THEORETICAL OUTLINE
When the relationship between the external load and defl ection experienced by a deformable system cannot be represented by a straight line, the system is said to be nonlinear. The natural frequency of such a system or structure is most often amplitude dependent, unlike the case of the perfectly linear system, where frequency remains constant regardless of amplitude. The usefulness of this, or the natural period concept in shock and impact analyses lies mainly in the fact that the defl ection from the initial position to the fi nal one can be regarded as taking place in one-half or one quarter of the natural period. For the system acted upon by a prescribed impulse, the main objectives are to determine the magnitude of that defl ection (pseudo-amplitude) as well as duration of motion. This chapter is devoted to calculating these quantities.
G
ENERALR
EMARKSABOUTN
ONLINEARITIESThe equation of motion of the basic oscillator:
Mu+Cu+ku=W t( ) (4.1)
is an example of a linear differential equation, because the unknown function u(t) as well as its derivatives appears in the fi rst power. When an equation of motion does not satisfy that condition, we call it a nonlinear equation and the system it describes is referred to as a nonlinear system. Most engineering objects show some deviation from linearity, but this is usually ignored for the practical reason of keeping computations simple. When defl ection amplitudes are small in comparison with dimensions of structural elements, it is often quite realistic to assume proportionality between defl ections and the elastic resistance. Yet, when those displacements signifi cantly change the manner in which the external loads act on a structure, the nonlinearity of the system may not be ignored.
A good example of a nonlinear system is a body like a wing of an airplane, subjected to aerodynamic forces that are proportional to the square of velocity. Another large class of problems relates to systems with a general resisting force R(u) so that we have
( ) ( )
Mu+Cu+R u =W t (4.2)
instead of the previous equation. The function R(u) is also referred to as a characteristic of a system. The fi rst example of an element with a nonlinear characteristic is shown in Figure 4.1a. It is a cantilever that is slightly curved upward in its unstressed state.
When subjected to a vertical downward force, the free length of the cantilever diminishes because the beam is gradually leaning against the rigid plane. In consequence, for every successive increment of displacement, a larger and larger increment of force W is needed.
Such elements in which the resistance grows faster than displacements are called hardening or stiffening elements.
Figure 4.1b gives an example of a softening element. This is the same cantilever as before, but loaded with a compressive force and having no additional restraint. As the defl ection grows, the process of deforming becomes easier, at least up to a point. This is because the effective arm of force W with respect to the point of fi xity is increasing. The characteristics of both types of element are plotted in Figure 4.2.
This example has illustrated an important principle, namely, that not only the element itself, but also its manner of loading and the magnitude of defl ections may cause nonlinear effects to appear.
Most structural materials begin to yield at a certain stress level. Once yielding starts, it makes the characteristic of an element deviate from the initial slope. This is called a material nonlinearity, as opposed to geometrical nonlinearity caused by the change of shape or constraints during the loading process. The impact problem typically involves both types of nonlinearity, because the change in contact area may be accompanied by yielding.
Unlike the case of the linear problems, very few exact, closed-form solutions are available, and various approximations must be employed. That makes the fi eld of nonlinear dynamics a diversifi ed and diffi cult branch of engineering science.
An important computational aspect of a nonlinear system is that the principle of super-position does not in general apply. When a system is subject to several dynamic loads acting at the same time, the entire loading must be simultaneously considered.
E
LASTIC, N
ONLINEARS
YSTEMSA system or a body is called elastic if there is a 1 : 1 relationship between displacement and load. If in addition to that, the resistance–defl ection curve is a straight line, the body is
W EI
(a)
W EI
(b) FIGURE 4.1 Hardening system (a) and softening system (b).
u Softening
R
Linear Hardening
FIGURE 4.2 Characteristics of nonlinear systems.
called linearly elastic;* otherwise, it is nonlinearly elastic. The latter type of elements is described in this Section.
In practical situations, a linear characteristic is often inapplicable, so the fi rst attempt is to replace it with straight line segments, making it piecewise-linear. A few such examples are shown in Figure 4.3. All are made of lines with two distinct slopes, but the term bilinear (BL) is usually applied only to the characteristic in Figure 4.3b. (The magnitude of slopes is indicated by k1 or k2.)
Elastic symmetry means that if the sense of displacement changes, so does the sense of the resisting force, but the magnitude of the latter remains unchanged. It is suffi cient to defi ne the resistance–defl ection curve for positive values of displacement. Among the graphs in Figure 4.3, only the fi rst one does not show the elastic symmetry.
