Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing Principle and Maximum Shearing Stresses

Transformation of Stress Recall the general state of stress at a point can be written in terms of 6 components: σx, σy, σz, τxy= τyx, τxz= τzx, τyz= τzy This general “stress state” is independent of the coordinate system used. The components of the stress state in the different directions do depend on the coordinate system. Principle and Maximum Shearing Stresses

Transformation of Stress Consider a state of plane stress: σz=τxz=τyz=0 Where does this occur? Thin plate Outer surface Principle and Maximum Shearing Stresses

Transformation of Stress What do we want to calculate? Principle stresses (σ maximum and σ minimum) Principle planes of stresses (orientation at which they occur) Slice cube at an angle θ to the x axis (new coordinates x’, y’). Define forces in terms of angle and stresses. Principle and Maximum Shearing Stresses

Transformation of Stress Sum forces in x’ direction. Sum forces in y’ direction. Principle and Maximum Shearing Stresses

Transformation of Stress Trig identities To get σy’, evaluate σx’ at θ + 90o. Principle and Maximum Shearing Stresses

Transformation of Stress Now, let’s perform some algebra: Constants (we can find these stresses). Variables Principle and Maximum Shearing Stresses

Principle and Max Shearing Stress Define Plug into previous equation Which is the equation of a circle with center at (σave,0) and radius R. Principle and Maximum Shearing Stresses

Principle and Max Shearing Stress We learned before that the principle stresses (maximum and minimum σ) occur when τx’y’ = 0. Principle and Maximum Shearing Stresses

Example Problem For the state of plane stress shown below, determine (a) the principal planes, (b) the principal stresses, (c ) the maximum shearing stress and the corresponding normal stress. Mohr's Circle

MAE 314 – Solid Mechanics Yun Jing Mohr’s Circle (7.4-7.6) MAE 314 – Solid Mechanics Yun Jing Mohr's Circle

Constructing Mohr’s Circle Given: σx, σy, τxy at a particular orientation Draw axes Plot Point O, center, at (σave,0) Plot Point X (σx,τxy), this corresponds to θ = 0° Plot Point Y (σy,-τxy), this corresponds to θ = 90° Draw a line through XY (passes through O): This is the diameter of the circle Draw circle t Y O s 2qP X Mohr's Circle

Sign Convention Mohr's Circle

Uses of Mohr’s Circle t s Find stresses on any inclined plane. Rotate by 2θ1 to point X’. This points gives the stress state (σx1, τxy1). The point Y’ gives (σy1, -τxy1). Find principle stresses. Q1 is the point with the highest normal stress. Q2 is the point with the smallest normal stress. S1 is the highest shear stress = R. S2 is the lowest shear stress = -R. S1 (save, tmax) t X’ (sx1, txy1) 2q1 2qQ1 s O Q2 Q1 (smin, 0) (smax, 0) 2qP (sy1, -txy1) Y’ 2qQ2 X S2 (save, tmin) Mohr's Circle

Example Problem For the state of plane stress shown below, determine (a) the principal planes, (b) the principal stresses, (c ) the maximum shearing stress and the corresponding normal stress. Mohr's Circle

Special Cases of Mohr’s Circle Uniaxial tension σmax = σx σmin = 0 τmax = σx / 2 Hydrostatic pressure σmin = σx τmax = 0 Pure torsion σmin = -σx τmax = σx Mohr's Circle

Example Problem For the given state of stress, determine the normal and shearing stresses after the element shown has been rotated (a) 25o clockwise and (b) 10o counterclockwise. Mohr's Circle

Example Problem For the element shown, determine the range of values of xy for which the maximum tensile stress is equal to or less than 60 Mpa. Mohr's Circle

3D Stress States Mohr’s circle can be drawn for rotation about any of the three principle axes. These axes are in the direction of the three principle stresses. The maximum shear stress is found from the largest diameter circle. x-c plane y-c plane x-y plane Mohr's Circle

Applied to Plane Stress Recall σz = 0 (z-axis is one of the principle axes) The third σ value is always at the origin. Where does τmax occur? 2 possibilities tmax is out of the x-y plane tmax is in the x-y plane Mohr's Circle

Example Problem For the state of stress shown, determine two values of σy for which the maximum shearing stress is 75 MPa. Mohr's Circle

Thin-Walled Pressure Vessels (7.9) MAE 314 – Solid Mechanics Yun Jing Thin-Walled Pressure Vessels

Thin-Walled Pressure Vessels Cylindrical vessel with capped ends Spherical vessel Assumptions Constant gage pressure, p = internal pressure – external pressure Thickness much less than radius (t << r, t / r < 0.1) Internal radius = r Point of calculation far away from ends (St. Venant’s principle) Thin-Walled Pressure Vessels

Cylindrical Pressure Vessel Circumferential (Hoop) Stress: σ1 Sum forces in the vertical direction. Longitudinal stress: σ2 Sum forces in the horizontal direction: Thin-Walled Pressure Vessels

Cylindrical Pressure Vessel There is also a radial component of stress because the gage pressure must be balanced by a stress perpendicular to the surface. σr = p However σr << σ1 and σ2 , so we assume that σr = 0 and consider this a case of plane stress. Mohr’s circle for a cylindrical pressure vessel: Maximum shear stress (in-plane) Maximum shear stress (out-of-plane) Thin-Walled Pressure Vessels

Spherical Pressure Vessel Sum forces in the horizontal direction: In-plane Mohr’s circle is just a point Thin-Walled Pressure Vessels

Thin-Walled Pressure Vessels Example Problem A basketball has a 9.5-in. outer diameter and a 0.125-in. wall thickness. Determine the normal stress in the wall when the basketball is inflated to a 9-psi gage pressure. Thin-Walled Pressure Vessels

Thin-Walled Pressure Vessels Example Problem A cylindrical storage tank contains liquefied propane under a pressure of 1.5MPa at a temperature of 38C. Knowing that the tank has an outer diameter of 320mm and a wall thickness of 3mm, determine the maximum normal stress and the maximum shearing stress in the tank. Thin-Walled Pressure Vessels

Thin-Walled Pressure Vessels Example Problem The cylindrical portion of the compressed air tank shown is fabricated of 8-mm-thick plate welded along a helix forming an angle β = 30o with the horizontal. Knowing that the allowable stress normal to the weld is 75 MPa, determine the largest gage pressure that can be used in the tank. Thin-Walled Pressure Vessels