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Basic Stress Equations Dr. D. B. Wallace Basic Stress Equations Centroid of Centroid of Internal Reactions: Cross Section Cross Section Shear Forces Bending Moments y y 6 Maximum (t ) (s) (3 Force Components & 3 Moment Components) x x "Cut Surface" "Cut Surface" V M x P z x T z Vy My Normal Force Torsional Moment (s) or Torque (t ) Force Components Moment Components Normal Force: Centroid y s P "Cut Surface" x s = l Uniform over the entire cross section. A l Axial force must go through centroid. P z Axial Force Axial Stress Shear Forces: Cross Section Point of interest "y" Shear Force y V LINE perpendicular to V through point of interest x b = Length of LINE on the cross section z Vy Aa = Area on one side of the LINE t y Centroid of entire cross section "x" Shear Force y Centroid of area on one side of the LINE Aa x t I = Area moment of inertia of entire cross section about an axis pependicular to V. Vx z y = distance between the two centroids V×bA × yg t = a I× b Note: The maximum shear stress for common cross sections are: Cross Section: Cross Section: Rectangular: tmax = 3 2 × V A Solid Circular: tmax = 4 3× V A I-Beam or Thin-walled H-Beam: flange web tmax = V A web tube: tmax = 2 × V A Basic Stress Equations Dr. D. B. Wallace Torque or Torsional Moment: Solid Circular or Tubular Cross Section: y r = Distance from shaft axis to point of interest R = Shaft Radius x T× r "Cut Surface" t = D = Shaft Diameter J z p× D4 p× R4 T J = = for solid circular shafts Torque t 32 2 16×T p× D 4 - D 4 t = for solid circular shafts e o i j max 3 J = for hollow shafts p × D 32 16×T× D t = o for hollow shafts max 4 4 p ×eDo - Di j Rectangular Cross Section: Cross Section: y t2 Centroid b "Cut Surface" x Note: t1 a a > b T z Torque Torsional Stress Method 1: 2 2 tmax = t1 = T×b3×a +1.8× bg ea ×b j ONLY applies to the center of the longest side Method 2: T a/b a1 a2 t1,2 = a ×a × b2 1.0 .208 .208 1,2 1.5 .231 .269 2.0 .246 .309 Use the appropriate a from the table 3.0 .267 .355 on the right to get the shear stress at 4.0 .282 .378 6.0 .299 .402 either position 1 or 2. 8.0 .307 .414 10.0 .313 .421 ¥ .333 ---- Other Cross Sections: Treated in advanced courses. 2 Basic Stress Equations Dr. D. B. Wallace Bending Moment "x" Bending Moment y x M × y M × x s = x and s = y M z s x Ix Iy "y" Bending Moment y where: Mx and My are moments about indicated axes x y and x are perpendicular from indicated axes z My Ix and Iy are moments of inertia about indicated axes s Moments of Inertia: b D b× h3 4 4 is perpendicular to axis p× D p × R c I = h c R I = = 12 64 4 h 2 I b× h I p× D3 p × R3 Z = = Z = = = c 6 c 32 4 Parallel Axis Theorem: new axis I = Moment of inertia about new axis Area, A d 2 I = Moment of inertia about the centroidal axis I = I + A ×d A = Area of the region centroid d = perpendicular distance between the two axes. Maximum Bending Stress Equations: M ×c M 32× M 6× M s = = smax = bSolid Circular g smax = aRectangularf max I Z p× D3 b × h2 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, r. The beam is assumed to be initially straight. The applied moment, M, causes the beam to assume a radius of curvature, r. Before: y s = E × After: r E = Modulus of elasticity of the beam material y = Perpendicular distance from the centroidal axis to the M r M point of interest (same y as with bending of a straight beam with Mx). r = radius of curvature to centroid of cross section 3 Basic Stress Equations Dr. D. B. Wallace Bending Moment in Curved Beam: Geometry: nonlinear so centroid stress centroidal axis y A distribution co A = cross sectional area e rn = dA r = radius to neutral axis ci n ro zarea r r = radius to centroidal axis si r n r r r e = eccentricity neutral axis i e = r - rn M Stresses: Any Position: Inside (maximum magnitude): Outside: -M × y M ×ci -M ×co s = si = so = e×A ×brn + yg e×A × ri e×A × ro Area Properties for Various Cross Sections: dA Cross Section r A zarea r Rectangle r h r F ro I ri + t ×ln t 2 G J h × t H ri K ri h ro Trapezoid r h ×bti + 2× to g ri + t + t r ro × ti - ri × to F ro I i o 3×bti + to g t - t + ×ln h × t t o i G J 2 i o h H ri K For triangle: r h i set ti or to to 0 ro Hollow Circle r a r L 2 2 2 2 O p×ea2 - b2 j r 2 × pNM r - b - r - a QP b 4 Basic Stress Equations Dr. D. B. Wallace Bending Moment in Curved Beam (Inside/Outside Stresses): Stresses for the inside and outside fibers of a curved beam in pure bending can be approximated from the straight beam equation as modified by an appropriate Centroidal curvature factor as determined from the graph below [i refers to the inside, and o Axis refers to the outside]. The curvature factor magnitude depends on the amount of c curvature (determined by the ratio r/c) and the cross section shape. r is the radius M of curvature of the beam centroidal axis, and c is the distance from the centroidal axis to the inside fiber. r M ×c Inside Fiber: s = K × i i I M M ×c Outside Fiber: s = K × o o I B A B A b/8 4.0 b/4 b c Values of K i for inside fiber as at A 3.5 B A B A U or T c 3.0 b/2 b Round or Elliptical A Curvature B 2.5 c Factor Trapezoidal b/3 b/6 b 2.0 B A B A I or hollow rectangular c Ki r 1.5 1.0 I or hollow rectangular Ko 0.5 U or T Round, Elliptical or Trapezoidal Values of Ko for outside fiber as at B 0 1 2 3 4 5 6 7 8 9 10 11 Amount of curvature, r/c 5.
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