Some simple structural arrangement can be used to obtain characteristics in Figure 4.3, such as the double-sided oscillator in Figure 4.4. First, consider the arrangement when there is no spring prestress and no gap. This may correspond to Figure 4.3a for springs of unequal
*The word “elastic” without a qualifi er usually means “linear elastic.”
R k1
FIGURE 4.3 Some simple nonlinear characteristics.
k2 k1
u
M
FIGURE 4.4 Double-sided oscillator.
stiffness. A discontinuity of slope, as in Figure 4.3c, may be caused by the springs being initially rigid and then beginning to yield at the R0 level. A gap on both sides will give rise to a horizontal discontinuity, as in Figure 4.3d. Finally, to get the behavior like in Figure 4.3b, an additional set of linear springs is needed.
Unlike in a linear case, the natural frequency of a system is now dependent on the amplitude of vibrations the system is performing. The hardening system will exhibit a decrease of the natural period with the increase in amplitude.
M
ATERIALN
ONLINEARITY, D
UCTILEM
ATERIALSOnce the yield point of material is exceeded, the basic linear elastic (LE) model, used so far, is no longer satisfactory. Several simple, useful material models, approximating true stress–strain relationships will now be presented. The most popular approximations in nonlinear dynamics research appear to be the elastic-perfectly plastic (EPP) and rigid-plastic (RP) models. (Figure 4.5) The latter one represents a material, which is assumed to be undeformable up to a certain stress level σ0, called here the fl ow strength.* When the level is reached, further elongation takes place with no increase of resistance. In EPP approximation, on the other hand, the elastic range is followed by deformation at the constant level of σ0, the effective yield strength, reached when strain attains the value of εy. In general, the EPP model offers a more realistic representation because it has one parameter more. Based on strain magnitude alone, simplifying the EPP to a RP model seems almost natural. Yet, one must keep in mind that when bending is involved, a signifi cant loss of accuracy may take place when EPP is replaced by a simpler RP model.
With reference to Figure 4.5 one may note that the curves are not drawn to scale, for the sake of clarity. The simplifi ed graphs should be constructed in such a way that the
*When reading technical papers, one of the items causing a frequent confusion is the word stress. Too often it is not known whether it means the applied stress or the material strength. Using the term fl ow strength makes it, beyond doubt, as the material property.
Elongation, % ε
εu ε
Stress
Original El. perf. plastic
Rigid-plastic Fu
Fy σ0 σ0
FIGURE 4.5 Original stress–strain curve and its two simple approximations.
area under the actual curve is preserved. The yield strength is denoted by Fy in general and by σ0 for materials with a constant yield stress level. There are two levels of σ0, one for the elastic, perfectly plastic and one for a RP model.* When straining is large enough to reach a multiple of εy, in most cases those two levels coincide, or nearly so.
Several of the remaining models are shown in Figure 4.6. They are: BL, rigid, strain hardening (RSH), and Ramberg–Osgood (RO). Only for BL some details are shown:
Young’s modulus E, plastic modulus Ep, and unloading portion of the curve, which for steels has the same slope as the initial modulus E. One of the models that is not illustrated, is the power law (PL) material. When graphed, it is quite similar to RO.
(When using RO form, n > 1. For PL, n < 1.) The equations, for the tensile stress are as follows:
LE:σ = εE (4.3a)
RP:σ = σ0 (4.3b)
EPP:σ = εE forε ≤ εy and
(4.3c)
y for y
σ =F ε > ε
BL:σ = εE forε ≤ εy and
(4.3d)
y p( y) for y
F E
σ − = ε − ε ε > ε
p y
or σ = + εY E forε > ε
*There is a point called proportionality limit, at which the σ–ε begins to diverge from the straight line. Our simplifi cations ignore this fact and we act as if the curve was linear all the way to the yield point.
σ
Y Fy
Ep
ε εy
E
(b) Stress
Elongation RSH
RO BL
1.0 Ep
E 1.0 Fy
(a)
FIGURE 4.6 The remaining simple material models (a) and BL model with an unloading segment (b).
y p model is very useful for steel working in high temperatures. One can often see a single-term stress–strain, relationships, written in a way similar to the second term of RO or PL above.
The BL model is used quite frequently; two alternative expressions were given for the plastic range.
The area under the stress–strain curve is a measure of material ability to absorb strain energy and will be referred to as material toughness.* More precisely, when the strain reaches its peak εu, the toughness Ψ is also attained. This is equivalent to the following defi nition of toughness:
For an RP material, where yielding takes place at σ0 or at τ0, the above expressions become very simple; Ψ = σ0εu and Ψ = τ0γu, where εu and γu are the ultimate strain and the distortion angle, respectively. In the listing of cases to follow, an expression for the strain energy Π is given as a function of the maximum strain reached, εm. When this is replaced with ultimate material strain εu, the equation for Ψ results.
