PROC ESO S OBJ ETO DE AUTOMAT IZAC IÓN

Introduction

The previous chapter has presented the application of energy principles to the analysis of systems of rigid bodies. These are systems in which the deformations of the bodies are insigni®cant when compared to the movements of the component bodies in the system. At that stage we illustrated this with the example of the collision of two billiard balls in which the deformations at the contact area are negligibly small compared to the distances moved by the balls. The rigid body approach allows us to predict the macroscopic, or global, movements of the components parts of the systems.

However, no body is ever completely rigid and we can consider the analysis of most systems at two scales; one is the macroscopic that, as described above, is concerned with the movements of the bodies as if they never change their shape. The other scale is rather microscopic in which we are concerned with the detailed changes of shape of the bodies comprising the system. In this scale the colliding billiard balls are `frozen' in macroscopic space and only the local deformations in the balls as they impact are considered. In reality, of course, systems operate simultaneously in the macroscopic and microscopic scales but it simpli®es analysis, usually without signi®cant loss of accuracy, to separate the analytical treatment of the two scales.

It must be noted that there are also many systems like, for example, rubber bands, which experience deformations on a large scale and whose analysis cannot be performed according to the earlier outlined procedures. These systems require a special treatment and there are several theories in structural mechanics which allow us to tackle such analyses. However, on account of their complexity, these methods of analysis are de®nitely out of the scope of the present book and the interested reader can ®nd excellent introductory textbooks to this fascinating area of mechanics.

Before moving on to formulate the energy involved in the deforma-tion of a body we need to examine how materials deform and to char-acterise the deformations, and the forces that cause the deformations.

The approach followed is to subject a sample of the material to a simple one-dimensional test and to generalise, using mathematical constructs, the results from the test to obtain a description of the prop-erties of the material for deformations in three dimensions.

3.1 Deformation properties in one dimension

The approach followed for most materials is exempli®ed here by refer-ence to testing of a typical structural carbon steel. Figure 3.1 shows a specimen machined to a suitable shape for the uniaxial test. The shape of the specimens, Figs 3.1(a) and (b), rectangular and round section, respectively, have evolved during a century of material testing. The objective of the shape is to ensure that the gauge length, l_g, has uniform conditions resulting from the axial force, P, applied to the specimen in the test machine and is not in¯uenced by the complex conditions of forces in the parts of the specimen gripped by the testing machine, as shown hatched in Fig. 3.1. The test section shown in Fig. 3.1(a) has thickness t and breadth b giving an area A.

During a typical test to evaluate the quasi-static properties of the material the axial tensile load is applied at a slow rate that eliminates any dynamic e€ects and causes the gauge length gradually to extend from l_gto l_g‡
l_g. The change of length is measured by an instrument

t

b

P P

l_g (a)

P P

l_g (b)

d

Fig. 3.1

that is called an extensometer and is attached to the surface of the test specimen. Several types of extensometers have been developed during the past century to measure the very small change of length, typically

lg 0:05 to 0:5 mm. The instrument may be based on mechanical magni®cation of the changes of the gauge length or it may use the change of electric resistance of a length of wire that is constrained to change its length proportionally with the change of the gauge length.

Figure 3.2 shows typical results from tests on specimens with di€erent values of cross-sectional area, say by increasing the thickness.

It is evident that the initial part of the curves is fairly linear and the load at which the curve reaches a maximum is dependent on the area.

Figure 3.3 shows comparable results from tensile tests specimens in which the area is maintained constant and the gauge length is changed.

Moreover the load at which the curves reach a maximum is unaf-fected by the gauge length but the extension is a€ected by the changes.

P

Δ1 Δ2 Δ3

Δlg

Fig. 3.2

P

Δlg l_g1 l_g2 l_g3

l_g3 > l_g2 > l_g1

Fig. 3.3

It is also evident from tests that if the ordinates of the graphs of the results are normalised, as shown in Fig. 3.4, by dividing the applied load by the area of the specimen at the gauge length and by dividing the extension by the gauge length, then all the curves coincide.

The normalisation de®nes two parameters that are used widely in structural analysis, namely,

strain; " 
l_g=l_g and stress;   P=A:

If in the above de®nition of strain the initial gauge length is used, then what is normally de®ned as the nominal strain is obtained. A di€erent measure of strain, called the true strain, can be obtained by dividing the elongation of the bar by the actual gauge length, which increases as the tensile load is applied. In the development of this book we will always make reference to the nominal strain.

