2. The steel bar shown (E=29x103 ksi) is made from two segments having cross-sectional areas of AAB = 1 in² and ABD = 2 in². Determine (a) the vertical displacement of end A and (b) the - displacement of B relative to C. Ay 15 kips A a.) 4 kips deformation 2 ft S=PL AE 4 kips Ans in Ksi pt.) Fy b.) B 8 kips 8 kips C D 1.5A + 1-Statics: Ay (15 mies) (24+) = 8 hips 30 AY Kies Ay=0.2 1 ft 6.246 Ay=3.20 UNIAXIAL STRESS-STRAIN Stress-Strain Curve for Mild Steel SSSS 40.000 20.000 MECHANICS OF MATERIALS Uniaxial Loading and Deformation G-P/A, where G-stress on the cross section P-loading cross-sectional area YIELD STRENGTH AT 0.2 PERCENT OFFSET A € = &L, where 8 -elastic longitudinal deformation 10.000 04 PERCENT 0.001 0.002 0.003 0.004 01 0.2 03 STRAIN The slope of the linear portion of the curve equals the modulus of elasticity. DEFINITIONS Engineering Strain E-AL/L, where € = engineering strain (units per unit) AL change in length (units) of member Loriginal length (units) of member Percent Elongation % Elongation - (4)× 100 Percent Reduction in Area (RA) The % reduction in area from initial area, 4, to final area, 4,, is: %RA-(4-4)× 100 Shear Stress-Strain Y-1/G, where L -length of member E=0/E=P/A PL True stress is load divided by actual cross-sectional area whereas engineering stress is load divided by the initial area. THERMAL DEFORMATIONS 8, = aL(T-T), where 8, deformation caused by a change in temperature temperature coefficient of expansion a L T - -length of member final temperature Tinitial temperature CYLINDRICAL PRESSURE VESSEL Cylindrical Pressure Vessel For internal pressure only, the stresses at the inside wall are: ₁ = P and 6,--P For external pressure only, the stresses at the outside wall are: and 6,--P., where γ shear strain Tshear stress o, tangential (hoop) stress G=shear modulus (constant in linear torsion-rotation o, radial stress relationship) P internal pressure G= E 2(1+v) where P. external pressure E modulus of elasticity (Young's modulus) ■ Poisson's ratio --(lateral strain)/(longitudinal strain) inside radius routside radius For vessels with end caps, the axial stress is: Go, and o, are principal stresses. Flinn, Richard A. and Paul K. Troj Engineering Materials & Their Applications 4th ed., Houghton Mifflin C, B1990 76 MECHANICS OF MATERIALS When the thickness of the cylinder wall is about one-tenth or less of inside radius, the cylinder can be considered as thin- walled. In which case, the internal pressure is resisted by the hoop stress and the axial stress. 6, Pr and d= where r-wall thickness and r STRESS AND STRAIN Principal Stresses For the special case of a two-dimensional stress state, the equations for principal stress reduce to The circle drawn with the center on the normal stress (horizontal) axis with center, C, and radius, R, where -(༡.-॰」, C=R= The two nonzero principal stresses are then: -C+R 0-C-R Tox + 0 0 The two nonzero values calculated from this equation are temporarily labeled σ, and σ, and the third value σ is always zero in this case. Depending on their values, the three roots are then labeled according to the convention: algebraically largest G, algebraically smallest = 0, other G. A typical 2D stress element is shown below with all indicated components shown in their positive sense. (0) The maximum inplane shear stress is t- R. However, the maximum shear stress considering three dimensions is always -=-- Hooke's Law Three-dimensional case: &- (1/E)[0,- (0,+0.)] £,- (1/E)[0,- (0,+0,)] -1G Y-1G = (1/E)[0,- 1(0,+0,)] 2-1G Plane stress case (0,= 0): &= (1/EX(0,-vo,) Mohr's Circle - Stress, 2D To construct a Mohr's circle, the following sign conventions are used. 1. Tensile normal stress components are plotted on the horizontal axis and are considered positive. Compressive normal stress components are negative. 