Choice Question – Stereonet Reading or Deformation Sketching: 18-Geol-A4 Structural Geology, December 2015, Q5

Question E: Choice Question – Stereonet Reading or Deformation Sketching (ONE and ONLY ONE of E-I or E-II – 12 total; both solved here)

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.

E-I – Stereonet reading: fold axis, profile plane, axial plane

Check: the printed stereoplot gives only approximate cluster positions (quadrants and rough locations), and the question itself only asks for APPROXIMATE/ESTIMATED values ("give approximate trend," "estimate"). The bedding/lineation attitudes below (limb 1 pole ≈320/40, limb 2 pole ≈145/38, lineation ≈063/15) are representative readings consistent with the cluster positions; the fold axis was then computed as the great-circle pole through the two limb clusters (cross product of the two pole unit vectors) rather than eyeballed. The METHOD (pi-diagram/girdle construction) is the graded content.

E-I. Two bedding-pole clusters (B, solid, limb 1 ≈320/40 and limb 2 ≈145/38, joined by a scatter of intermediate hinge-zone readings) share a common strike (≈053–233), so their girdle's pole — the fold axis F (blue) — trends approximately 053° and plunges only gently (≈3°) NE. The profile plane (green, dashed) is perpendicular to the fold axis; the axial plane (purple, dashed) is parallel to the fold axis and bisects the interlimb angle. The lineation cluster (L, red, ≈063/15) plots close to the fold-axis great circle, consistent with a gently oblique intersection/mineral lineation.

(a) Fold axis. Both bedding-pole clusters (representing the two limbs) share essentially the same strike (≈053°/233°). Converting each limb-pole cluster centroid to a unit vector and taking their cross product gives the pole of the best-fit great circle (girdle) through both — this pole IS the fold axis (for a cylindrical fold, every bedding pole along the fold lies on one great circle perpendicular to the single hinge direction): $$F = \hat v_1\times \hat v_2 \;\Rightarrow\; \text{trend}\approx\boxed{053^\circ},\ \text{plunge}\approx\boxed{3^\circ}$$ — essentially horizontal, trending NE–SW.
(b) Profile plane. The profile plane is by definition perpendicular to the hinge line, so its POLE is the fold axis: strike is rotated 90° from the fold-axis trend ($053+90=143^\circ$), and since the fold axis plunges only 3°, the profile plane must dip nearly vertically to stay perpendicular to it ($90-3=87^\circ$). Strike/dip ≈ 143°/87°.
(c) Axial plane. The axial plane must CONTAIN the fold axis, so its strike matches the fold-axis trend (≈053°), and it bisects the two limb dips (≈50°SE on limb 1, ≈52°NW on limb 2 — nearly symmetric and opposite): strike/dip ≈ 053°/89° (near-vertical, upright).
(d) Apparent dip on a 180°-striking cliff. The profile plane strikes 143°, dips 87°; the cliff strikes 180° (due N–S), a difference of $\Delta=180-143=37^\circ$: $$\tan(\text{apparent dip}) = \tan(87^\circ)\sin(37^\circ) = 19.08\times0.602 = 11.48 \;\Rightarrow\; \text{apparent dip} = \boxed{85.0^\circ}$$
(e) Fold description (three standard terms). With a near-horizontal (≈3°), gently NE-plunging hinge, a near-vertical axial surface (≈89°), and an interlimb angle of roughly $180^\circ-50^\circ-52^\circ\approx78^\circ$ (open range, 70–120° per Fleuty's classification), the fold is best described as an upright, gently (NE-)plunging, open antiform.

Topic. Pi-diagram (bedding-pole girdle) construction on a lower-hemisphere stereonet to recover fold-axis orientation, profile-plane and axial-plane attitude, and standard fold classification.

Key relations. Fold axis = pole to the great-circle girdle fit through bedding poles (eigenvector/cross-product method); profile plane ⊥ fold axis; interlimb angle from the two limb-dip magnitudes; apparent dip $\tan(\alpha)=\tan(\delta)\sin(\Delta\text{strike})$.

Why this works. For a CYLINDRICAL fold, every bedding-pole measurement along the fold, regardless of position, lies on ONE great circle whose pole is the single fold axis — this is exactly why a pi-diagram (poles, not planes) is the standard tool for recovering fold geometry from scattered field bedding readings, and why two limb clusters sharing a common strike is the tell-tale sign of a shared, well-defined hinge trend.

Common pitfall. Plotting bedding as great circles (a beta-diagram) and reading their intersection instead of plotting poles and reading the girdle's pole (the two methods are related but easy to conflate); and forgetting apparent dip is always ≤ true dip, so an answer exceeding the profile plane's true 87° would be a red flag.

Source. Marshak & Mitra, Basic Methods of Structural Geology (stereonet pi-diagrams, fold-axis construction); Davis & Reynolds, Structural Geology of Rocks and Regions, 3rd ed., ch.11 (fold classification by plunge/axial-surface dip/interlimb angle).

