FE Review Mechanics of Materials

FE Review Mechanics of Materials

F Stress V M N N = internal normal force (or P) V = internal shear force M = internal moment Double Shear Normal Stress = N = = A Average Shear Stress = τ = P A V A F/ F/ V = F/ V = F/ τ = F A F F

Strain Normal Strain FE Mechanics of Materials Review L L L0 δ ε = = = L L L 0 0 0 Units of length/length ε = normal strain L = change in length = δ L 0 = original length L = length after deformation (after aial load is applied) Percent Elongation = L 100 L0 A A i f Percent Reduction in Area = 100 Ai A i = initial cross- sectional area A f = final cross-sectional area

Strain Shear Strain = change in angle, usually epressed in radians y B γ y B' θ

Stress-Strain Diagram for Normal Stress-Strain

Hooke's Law (one-dimension) = Eε = normal stress, force/length^ E = modulus of elasticity, force/length^ = normal strain, length/length ε τ = Gγ τ = shear stress, force/length^ G = shear modulus of rigidity, force/length^ γ = shear strain, radians

G = E (1 + ν ) FE Mechanics of Materials Review ν = Poisson's ratio = -(lateral strain)/(longitudinal strain) ε ν = ε lat long δ ' ε lat = change in radius over original radius r δ ε long = change in length over original length L

Aial Load E If A (cross-sectional area), E (modulus of elasticity), and P (load) are constant in a member (and L is its length): = PA PL δ = ε = δ L AE Change in length If A, E, or P change from one region to the net: δ = PL AE Apply to each section where A, E, & P are constant δ A/ B δ A = displacement of pt A relative to pt B = displacement of pt A relative to fied end

-Remember principle of superposition used for indeterminate structures - equilibrium/compatibility

Thermal Deformations FE Mechanics of Materials Review δ α α t = ( T) L= ( T T0 ) L δ t = change in length due to temperature change, units of length α = coefficient of thermal epansion, units of 1/ T = final temperature, degrees T 0 = initial temperature, degrees

Torsion Torque a moment that tends to twist a member about its longitudinal ais τ γ Shear stress,, and shear strain,, vary linearly from 0 at center to maimum at outside of shaft

τ = Tr J τ = shear stress, force/length^ T = applied torque, force length r T φ = TL JG r = distance from center to point of interest in cross-section (maimum is the total radius dimension) J = polar moment of inertia (see table at end of STATICS section in FE review manual), length^4 φ = angle of twist, radians L = length of shaft G = shear modulus of rigidity, force/length^ τ = Gγ = Gr( dφ/ dz) φz φz ( dφ / dz) = twist per unit length, or rate of twist

Bending Positive Bending Makes compression in top fibers and tension in bottom fibers Negative Bending Makes tension in top fibers and compression in bottom fibers

dv Slope of shear diagram = negative of distributed loading value = q ( ) d Slope of moment diagram = shear value dm V d =

Change in shear between two points = neg. of area under distributed loading diagram between those two points Change in moment between two points = area under shear diagram between those two points V V1 = [ q( )] d 1 M M1 = [ V( )] d 1

Stresses in Beams = My I = normal stress due to bending moment, force/length^ y = distance from neutral ais to the longitudinal fiber in question, length (y positive above NA, neg below) I = moment of inertia of cross-section, length^4 Mc ma =± c = maimum value of y; I ε From distance from neutral ais to etreme fiber = y ρ ρ = radius of curvature of deflected = Eε = E = My I y ais of the beam ρ and 1 M ρ = EI

S Then FE Mechanics of Materials Review = I c S = elastic section modulus of beam Mc M ma =± =± I S Transverse Shear Stress: Transverse Shear Flow: Q VQ τ = It VQ q = I = y ' A' t = thickness of cross-section at point of interest t = b here

Thin-Walled Pressure Vessels (r/t >= 10) Cylindrical Vessels pr t = = 1 t p 1 r t a = hoop stress in circumferential direction = gage pressure, force/length^ = inner radius = wall thickness pr = = = aial stress in longitudinal direction t See FE review manual for thick-walled pressure vessel formulas.

-D State of Stress Stress Transformation + y y ' = + cos θ + τ y sin θ + y y y' = cosθ τ y sin θ y τ ' y' = sin θ + τ y cosθ Principal Stresses + y y 1, = ± + τ y tan θ p = y ( τ ) y No shear stress acts on principal planes!

Maimum In-plane Shear Stress ma y τ in plane = + ( τ ) y avg = + y tan y θ s = / τ y

Mohr's Circle Stress, D FE Mechanics of Materials Review + y Center: Point C( avg =,0) τ, positive to the right tau, positive downward! R = ( ) + ( τ ) avg y = + R = 1 avg a = R = avg b τ ma in plane = R +τ A rotation of θ to the ais on the element will correspond to a rotation of θ on Mohr s circle!

Beam Deflections FE Mechanics of Materials Review + - Fig. 1- Inflection point is where the elastic curve has zero curvature = zero moment 1 ε ρ = y My Alsoε = and = E I 1 M ρ = radius of curvature = ρ EI of deflected ais of the beam

1 M d y ρ = EI = d d y M( ) = EI d from calculus, for very small curvatures dm ( ) V = d V( ) 3 d y EI d 3 = for EI constant dv ( ) w ( ) = w ( ) = EI = q for EI constant d 4 d v d 4 Double integrate moment equation to get deflection; use boundary conditions from supports rollers and pins restrict displacement; fied supports restrict displacements and rotations

d y M( ) = EI d y = [ M ( dd ) ] EI For each integration the constant of integration has to be defined, based on boundary conditions

Column Buckling P cr r FE Mechanics of Materials Review π EI = Euler Buckling Formula P cr I I A (for ideal column with pinned ends) = critical aial loading (maimum aial load that a column can support just before it buckles) = unbraced column length = = radius of gyration, units of length I r = I = r A A / r = the smallest moment of inertia of the cross-section Pcr π E cr = = A ( / r) = critical buckling stress = slenderness ratio for the column

Euler s formula is only valid when cr yield. > When cr yield, then the section will simply yield. For columns that have end conditions other than pinned-pinned: P cr π EI = K = the effective length factor (see net page) ( KL) KL = L e = the effective length = cr π E ( KL / r) KL/r = the effective slenderness ratio

Effective Length Factors FE Mechanics of Materials Review