Finding Maximum Deflection of Bending Curve

[teng125]

if i have a bending curve eqn of w'(x) = (q/6)<x-l>^3 - ql/4 <x-0>^3 - 3ql/4 <x-l>^2 + (9ql^3)/24

suppose to find the deflection of maximum of the beam, we have to set w'(0)=0 .Am i right??

then if it is right,how can i find the value for x on which the max bending occur because i don't know how to factorize the heaviside function such as <x-l>^3


anybody pls help
thanx

 

[Meson]

All right, you can look for the explanation in PDF format by clicking the link below:

https://www.physicsforums.com/attachment.php?attachmentid=7405&stc=1&d=1154053179"

Good luck.

 

[Pyrrhus]

if $\nu$ represents the deflection function then

$$\nu (x) = f(x)$$

therefore by using you knowledge on differential calculus, in order to look for a max or min you must differentiate $\nu (x)$ (coincidentially the rotational angle or slope) and set it to 0, find the x and substitute in your deflection function. This is the way for the integration of the deflection differential equation.

 

[teng125]

ya,that's exactly what i stated above.But,i don't know how to find x as ican't find a way to factorize

pls show me
thanx

 

[Meson]

In the attachment above I stated that it is not necessary to find the x and I stated the reasons and also an alternative to solve it. Didn't you read it, teng125?

 

[teng125]

the link is invalid

 

[Meson]

All right then, I rewrite it here.

I have corrected your original bending curve by inserting the factor of 1/EI where EI is the bending stiffness of the beam I assumed.

$$w'(x)=\frac{1}{EI}[\frac{q_{0}}{6}(x-l)^3-\frac{q_{0}l}{4}(x-l)^3-\frac{3q_{0}l}{4}(x-l)^2+\frac{9q_{0}l^3}{24}]$$

To find the position along the beam at which the deflection is maximum, w' (x) = 0, NOT w' (0) = 0.



Hence, expanding the unitstep or heaviside functions as if they were normal polynomial functions gives

$$w'(x)=\frac{1}{EI}[\frac{q_{0}}{6}(x-l)^3-\frac{q_{0}l}{4}(x-l)^3-\frac{3q_{0}l}{4}(x-l)^2+\frac{9q_{0}l^3}{24}]$$

$$= \frac{1}{EI}[\frac{q_{0}{6}(x^3-l^3+2l^2x-2lx^2)-\frac{q_{0}l}{4}(x^3\mbox{...})\mbox{...}$$

$$=\mbox{...}$$

As you can imagine, it is very complicated. I doubt you would have time solving for x in the examination.



However, you can predict the position along the beam at which the deflection is maximum by observing the moment diagram M(x). Just look at the stationary point(s) and the corresponding position x yields the maximum deflection.

Anyway, our professor has told us that this kind of problem will not be asked in the examination. If it were asked, you just need to write

$$w'(x)=\frac{1}{EI}[\frac{q_{0}}{6}(x-l)^3-\frac{q_{0}l}{4}(x-l)^3-\frac{3q_{0}l}{4}(x-l)^2+\frac{9q_{0}l^3}{24}]=0$$
solving for x gives the position along the beam at which maximum deflection occurs.

for the examination is to test your knowledge and skills in mechanics, not elementary pure mathematics (leave this task to the mathematicians).


some of the latex appears to be incorrect, they refuse show what I wanted to display.

 

[teng125]

okok...thanx very much...besides,do u know any websites or materials that shows or teach students how to draw bending moment diagrams for the chapter above??

 

[Meson]

Refer to your lecture or exercise notes perhaps? If you have time, you can make a small table listing all the key points. For example:

   x	| V(X)	|  M(x)	|
	|	|	|
-------------------------
   l	| 1/2ql	| ql^2	|
	|	|	|
-------------------------
  2l	| ... 	| ...	|
	|	|	|
-------------------------
 3/2l   | ...	| ...	|
	|	|	|
-------------------------

By identifying the nature of the functions within each segment ((1/2)l, l, 2l etc), that is to check if they are linear, quadratic or cubic, then plot the curve. Of course, you will have to know the basic shape of the curves of each type of function.

 

[teng125]

for the basic shape of the curves of each type of function, do u have any materials for them??or any recommended websites??

thanx

 

[Meson]

I hope that you were joking. You know the linear (a line), quadratic curves and cubic curves, don't you?

Anyway, I have prepared the shear and bending moment diagrams for the present problem. As you expected, they are piecewise-connected functions due to the fact that you have defined them using unitstep/singularity/heaviside functions.

Shear Force, V(x)

Bending Moment, M(x)

Note that the y-axis of the moment diagram is reflected about the x-axis by convention. Now you see that the maximum bending moment occurs at $$x=\frac{l}{2}$$. Hence, inserting this value of $$x$$ into the deflection line/curve above yields the maximum deflection.