Exam Friday January 31th

# Matplotlib compatibility patch for Pyodide
import matplotlib
if not hasattr(matplotlib.RcParams, "_get"):
    matplotlib.RcParams._get = dict.get

Exam Friday January 31th#

Today you’ll make the second exam assignment covering Continuuum mechanics including its prerequisites and/or the first exam assignment covering Buckling including its prerequisites. For more information about the exam see the assessment information in course information

Exam assignment 2 Continuum mechanics#

Your own submission and its grading will be available on ANS exam assignment Continuum mechanics 2 after the exam.

Given is the following structure:

../../_images/structure10.svg

  • Find the stress state with maximum shear stresses in point \(\text{D}\) and draw the (rotated) stress element for this stress state

  • Find the principal stresses in point \(\text{D}\) and draw the (rotated) stress element for this stress state.

  • Draw the required yield stress envelope for Tresca’s yield criterion in the relevant principal stress plane including values.

Solution

In \(\text{D}\) the shear stress \(0\).

Relevant cross-sectional properties are:

  • \(A = 20250 \text{ mm}^2\)

  • From top fibre downwards, the normal force centre is at \(70 \text{ mm}\)

  • \(I_{zz} = 37462500 \text{ mm}^4\)

The normal stress is \(-25 \text{ MPa}\) with no shear stresses.

../../_images/xy.svg

As this is the only stress, the principal stresses are: \(\sigma_1 = -279 \text{ MPa}\), \(\sigma_2 = \sigma_3 = 0 \text{ MPa}\).

Because of \(\sigma_2 = 0 \text{ MPa}\), the maximum shear stress is \(\cfrac{279}{2} = 139 \text{ MPa}\) at \(45^{\text{o}}\). Or it can be found by rotating the right plane \(\cfrac{1}{4}\pi \text{ rad}\):

  • \(\sigma_{\bar x \bar x} = -279 \cos \left(\cfrac{1}{4}\pi\right)^2 = -139 \text{ MPa} \)

  • \(\tau_{\bar x \bar y} = -279 \sin \left(\cfrac{1}{4}\pi\right) \cos \left(\cfrac{1}{4}\pi\right) = 139 \text{ MPa}\)

../../_images/rotated.svg

This can also be found using Mohr’s circle:

../../_images/mohr4.svg

The Tresca yield criterion looks as follows:

../../_images/tresca.svg

Exam assignment 1 Buckling#

The exam assignment was provided as shown here

Solution

The solution is shown here

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