Q3[A] prove that principal stresses are all REAL
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Q3[A]
prove that principal stresses are all REAL
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- Q3: If the stresses system ox ay =60 MPa 0₂ -15 MPa, and Txy= 45 MPa, Tyz Txz = 30 MPa. Determine principal stresses and verify your results?Question: Find schematically the normal and shear stresses in the plane when the element in the plane stress state is rotated 30 degrees counterclockwise with the Mohr circle.A 30 ?? square plate is subjected to uniform pressure as shown in Figure.1. The 2D loading stressesare described. If the modulus of elasticity is 200 ??? and the lateral constant (Poisson's ratio ? = 0.3). Estimatethe change in length in the vertical, horizontal and diagonal sides as well as determine the volumetric strain.Illustrate the effect of two-dimensionalstrain by constructing appropriate sketches (before and after deformation).Repeat the analysis for lower value of pressure and higher value in the range of [20 -100 MPa] and comment onthe overall results.
- (a) Construct Mohr's circle for an element in a uniaxial state of stress (b) Use this Mohr's circle to derive the equation for the normal stress On and shear stress En on the n-face. The equations should depend on Ox and θ (c) Use the Mohr's circle to determine the planes on which the maximum shear stress acts. Sketch a properly oriented maximum-shear-stress element and indicate the normal and shear stresses acting on its faces. Ơn nt o, Note: for (b), the angle θ is between 0 and 90° and is positiveConsider a point in a structural member that is subjected to plane stress. Normal and shear stress magnitudes acting on horizontal and vertical planes at the point are S, - 10 ksi, S, -39 ksi, and 5x, -24 ksi. (a) Determine the principal stresses (0,1 > 2) and the maximum in-plane shear stress Tax acting at the point. (b) Find the smallest rotation angle #, (counterclockwise is positive, clockwise is negative) that will rotate to principal directions. Then show these stresses in an appropriate sketch (e.g., see Figure 12.15 or Figure 12.16) Answers: i Opl = p2 = free m i 0₂ = i O ksi. ksi. ksi.Calculate the principal stresses to the given state. Calculate the angle of rotation in the x-y plane. Calculate the extreme shear stresses. Are the extreme shear stresses the maximum shear stresses? Explain, Why or why not?
- Current Attempt in Progress Consider a point in a structural member that is subjected to plane stress. Normal and shear stress magnitudes acting on horizontal and vertical planes at the point are S= 26 MPa, S, = 19 MPa, and Sy = 30 MPa. (a) Determine the principal stresses (o,1 > o,2) and the maximum in-plane shear stress Tmax acting at the point. (b) Find the smallest rotation angle 0, (counterclockwise is positive, clockwise is negative) that will rotate to principal directions. Then show these stresses in an appropriate sketch (e.g., see Figure 12.15 or Figure 12.16) Answers: Opl = i MPa. Op2 = i MPa. Tmax = MPa. Op = iConsider the following plane stress state: Ox=30 MPa, y= -60 MPa, Txy= 30 MPa cw Calculate the following: 1. The coordinates of the center of the Mohr's circle C The location of the center of the Mohr's circle Cis ( 2. Principal normal stresses (01, 02) The principal normal stresses are σ₁ = 39.08 3. Maximum shear stress (7) The maximum shear stress is 54.08 MPa. 4. The angle from the x axis to 0₁ (p) The angle from the x axis to 0₁ (p) is -16.85 5. The angle from the x axis to 7 (s) The angle from the x axis to T (s) is 28.15 6. The radius of the Mohr's circle The radius of the Mohr's circle is 54.08. ✰ MPa. MPa and 02 = -69.08 MPa. O -15 MPa, CW CCW O MPa).Consider a point in a structural member that is subjected to plane stress. Normal and shear stress magnitudes acting on horizontal and vertical planes at the point are Sy= 41 MPa, Sy = 11 MPa, and S=-44 MPa. (a) Determine the principal stresses (o,l > 02) and the magnitude of the maximum in-plane shear stress Tmax acting at the point... (b) Find the smallest rotation angle 0, (counterclockwise is positive, cločkwise is negative) that will rotate to principal directions. Then show these stresses in an appropriate sketch (e.g., see Figure 12.15 or Figure 12.16) Answers: OpI = MPa. Op2 = MPa. Tmax | MPa.
- Q4 (a) A thin eylinder 95 mm internal diameter, 500 mm long with walls 4.5 mm thick is subjected to an internal of 9 MN/m2. If E = 200 GPa and v = 0.25, and assuming rigid end plates. Calculate: (i) The change in volume. (ii) The change of internal diameter and change in length. (iii) The value of hoop and longitudinal stresses.Problem 4 (6-6 , Eringen) In a linear, isotropic, elastic solid, show that the stress, t, satisfies the Beltrami-Michell equations Orsikk +HvO kk rs +, Fkik Ors + Fr,s +F,,r= 0 1+v Ser where F =p(f - a).Give me right solution with clear calculations. Determine the normal and shear stresses that occur in the n-n plane of the figure. Present a diagram showing such stresses on the n-n plane.