# Maximum stress point

I want to determine the point where the maximum stress is located. Also, I want to know how this kind of questions is answered.

## 2 Answers

let's say the coefficient of bounding between the plastic rod and concrete is q and the diameter of the plastic rod is d, meaning every square inch of the plastic embedded in the concrete has q lbs adhesion, thus $$q\pi d$$ for every inch length of embedment.

Therefore the bound between the plastic rod and concrete starts from zero at point A and linearly increases to the level where the bound becomes equal to maximum tension the plastic rod can take. Assuming the maximum tensile strength of the plastic rod $$F_p$$ this point is located at an embedment length of:

$$L_{embement}=\frac{F_p*Area\ of\ rod}{q\pi d}= \frac{F_p\pi d^2/4}{q\pi d}=\frac{F_pd}{4q}$$

This point is independent of the points B, C, D on the graph. It is pre-determined by the properties of the plastic rod. but it certainly is not at point A, because there the concrete stress is zero.

I attach a graph of concrete bounding stress here. The stress in rod depends on its embedment length.

The axial tensile stress in the rod at point A is the sum of cylindrical shear stress plus the cone shear stress.

. EDIT

I added the narrowing of the rod under the Poisson's effect to clarify why shear stress at point A is zero. Notice that due to axial elongation under tension the rod gets a bit narrower through a transition region from the Point A inward.

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• Your diagram does not show a concrete stress of zero at A. The axial tension stress is zero, but the adhesion shear stress is not. – ingenørd Jan 25 '20 at 6:10
• @ingenørd, the axial tensile stress in the rod at point A is the sum of cylindrical shear stress plus the cone shear stress. I guess I should add this to my answer. – kamran Jan 25 '20 at 7:03

The answer is A. It's mainly because of the elastic modulus of those materials. The plastic is very flexible and can't transfer the force to the end of the hole. So the force is transfered at the outer end. If the materials would be interchanged, the maximum stress would be located at the end of the hole.

• The stress at A is not zero using your assumptions, because there is a shear force acting on the inside of the hole. But I agree that unless you make some assumptions which the OP doesn't mention, the answer could be anything. Actually, I would go for D, because for a perfectly sharp reentrant corner the stress around the edge of the end of the hole would theoretically be infinite. – alephzero Jan 25 '20 at 1:51
• @alephzero, I drew a diagram of concrete shear stress showing concrete stress at A is zero. The OP doesn't need to give much more info. He asks for a typical method of handling the embedment of a bar in concrete. This is the typical method engineering codes handle it. – kamran Jan 25 '20 at 3:17
• @kamran, The shear stress at point A can't be zero. The elastic properties of both materials has to be taken into account to sovle this. – teshim Jan 25 '20 at 15:46
• @teshim, the plastic rod under the tension will strain and elongate. At the same time, it will narrow down by Poisson's, v ratio. so during a transition length starting at A into concrete, there is a gradient of shear from zero to fully stressed shown by the cone. – kamran Jan 25 '20 at 16:11
• @kamran I agree on the first part but the maximum is at point A and the shear stress will decrease further inwards. You can easily see this by FE model. – teshim Jan 25 '20 at 17:12