00:04
In this problem, we are given a linearly decreasing low density between some unknown density force downward of w0 at the, at one end.
00:28
At the opposite end, there is an 800 pounds per foot density force density value.
00:37
We are asked to find the reaction force at supports a and b, given their location along the beam at 5 feet from left and 12 feet from left.
00:55
Or the span between them is 7 feet.
01:02
Later, we're given that, or this program also gives us the value for w suburb.
01:10
Not, our force density value at the extreme left -hand side of the beam, we're given a value of 400 pounds per foot.
01:22
The approach we're going to use to solve for these reaction forces at each of the supports is to first write out the load density, this linearly varying relationship, as a function of the beam coordinate x, that's the span from the leftmost, point.
01:45
And in terms of the arbitrary parameter that we are given, but for the next problem, it's better that we keep it arbitrary as long as we can.
01:55
This parameter we will symbolically represent as w -sub -not.
02:01
Next, we'll integrate that function to find the total load, integrated over the span coordinate x, to find the total load that we'll denote here as f total and it's going to be pointing downward from the load.
02:17
The reaction has to exactly oppose that.
02:24
We will also calculate the centroid location on the beam using the function we derived in step one.
02:34
Finally, we will insert the specific value we're given of 400 pounds per foot for w -0, and then we'll solve for the reaction forces at each of the support.
02:45
Using the usual statics equations that were given.
02:52
If something is static, it cannot move linearly in space.
02:58
It can't have any translations.
03:00
So the opposing forces of the supports, the sum of those forces will exactly cancel the total force of the load.
03:18
And the other thing is, since it's not rotating in space, the torques about the centroid must be equal and opposite.
03:28
So this total torque is equal to zero.
03:32
So for step one, again, we're going to write out an equation for our low density function along the span of x.
03:41
We have our endpoint at 800 pounds per foot.
03:47
We have our starting point as w sub not.
03:56
So we're going to subtract, and that should be.
04:06
My apologies.
04:08
I just double -checked the problem, and i miswrote the value here.
04:14
It's not 800 pounds per foot.
04:16
It is 300, as is correctly stated in my equation.
04:24
Fix that real quick.
04:26
Sorry about that.
04:27
So going back to our derivation of the four -stance -the - equation, we have 300, the y -coordinate at the final location, minus w -s -s -not, the y -coordinate at the initial location, over run of 12 feet.
04:43
So a total of 12.
04:44
That's our slope.
04:46
Then for a linear equation, we multiply the slope by our dependent variable x, and then we add the offset and the offset is at x equals zero we're given a value of w sub not for our offset thus we have our equation for the load density over the span of x our dependent variable here next step is to integrate over the span of x in order to get the total load so we have our equation just derived above as a function of x times the infinitesimal dx that we sum over the span perform the integration we are and we're integrating from 0 to 12 the total span of the beam our independent variable x goes to x squared over 2 rather that was x d x goes to x squared over 2 and w said not d x goes to wx.
06:03
We evaluate from 0 to 12 and we are left with this equation for our total load in terms of w sub not.
06:17
Secondly, we can find the centroid by this time integrating the product of our x coordinate times our force density function performing and then our infinitesimal dx performing that integration, evaluating from 0 to 12.
06:40
And i'm sorry, i went too fast there, that that numerator, the integration of the product of xw sub -x, dx, is over our total load, which we just calculated or worked out in the step above.
07:02
So evaluating our numerator, we have 300 minus w -sub -0 over 12 times x -cubed over 3 plus w -sub -not x squared over 2.
07:17
That numerator, the both terms evaluated from 0 to 12, and then dividing by our total load relationship below, this can be expressed as follows here.
07:34
Our step three, we will use those two equations from statics.
07:40
Forces must sum to the total load, the reaction forces, that is.
07:45
And the torques must sum to zero.
07:49
The magnitude of the total load by plugging in are a w -sabnot of 400 pounds per foot.
07:57
If you plug that into this equation here, you should get 4 ,200 pounds for our total load.
08:06
We also know that it's not rotating in space, so the opposing torques must exactly cancel each other...