# Non-intuitive FEM simulation of a 4 point bend

I am conducting some finite element analyses of beams under 4 point bend mechanisms. The boundary conditions are set up such that there are 2 fixed locations, 2 displacement driven locations, and the rest are free. I have attached 2 figures below of my FEA. This is non-dimensional, and I am only interested in studying the trends.

For reference: http://www.unitedtest.com/products/cp01/cp017/cp103/123.html This link contains a 4 point bend figure. My model is slightly different in that instead of the 2 bottom force locations, I used a fixed boundary condition. And instead of the 2 top force locations, I use a displacement-driven boundary condition.

The first figure shows the contours of the displacement in the vertical direction. The second figure shows the contours of the strain. This strain is $$\epsilon_{yy} = \frac{dv}{dy}$$.

The first figure makes sense to me, and if you plot the displacement as a function of the length of the beam, you would see a bell-like curve.

However, this second contour of the strain is puzzling to me, and perhaps that is due ignorance on my part or a lack of intuition. I was expecting the strain to mimic the displacement in terms of trends. That is, the center of the beam should exhibit the largest strain. But this is clearly not the case. The computed strain here is ~0 except for where the displacement driven boundary condition is applied.

Is my intuition wrong or is this indeed a strange result? If my intuition is wrong, what should the trend resemble?

• If you are trying to plot the "bending strain" you want to plot $\epsilon_{xx}=\frac{\partial u}{\partial x}$. – Bill Greene Dec 27 '18 at 18:32
• Is $u$ the displacement in the direction of the length of the beam? If so, I am confused why this is the bending strain. I think this may be a semantics issue. I am interested in answering the question: where, along the thickness direction (y-direction in this case), does the beam elongate the most? I think it would be $\epsilon_{yy}$ that would best answer this question? – structuralengineer Dec 27 '18 at 18:39
• Yes, I am following the coordinate system in your plots so $u$ is along the beam length. In classical beam theory, $\epsilon_{xx}$ is the only non-zero strain. $\epsilon_{yy} is just a local compressive stress in the immediate vicinity of your applied displacements, as your plot shows. – Bill Greene Dec 27 '18 at 19:08 • That makes sense! My modeling is based on an experimental study, in which a strain gauge was placed on the top surface (-y in the above plots) right at the center. I was thinking this strain gauge was measuring$\epsilon_{yy}$, but after reading what you said, that doesn't appear to be so because the measured value is non-zero. Is this strain gauge actually measuring the$\epsilon_{xx}$component? – structuralengineer Dec 27 '18 at 19:26 • Yes, your strain gauge is measuring$\epsilon_{xx}\$. – Bill Greene Dec 27 '18 at 19:56