Question:
Which one of the following stress conditions represents the state of pure shear stress?
Which one of the following stress conditions represents the state of pure shear stress?
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Pure shear needs zero normal stress on every face and a complementary, self-balancing arrangement of equal shear arrows all around the element.
Updated On: Jul 22, 2026
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The Correct Option is A
Solution and Explanation
Step 1: Define pure shear.
A stress element is in a state of pure shear when only shear stresses act on its faces, with zero direct (normal) stress on every face, and the shear stresses on adjoining faces are equal in magnitude (here 40 units on every face) and arranged as complementary pairs, i.e. \(\tau_{xy}=\tau_{yx}\).
Step 2: Apply rotational equilibrium.
For any real stress element, the couple produced by the shear stresses on one pair of faces must be balanced by an equal and opposite couple on the other pair of faces, otherwise the element would spin under a net moment, which cannot happen for an element in equilibrium. This complementary shear stress rule means the shear arrows must point either both toward a common edge or both away from it on adjacent faces, consistently all around the element.
Step 3: Compare the four figures.
In figure (A) the arrows of magnitude 40 on all four faces act tangential to each face with the complementary pairing that satisfies rotational equilibrium, and no arrow acts perpendicular (normal) to any face, exactly the signature of pure shear. In figures (B) and (C) the sense of one or more arrow pairs is reversed relative to (A); reversing the sense on only some faces breaks the complementary pairing and leaves a net unbalanced moment, which cannot represent a valid, self-equilibrated stress state. In figure (D) the arrows point straight out of (or into) each face, normal to that face, describing a biaxial direct (normal) stress state, not shear at all.
Step 4: Conclusion.
Only figure (A) shows four equal-magnitude shear stresses acting tangentially with the correct complementary sense, satisfying equilibrium and containing no normal stress component.
Final Answer:
Figure (A) represents the state of pure shear stress. \[ \boxed{\text{Option (A)}} \]
A stress element is in a state of pure shear when only shear stresses act on its faces, with zero direct (normal) stress on every face, and the shear stresses on adjoining faces are equal in magnitude (here 40 units on every face) and arranged as complementary pairs, i.e. \(\tau_{xy}=\tau_{yx}\).
Step 2: Apply rotational equilibrium.
For any real stress element, the couple produced by the shear stresses on one pair of faces must be balanced by an equal and opposite couple on the other pair of faces, otherwise the element would spin under a net moment, which cannot happen for an element in equilibrium. This complementary shear stress rule means the shear arrows must point either both toward a common edge or both away from it on adjacent faces, consistently all around the element.
Step 3: Compare the four figures.
In figure (A) the arrows of magnitude 40 on all four faces act tangential to each face with the complementary pairing that satisfies rotational equilibrium, and no arrow acts perpendicular (normal) to any face, exactly the signature of pure shear. In figures (B) and (C) the sense of one or more arrow pairs is reversed relative to (A); reversing the sense on only some faces breaks the complementary pairing and leaves a net unbalanced moment, which cannot represent a valid, self-equilibrated stress state. In figure (D) the arrows point straight out of (or into) each face, normal to that face, describing a biaxial direct (normal) stress state, not shear at all.
Step 4: Conclusion.
Only figure (A) shows four equal-magnitude shear stresses acting tangentially with the correct complementary sense, satisfying equilibrium and containing no normal stress component.
Final Answer:
Figure (A) represents the state of pure shear stress. \[ \boxed{\text{Option (A)}} \]
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