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Mechanics of Materials

Stress State: Dive into the World of Forces and Stresses!

Are you ready for an exciting journey into the world of physics? Then buckle up and discover the secrets of the stress state with us!

What is stress? Imagine you are building a giant Lego structure. The individual blocks push and pull against each other - that's exactly what stress is! In this course, you will learn how to calculate and understand these forces.

Stress components: Break down stress into its individual parts and discover how they interact. Just as a puzzle consists of many pieces, stress is also made up of different components.

Calculation: Crack the code of stress calculation! With a few clever formulas and tools, you can determine the forces in any component.

Transformation: Stresses change depending on the perspective. Learn how to transform them into different cutting planes and thus make the whole story of the load in the component visible.

Maximum stresses: Where does the greatest danger lurk? Find out where the stresses are highest in the component and how you can minimize them.

Mohr's circle of stress: This ingenious tool helps you to visualize stresses and to grasp important information at a glance.

Discover the fascination of the stress state! In this course you will not only learn dry knowledge, but also immerse yourself in the world of engineering. With good explanations and exciting application examples, the stress state becomes child's play.

Together we are strong! We will accompany you on your journey and help you to understand the complex concepts of the stress state. With our support you will master every challenge and become an expert for stable constructions.

So what are you waiting for? Start your journey into the world of stress now!

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Table of Contents

1.6 Associated Shear Stresses

Shear Stress Buddies: The Thrilling Pair-Up!

Hey, ready for some stress? No, not the "I'm late for work" kind of stress, but the associated shear stresses kind! It may sound complicated, but trust me, it's a piece of cake.

Picture a rectangular block. Boring, right? But inside that block lies a thrilling secret – shear stress buddies.

This Figure 1.1.5 illustrates the derivation of the moment equilibrium.
Fig. 1.1.5: Derivation of the moment equilibrium

These buddies are like peas in a pod. They always come in pairs, with the same magnitude and orientation. Think BFFs in the world of stress.

How do we find these pairs? Easy, with a little mathematical magic.

Moment equilibrium is the magic word. We calculate the moment equilibrium about an axis parallel to the z-axis through the center of the block, and voila:

$$ \begin{aligned} \curvearrowleft M = 0 &= (\tau_{xy} \cdot dz \cdot dy) \cdot \dfrac{dx}{2} - (\tau_{yx} \cdot dz \cdot dx) \cdot \dfrac{dy}{2} + (\tau_{xy} \cdot dz \cdot dy) \cdot \dfrac{dx}{2} - (\tau_{yx} \cdot dz \cdot dx) \cdot \dfrac{dy}{2} \\ 0 &= 2 \cdot (\tau_{xy} \cdot dz \cdot dy) \cdot \dfrac{dx}{2} - 2 \cdot (\tau_{yx} \cdot dz \cdot dx) \cdot \dfrac{dy}{2} \\ 0 &= \tau_{xy} \cdot dz \cdot dy \cdot dx - \tau_{yx} \cdot dz \cdot dx \cdot dy \\ 0 &= (\tau_{xy} - \tau_{yx}) \cdot dx \cdot dy \cdot dz \\ 0 &= \tau_{xy} - \tau_{yx} \\ \tau_{xy} &= \tau_{yx} \end{aligned} $$

Note: Since equilibrium statements apply only to forces, we need to multiply the stresses by the area elements on which they act!

What does this mean?
  • Shear stresses in perpendicular cuts are tight as thieves.
  • They have the same orientation regarding the common edge.
  • They're inseparable.
  • Their indices can be swapped.
The same goes for the other shear stress buddies:
$$ \begin{aligned} \tau_{xy} = \tau_{yx} \qquad \tau_{yz} = \tau_{zy} \qquad \tau_{zx} = \tau_{xz} \end{aligned} $$

(1.2)

So remember: Shear stress buddies in perpendicular cuts are always equal and have the same orientation.

Are you ready for the next round of brain jogging?