Question

Learning Goal: To be able to identify the initial stresses and the angle of rotation, including the correct signs, and use these in the stress-transformation equations to find the stress on a plane or element at a different angle than the original. The method of calculating the state of stress on an inclined plane is tedious, prone to error, and incomplete—if we calculate σx′ and τx′y′, we have to do a separate calculation to determine σy′. Consider the stress element (a) shown in the picture below. (Figure 1) After being rotated through an angle θ, the stress elements σx′ and τx′y′ can be calculated by the inclined-plane method by letting the inclined plane be the y′ axis. (Figure 2) Balancing the sums of the forces in the primed coordinate system yields two equations: σx′=σxcos2θ+σysin2θ+τxy(2sinθcosθ) τx′y′=(σy−σx)sinθcosθ+τxy(cos2θ−sin2θ) Using the trigonometric identities 2sinθcosθ=sin(2θ), sin2θ=(1−cos(2θ))/2, and cos2θ=(1+cos(2θ))/2 and the fact that the +y′ axis is always 90∘ counterclockwise from the +x′ axis, we can derive the equations: σx′=σx+σy2+σx−σy2cos(2θ)+τxysin(2θ) σy′=σx+σy2−σx−σy2cos(2θ)−τxysin(2θ) τx′y′=−(σx−σy2)sin(2θ)+τxycos(2θ) These stress-transformation equations allow us to eliminate the geometric work that was involved in the incline-plane method of stress transformation and provides us with all three stresses in the primed coordinate system. In deriving these equations, we have used the standard sign convention that a normal stress, σ, is positive if it points outward from the stress element, a shear, τ, is positive if, on the face through with the +x axis passes, it points in the +y direction, and angles, θ∈(−180∘,180∘], are positive if they are counterclockwise from the +x axis.

figures numbered from left to right and bottom is 6th figure

1. The state of stress at a point in a member is shown on the rectangular stress element below; the magnitudes of the stresses are |σx|=70 MPa,|σy|=36 MPa, and |τxy|=29 MPa.

(Figure 3)Using the stress-transformation equations, determine the state of stress on the inclined plane AB.

2. The state of stress at a point in a member is shown on the rectangular stress element below where the magnitudes of the stresses are |σx|=174 MPa and |τxy|=54 MPa.(Figure 4)

Determine the state of stress on an element rotated 40∘ clockwise from the element shown.

3. The state of stress at a point in a member is shown on the rectangular stress element below where the magnitudes of the stresses are |σx|=10 ksi,|σy|=29 ksi, and |τxy|=5 ksi.(Figure5)Determine the state of stress on an element rotated 60∘ counterclockwise from the element shown.

60° 15° xi

Answer 1

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版< 116; 23 -3.5 + 16.8s1 +8.5 6+23) Cos120 11 Sin 126 一9.15-14-722 个30 20.35-esi