Comprehensive Study Notes: Shear Stress and Strain
Comprehensive Study Notes: Shear Stress and Strain
Welcome to this essential
study module for your Strength of Materials coursework.
1. Introduction to Shear Stress
(tau)
When a body is subjected to two equal and opposite forces acting
tangentially across the resisting section, the body tends to shear off across
that section. The stress induced in this condition is called **Shear Stress** or
**Tangential Stress**. Unlike tensile or compressive stresses which act *normal*
(perpendicular) to the cross-section, shear stress acts *parallel* to the
surface.
Formula - If
P is the tangential force applied and
A is the
cross-sectional area resisting the force,
The average shear stress (tau) is
given by: ta = \frac{P}{A} $$ **Units:** The standard SI unit is N/m^2 (Pascals, Pa), though N/mm^2 (MPa) is highly preferred in mechanical
engineering problems.
2. Shear Strain (gamma or phi)
When a shear
stress acts on a body, it causes a distortion in its shape rather than a change
in its length. **Shear Strain** is defined as the measure of this angular
distortion.
Imagine a rectangular block fixed at its bottom face. If a
tangential force is applied to the top face, the block deforms into a
parallelogram. The angle (in radians) through which the vertical faces shift is
the shear strain.
Formula gamma = tan(phi) \approx \phi
*(Since the
elastic deformation is extremely small, $\tan(\phi)$ is approximately equal to
$\phi$ in radians).* **Units:** Dimensionless (expressed in radians). ---
3.
Hooke's Law for Shear & Modulus of Rigidity (G or C)
Just as normal stress
is proportional to normal strain within the elastic limit, **shear stress is
proportional to shear strain** within the elastic limit.\tau \propto \phi \tau = G \cdot \phi
Where **$G$** (also denoted by $C$ or $N$ in
Ramamrutham's texts) is the constant of proportionality known as the **Modulus
of Rigidity** or **Shear Modulus**. ### Formula $$ G = \frac{\tau}{\phi} $$
**Significance for Exams:** You will frequently need to use $G$ to find the
angle of twist in torsion problems (e.g., shafts in power transmission). ---
4. Principle of Complementary Shear Stress
A highly frequently asked exam
concept is the **Principle of Complementary Shear**. **Statement:** *A shear
stress acting on a given plane is always accompanied by an equal shear stress
acting on an orthogonal (perpendicular) plane, and in the opposite sense (to
maintain equilibrium).* **Why does this happen?** If a shear stress $\tau$ acts
on the top and bottom faces of a small cubic element, it creates a couple (a
turning moment). For the element to remain in static equilibrium, an equal and
opposite balancing couple must be generated by shear stresses on the vertical
faces. Therefore: $$ \tau_{xy} = \tau_{yx}
5. State of Simple Shear
A
body is said to be in a state of **simple shear** if it is subjected only to
mutually perpendicular and equal shear stresses (a primary shear and its
complementary shear), with no normal (tensile or compressive) stresses acting on
those planes. **Important Exam Derivation Fact:** In a state of simple shear,
the maximum normal stresses (principal stresses) occur on planes inclined at
$45^\circ$ to the planes of pure shear, and their magnitude is exactly equal to
the magnitude of the shear stress (one is tensile, the other is compressive).
Quick Exam Checklist & Tips * **Always draw the element:** When
explaining Complementary Shear Stress, always sketch a 2D square element showing
the four shear arrows head-to-head and tail-to-tail. Examiners look for this
specific diagram! * **Unit Conversions:** Watch out for force in $kN$ and area
in $cm^2$. Always convert to Newtons and millimeters to consistently get your
stress in $MPa$ ($N/mm^2$). * **Definitions:** Rote learn the definitions of
Shear Modulus ($G$) and Complementary Shear exactly as stated, as they are
standard 2-mark questions. *** *End of Notes. Navigate back to the Strength of
Materials subject tab to continue your studies.* shear-stress-strain-notes.md
Displaying shear-stress-strain-notes.md.
