calculating slope of beam

Professional Engineer & PE Exam Forum

Help Support Professional Engineer & PE Exam Forum:

This site may earn a commission from merchant affiliate links, including eBay, Amazon, and others.
K

ketanco

Well-known member
Joined
Oct 28, 2010
Messages
157
Reaction score
0
hello,

i am studying beam deflection methods such as double integration or moment area method or conjugate beam method etc right now.

how can we know where the slope is 0 for a simply supported beam, with any distributed or point loads on it.

is the beam slope 0 where moment diagram is at its maximum or minimum ? if not is there a shortcut of how to find where slope is 0?

 
The moment is zero at the inflection point. Therefore it's a safe assumption that the slope of the deflected beam on a horizontal simply supported beam is zero at the inflection point

 
At the inflection point, the CURVATURE is zero, not the slope. A simply supported beam with downward loads only is bent entirely in single curvature, there is no inflection

 
you are correct and I apologize for my misuse of the terminology. However, when the curvature is zero wouldn't the slope also be zero? Deflection decreases and starts to return back to the support.

 
civilized_naah said:
No, when the curvature is zero, the rate of change of the slope is zero, but the slope could be no-zero
Click to expand...
call me simple-minded but you need to draw that for me. I'm just not seeing it.....unless the supports are not in the same horizontal plane.

 
Imagine a simple span beam in reverse curvature (two like moments - either both clockwise or both anticl) applied at the two ends. The deflected shape is sort of (only looks like, not really) a complete sine wave. At the midpoint, curvature is zero, slope is not. This is an inflection point. At the quarter and three-quarter points, deflection is maximum/minimum, curvature is not zero, slope is zero

Simpler case - simple span with distr load. At midpoint, bending moment max, curvature max, slope zero, deflection max

 
ok so bottom line, and back to my original question, is there a simple way to determine where the SLOPE is zero?

 
Let Y = deflection. Y(x) = deflection as a fucntion of x. The first derivative of deflection is slope Y'(x) or dy/dx. Therefore the beam slope would be zero whenever the beam deflection is at a maximum or minimum. Imagine the deformed profile of the beam. The slope is zero wherever the line tangent to the profile is zero. For a simply supported beam this will always be whereever the deflection is the maximum (at the centroid of the load). This is probably the easiest way to see it. For a cantilevered beam, slope will be zero at the fixed end. For a fixed-end beam slope will be zero at the fixed ends and also at the point of highest deflection.

The second derivative of deflection (or first derivative of slope) is called the curvature y''(x) or d^2y/dx^2. If you multiply the curvature by the constant EI, you get M(x)... that is the bending moment. So if you had a bending diagram you would have divide it by EI (although you can skip this if all you care about is where the slope is zero) to visually integrate it and apply the proper boundary conditions. Then see where that is zero.

 
Last edited by a moderator:
Well, putting everything in a simplistic way all you need to do is to draw a shear diagram and if it is zero anywhere you'll be at a zero slope.

 
kevo_55 said:
Well, putting everything in a simplistic way all you need to do is to draw a shear diagram and if it is zero anywhere you'll be at a zero slope.
Click to expand...
Does this account for fixed ends though? It seems that here the shear is non-zero, but the slope is. Looks like this might capture some, but not all places.

 
Last edited by a moderator:
^^ For the Civil PE level of question, I would argue yes.

Again, we're not in the SE forum.

 
kevo_55 said:
Well, putting everything in a simplistic way all you need to do is to draw a shear diagram and if it is zero anywhere you'll be at a zero slope.
Click to expand...
this may be ok for some conditions but doesnt seem correct to me for all cases. i know that you mean when the shear is zero, it is transitioning from positive to negative or the opposite, but what if it stops at zero, continues like that for a long distance, and then again becomes some other value. for some load combinations this is possible. and at those zero shear locations, all along that length, the slope is not zero.

 
palvarez83 said:
Let Y = deflection. Y(x) = deflection as a fucntion of x. The first derivative of deflection is slope Y'(x) or dy/dx. Therefore the beam slope would be zero whenever the beam deflection is at a maximum or minimum. Imagine the deformed profile of the beam. The slope is zero wherever the line tangent to the profile is zero. For a simply supported beam this will always be whereever the deflection is the maximum (at the centroid of the load). This is probably the easiest way to see it. For a cantilevered beam, slope will be zero at the fixed end. For a fixed-end beam slope will be zero at the fixed ends and also at the point of highest deflection.

The second derivative of deflection (or first derivative of slope) is called the curvature y''(x) or d^2y/dx^2. If you multiply the curvature by the constant EI, you get M(x)... that is the bending moment. So if you had a bending diagram you would have divide it by EI (although you can skip this if all you care about is where the slope is zero) to visually integrate it and apply the proper boundary conditions. Then see where that is zero.
Click to expand...
thanks but how do you know where the deflection is maximum or minimum? i know that where the deflection is max or min the slope is zero... again i am asking for a simple way... if there is a shortcut etc... of course we can ultimately determine where the slope is zero by following the methods as you described in second paragraph... i only wanted to know a shortcut to see where slope is zero.

 
Last edited by a moderator:

Similar threads

W
Bending device Calculate required forces
Replies
1
Views
282
Lumber Jim
Lumber Jim
EBAT75
STEEL MANUAL - TABLE 3-10
2 3
Replies
49
Views
10K
EBAT75
EBAT75
S
Concrete Beam HW HELP!
Replies
2
Views
2K
MA_PE
MA_PE
O
Structural Beam Calculator
Replies
0
Views
1K
Optimal Beam
O
A
Suggestions for Passing SE Exam Part 1 – by Andy Liu
Replies
1
Views
5K
Serg305
S

Latest posts

Back
Top