# . Series Coupling of N Equal Strain Softening Elements

Consider now a chain of N nominally identical softening elements (Fig. 8.1.4a). Following the same reasoning as in the previous analysis, it is immediately obvious that after reaching the peak, only one of the elements, say element 1, will soften, while the remaining N — 1 will unload. Keeping this in mind, we consider how the mean strain will evolve as a function of N.

The mean strain of the whole chain is

where і indicates the element number, and c; the strain of that element. Expressing the fact that on the hardening branch all the strains are identical and equal to Eu and that on the softening branch the strain of the first element is є5| while the strain of the remaining N — 1 elements is e„i, we get the following result for the mean stress-strain curve:

f Shi for hardening (8 14)

6 ful + jtr(esi – Sal) forsoftening

Fig. 8.1.4b plots the foregoing analytical results for N–1, 2, 4, 8, and oo based on the curve of Fig. 8.1.3b (note that the horizontal scale has been expanded). The construction of the softening branch is very easy to perform graphically: at each stress level, take the segment UH where U and H are the points, respectively, on the unloading and softening branches for a single element. Then, take a segment N times smaller with origin at U. The other end of the segment determines the point of the softening branch of the series coupling of N elements.

One essential result of this analysis is that, while the peak load does not change with the number of elements, the brittleness does so in the sense that the larger the number of elements, the steeper the softening branch gets. In the limit of an infinite number of elements, the behavior is perfectly brittle.

Exercises

8.1 Analyze the response of a series coupling of two equal elements whose load-displacement curve shows a perfect plateau at peak load. For simplicity, assume that the load-elongation curve has the shape of a trapezium, rising linearly from (0,0) to (Pu, щ), then extending horizontally to (Ри, щ), and finally descending linearly to (0,u2), where ua < ui < и2.

8.2 Consider the series coupling of elements that have a triangular load-displacement curve and are identical except for small imperfections. Assume that for one element the peak occurs at 1.2 kN for an elongation of 5 /rm, and that a zero load is reached for an elongation of 200 gm. Determine and make a sketch of the load-displacement curve for (a) 2 elements, (b) 10 elements, (c) 100 elements, and (d) determine the number of elements for which the load drops verticttlly just after the peak.

8.3 Consider the series coupling of elements with exponential softening. The load-displacement curve for one single element is given by the equations

(8.1.5)

in which Co = 1.1 /ші/kN, Pu – 3.1 k. N, no — 68.2 gin. Determine: (a) the energy required to break one element, (b) the load-elongation curve for a coupling of 10 elements, (c) same for 100 elements (draw the curve), (d) Determine also the lowest number of elements for which the resulting softening branch displays a vertical tangent.