calculating slope of beam

[ketanco]

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?

 

[MA_PE]

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

 

[civilized_naah]

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

 

[MA_PE]

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]

No, when the curvature is zero, the rate of change of the slope is zero, but the slope could be no-zero

 

[MA_PE]

No, when the curvature is zero, the rate of change of the slope is zero, but the slope could be no-zero

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.

 

[civilized_naah]

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

 

[MA_PE]

fair enough

 

[ketanco]

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

 

[palvarez83]

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.

 

[kevo_55]

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.

 

[palvarez83]

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.

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.

 

[kevo_55]

^^ For the Civil PE level of question, I would argue yes.

Again, we're not in the SE forum.

 

[ketanco]

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.

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.

 

[ketanco]

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.

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.

 

[Ble_PE]

^The only shortcut is if your beam matches one of the beams from the beam tables in the AISC manual. If so, just look it up.

 

[palvarez83]

^What he said.