MATH7019: Week 11

Assignment 1

Question 1

(g) In this question you were asked to use your model

to extrapolate the consumption for this year and 2020. Your model, for Sweden, was

.

This is the consumption years after 2005. The is like the — it denotes that depends on — it does not mean .

Question 2

(j) In this question you were asked to find the slope of your parabola at . Now should have worked with

as you had found but instead you differentiated

.

In this case was a constant and so the derivative should have been

not as you had.

Question 3

(e) You are given (span,max deflection) data and are asked to to a log-linear least squares fit. First you linearised the relationship successfully as

Note this is in the form

,

and so if you plot vs you get data that looks like a straight line to which you can fit a line and so find and .

Instead of doing least squares with you just did least squares with the original data… the original data does not have a straight-line relationship and so what you do doesn’t really make any sense.

The examples on p.37 to 42 of the notes show what you need to do with these.

Question 4

(h) You were supposed to use your model to find the grain size such that . Similarly to the issue with Q. 1(g), just means that depends on — . You needed to find the such that was equal to fifty. Your model was

so you needed to solve

.

Assignment 2

Constant Load on a Simply Supported Beam

Question 3 (c)

To see why for simply supported please see the bottom of p.99

Question 6 (f)

In your comment to your graph you should have noted that the maximum bending moment takes place where the shear crosses the -axis — aka where the shear was zero. You could have also said that the bending moment was zero on the endpoints.

Linear Load on Fixed Ends Beam

Question 5

You find

and

You take — correctly as it is fixed ends.

Now you calculate and (with rounding errors too) and you get

as I said you have rounding errors and you should use fractions.

Now you went on to solve these as simultaneous equations… but these aren't equations at all so you were bound to go wrong and you did.

See

Now we know that both and and so and … so what you should have had was, remembering these are and

These are genuinely equations. See the marking scheme on p.93 of the notes to see this done properly.

Question 6

First of all you wrote are equal to zero… is not the same as 'and'! You should write and are equal to zero or not … you can have with neither equal to zero…

Your picture of a beam with fixed ends and on the left and on the right showed a lack of understanding. Please see the discussion starting on p.99 and running into p.100 to understand this properly.

Question 7 (a)

You were supposed to find the location of the maximum bending moment. You found it to be at or — yet your beam had a length of only four and so neither of these are on the beam… this should have told you that something was wrong.

Question 8 (a)

In your comment to your graph you should have noted that the maximum bending moment takes place where the shear crosses the -axis — aka where the shear was zero. You could have also said that the bending moment was not zero on the endpoints as the beam was fixed ends.

Question 8(b)

Your graph is clearly wrong… have a look at it. It should look like Figure 2.26 on p.91 of the notes.

Cantilever with Point Load(s)

Question 3

… why would you think this?

Non-Integrable Load

Question 1

It wasn't a UDL. There is even a picture of the load on the previous page.

Question 9

You never used the chain rule. You had

.

To differentiate this you can fix the constant and then when you are differentiating look on the function as:

Using the Chain Rule:

.

The derivative of the inside can be carried out with a second chain rule or else after multiplying out.

This is first year material but as I have discussed in class semesterisation and other issues means that ye have forgotten a lot of this stuff.