First year civil engineering, stress of a curved beam

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First year civil engineering, stress of a curved beam 04/12/15 (Sun) 03:35:07 No.2084

Okay /sci/, please help a first year engineering student understand stress and strain in beams. We're covering Euler-Bernoulli beam theory, and they (the lecturers) have stated that curvature is the rate of change of the slope of a beam in deformation. That I understand. I understand how curvature can be related to the radius of a circle, given that we assume the deformation to be in that shape (i.e.: a minimal deformation).

We arrive at
Stress = Modulus X Strain = Modulus X Curvature X Distance from the neutral axis
(Picture 1)

Then this comes up:
>The force acting on the cross-sectional area is equal to the integral of the stress on the cross-sectional area (Picture 2)
I don't understand how they got to that. Can anyone explain?

04/12/15 (Sun) 03:36:15 No.2085

>>2084
Specifically, why integrate?

10/03/15 (Sat) 23:21:06 No.3132

>>2085

Because you're summing the stresses along the y axis.

10/03/15 (Sat) 23:24:08 No.3133

>>3132

Wow, I just noticed how old this thread is.

10/05/15 (Mon) 07:14:17 No.3144

>>2085

>why integrate?

m8 if you're working with a continuous object and not just a one-dimensional stick you have no choice but to integrate

>>2084

>>The force acting on the cross-sectional area is equal to the integral of the stress on the cross-sectional area (Picture 2)

>I don't understand how they got to that. Can anyone explain?

That's the definition of stress; stress is a tensor whose three components represent stretching and torsional forces in a continuum per unit of surface, in all three directions. Therefore, if you pick a cross-section, to get a force acting on it, all you need to do is integrate that component of the stress tensor along entire cross-section.