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8a-ex. Stress Transformation Examples Ex. 8a.1 Ex. 8a.2

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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 thevertical.

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 (1cos2q)

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

x. Examples http://strengthandstiffness.com/8_transformation/page_8a-ex.ht

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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 positivetmax means the shear stress causes a counterclockwise

rotation on the face defined by qs. ... the shear stress on the xs

face is positive.


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A negative tmax means the shear stress causes a clockwise

rotation on the face defined by qs... the shear stress on the xsface is negative.

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Updated: 05/23/09 DJD


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