To complete this description of ductile materials, one should include the stress–strain curve of a mild steel, a commonly used material with Fy between 200 and 350 MPa and illustrated in Figure 4.7. Apart from the plateau at the yield strength level Fy, the material also displays the upper yield point Fy′. The proportionality limit Fp is the stress where the characteristics shows a deviation from a straight line, as mentioned before.† The ultimate strain εu is the quantity specifi ed by the manufacturer. There are many references in literature to the rupture strain, suggesting that this may refer to the strain associated with failure, as shown in the fi gure, but it appears that in many cases they really refer to εu. The falling part of the curve, between εu and εr is associated with the decrease in section area of the sample past the peak point.
*This is not to be confused with the toughness defi nition used in fracture mechanics.
† The knowledge of Fp is needed only in exceptional circumstances.
The discussion so far related to the results and simplifi cations of a tensile test. If a short sample is subjected to compression, the small-strain response is similar to that in tension. For larger strains there is swelling of the sample and other secondary effects.
Presenting both tensile and the compressive stress in terms of logarithmic strain makes both curves more similar. When deformation is advanced, circumferential cracks appear in a compressed sample, which is not of great relevance with regard to com-pressive properties. It is not possible to induce a comcom-pressive failure, analogous to that in tension.
A selection of stress–strain curves was presented above, but it does mean that they are readily available. Instead, they must be constructed on the basis of experimental data.
Typically, when working with metals in ambient temperature, one has only three numbers characterizing the material strength: Fy, Fu, and εu. It is natural to construct a BL character-istic based on these data, as well as E and ν, unless we have a reason to expect a different shape. Also, if the material is simplifi ed to resemble EPP, for example, then the yield point must be so selected that toughness Ψ is preserved.
S
HEARANDT
WISTINGOFS
HAFTSMaterial testing is usually done on tensile samples, but the same material often works in shear, so the question of material constants arises. As it is shown later, in Chapter 6, for metallic materials the yielding and the ultimate strength in pure shear, τy and τu, are related to the tensile properties as follows:
y
y 3
τ = F (4.5a)
u
u 3
τ = F (4.5b)
where the fi rst equation results from Huber–Mises theory and the second is a custom-ary assumption related to the ultimate strength.* A determination of stress–strain plot
*The reader should keep in mind that the above equations are based on small defl ection theory.
Fu
εu εr Fp
Fy Fy΄
FIGURE 4.7 Stress–strain curve of a mild steel.
parameters in shear is presented for a BL material as Case 4.8. When Equations 4.5a and b are augmented by putting a limit on shear strain, namely γ =u 3εu the relation between the tensile and the shear characteristics becomes complete. Its example for a RSH material is shown in Figure 4.8 While the above limit on shear strain has no theoretical justifi ca-tion, it gives a simple relationship between the moduli in plastic range, Gp = Ep/3, which is consistent with ν = 0.5, generally assumed for this range. This ratio of Gp and Ep holds strictly true for the RHS material only, but under well-developed plasticity may be a good approximation for many real materials.
Twisting of shafts is detailed in Case 4.29. In general, this mode of deformation is more complex than tension, because of a nonuniformity in stress distribution over a cross section.
The process of a gradual yielding of a solid, circular shaft begins in the outside fi bers and continues until the near-center fi bers reach yielding, in an asymptotic way. If an EPP model is used, the uniform shear stress distribution is a limiting condition. In practice it is approached only after the angle of twist is at least an order of magnitude larger than twist-ing associated with the initiation of yieldtwist-ing. In Case 4.29 the approximation is based on ignoring the angle of twist associated with the stress redistribution. Owing to this, a shaft made of a RSH material as in Figure 4.6a, can have its characteristic represented by a simi-lar, trapezoidal shape. A thin-wall shaft is an exception. Its characteristic is simply a scaled material characteristic, as no redistribution takes place.
P
ROPERTIESOFC
ONCRETEConcrete has a pronounced nonlinearity in its stress–strain curve, as Figure 4.9 illustrates.
