using the concentration factor k and the r/D and D/d ratios, with tau max at the fillet base.

Analyze a steel step shaft with sudden cross-section change to find the torque under eight megapascals of shear stress using tau max equals k times t c over j.

Calculate the power the built up shaft can transmit by converting 450 rpm to 47.12 rad/s and using the allowable shear stress with a stress concentration factor to find torque.

Explore bending stress in slender beams with linear elastic material and master shear and moment diagrams using the equation method, covering simply supported, cantilever, and overhanging beams and sign conventions.

Derive and plot shear and moment diagrams for a cantilever beam with distributed and point loads plus an end moment, and solve reactions and v(x), m(x) in two beam segments.

Analyze a three-section overhanging beam in mechanics of materials, deriving v and m equations, building shear and moment diagrams, and identifying maximum shear and moment.

This lecture analyzes a beam with triangular distributed load using left-section analysis and similar triangles to locate centroid, deriving shear and moment and showing moment 3000 ft and shear 900.

Apply a graphical method to build shear and moment diagrams from distributed and concentrated loads. Use dv/dx = w(x) and dm/dx = v to find changes and jumps.

Use the graphical method to construct shear and moment diagrams for a beam with roller and pin supports and a 30 kip-ft moment, plus a 5 ft distributed load.

This lecture demonstrates solving a shear and moment diagram using the graphical method. It covers computing external reactions and constructing the diagrams to locate maximum values.

Develop understanding of solving a beam with upward and downward distributed loads on a fixed support by calculating reactions, building V and M diagrams, and interpreting sign conventions.

Compute the free body diagram and reactions at A and B for a beam with a 150 lb/ft distributed load over 6 ft, then draw the shear and moment diagrams.

Work through an overhanging beam with triangular and uniform distributed loads to determine support reactions and construct shear and moment diagrams using section analysis.

explain bending deformation in symmetric cross-section beams, identify the neutral surface and neutral axis, and derive the normal strain relation in terms of distance from the neutral axis.

Relate bending stress to moment with flexure formulas; sigma max = M c / I and sigma = minus M y / I.

Determine max bending stress by locating the neutral axis and centroid, computing the moment of inertia, and applying sigma = M c / I, plus a three-dimensional stress sketch.

Determine the minimum cross-section dimension D for a beam by finding moment from shear-moment diagrams, locating the centroid and neutral axis, and enforcing sigma max = M c / I.

Explore unsymmetric bending by decomposing a moment into My and Mz components along principal axes, applying flexure formulas to obtain sigma, and locating the neutral axis via a tangent relation.

Determine the maximum bending stress and neutral axis orientation by locating the centroid, computing I_y and I_z (with parallel axis corrections), and evaluating edge stresses on the beam.

Using the unsymmetric bending equation with moments from a 30-degree axis, this example finds the maximum bending stress and the neutral axis at 66.6 degrees from negative z.