Basic Stresses

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Dr. D. B. Wallace Basic Stress Equations Internal Reactions: 6 Maximum (3 Force Components & 3 Moment Components) Cut Surface Centroid of Cross Section y Centroid of Cross Section Shear Forces (τ ) x Cut Surface z y Bending Moments (σ) x Vx Vy P Mx My T z Normal Force (σ) Force Components Moment Components Torsional Moment or Torque ( τ ) Normal Force: Centroid y Cut Surface x z P Axial Force σ σ= P A l Uniform over the entire cross section. l Axial force must go through c
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   Dr. D. B. Wallace Basic Stress Equations Internal Reactions: 6 Maximum (3 Force Components& 3 Moment Components) Normal Force) ττ () σσ (Shear Forces zxy PV y V x Torsional Moment) ττ () σσ (Bending Moments zxy TM y M x or Torque  Force Components Moment Components Cut Surface Cut Surface Centroid of Cross SectionCentroid of Cross Section Normal Force: Axial Force zxy P Centroid σσ Axial Stress Cut Surface σ = PA l Uniform over the entire cross section. l Axial force must go through centroid. Shear Forces: Cross SectionyAa Point of interestLINE perpendicular to V through point of interest= Length of LINE on the cross section= Area on one side of the LINECentroid of entire cross sectionCentroid of area on one side of the LINE= distance between the two centroids= Area moment of inertia of entire cross sectionabout an axis pependicular to V. VbAayI y Shear Force zxy V y x Shear Force zxy V x ττττ τ =⋅ ⋅⋅ VAyIb a b g  Note : The maximum shear stress for common cross sections are: Cross Section:Cross Section: Rectangular:  τ max = ⋅ 32VA Solid Circular: τ max = ⋅ 43VA I-Beam orH-Beam:webflange τ max = VA web Thin-walledtube: τ max = ⋅ 2VA   Basic Stress EquationsDr. D. B. Wallace Torque or Torsional Moment: Solid Circular or Tubular Cross Section: r = Distance from shaft axis to point of interestR = Shaft RadiusD = Shaft Diameter JDRJDD forsolidcircularshaftsforhollowshafts oi =⋅=⋅=⋅ −π ππ 4444 32232 e j Torque zx   y T Cut Surface ττ τ =⋅ TrJ τπτπ maxmax =⋅⋅=⋅ ⋅⋅ − 1616 344 TDTDDD forsolidcircularshaftsforhollowshafts ooi e j Rectangular Cross Section: Torque zxy T   Centroid ττ Torsional Stress Cut Surface ττ 12 abNote:a > bCross Section: Method 1: τ τ max . = = ⋅ ⋅ + ⋅ ⋅ 122 318Tabab b g e j ONLY applies to the center of the longest side Method 2: τα 12122  , , =⋅ ⋅ Tab a/b αα αα 1.0.208.2081.5.231.2692.0.246.3093.0.267.3554.0.282.3786.0.299.4028.0.307.41410.0.313.421 ∞ .333----Use the appropriate αα from the tableon the right to get the shear stress ateither position 1 or 2 . Other Cross Sections: Treated in advanced courses.2   Basic Stress EquationsDr. D. B. Wallace Bending Moment x Bending Moment zx   yzxy M x σσσσ M y y Bending Moment σ σ=⋅=⋅ MyIandMxI xxyy where: Mx and My are moments about indicated axes y and x are perpendicular from indicated axes Ix and Iy are moments of inertia about indicated axes Moments of Inertia: hcbDcR IbhhZIcbh isperpendiculartoaxis =⋅= =⋅ 32 126IDRZIcDR =⋅=⋅= =⋅=⋅π ππ π 4433 644324 Parallel Axis Theorem: I = Moment of inertia about new axis IIAd = + ⋅ 2 centroiddnew axisArea, A I = Moment of inertia about the centroidal axisA = Area of the regiond = perpendicular distance between the two axes. Maximum Bending Stress Equations: σπ max =⋅⋅ 32 3 MD SolidCircular b g σ max =⋅⋅ 6 2 Mbh Rectangular a f σ max =⋅= McIMZ The section modulus, Z , can be found in many tables of properties of common cross sections (i.e., I-beams,channels, angle iron, etc.). Bending Stress Equation Based on Known Radius of Curvature of Bend, ρ   ρ . The beam is assumed to be initially straight. The applied moment, M , causes the beam to assume a radius of curvature, ρρ . Before:After:M   M ρρ σρ= ⋅ Ey E = Modulus of elasticity of the beam material y = Perpendicular distance from the centroidal axis to thepoint of interest (same y as with bending of astraight beam with Mx ). ρρ = radius of curvature to centroid of cross section3   Basic Stress EquationsDr. D. B. Wallace Bending Moment in Curved Beam: Geometry:rrerr n σσσσ oi centroidcentroidalneutral axisaxis oi y nonlinearstressdistribution Mcc io ρρ rAdAerr narean == − z  ρ A = cross sectional area r n= radius to neutral axis r = radius to centroidal axise = eccentricity Stresses:   Any Position:Inside (maximum magnitude) :Outside: σ =− ⋅⋅ ⋅ + MyeAry n b g σ iii MceAr =⋅⋅ ⋅σ ooo MceAr =− ⋅⋅ ⋅ Area Properties for Various Cross Sections:   Cross Section rdA area ρ z  A thr i r o r Rectangle ρ rh i + 2trr oi ⋅ F H GI K J  ln ht ⋅ thr i r o r Trapezoid ρ o t i rhtttt iioio +⋅ + ⋅⋅ + 23 b gb g  For triangle: set tior toto 0 ttrtrthrr oioiiooi − +⋅ − ⋅⋅ F H GI K J  ln htt io ⋅+ 2r Hollow Circle ρ abr 2 2222 ⋅ − − − LNMOQP π rbra π⋅ − ab 22 e j 4
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