I Finding yield strength with a load/displacement curve

[Ferdiss]

Hi! I have done a simulation where I got out the load/displacement curve of a steel beam that is simply supported with an equally distributed load across the whole length (10 m). I want to field the point at which the steel yields. I am used to use the 0,2% value on a stress/strain curve, but how do I do it at a load/displacement curve? I have the units N on y-axis and m on x-axis.

Is it correct to say that strain is displacement(in z-direction, in the lateral direction of the beam) / l0(10 meters) ? Even though the displacement is in 'another direction' than the beam length?

So strain = 0,002 * 10 meters = 0,02 m?

Thank you!

 

[Lnewqban]

Welcome, @Ferdiss !

Sorry, I can't understand your description of the task at hand.
Do you have any specification of material and type of that beam?

 

[Ferdiss]

Hi, it is a steel beam S355. A simply supported beam with a uniformly distributed load :)

 

[Lnewqban]

Please, see:
https://mathalino.com/reviewer/mech...tion-to-problem-606-double-integration-method

https://mathalino.com/reviewer/mechanics-and-strength-of-materials/chapter-6-beam-deflections

 

[deajohn]

Hi @Ferdiss

No, you cannot compute strain that way. Unlike a bar in pure tension under axial load, this is a beam bending problem where strain is maximum at the outer fibers of the beam at the beam center location. If you have a plot of load vs displacement z, you can get the 0.002 z yield by drawing ne parallel from .002 z to the linear portion of curve and find the yield load and hence z. load. To convert to strain you need to compute z by the defection equation of the simple support beam (proportional to L^3/EI) and computes tress from the maximum moment (Mc/I) where c is rhe distance neutral axis to outer fiber. Then you compute strain using Hooke Law ( stress = E x strain) and find strain as proportional to zc/L^2 . This max strain at beam center.

 

[mfig]

In a tensile test, the onset of yielding will show on practically the entire cross-section. So, spotting the point at which the force-displacement curve deviates from linear is relatively easy. For bending, the situation is different. In bending, yield begins at the portions of the beam furthest from the neutral axis and spreads to the entire cross section as the load increases. Thus, the deviation of the force-displacement curve from linearity is more subtle.

To find an estimate of the yield strength from a force-displacement curve, where the force here is the value of the distributed load q and the displacement is that of the beam at L/2, look for the first sign of deviation from linear. At that load, you can calculate the stress by using the normal stress in bending equation.

$$\sigma = \frac{Mc}{I}$$

and the maximum moment on the midpoint cross-section is

$$M = \frac{qL^2}{8}$$.

Plugging in the expression for M into the stress equation, with q equal to the load at deviation from linearity, is your estimate of the yield stress.