Stress is a parameter that can be derived simply in the analysis of the results of a tensile test by measuring the applied load and dividing it by the corresponding initial value of the area of the cross-section of the gauge length, which gives the nominal stress (a more exact value of the axial stress, known as the true stress, can be calculated by using the actual area of the bar, which can become signi®cantly less than the initial one). The de®nition of stress depends on the assumption that the load is uniformly spread across the cross section of the gauge length. In general, the state of stress is complex in practical structures and since it cannot be measured directly it must be inferred from measured values of strains and deformations using assumed or measured values of material properties. Alternatively, strains can be measured directly on the surface of a structure by arranging the read-ings from extensometers to be normalised by dividing the measured

P/A

Δlg/ l_g Fig. 3.4

elongation of the gauge length by the gauge length of the instrument.

Modern electrical resistance strain gauges can have very short gauge lengths, currently as small as lg 2 mm, and the electronic monitoring of the changes in resistance of the gauges directly provide strain values, usually up to a maximum value of strain of 0:05, i.e. 5%. An example of the measurement of strains on a tensile test specimen using micro-gauges is shown in Fig. 3.5.

Other methods, such as the use of laser interferometry, also enable the direct measurement of strains in engineering structures.

Figure 3.6 illustrates the full range of results that are obtained from a tensile test on a structural grade carbon steel.

The linear part of the graph extends up to a strain value of about 0:002, that is 0:2%, and thereafter the stress±strain graph becomes non-linear; this is called yielding. The ®gure shows the stress increasing gradually with increasing strain up to a strain level of about 0:04, i.e.

Fig. 3.5

4%, this is called strain hardening. In some structural steels the stress does not increase with increasing strain following the onset of yielding, up to a strain of about 2%. This is called the yield plateau, or LuÈder's plateau after the famous German physicist. The maximum load is reached at a strain of about 8±10%, as is shown in Fig. 3.6(b), at which stage the cross-sectional area of the tensile specimen reduces rapidly and the specimen fractures, this is sometimes called necking.

The ultimate load, at which necking begins, is usually about 10±

20% greater than the load at which yielding is initiated. Fracture

Stress, σ (N/mm2) 600

500

400

300

200

100

00 1·0000 2·0000 3·0000 4·0000

Strain, ε (%) (a)

Stress, σ (N/mm2) 600

500

400

300

200

100

00 2·0000 4·0000 6·0000 8·0000

Strain, ε (%) (b)

10·0000 12·0000 14·0000 16·0000 18·0000 20·0000

Fig. 3.6

occurs usually at about a strain of 18±25% and is a measure of the ductility of the steel.

Close to the yield strain the test specimen completely recovers its original geometry when the load is removed. In other words the material properties are reversible; this portion of the stress±strain properties are classi®ed as elastic. Thus for the elastic condition, as shown in Fig. 3.7(a), provided that the load is kept to less than the value at which linearity is lost (that is the proportional limit of strain, "p), the variation of load, increasing and decreasing, is directly related to the strain.

In many structural steels the proportional limit and the loss of elas-ticity due to the onset of yielding are virtually coincident and thus the

Stress, σ σP B

C E

A εP ε0 D

σ0

Strain, ε (a)

Stress, σ

A Strain, ε

(b)

D C

B σ0

loading

(i)

unloading (ii)

Fig. 3.7

limiting elastic condition for design assessment is taken to be the yield strain, or stress whichever is appropriate. An approximation to the stress±strain curve is made that considerably simpli®es analysis and is particularly relevant to the methods presented in this text. This is that stress±strain curve up the yield strain is modelled as being linear, and this gives rise to the linear elastic conditions that underlie the later developments in this text.

If the load in Fig. 3.7(a) is reduced during the test, at a strain greater than the yield strain, "₀, the graph follows the line (ii) in Fig. 3.7(a) until the load is completely removed at point D. In other words, after the yield strain is reached the material no longer has a unique stress±strain relationship and the loading and unloading paths are not reversible back to the original material condition of zero stress and strain. If the test specimen is reloaded from D it will follow the slope of the original elastic line, A±B, until there is a loss of proportionality and the stress±strain graph intersects with the yield line at E. Thus, provided the change of strain due to external loading, starting from zero load value, is less than the yield strain for the material, the material may be regarded as elastic no matter what its previous history of loading may have been. Tests show that for many elastic materials the stress±strain properties are the same in com-pression as in tension.