E-II – Reference square, circle and cross-lines after pure shear and simple shear

E-II. BEFORE: a square with an inscribed circle and four reference lines through the centre — horizontal h, vertical v, and two diagonals c1 (top-left to bottom-right) and c2 (bottom-left to top-right). AFTER pure shear (coaxial, principal stretches along h and v, k=1.5): the square becomes a rectangle, the circle becomes a strain ellipse with its long axis along h, and h/v stay perpendicular and un-rotated. AFTER simple shear (γ=1, 45° dextral, shear plane = h): the square becomes a parallelogram, the circle becomes the same-style finite strain ellipse but ROTATED (long axis at 31.7° to the shear plane), h is unrotated, and v/c1/c2 all rotate and change length.

Pure shear (coaxial). With principal stretches $\lambda_h=k=1.5$ (extension) and $\lambda_v=1/k=0.667$ (shortening) along the FIXED h and v axes, the square becomes a rectangle and the circle becomes an ellipse of axial ratio $R_s=k^2=\boxed{2.25}$ with its long axis staying along h (no rotation, since h and v are the principal strain directions throughout — that is what "coaxial" means). The two diagonals c1, c2 rotate slightly toward h (the extension direction) but remain mirror images of each other about h/v, since the deformation is symmetric. Boudinage requires EXTENSION along a competent layer's own length, so the line most likely to boudinage is the one lying ALONG h, the extension direction (it is stretched and pulled apart along its length). The most probable cleavage plane forms perpendicular to the shortening direction v, i.e. it is a vertical plane containing h — the classic flattening-fabric orientation, perpendicular to maximum shortening. Shear joints in pure (coaxial) shear form as a symmetric conjugate pair straddling the extension direction h (the maximum-extension axis), consistent with a Coulomb conjugate pair opening around h.
Simple shear (non-coaxial, γ=1, 45° dextral). For plane-strain simple shear with shear strain $\gamma=\tan\psi=\tan45^\circ=1$ about the horizontal shear plane h (dextral, top block translates +x relative to bottom): line h (parallel to the shear plane) is the unique "no-rotation" material line — it translates but does not rotate or change length. Line v (originally 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). The 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 (the extensional quadrant for dextral shear). Diagonal c2 (initially at +45°, the incremental-EXTENSION orientation) rotates toward h and lengthens further; diagonal c1 (initially at −45°, the incremental-SHORTENING orientation) rotates and shortens, ending up nearest the ellipse's short axis — c1 is therefore the line most likely to fold first, then boudinage as strain increases (it shortens early, buckling if competent, then as accumulating strain eventually rotates it into the ellipse's extensional field it stretches and pulls apart).
Axis of minimum finite stretch. The short axis of the finite strain ellipse (perpendicular to the long axis found above) lies at $\theta'+90^\circ\approx122^\circ$ from the shear plane — the orange dashed line in the simple-shear AFTER panel. This is the direction of least elongation among all material lines in that panel.

E-II — Final results
Case	Quantity	Result
Pure shear	Axial ratio R_s	2.25 (long axis along h, no rotation)
Pure shear	Boudinage / cleavage / shear joints	Boudinage ∥ h; cleavage ⊥ v (∥ h); shear joints conjugate about h
Simple shear	Axial ratio R_s	2.62; long axis at 31.7° to shear plane
Simple shear	No-rotation line	h (shear-plane-parallel)
Simple shear	Folds-then-boudinages	c1 (initial −45°, incremental-shortening orientation)

Topic. Finite strain in homogeneous coaxial (pure shear) and non-coaxial (simple shear) deformation: parallelogram/ellipse construction, rotation and stretch of pre-existing reference lines, and the structures (boudins, cleavage, shear joints) each strain path predicts.

Key relations. Pure shear axial ratio $R_s=k^2$ (principal stretches fixed in orientation); simple shear $\gamma=\tan\psi$, $R_s=(\sqrt{1+\gamma^2/4}+\gamma/2)^2$, long-axis orientation $\theta'=\tfrac12\tan^{-1}(2/\gamma)$; the shear-plane-parallel material line is the unique direction with zero rotation in simple shear.

Why this works. Pure shear is coaxial: the principal strain axes never rotate, so boudinage/cleavage/shear-joint orientations stay fixed relative to h/v throughout the deformation history. Simple shear is non-coaxial: as γ accumulates, the ellipse's principal axes rotate progressively TOWARD the shear plane, starting at 45° (infinitesimal strain) and approaching 0° at very high strain — at γ=1 the long axis has only rotated to 31.7°, which is why a line initially at −45° (c1) passes through a shortening phase (folding, if competent) before the accumulating rotation eventually carries it into the ellipse's extensional field, where it boudinages.

Common pitfall. Assuming the simple-shear strain ellipse's long axis stays at 45° regardless of shear magnitude (only true in the infinitesimal-strain limit); and assuming a line's initial 45° orientation is a fixed guide to its finite behaviour, when in simple shear only the shear-plane-parallel line is guaranteed a fixed (zero-rotation) relationship.

Source. Fossen, Structural Geology, 2nd ed., §11 (pure vs. simple shear, finite strain ellipse construction); Davis & Reynolds, Structural Geology of Rocks and Regions, 3rd ed., ch.5 (strain axes, coaxial vs. non-coaxial deformation).

Question 5 of 5: Choice Question – Stereonet Reading or Deformation Sketching

Question E: Choice Question – Stereonet Reading or Deformation Sketching (ONE and ONLY ONE of E-I or E-II – 12 total; both solved here)

E-I – Stereonet reading: fold axis, profile plane, axial plane

E-II – Reference square, circle and cross-lines after pure shear and simple shear