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Hooke's Law: Nonlinear Generalization and Applications
Part 3: Distribution of Stress in Prisms and Beams

Prof. Dr. Hans Schneeberger
Institute of Statistics, University of Erlangen-Nürnberg, Germany
2021, 26^th of October

Introduction

As stated in [5], part II, the distribution of stress σ in the bending pressure zone of a beam is identical with the distribution of stress in a non-centric loaded prism in such a manner, that the strain ε₂, remote to the loading, is zero (see figure 1).

In a huge number of experiments with the variables
ε₁ (⁰/₀₀ = ε₀ (in contrast to ε₂=0)

x=1-β₀, the center of loading; NB! The width of the prism d=150cm (see [5], part II, figure 2) is standardized to 1: 0 ≤ x = ≤ 1 = d/150,

, the (relative) loading; 𝐾_b(kg/cm²) is the utmost strength of the prism with centric loading - Generally: Index 0 means „experiment with ε₂=0“,

in [2] the experimental results are given in dependence on strength of cube W=W²⁰ (kg/cm²; cube length 20 cm) and (B for break). See also tables 1, 2, 3 and figures 4, 5 and 6 of [5], part II.

With these experimental data 3 hypothesis were postulated in II (hypothetical values are signed as in contrast to experimental values – as it is usus in statistics).

The good coincidence of experiments and hypotheses can be seen in the tables and figures of [5], part II.

Distribution of stress σ in an eccentric pressed prism (with ε₂=0)

The (relative) stress y(x)=σ(x)/Κ_b is

Hypothesis 4: y(x)=Axe^-Bx

see figure 1. That means: The stress-function σ(x) for eccentric pressed prisms is of the same mathematical form as for centric pressed prisms, as Κ_b, as well as W, is a constant for a fixed experiment.

The two parameters A and B are estimated by minimizing according to the method of least squares of Gauss

together by minimizing the objective function D=D₁+D₂.

The values of and are given in [5], part II, tables 2 and 1 for a series of values of W and of κ

For example for W=300 and κ=0,9 we have 0.653 and 0.374.

The minimum of D is found by the iterative nonlinear Simplex-method of Nelder and Mead [1]. The resulting optimal parameters A and B of hypothesis H₄ are registerd together with figure 2 for W=80, figure 3 for W=160, figure 4 for W=300, figure 5 for W=450 and figure 6 for W= 600 (kg/cm²). To every value of W seven values of κ= 0.3, 0.4, … 0.9 were chosen for stress-lines. For κ>0.9 and W< 80 I do not put my hand in the fire with my formulae. This may already be seen in figure 5b of [5], part II for κ=0.9 and small values of W.

In the left part of the figures the disrtribution of stress y is given.

The strain ε is zero for x=0 according to the experiments. For x=1 strain ε=ε₀ can be computed with formulae =aε₀e^-bε₀ of [5], part II. This is shown in the right part of the figures. For example in the experiment with W=300 and some values of we get (see also figure 4):

	.3	.4	.5	.6	.7	.8	.9
ε0	.39	.55	.72	.92	1.16	1.45	1.86

Distribution of stress σ in the bending pressure zone of a beam

According to [5], part II the distribution of the stress in the bending pressure zone of a beam is identical with that of a corresponding eccentric pressed prism with ε₂=0: The side of the prism with ε=0 (y-axis in figure 1) is equivalent with the neutral axis of the beam (see figure 7).

In figures 8, ..., 12 the lower part give the stress-lines in the bending pressure zones of the beams. They simply are the mirror-images of the stress lines of the corresponding prisms, reflected at the mirror with y=x.

The curves =aε₀e^-bε₀ in the upper part of the figures are the same as those of the corresponding prisms (in a somewhat modified scale).

Comment

The experiments, the results of which are used in this analysis, were directed by Rüsch [2] at the University of Technology Munich. The evaluation of the data was done under the direction of Scholz, who also made two hypotheses on the distribution of the stress in beams ([3] and [4]).

References