Figure 3.7(b) shows the stress±strain curve for a material, typi®ed by an aluminium alloy, that has a much more marked non-linear stress±

strain curve: from points A to B the relationship is fairly linear, but from B to C there is a distinct curve. The limit of proportionality at B does not necessarily indicate the limit of elastic behaviour of the material. The material may load and unload elastically along the line A±C±A. A pseudo elastic property is often constructed for such a non-linear elastic material, in that the stress±strain relationship is taken to be linear up to the load corresponding to point C, as shown by AD in Fig. 3.7(b).

The analysis methods developed in this book are concerned only with the linear elastic part of the stress±strain curve of the material and with predicting the limiting levels of loading at which the material no longer remains elastic. The methods therefore apply to the great majority of engineering analyses that are performed as a basis for design. The fact that the material does not fracture at the yield strain and that the engineering structure can carry increased load before material fracture actually occurs is used by structural engineers as a measure of in-built safety for their structure. The elastic material model that is used as the basis for the analysis methods developed here is shown in Fig. 3.8.

The strain is directly proportional to the load applied in the tensile test, the constant of proportionality is called the material modulus, or Young's modulus.¹ The material modulus is given the symbol E. The limiting allowable level of strain for the elastic analysis is the yield strain, "0. Thus the strain is related to the applied load in a simple uniaxial loading test by

" ˆ P AEˆ 

E …3:1†

This equation is often referred to as Hooke's law, after Hooke,² who

®rst proposed this load±de¯ection proportionality.

The values of E for a typical carbon steel range from 195 000 to 210 000 N/mm² and commonly a value of 205 000 N/mm² is used in design analyses. The yield strains for steels widely used in structural applications vary from 0:12% to 0:25% depending of the grade of material. Special heat-treated steels can have yield strains of up to 0:5%.

Stress conventionally plays a central role in structural analysis. Many text books in strength of material and in structural mechanics make great use of stress in the development of the theory of structures and in the presentation of the limiting strength of materials. Mathematical descriptions of the equilibrium of components or complete structures

α

ε σ

σ0

ε0 E = tanα

Fig. 3.8

1Young, Thomas (Milverton, Somerset, 1773 ± London, 1829), English natural philo-sopher.

2Hooke, Robert (Freshwater, Isle of Wight, 1635 ± London, 1703), English physicist.

are often based on the components of stress in the structure, etc. A structural analysis then has the objective of evaluating the magnitudes of these stresses throughout the structure and comparing the greatest magnitude with the maximum allowable stress for the particular material from which the structure is composed. This comparison then determines the maximum allowable value of the loading that may be applied to the structure. In this text, which is primarily concerned with linear elastic structural analysis using energy methods, the basis of the analysis methods is not stress but the deformation, and therefore strain ®elds, that exist in structure and their components. The arbiter of the allowable loading is not the level of the maximum stress, but the level of the maximum mechanical strain in comparison with the allow-able maximum strain for the material. This text therefore does not use the concept of stress widely, not for any reason other than that from this standpoint in most cases there is no real need to make resort to it.

3.2 One-dimensional thermal strain

All structural materials tend to expand when heated, as illustrated in Fig. 3.9.

The rollers in the ®gure represent that the specimen is not restricted from axial movement. The thermal strain "tis de®ned as the ratio of the extension of a gauge length to the original gauge length, that is

"tˆ
l_g

l_g …3:2†

The ratio of the change of length to the original length is found experimentally to be linearly related to the change of temperature, for temperature ranges up to about 100 8C. Thus,

"_tˆ 
T i:e:
l_gˆ l_g
T …3:3†

where  is the coecient of linear expansion of the material.
T is the change of temperature.

The coecient of linear expansion varies slightly with temperature, but is taken to be constant in the analysis methods developed in this

l_g Δlg

ΔT

Fig. 3.9

book. The value of the coecient for a structural steel in the range 0 < T < 100 8C is usually taken as

 ˆ 1:25  10^ÿ5mm=mm=8C …3:4†

The actual variation of the coecient of expansion with the tempera-ture is shown in Fig. 3.10 for a high strength carbon steel.

