advanced structural behavior

8/11/2019 Advanced Structural Behavior

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'nder certain circumstances structures will fail by elastic instability. Thisoccurs when the load produces a ending or twisting moment that isproportional to the deformation. The most common example is the Eulercolumn. #n this case, a slender column is loaded axially. The load results insmall de(ections, up to a critical load value. t this magnitude of load, themember collapses. )ailure occurs even though the stress in the memberremained below the yield values. The mathematical equation for elasticbuc!ling of a long column is*

+here P total load, area of section, -r slenderness ratio, and $ isthe coe/cient of constraint %depends on end constraints 00 usually 1 or 2 isused for end conditions that occur in practice. 3eference boo!s havetabulated equations that cover a wide variety of geometric cases %non0uniform cross0sections, tapered bars, etc.&, loadings, and end constraints.4ther examples of this phenomenon are thin plates in compression, thincylinders under compression, etc. Elastic buc!ling behavior is only seen instructures that are much longer in one or two directions than the other oneor two directions.+hen columns %and thin plates, etc.& are short %or thic!& enough, thesestructures can exhibit behavior that re(ects a combination of eects. Theslenderness ratio where this occurs starts roughly between 125 and 165. Theequation given no longer holds for these cases. )ormulas exist for nding thecritical loading for these structures as well, but these equations are more

complicated, and contain terms that have been empirically ad"usted to agreewith testing.

Dynamic effects

+hen a loads on a system result in the lac! of static equilibrium %where thesum of all forces are 7ero&, we can say that dynamic eects are important.4ften the inertial eects of the structure itself are important, when it moveswith appreciable velocity.ll structures tend to vibrate with specic mode shapes, at specicfrequencies of vibration. These tendencies are termed the natural

frequencies of the structure, and are dependent on stiness and massdistribution.+hen a structural member is excited with a dynamic load that acts with afrequency near one of its8 natural frequencies, the resultant de(ections %andstresses& can be very large. The inertia forces in the structure may becomeimportant, as the spring0bac! forces of the structure tend to add to the eectof the dynamic loads4ften, the rst step in attac!ing a dynamic structural problem is todetermine the natural frequencies of the structure involved. This can be doneby hand calculations, or more li!ely, through nite element analysis. Thenatural frequencies are dependent on geometry, material, and constraintsonly. They are independent of loading. fter nding the natural frequencies,it then must be determined whether or not the dynamic loads will tend toexcite these frequencies within the structure. #t is often valuable simply tocompare the frequencies of the loads to the natural modes directly.


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The equilibrium equation for forces in an undamped simple harmonicoscillator system is shown below*

m x88%t& 9 ! x%t& 5The solution to this problem is given by

x%t& cos %t&where is a constant that can be determined from the initial conditions. #tcan be written that*

% 0m29 ! & cos %t& 5This equation is good for any time t, only if the term in parenthesis is equal

to 7ero. )or this to be true, it means that the value of is given by

%!m&12

where is !nown as the natural frequency, or fundamental frequency of thesystem. :ote the dependence of the natural frequency on mass andstiness.

Temperature effects

$hanges in temperature create strain, as given by the equation

T

where is the thermal expansion coe/cient of the material, and T is thechange in temperature seen.#f the member in question is restrained against expansion %or contraction&due the temperature change, stress is created. This stress is equal to thestress that would be created if an external force were to create thedeformation from the stress0free state to the constrained state.

;aterial properties often behave as functions of temperature.


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structure& from some reference temperature to the temperature of theapplied distribution. The load causes deformation goverenced by theequation given above.

Fatigue

ll materials will brea! under numerous repetitions of a stress less than the

stress required to produce immediate failure. #t is common for structuralmembers to be sub"ected to load cycles repeated thousands or millions oftimes. #n such cases rupture can occur at a stress much lower than the staticbrea!ing strength 0 this is !nown as fatigue. fatigue failure is of a brittlenature, even for materials that are normally ductile.The number of loading cycles required to cause failure may be determinedexperimentally. #f a number of tests are conducted using dierent levels ofstress, the resulting data can be plotted to create a useful graph. =uch a

graph is !nown as a 0n curve.

typical 0n curve for steel is shown below. luminum is also plotted on the

graph. The cycles n are plotted on a logarithmic scale because of the largenumber of cycles required for failure.

3ecently, the microscopic cause of fatigue damage has been dened withcrac! initiation and crac! propagation theories.

Plasticity

Elastic deformation represents a change in the relative position of molecules.Plastic deformation represents a permanent change in the relative positionsof molecules.#n crystalline materials this permanent rearrangement consists largely ofgroup displacements of atoms in the crystal lattice slipping past one another0 brittle fracture.>uctile materials show the rearrangement in the shifting of molecularpositions, with the release of heat. (ow of material is seen, much li!e aviscous liquid.Plastic deformation before failure is much more pronounced in uniaxialtension than in a ?> stress state. #n a uniaxial state, plastic deformationbehavior can be seen in the standard stress0strain diagram.


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=ome )E codes allow for the use of material properties that are functions of

stress. 4ne common way this is handled is with a two0sloped curve, asshown*

The stress0strain curve is approximated by two straight lines. The solutionusing this material property must be an iterative one, because the materialproperty depends upon a result quantity, the strain. The solution isperformed in steps, using incremental amounts of load in each step. @eforeeach new step, a chec! is made for each element that has a stress valuehigh enough to get assigned the plastic modulus.#n reality, the stress strain curves of most materials are di/cult to ndtabulated in the nonlinear range. The solutions are generally much less

accurate %than standard linear solutions& because of this.

Creep

Experience has shown that for the design of equipment sub"ected tosustained elevated temperatures, little reliance can be placed on the short0term properties measured at those temperatures.'nder the application of constant load at elevated temperatures, materialsshow a gradual (ow or creep even for stresses below the proportional limit.=imilar eects are seen in low0melting metals such as lead at roomtemperature. graph of strain vs. time for any common metal shows that the creep graphhas three distinct sections, after an initial instantaneous elongation. Theseillustrate the dierent processes present with creep, !nown as primary,secondary, and tertiary creep.


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=train0hardening of metal decreases creep rate in the primary creep region.#n the secondary creep region, the material elongates at a steady rate. #n thenal phase, the tertiary creep region, elongation proceeds at an increasingrate until failure %due to nec!ing and void formation&.

advanced structural behavior

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