Lesson October 25th

Lesson October 25th#

During today’s lesson you’ll work on two complex exercises on the topic of Temperature influences. Please ask your questions regarding the homework as well!

Exercise 1#

Gegeven is de volgende constructie:

../../_images/structure31.svg

Fig. 185 Constructie, \(EI = \cfrac{800}{3} \ \rm{kNm^2}\)#

We gaan deze constructie doorrekenen met behulp van differentiaalvergelijkingen

Exercise

Wat is \(\kappa_{\rm{T}}\)?

Solution

\[\kappa^T = -\cfrac{\alpha \cdot T}{h} = - \cfrac{0.0001 \cdot 30}{0.2} = -0.015 \ m^{-1}\]

Exercise

Bepaal met behulp van de differentiaalvergelijkingen de uitdrukkingen voor de snedekrachten en verplaatsingen. Merk op dat twee randvoorwaarden direct twee integratieconstantes geven.

Solution

Voor deze constructie gelden den onderstaande randvoorwaarden:

\[ w \left( 0 \right) = 0 \]
\[ M \left( 0 \right) = +6 \rm{kNm} \]
\[ w \left( 8 \right) = 0 \]
\[ \varphi \left( 8 \right) = 0 \]

Hieruit volgt voor de snedekrachten en verplaatsingen:

\[ V\left( x \right) = C_1 \]
\[ M\left( x \right) = C_1 \cdot x + 6 \]
\[ \kappa \left( x \right) = \cfrac{M}{EI} = \cfrac{3}{800} \cdot C_1 \cdot x + \cfrac{6 \cdot 3}{800} - 0.015 = 0.00375 \cdot C_1 \cdot x + 0.0075 \]
\[ \varphi \left( x \right) = 0.001875 C_1 \cdot x^2 + 0.0075 \cdot x + C_3 \]
\[ w \left( x \right) = -0.000625 \cdot C_1 \cdot x^3 -0.00375 \cdot x^2 - C_3 \cdot x + 0 \]

Exercise

Bepaal de waardes van de integratieconstantes

Solution

\[ C_1 = -0.375 \]
\[ C_3 = -0.015 \]

Exercise

Solution

Als de temperatuur verder toeneemt, dan wordt de absolute waarde van maximale verplaatsing eerst kleiner dan groter.

Exercise

Waar is het buigpunt?

Exercise 2#

Given the following structure.

../../_images/struct.svg

Exercise

Find the curvature and strain due to the temperature influence

Exercise

Create a statically determinate system which uses the following displacement constraint: \(w_{\rm{C}}^{\rm{BC}} = w_{\rm{C}}^{\rm{CD}}\) with \(N_{\rm{CD}}\) as the statical indeterminate force. It is shown in the figure below without equivalent forces for the temperature influence.

../../_images/SD2.svg

What are the force distributions and displacements due to the distributed load as a function of \(N_{\rm{CD}}\)? So don’t include kinematical equivalent forces due to the temperature influence.

Exercise

What are the force distributions and displacements including temperature influences and as a function of \(N_{\rm{CD}}\)?

Exercise

Use the displacement constraint to solve for \(N_{\rm{CD}}\)

Exercise

What are the bending moments?

Exercise

What is the curvature?

Exercise

What are the displacements?

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