Notice that the thermal strain equals the yield strain at a tempera-ture change of

T ˆ"₀

 …3:5†

that is, for a medium grade structural steel

T ˆ 1:5  10^ÿ3

1:25  10^ÿ5ˆ 120 8C …3:6†

The foregoing remarks relate to changes of length that result from changes of temperature where there are no constraints or applied forces. The situation illustrated in Fig. 3.11 is one where the change of temperature occurs simultaneously with the application of an external force P.

The strain " induced in the bar is the sum of the mechanical strain due to the force and the thermal strain due to the change of temperature.

16

14

12

0 100 200 300

Temperature °C Coefficient of thermal expansion α, 10–6 °C–1

Fig. 3.10

ΔT

P P

Fig. 3.11

Thus

" ˆ P

EA‡ 
T …3:7†

The constraint can take the form of displacement restrictions on the boundaries of the body. An extreme example is illustrated in Fig. 3.12 in which the ends of a bar are prevented from movement, i.e.
l ˆ 0 and therefore the total strain in the bar is zero.

P

EA‡ 
T ˆ 0 and P ˆ ÿEA
T …3:8†

The negative sign implies that the force is compressive to maintain zero total strain for the increase in temperature.

3.3 Three-dimensional strain

Section 3.1 has presented the observations of the change of length of the gauge length of a tensile test specimen when viewed in the direction of the applied load. However, test observations show that the cross-section of the gauge length also changes its dimensions. That is, as illu-strated in Fig. 3.13, the application of the load P causes changes in the dimensions normal to the direction of the load.

The thickness t changes to t ‡
t and the breadth changes from b to b ‡
b. If we de®ne a set of orthogonal coordinates, x, y and z as

Fig. 3.12

P

t t + Δt P

y

x z

l_g l_g + Δlg b b + Δb

Fig. 3.13

shown in Fig. 3.13, the strains in the three directions are

"_x ˆ
lg

lg ; "_y ˆ
b

b ; "_zˆ
t

t …3:9†

Observations show that that
t and
b are negative for positive values of
l. The test results show that the strain normal to the direction of the applied load is related to the strain in the direction of the load by a constant, called Poisson's ratio, , after S. D. Poisson.³ Thus,

 ˆÿ"y

"_x ˆÿ"z

"_x …3:10†

The values of Poisson's ratio measured in tests for the elastic part of the stress±strain curve of a typical carbon steel are about 0:27±0:31 and 0:3 is commonly used in design analyses for steel structures.

Poisson's ratio increases to about 0:5 for strains much greater than the yield strain.

During the application of the load P the volume of the gauge length will change due to the elastic strains. The change of volume
V is

V ˆ l…1 ‡ "_x†b…1 ‡ "_y†t…1 ‡ "_z† ÿ lbt

ˆ lbt…"_x‡ "_y‡ "_z† ‡ f …"²; "³; . . .† …3:11†

Since the strain is very small for elastic deformations, the higher order terms in the equation can be ignored as being insigni®cantly small and the volumetric strain "vthat is the ratio of the change of volume to the original volume, is

"_v 
V

V  "_x‡ "_y‡ "_zˆ "_x…1 ÿ 2† …3:12†

Uniaxial tests are commonly carried out to establish relevant material properties for use in design analyses; typical results have been discussed above. It is possible to generalise these results to describe the possible e€ects of applying simultaneously load in the three orthogonal directions. The practical diculties of ensuring that the loads are applied is such a manner that the test specimen has uni-form conditions throughout its gauge volume have meant that there is only a limited range of experimental con®rmation of the theoretical development that now follows. However, the application of the theory to the analysis of real structures and the measurement of strains

3Poisson, SimeÂon-Denis (Pithiviers, Loiret, 1781 ± Paris, 1840), French mathemati-cian, physicist and astronomer.

on the structures have provided a high degree of con®dence that in fact the generalisation of uniaxial tests to triaxial loading is valid.

Figure 3.14 shows a test specimen subjected to three loads, P_x, P_y and P_z applied in the x, y and z directions such that the conditions are uniform throughout the gauge volume.

Figure 3.14 shows a test specimen subjected to three loads, P_x, P_y and P_z applied in the x, y and z directions such that the conditions are uniform throughout the gauge volume.

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