Uniaxial State of Stress - Any Section Angle, Step 1

Normal and Shear Stresses at an Arbitrary Angle

Step 2: Determining the Normal Force

N_{ξ}

as a Function of the Angle

φ

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Practice Exercises

Table of Contents

Step 1: Determining the Cross-Sectional Area $A^{*}$ as a Function of the Cutting Angle $φ$

Slice and Dice with Angles

Ready for some geometry fun?

Today, we're going to explore the cross-sectional area $A^{*}$ , but not just any area, the area as a function of the cutting angle $φ$ . Sounds complicated? It's not, I promise!

First, let's sketch two cross-sections: a perpendicular one and an arbitrary, oblique one.

This figure shows the dimensioning of the cross-sectional area A* with the width b and length s, as well as the dimensioning of the cross-sectional area of the perpendicular cut A with the width b and height h. — Fig. 1.2.8: Determination of the Cross-Sectional Area $A^{*}$

$A$ and $A^{*}$ look like rectangles, and we can calculate their areas with the formulas:

\begin{aligned} (1) & A & = h \cdot b & (perpendicular) \\ (2) & A^{*} & = s \cdot b & (oblique) \end{aligned}

But wait! The oblique area $A^{*}$ depends on the angle $φ$ . The more oblique the cut, the larger or smaller the area becomes.

How do we find the connection?

Easy peasy with a triangle!

This figure shows a triangle with the height h of the perpendicular cross-section, the length s of the arbitrary cross-section, and the angle Phi. — Fig. 1.2.9: Geometric Relationships at the Cross-Section

The height of the triangle is $h$ (the length of the perpendicular cross-section), the hypotenuse is $s$ (the length of the oblique cross-section), and the angle $φ$ is, well... the cutting angle $φ$ .

Using trigonometry (yes, I know, math) we can calculate $s$ as a function of $φ$ :

\begin{aligned} (3) & \cos (φ) & = \frac{adjacent}{hypotenuse} \\ (4) & \cos (φ) & = \frac{h}{s} \end{aligned}

This allows us to express $s$ as:

\begin{array}{r} (5) & s = \frac{h}{\cos (φ)} \end{array}

What's that?

Don't worry, it's just the cosine! It tells us the length of the hypotenuse ( $s$ ) relative to the adjacent side ( $h$ ).

Now we have everything we need!

The cross-sectional area $A^{*}$ as a function of $φ$ :

\begin{array}{r} (6) & A^{*} = s \cdot b = \frac{h}{\cos (φ)} \cdot b \end{array}

Remember:

$A^{*}$ gets larger when you cut more obliquely (larger $φ$ $\Rightarrow$ smaller $\cos φ)$ ).
$A^{*}$ gets smaller when you cut more perpendicularly (smaller $φ$ $\Rightarrow$ larger $\cos (φ)$ ).

By the way:

If we combine the product $h \cdot b = A$ , we can express $A^{*}$ as:

\begin{array}{r} A^{*} = \frac{A}{\cos (φ)} \end{array}

(7)

Practice Exercise

Practice Exercise F-1.1.4

The illustration shows a beam with a square cross-section, fixed on the left side. In the drawing, three sectional planes a, b, and c are indicated. Sectional plane a is perpendicular to the beam axis. Sectional plane b runs from the upper left to the lower right, with the given angle Beta describing the smaller angle at the top. Sectional plane c runs from the lower left to the upper right, with the given angle Gamma describing the smaller angle at the bottom. The side length of the cross-section is denoted as d. A tensile force F acts at the right end of the beam at the center of the cross-section.

Normal and Shear Stress at an Arbitrary Section Angle

A clamped beam with a square cross-section (side length $d = 20 mm$ ) is subjected to a tensile force $F = 10 kN$ along the beam axis.

Determine the average normal stress and the average shear stress...

...acting in cross-sectional plane a.
...acting in cross-sectional plane b ( $β = 50 °$ ).
...acting in cross-sectional plane c ( $γ = 40 °$ ).

Mechanics of Materials

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

Step 1: Determining the Cross-Sectional Area $A^{*}$ as a Function of the Cutting Angle $φ$

Normal and Shear Stress at an Arbitrary Section Angle

Mechanics of Materials

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

Step 1: Determining the Cross-Sectional Area A∗ as a Function of the Cutting Angle φ

Normal and Shear Stress at an Arbitrary Section Angle

Step 1: Determining the Cross-Sectional Area $A^{*}$ as a Function of the Cutting Angle $φ$