GG303 Lab 10 10/26/09 1. Stresses

Transcription

1 GG303 Lab 10 10/26/09 1 Stresses Eercise 1 (38 pts total) 1a Write down the epressions for all the stress components in the ' coordinate sstem in terms of the stress components in the reference frame. The epression for σ is given as a guide.! = a a! + a a! + a a! + a a! " # # = " # # = " # # = 1b Using the picture below, describe the meaning of terms a, a, a', a'. ' 1c On each bo below draw and label positive normal stresses and positive shear stresses using the tensor ("on-in") convention. ' Stephen Martel (Fall 2009: Conrad) Lab10-1 Universit of Hawaii

2 GG303 Lab 10 10/26/09 2 1d On each bo below draw positive normal tractions and positive shear tractions. ' 1e Assuming that σ = 4 σ (both etensive) and σ = σ =0, draw Mohr s circle in the space below. 6pts 1f What is the angle between the and aes for the following primed coordinate sstems? Denote both angles (θ RL and θ LL ) on the Mohr s circle above (and be careful with sign conventions!): The primed sstem with the maimum right-lateral shear tractions: θ RL = The primed sstem with the maimum left-lateral shear tractions: θ LL = Stephen Martel (Fall 2009: Conrad) Lab10-2 Universit of Hawaii

3 GG303 Lab 10 10/26/09 3 Eercise 2: (72 pts total) Suppose σ = -3MPa, σ = 5MPa, σ = 5MPa, σ = -27MPa. 2a 2b On a separate piece of paper, draw a square bo about 2" on a side showing the stresses with arrows acting on the sides of the square. Show arrows in the positive direction, and label them with the correct magnitudes. Put a north arrow on our bo parallel to the - ais. Make sure the bo will be in equilibrium. On the same piece of paper, draw a new square bo about 2" on a side showing the tractions (τn, τs, τn, and τs) with arrows acting on the sides of the square. Remember that the n- and s-directions for a traction are normal and parallel, respectivel, to the side of the bo the traction acts on. For each side of the bo, show arrows in the positive n- and s-directions, and label them with the correct magnitudes. Put a north arrow on our bo parallel to the -ais. 2c Inside the bo draw a line representing a vertical fault that strikes d Guess whether this fault will tend to slip left-laterall or right-laterall. 2e Plot a Mohr circle describing this state of stress, labeling the point on the Mohr circle corresponding to the TRACTION COMPONENTS (see 2b) that act on the face that has a normal along the -ais (i.e., τn, τs) and the point on the Mohr circle corresponding to 2f the TRACTION COMPONENTS that act on the face that has a normal along the -ais (i.e., τn, τs). Using the Mohr circle, find the magnitude of τ1, the most tensile (i.e., most positive) traction, and τ2, the least tensile (i.e., most negative) traction. 2g Mark on the Mohr circle the points corresponding to τ1 and τ2. 2h 2i 2j Augment the point marked τ1 with a second label τn', τs' while retaining the original τ1 label. Augment the point marked τ2 with a second label τ'n', t's' while retaining the original τ2 label. Use the Mohr circle to find the negative double angles between the -ais and the -ais, and between the -ais and the '-ais, and label the angles 2θ and 2θ ', respectivel. On another sheet of paper draw new,, and ' aes in their correct orientation. Then draw a 2" square with sides normal to the and ' aes. Then show the principal stresses σ and σ'' acting on the sides of the square. Using the angle of (2h) and the formulas on the first page, calculate σ, σ', σ', and σ'' to check our answer of (2f) are the consistent? 16pts 2k Inside this square draw a line representing a vertical fault that strikes l Make a guess as to whether this fault will tend to slip left-laterall or right laterall. Is this guess the same as our first guess? 2m On one more piece of paper, draw the fault and surround it with a bo whose sides are parallel and perpendicular to the fault. Add an " ais that is parallel to the fault and a " ais perpendicular to the fault; make sure the are right-handed. 2n Measure the angle from the -ais to the " ais and label it +θ ". 2o Using the corresponding negative double angle, plot on the Mohr circle the point representing (τ"n", τ"s"). Label the negative double angle 2θ ". 2p Will the fault slip left-laterall or right laterall? How does this compare with (2d) and (2l)? 2q Use the procedure of (2j) to calculate σ"" and σ"" to check our results. 8pts 2r Use the Matlab command [V,D] = eig and the stress components at the top of the page to find the principal stresses to check our results (2). Draw a picture showing the principal stresses acting on the sides of a square (2) and the two sets of direction cosines defining the eigenvectors that give the orientation of the two principal stresses (2+2). Include a printout of our Matlab results. 8pts Stephen Martel (Fall 2009: Conrad) Lab10-3 Universit of Hawaii

4 GG303 Lab 10 10/26/09 4 The following page might prove helpful in organizing our work σ = τn = σ = τs = σ = τn = σ = τs = σ = τ n = σ = τ s = σ = τ n = σ = τ s = σ = τ n = σ = τ s = σ = τ n = σ = τ s = Stephen Martel (Fall 2009: Conrad) Lab10-4 Universit of Hawaii

5 GG303 Lab 10 10/26/09 5 Stephen Martel (Fall 2009: Conrad) Lab10-5 Universit of Hawaii