The nominal Young’s modulus is taken as a secant modulus Ec, which may be related to its compressive strength as follows:
c 5000 cm
E = F′ (4.6)
where F′c ~ MPa is the nominal compressive strength and Fcm′ ≈ 1.1Fc′ for most of industrial strength grades.* The σ–ε curve has a continuously decreasing slope with the maximum
*F is the average 28 days compressive strength.c′ Fu
Fy
Ep σ
εu ε (a)
Gp
γu γ τ
τy
τu
(b)
FIGURE 4.8 Direct stress–strain (a) and the corresponding shear stress vs. shear angle (b) for a RSH material.
tangent of Eci at the origin. The strain at failure εu is usually assumed to be 0.002 for design purposes. The following approximation is also useful for cement-based materials as well as hard rocks:
c 20 0.25 c( GPa if c MPa)
E = + F E∼ F ∼ (4.7)
A more detailed presentation of concrete stress–strain properties is given in Case 4.30. The strain-rate effect is also included there, but the situation is not quite clear with this variable.
For a broader discussion of the subject the reader should refer to Chapter 6.
The above results describe the behavior of a cylindrical concrete sample under axial compression. There is also another test, called triaxial compression, which illustrates an important concrete property, namely sensitivity to confi ning pressure. The sample is placed in a device, which makes it possible to apply a uniform (hydrostatic) pressure to the side surface, in addition to axial compression. The larger the confi ning pressure, the larger the apparent axial strength. One of the simplest relationships was proposed by Reinhardt [71]:
cc c 3 c
F =F′+ p (4.8)
where
F′c is the nominal strength
Fcc is the confi ned strength under lateral hydrostatic pressure pc A similar relationship can be written for a hard rock.
When describing concrete, the attention is usually focused on its compressive properties.
This material is weak in tension, as it can withstand only the stress level of
t 0.6 c
F = F′ (4.9)
0.85Fc΄ Eci
Ec Fc΄/2
1
εu εr ε
σ Fc΄
FIGURE 4.9 Typical, static stress–strain for concrete showing the initial Eci and nominal Ec moduli.
where F′c is the nominal compressive strength, with both quantities in MPa.* Yet, the tensile strength is quite important, as it dictates the location of the fi rst crack as well as subsequent breakup.
S
TRESSANDS
TRAININAL
ARGED
EFORMATIONR
ANGEConsider a stretched bar in Figure 4.10. When extension is as large as the scale of the fi gure indicates, there is a need to distinguish between engineering strain, ε and the natural (or logarithmic) strain† e–. The fi rst relates to the change of length with respect to the initial length L and the second uses the current length l:
l L u
L L
ε = − = (4.10a)
d dl
−ε = l (4.10b)
The total natural strain e– is the sum of elementary increments:
d ln ln 1
l
L
l l u
l L L
⎛ ⎞
−ε =
∫
= = ⎜⎝ + ⎟⎠ (4.11a)and
ln(1 )
−ε = + ε (4.11b)
In this book, mostly, the strains are presumed small and there is no need to distinguish between e– and ε. For this reason strain is designated usually as ε, because e– ≈ ε, as long as
*Ft as given above is the fl exural strength, a typical application. Pure tension on concrete members is quite rare.
† There also is a concept of “true” strain, (l − L)/l. The same term is applied by some authors to the logarithmic strain defi ned above. Once we agree that specifi cation of strain is only a matter of convention, then it is obvious the term “true strain” is a misnomer.
u
P
l (a)
(b)
L
P A0
A
FIGURE 4.10 An axial bar, unstretched in (a) and stretched with a force P in (b).
ε does not exceed a few percent.* For example, if ε = 0.20 then e– = 0.1823, already a noticeable difference.
If defl ections are as large as suggested by Figure 4.10, this means, for most materials, that they are well outside of the elastic range. It is usual to assume, in such a circum-stance, that the volume of material remains invariant, i.e., A0L = Al, where A and l des-ignate the current area and length. As stretching progresses, the section area decreases according to
0
0 exp
L A
A A
= l = −ε (4.12a)
or
0
1 A= A
+ ε (4.12b)
The tensile test results are usually prepared by plotting engineering stress, or stretching force divided by the original sample area, σ = P/A0, vs. ε. If a true stress S is defi ned by dividing that force by the current area A, then
0
(1 )
P P
S= A= A + ε (4.13a)
or
(1 )
S= σ + ε (4.13b)
A conventional σ−ε curve, which for metals is typically convex upward will become less convex (or may even become concave) when converted to the true strain. This is quite an important point when large distortions are involved.
There are several defi nitions of strain available. Which of them is best depends on the purpose at hand. If the objective of the investigation is to determine the load–defl ection curve of a tensile member, then the engineering stress and strain are the most appropriate
There are several defi nitions of strain available. Which of them is best depends on the purpose at hand. If the objective of the investigation is to determine the load–defl ection curve of a tensile member, then the engineering stress and strain are the most appropriate