ENGI 1331代做、代写R程序语言

日期：2024-04-27 08:39

ENGI 1331: Project 3 - Problem 3 Sample Calculations

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Given:

A simply supported beam of length L subject to a force F. The deflection of

the beam y is characterized by the deflection equation

where I is the moment of inertia of the cross section, R is the reaction force on the beam at the left end, 𝜃 is the

clockwise rotational angle of the beam at the left end, and E is the Young’s modulus of the beam’s material. I, R,

𝜃 and E are all geometry- or material-based constants. E is found in a table (MaterialElasticity.mat), and the

other values are found with the following (already derived) equations

Since all the properties are either given or can be directly calculated, to find deflection at a point simply find the

coefficients I, R, 𝜃 and E and then plug them into Equation 1.

Case 1: Single force

For the beam shown with width w = 0.2 [m], height h = 0.2 [m], and modulus

E = 190 * 10^9 [Pa], calculate the beam deflection at x = 2 [m] and x = 5 [m].

First, calculate the constants:

(2𝐿 − 𝑎)(𝐿 − 𝑎) =

(500)(3)

6(190∗109)(1.33∗10−4)(10)

(17)(7) = 1.682 ∗ 10−5

[rad]

For 𝑥 = 2 (𝑥 ≤ 𝑎), use the first half of Eq. 1:

𝑦(2) = −𝜃(2) +

𝑅(2

3)

6𝐸𝐼

= −(1.682 ∗ 10−5

)(2) +

(350)(2

3)

6(190∗109)(1.33∗10−4)

→ −1.517 ∗ 10−5

[m] 𝑜𝑟 ~0.015 [mm]

For 𝑥 = 5 (𝑥 > 𝑎), use the second half of Eq. 1:

𝑦(2) = −𝜃(5) +

𝑅(5

3

)

6𝐸𝐼

−

𝐹

6𝐸𝐼

(𝑥 − 𝑎)

3 = −(1.682 ∗ 10−5

)(5) +

(350)(5

3

)

6(190∗109)(1.33∗10−4)

−

500

6(190∗109)(1.33∗10−4)

(2)

3 →

−1.781 ∗ 10−4

[m] 𝑜𝑟 ~0.18 [mm]

This application can be generalized with x as a vector instead of a single value to find the deflection at all points

on the beam.

=3 [m]

=500 [N]

=10 [m]

ENGI 1331: Project 3 - Problem 3 Sample Calculations

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Case 2: Multiple forces

Similar case, but with more than one force. Simply treat the problem as

two instances of Case 1. Calculate the deflection caused by force F at

distance a, then calculate the deflection caused by force F2 at distance a2.

The sum of those deflections will be the total deflection across the beam.

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