CEEN 311 Mechanics of Materials | Colorado School of Mines | Professor Susan Reynolds | [email protected]

LE04 Shear Stress and Strain

​ ​ In LE01, LE02, and LE03, we discussed average normal stress (훔) and average normal strain (훆). ​ ​ ​ ​ Today, we will introduce average shear stress (흉) and average shear strain (휸). ​ ​ ​ ​

These images illustrate the differences between normal stress & strain and shear stress & strain: ​ ​ ​ ​

Fig.04a - Normal & Shear Stress (UNDEFORMED) Fig.04b - Normal & Shear Stress (DEFORMED) ​ ​ ​ ​ (image credit: tec-science.com) (image credit: tec-science.com) ​ ​ ​ ​ OBSERVATIONS: OBSERVATIONS: ​ ​

SOME FUN FACTS TO TELL YOUR FRIENDS ABOUT SHEAR STRESS: ​ ● it’s the tendency of 2 parallel planes to slip past each other (think of the cutting tool called “shears”) ● shear stress is always in-plane (parallel) to the theoretical cutting plane (usually the cross-section) ​ ​ ● shear stress is force per area (the shear force transferred through that plane over the shearing area)

LE04 Shear Stress and Strain | copyright Prof. Susan Reynolds 2020-21 | [email protected] | page 1 of 4

EQUILIBRIUM OF THE STRESS ELEMENT / STRESS CUBE

Remember: if a structure is in static equilibrium, then we can cut any FBDs we want and they are also in equilibrium, provided we add in the effects of whatever we cut away. These “effects” can be (1) forces and ​ ​ moments, or (2) an equivalent system of stresses. Let’s consider a stress element in the sheared plane above.

Step 1: Draw an arrow on the positive y-face in ​ the x-direction. That represents the direct shear stress (흉) between the hand and the solid material. It’s just a force / area.

Step 2: Do you think the material experiences ​ any normal stresses (흈x, 흈y, or 흈z)? ​ ​ ​ ​ ​ ​ Step 3: Force equilibrium in the x-direction: ​

Step 4: Moment equilibrium (ΣMA = 0): ​ ​ ​ Step 5: Force equilibrium in the y-direction: ​

Step 6: Verify moment equilibrium (ΣM0 = 0): ​ ​ ​

Fig.04c - Shear Stress on a Stress Cube Fig.04d - Shear Stress on a Stress Element ​ ​

Here is the big takeaway from this activity. It’s important. Whenever you find any ONE shear stress, then there are three more shear stresses on other planes. You can find them by taking a FBD of a stress cube (or stress element) and applying the equations of equilibrium. This will be extremely important a little later in this course.

DEFORMED SHAPE OF THE STRESS ELEMENT / STRESS CUBE

Let’s visualize the deformation of this stress element. We’ll use 휸 (gamma) to measure the warping of angles.

By the way, this state of stress has a special name: “planar shear stress” or “in-plane shear stress.” ​ ​ ​ ​

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THE TRICKY SIGN CONVENTION FOR SHEAR STRESS (WHICH IS DIFFERENT FROM THE SIGN CONVENTION FOR SHEAR FORCE) ​…

The sign convention for shear stress always causes problems for students. ALWAYS. In fact, ​ ​ my most popular YouTube video (GoStructuresGo ​ -> Muscadine) deals with this very topic. ​ ​

This image shows ALL possible stresses. They are all drawn in the positive direction. ​

● for the 3 normal stresses (훔), we use a single subscript (x, y, or z) ● for the 6 shear stresses (흉), we use two subscripts: ○ the first subscript is for the PLANE ○ the second subscript is for the DIRECTION OF THE SHEAR STRESS Fig.04e - All possible stresses in 3D (this is as complicated as it gets!) ​ ​

SHEAR STRESS (흉), SHEAR STRAIN (휸), & THE SHEAR MODULUS (G) ​ ​

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Example problem for shear strain (and clarification on sign convention):

Forces have been applied in-plane to a very flexible plate in a laboratory setting. Rectangle A-B-C-D depicts the undeformed geometry. Polygon A’-B’-C’-D’ depicts the deformed geometry. What is the shear strain at point C?

Clarification on sign convention: using the definition of shear strain above, an angle that becomes more acute is positive and an angle that becomes more obtuse is negative. You may be tested on this sign convention on the ​ ​ Fundamentals of Engineering (FE exam). In practice, it is MUCH more common to report shear strain as a magnitude and illustrate the directionality on a stress element or stress cube, just like shear stress.

SUMMARY: (1) There are two simple stresses: normal stress and shear stress. ​ (2) Normal stress causes elongation/shortening; shear stress causes warping. (3) We quantify shear through shear stress (흉), shear strain (휸), and the shear modulus (G). ​ ​ (4) If you can identify 1 shear stress, you know that there are 3 others due to equilibrium. (5) We always report shear stress as a MAGNITUDE. It never gets a negative sign. (6) We show the DIRECTIONALITY of the shear stress by drawing a stress element / cube.

(7) The average shear stress equation (흉avg = V/A) only applies to a few specialized situations. ​ ​ ​ ​ ​ ​ For now, please do not use that equation. (We’ll use it much later in the course ) …

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