2. For constructing Mohr's circle only, shearing stresses are plotted above the normal stress axis when the pair of shearing stresses, acting on opposite and parallel faces of an element, forms a clockwise couple. Shearing stresses are plotted below the normal axis when the shear stresses form a counterclockwise couple. - (1/EX0,-vo) &=-(1/EX(vo,+10) F 1 0 0 Uniaxial case (6,-6,-0): 6,-Eɛ, or G-Eɛ, where . . Ç - normal strain 0,0,0,- normal stress Y Y Y shear strain shear stress E-modulus of elasticity G-shear modulus v-Poisson's ratio Crandall, S3, and NC. Dal Action to Mechanics of Solids McGraw-Hil New York, 1959 77 MECHANICS OF MATERIALS TORSION Torsion stress in circular solid or thick-walled (t>0.1r) shafts: The relationship between the load (w), shear (V), and moment (M) equations are: x) where J-polar moment of inertia TORSIONAL STRAIN The limit (A/A) = r(didz) The shear strain varies in direct proportion to the radius, from zero strain at the center to the greatest strain at the outside of the shaft.do/dz is the twist per unit length or the rate of twist. G Gr(dbidz) T = G(db) fr³d = G(d) =d=y, where total angle (radians) of twist T-torque L-length of shaft To gives the twisting moment per radian of twist. This is called the torsional stiffness and is often denoted by the symbol k or c. For Hollow, Thin-Walled Shafts T=where 2.4 -thickness of shaft wall =(x) y= --L[-w(x)]ate M-M-(x)de Stresses in Beams The normal stress in a beam due to bending 0,--My-Z. where M-the moment at the section I - the moment of inertia of the cross section y the distance from the neutral axis to the fiber location above or below the neutral axis The maximum normal stresses in a beam due to bending 0,-± Mc/1, where e-distance from the neutral axis to the outermost fiber of a symmetrical beam section. 0,--Mis, where se: the elastic section modulus of the beam. Transverse shear stress: -Qub), where V-shear force Q-xy, where -the total mean area enclosed by the shaft measured to area above the layer (or plane) upon which the BEAMS the midpoint of the wall. Shearing Force and Bending Moment Sign Conventions 1. The bending moment is positive if it produces bending of the beam concave upward (compression in top fibers and tension in bottom fibers). 2. The shearing force is positive if the right portion of the beam tends to shear downward with respect to the left. desired transverse shear stress acts -distance from neutral axis to area centroid B-width or thickness or the cross-section Transverse shear flow: 9-101 Tik S. ad Gleason I MacCullough Elements of Strength of Material POSITIVE BENDING NEGATIVE BENDING POSITIVE HEAR NEGATIVE SHEAR 78 MECHANICS OF MATERIALS Deflection of Beams Using 1/p=M/(EI). El-M, differential equation of deflection curve E dd dr -=dM(x)/dx=V Determine the deflection curve equation by double integration (apply boundary conditions applicable to the deflection and or slope). El (dy/dx) = {M(x) dx Ely-[M(x) dx] dx The constants of integration can be determined from the physical geometry of the beam. Composite Sections The bending stresses in a beam composed of dissimilar materials (material I and material 2) where E,> E, are: 0,--Myl 0,--Myl, where I, the moment of intertia of the transformed section -the modular ratio E/E E,-elastic modulus of material 1 E-clastic modulus of material 2 The composite section is transformed into a section composed of a single material. The centroid and then the moment of inertia are found on the transformed section for use in the bending stress equations. COLUMNS Critical axial load for long column subject to buckling: Euler's Formula P-E where (Ke) ' -unbraced column length K-effective-length factor to account for end supports Theoretical effective-length factors for columns include: Pinned-pinned, K-1.0 Fixed-fixed, K=0.5 Fixed-pinned, K-0.7 Fixed-free, K=2.0 Critical buckling stress for long columns: T E (KU) where -radius of gyration √TA Kur effective slenderness ratio for the column ELASTIC STRAIN ENERGY If the strain remains within the elastic limit, the work done during deflection (extension) of a member will be transformed into potential energy and can be recovered. a If the final load is P and the corresponding elongation of tension member is &, then the total energy U stored is equal to the work done during loading. U-W-P82 COMPOSITE SECTION MATERIAL 1 EA MATERIAL 2 E₂ A TRANSFORMED SECTION NEUTRAL AXIS The strain energy per unit volume is -U/AL-6/2E (for tension) 79 MECHANICS OF MATERIALS