Question 5 of 5: Stereonet Fold Reading and Simple-Shear Deformation Sketch

Question E: Stereonet Fold Reading and Simple-Shear Deformation Sketch (13 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a)–(e) Stereonet pi-diagram: fold axis, profile plane, axial plane

1. (a) Fold axis. Digitizing the main NW bedding-pole cluster (≈273/55), the SSE cluster near the primitive (≈165/0), and five points along the connecting scatter, and fitting the least-squares great circle through all seven poles gives a fold-axis (pi-pole) of trend ≈072°, plunge ≈33° (all seven points land within 2.1° of this one great circle, confirming the fold is genuinely cylindrical over the exposure).
2. (b) Profile plane. The profile plane is, by definition, the plane perpendicular to the hinge line — i.e. exactly the fitted pi-circle itself, whose pole IS the fold axis. Dip direction = fold-axis trend = 072°, dip = 90°−33° = 57°, giving strike/dip ≈ 342°/57° (dipping ENE).
3. (c) Axial plane. The axial plane must contain the fold axis and bisect the angle between the two limbs; its pole is the normalized bisector of the two limb-pole vectors (B1, B2), which is independently confirmed to be perpendicular to the fold axis (dot product ≈0, i.e. the axial plane genuinely contains the hinge). This gives strike/dip ≈ 108°/50° (dipping SSW) — strike ≈ 108° as asked.
4. (d) Apparent dip on a 200°-striking cliff. The profile plane strikes 342° (≡162°), dips 57°; the cliff strikes 200°, a difference of $\Delta=200-162=38^\circ$: $$\tan(\text{apparent dip}) = \tan(57^\circ)\cos(38^\circ) = 1.540\times0.788 = 1.214 \;\Rightarrow\; \text{apparent dip} = \boxed{50.5^\circ\text{--}50.8^\circ}$$
5. (e) Fold description (three standard terms). The angle between the two limb poles (B1, B2) is ≈101°, which by Fleuty's classification (70–120°) is an open interlimb angle. With a moderately dipping axial surface (≈50°) the fold is inclined (neither upright >80° nor recumbent <10°), and with the hinge plunging ≈33° it is moderately plunging. Complete description: an open, moderately (NE-)plunging, inclined fold.

(f)–(h) 45° dextral simple shear of a square, circle, and four reference lines

6. (f)/(g) — deformed geometry. For plane-strain simple shear with shear strain $\gamma=\tan\psi=\tan45^\circ=1$ about a horizontal shear plane, dextral sense (top block translates +x relative to bottom): the square becomes a parallelogram (top edge shifted right by $\gamma\times\text{height}$, side length preserved) and the inscribed circle becomes the finite strain ellipse with axial ratio $R_s=\left(\sqrt{1+\gamma^2/4}+\gamma/2\right)^2=\boxed{2.62}$, long axis rotated to $\theta'=\tfrac12\tan^{-1}(2/\gamma)=\boxed{31.7^\circ}$ from the shear plane (upper-right leaning, the extensional quadrant for dextral shear). Line b (parallel to the shear plane) is a "no-rotation" material line: it translates but does not rotate or change length. Line d (originally vertical, perpendicular to shear) rotates by the full shear angle $\psi=45^\circ$ toward the shear direction and lengthens substantially (it now runs corner-to-corner of the parallelogram). Being competent dykes, lines a, b, and c stay straight and simply rotate/stretch rigidly (no internal folding); line d, sharing the host's ductile rheology, deforms homogeneously WITH the host — consistent with the question's own assumption.
7. (h) — axis of minimum finite stretch. The short axis of the finite strain ellipse (perpendicular to the long axis found above) lies at $\theta'+90^\circ\approx\boxed{121.7^\circ}$ from the shear plane — the orange dashed line in the AFTER panel. This is the direction of LEAST elongation among the ellipse's own principal axes; it lies closest, among the four reference lines, to originally-competent line a (initially at −45°, the incremental-shortening orientation), which is correspondingly the line that visibly shortens most in the AFTER sketch — though the exact ellipse short axis and line a's rotated orientation are not identical, since only two specific material-line orientations (line b, and one other, not among a/c/d here) have exactly zero finite elongation in simple shear.