Chapter 2 Review of Forces and Moments - Brown University
Chapter 2 Review of Forces and Moments - Brown University
Chapter 2 Review of Forces and Moments - Brown University
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Example 3. The structure shown is subjected to<br />
a force T acting at E along the line EF.<br />
Calculate the moment <strong>of</strong> T about points A <strong>and</strong> D.<br />
[( /2) ( /2)tanθ<br />
] [ ]<br />
{ θ}<br />
k<br />
MB = r× F= −L i− L j × Pi −Wj<br />
= ( L/2) W + Ptan<br />
This example requires a lot more work. First we<br />
need to write down the force as a vector. We<br />
know the magnitude <strong>of</strong> the force is T, so we only<br />
need to work out its direction. Since the force<br />
acts along EF, the direction must be a unit vector<br />
pointing along EF. It’s not hard to see that the<br />
vector EF is<br />
<br />
EF = ai − 3aj + 2ak<br />
We can divide by the length <strong>of</strong> EF ( a 14 ) to<br />
find a unit vector pointing in the correct<br />
direction<br />
e = ( i− EF<br />
3 j+<br />
2 k )/ 14<br />
The force vector is<br />
F= T ( i− 3j+<br />
2 k )/ 14<br />
Next, we need to write down the necessary position vectors<br />
Force: r = 2ai+<br />
3aj<br />
Point A: rA<br />
=−2ai<br />
Point D: rD<br />
= 4aj<br />
Finally, we can work through the necessary cross products<br />
MA<br />
= ( r− rA)<br />
× F<br />
MD<br />
= ( r− rD)<br />
× F<br />
= (4ai+ 3 aj) × T( i− 3j+<br />
2 k) / 14 = (2 ai− aj) × T( i− 3j+<br />
2 k) / 14<br />
i j k<br />
i j k<br />
=<br />
( Ta )<br />
/ 14 4 3 0<br />
1 −3 2<br />
( i j k)<br />
= Ta 6 −8 −15 / 14<br />
Clearly, vector notation is very helpful when solving 3D problems!<br />
( Ta )<br />
= / 14 2 −1 0<br />
1 −3 2<br />
( i j k)<br />
= Ta −2 −4 −5 / 14<br />
Example 4. Finally, we work through a simple problem<br />
involving distributed loading. Calculate expressions for the<br />
moments exerted by the pressure acting on the beam about<br />
points A <strong>and</strong> B.<br />
j<br />
A<br />
x<br />
dF = p dx<br />
dx<br />
p (per unit<br />
length)<br />
B<br />
i<br />
L