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Plane stress

Plane stress is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Plane stress rather than just read about it. In short: In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2 (representable as a 2×2 matrix rather than 3×3).

Plane stress — main illustration
Plane stress — illustration

Key takeaways

  • Plane stress belongs to physics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Plane stress to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Plane stress from memory before moving on to harder problems.

Reference excerpt

In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2 (representable as a 2×2 matrix rather than 3×3). A related notion, plane strain, is often applicable to very thick members. Plane stress typically occurs in thin flat plates that are acted upon only by load forces that are parallel to them. In certain situations, a gently curved thin plate may also be assumed to have plane stress for the purpose of stress analysis. This is the case, for example, of a thin-walled cylinder filled with a fluid under pressure. In such cases, stress components perpendicular to the plate are negligible compared to those parallel to it. In other situations, however, the bending stress of a thin plate cannot be neglected. One can still simplify the analysis by using a two-dimensional domain, but the plane stress tensor at each point must be complemented with bending terms.

Mathematical definition Mathematically, the stress at some point in the material is a plane stress if one of the three principal stresses (the eigenvalues of the Cauchy stress tensor) is zero. That is, there is Cartesian coordinate system in which the stress tensor has the form

σ = [ σ 11 0 0 0 σ 22 0 0 0 0 ] ≡ [ σ x 0 0 0 σ y 0 0 0 0 ] {\displaystyle \sigma ={\begin{bmatrix}\sigma _{11}&0&0\\0&\sigma _{22}&0\\0&0&0\end{bmatrix}}\equiv {\begin{bmatrix}\sigma _{x}&0&0\\0&\sigma _{y}&0\\0&0&0\end{bmatrix}}}

For example, consider a rectangular block of material measuring 10, 40 and 5 cm along the x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} , that is being stretched in the x {\displaystyle x} direction and compressed in the y {\displaystyle y} direction, by pairs of opposite forces with magnitudes 10 N and 20 N, respectively, uniformly distributed over the corresponding faces. The stress tensor inside the block will be

σ = [ 500 P a 0 0 0 − 4000 P a 0 0 0 0 ] {\displaystyle \sigma ={\begin{bmatrix}500\mathrm {Pa} &0&0\\0&-4000\mathrm {Pa} &0\\0&0&0\end{bmatrix}}}

More generally, if one chooses the first two coordinate axes arbitrarily but perpendicular to the direction of zero stress, the stress tensor will have the form

… excerpt ends here. Continue reading the full article.

Illustrations

Plane stress: Figure 7.1 Plane stress state in a continuum.
Figure 7.1 Plane stress state in a continuum.
Plane stress: Figure 7.2 Plane strain state in a continuum.
Figure 7.2 Plane strain state in a continuum.
Plane stress: Figure 8.1 - Stress transformation at a point in a continuum under plane stress conditions.
Figure 8.1 - Stress transformation at a point in a continuum under plane stress conditions.
Plane stress: Figure 8.2 - Stress components at a plane passing through a point in a continuum under plane stress conditions.
Figure 8.2 - Stress components at a plane passing through a point in a continuum under plane stress conditions.
Plane stress: Figure 8.3 - Transformation of stresses in two dimensions, showing the planes of action of principal stresses, and maximum and minimum shear stresses.
Figure 8.3 - Transformation of stresses in two dimensions, showing the planes of action of principal stresses, and maximum and minimum shear stresses.

Worked examples

Example 1 — a first encounter with Plane stress

Start with the simplest possible case. Write down what Plane stress claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Plane stress before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Plane stress ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Plane stress

In research
Plane stress appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Plane stress in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Plane stress is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mechanical engineering, Metallurgy, so understanding it makes those chapters shorter.
In everyday life
Look for Plane stress outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Plane stress in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Plane stress means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Plane stress out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Plane stress in simple terms?

In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can…

Why does Plane stress matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Plane stress?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Plane stress.

Tags

  • Mechanical engineering
  • Metallurgy

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