8a-ex. Stress Transformation Examples • Ex. 8a.1 • Ex. 8a.2 Back | Index | Next

 Example 8a.1 Given: A welded plate supports a force of P = 50 kN. The width of the plate W = 200 mm, and the thickness, t = 50 mm. The weld is at 30° from the vertical. Req'd: Determine the Normal Stress and Average Shear Stress in the weld. Sol'n: Step 1. The axial stress in the plate is: sx = P/A = P/(Wt) = 5.0 MPa
 Step 2. The Stress Transformation Equations are: Here, sy and txy are both zero, so the equations simplify to:        sx' = 0.5 sx(1+cos2q) = stress normal to weld        sy' = 0.5 sx (1–cos2q)     tx'y' = –0.5 sx (sin2q) = Ave. shear stress parallel to weld Therefore : sx' = 3.75 MPa,   sy' = 1.25 MPa,   tx'y' = –2.17 MPa Or, the normal and shear stresses acting on the weld are: sw = 3.75 MPa,  |tw| = 2.17 MPa Note: the Principal of Invariance for the normal stresses is satisfied: sx' + sy'  =  sx + sy =   3.75 MPa + 1.25 MPa = 5.00 MPa

Example 8a.2

Given: An element is subjected to the following stress:
sx = 10 ksi;  sy = 20 ksi;  txy = 5 ksi.

Req'd:
(a) If the element is rotated q = 15°, determine the new stresses.
(b) Determine the Principal Stresses and their Angles.
(c) Determine Maximum In-Plane Shear Stress, tmax, the angles of the vectors that are normal to the faces on which they act, and the associated normal stresses.

#### Stress Element

 Sol'n: Step 1. For the element rotated by 15° : sx' = 13.2 ksi,   sy' = 16.8 ksi,  tx'y' = 6.83 ksi
 Step 2. Principal Stresses and Principal Angles. The Principal Stresses are: and occur at angles rotated byqp: For the element: sI  = 22.1 ksi at qI = 67.5°       sII = 7.93 ksi at qII = 113°
 Step 3. Maximum Shear Stress. The Maximum In-Plane Shear Stress is: and act on element faces that have outward pointing vectors rotated by qs from the x-axis: Thus: tmax,1 = 7.07 ksi on face:   qs,1 = 22.5° tmax,2 = –7.07 ksi on face:   qs,2 = 113° ss,x = ss,y = save = 15 ksi A positive tmax means the shear stress causes a counterclockwise rotation on the face defined by qs. ... the shear stress on the xs face is positive. A negative tmax means the shear stress causes a clockwise rotation on the face defined by qs... the shear stress on the xs face is negative.

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Updated: 05/23/09 DJD