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| The '''Föppl–von Kármán equations''', named after [[August Föppl]]<ref>Föppl, A., "Vorlesungen über technische Mechanik", ''B.G. Teubner'', Bd. 5., p. 132, Leipzig, Germany (1907)</ref> and [[Theodore von Kármán]],<ref>von Kármán, T., "Festigkeitsproblem im Maschinenbau," ''Encyk. D. Math. Wiss.'' '''IV''', 311–385 (1910)</ref> are a set of nonlinear [[partial differential equation]]s describing the large deflections of thin flat plates.<ref>E. Cerda and L. Mahadevan, 2003, "Geometry and Physics of Wrinkling" [http://prola.aps.org/abstract/PRL/v90/i7/e074302 Phys. Rev. Lett. 90, 074302 (2003)]</ref> With application ranging from the design of submarine hulls to the mechanical properties of cell wall,<ref>http://focus.aps.org/story/v27/st6</ref> the equations are notoriously difficult to solve, and take the following form: | | The S&P 500 closed at a record on Thursday and ended in positive territory for the year after Federal Reserve Chair Janet Yellen said harsh weather seems to be to behind recent U.S. economic softness.<br>That gave some relief to investors who supported the view that heavy snowstorms and unusually cold weather - and not worsening fundamentals - were to blame for weak U.S. employment, retail sales and other data.<br>The advance lifted the S&P 500 above its 2013 year-end closing level of 1,848.36, which has served as resistance in recent sessions.<br>"The market was worried. She could have excluded weather and perhaps talked more about the soft patch," said Quincy Krosby, market strategist at Prudential Financial, which is based in Newark, New Jersey.<br>"I think she gave the market some comfort that she thought it was probably mostly due to weather-related issues."<br>Testifying before the Senate Banking Committee, Yellen also said the Fed would watch carefully to make sure weather was indeed behind the recent weakness. But she said it would take a "significant change" to the economy's prospects for the central bank to put plans to reduce its bond-buying program on hold.<br>Some retailers scored sharp gains for a third session, with the shares of J.C. Penney Co Inc (JCP.N) and others jumping after the companies posted strong results.<br>Mylan Inc (MYL.O) gave one of the biggest boosts to both the S&P 500 and Nasdaq. Mylan's shares shot up 9.4 percent to end at $56.27 after the U.S. generic drugmaker gave a 2014 forecast above Wall Street's estimates. Mylan also said it plans to make a "substantial" transaction this year that would add to future earnings.<br>The Dow Jones industrial average .DJI rose 74.24 points or 0.46 percent, to end at 16,272.65. The S&P 500 .SPX gained 9.13 points or 0.49 percent, to finish at 1,854.29, surpassing its previous record closing high set on January 15.<br>The Nasdaq Composite .IXIC added 26.869 points or 0.63 percent, to close at 4,318.933.<br>For the year, the S&P 500 index is now up 0.3 percent.<br>After the bell, Gap Inc GSP.N shares slid 1 percent to $43.24 after the clothing retailer reported results. Shares of Deckers Outdoor Corp (DECK.O) tumbled 12.5 percent to $74.05 after the company, whose brands include [http://tinyurl.com/ku6vjks ugg boots sale] [http://tinyurl.com/ku6vjks ugg boots sale] and Teva sandals, posted earnings.<br>During the [http://Answers.yahoo.com/search/search_result?p=regular&submit-go=Search+Y!+Answers regular] session, J.C. Penney shares surged 25.3 percent to $7.47, a day after the U.S. department store chain forecast more improvement in its comparable sales and gross profit margin this fiscal year.<br>The S&P retail index .SPXRT has climbed 4.2 percent for the week so far, including its slim gain of 0.1 percent on Thursday.<br>Among other retailers, Best Buy Co Inc (BBY.N) reported a better-than-expected profit on Thursday. The stock rose as high as $28.19 before ending at $25.57, down 1 percent.<br>Sears Holding Corp (SHLD.O) reported a quarterly loss that narrowed from the year-ago period, sending its stock up 6.5 percent to $43.01. Kohl's Corp (KSS.N) said it expected modest sales gains in its new fiscal year and reported a lower fourth-quarter profit. Shares of Kohl's rose 2.4 percent to $55.74.<br>The day's economic data added to the positive tone, with orders for long-lasting U.S. manufactured goods excluding transportation, or durable goods excluding transportation, and a gauge of business spending unexpectedly rising in January.<br>About 6.5 billion shares changed hands on U.S. exchanges, below the 7 billion average so far this month, according to data from BATS Global Markets.<br>Advancers beat decliners on the New York Stock Exchange by a ratio of 2 to 1. On the Nasdaq, 17 stocks rose for every nine that fell.<br>(Editing by Jan Paschal)<br>Tweet this <br>Link this <br>Share this<br>Digg this <br>Email<br>Print<br>Reprints<br>We welcome comments that advance the story through relevant opinion, anecdotes, links and data. If you see a comment that you believe is irrelevant or inappropriate, you can flag it to our editors by using the report abuse links. Views expressed in the comments do not represent those of Reuters. For more information on our comment policy, see website Comments (3) Simplerman wrote: It�s clear the US has recently entered yet another recession evidenced by existing home sales and the Christmas (sorry - Holiday) season retail bust. Prepare accordingly.<br>Feb 27, 2014 10:51am EST -- Report as abuse Simplerman wrote: JC Penny will now accept EBT?? What did I miss?<br>Feb 27, 2014 11:35am EST -- Report as abuse TRADEEDGE wrote: [http://tinyurl.com/ku6vjks ugg boots usa] Nice so now they will sleep over this again in the name of continuous monitoring till the its too late. <br>Feb 27, 2014 8:34pm EST -- Report as abuse This discussion is now closed. We welcome comments on our articles for a limited period after their publication. See All Comments �<br> |
| <ref name="ld">"Theory of Elasticity". L. D. Landau, E. M. Lifshitz, (3rd ed. ISBN 0-7506-2633-X)</ref> | | Read<br> |
| | Obama makes rare campaign trail appearance, people leave early 19 Oct 2014<br><br> |
|
| |
|
| :<math>
| |
| \begin{align}
| |
| (1) \qquad & \frac{Eh^3}{12(1-\nu^2)}\Delta^2 w-h\frac{\partial}{\partial x_\beta}\left(\sigma_{\alpha\beta}\frac{\partial w}{\partial x_\alpha}\right)=P \\
| |
| (2) \qquad & \frac{\partial\sigma_{\alpha\beta}}{\partial x_\beta}=0
| |
| \end{align}
| |
| </math>
| |
|
| |
|
| where {{math|''E''}} is the [[Young's modulus]] of the plate material (assumed homogeneous and isotropic), {{math|''υ''}} is the [[Poisson's ratio]], {{math|''h''}} is the thickness of the plate, {{math|''w''}} is the out–of–plane deflection of the plate, {{math|''P''}} is the external normal force per unit area of the plate, {{math|''σ''<sub>''αβ''</sub>}} is the [[Cauchy stress tensor]], and {{math|''α'', ''β''}} are [[Einstein notation|indices]] that take values of 1 or 2. The 2-dimensional [[biharmonic equation|biharmonic operator]] is defined as<ref>The 2-dimensional [[Laplacian]], {{math|Δ}}, is defined as
| | Turkey to let Iraqi Kurds reinforce Kobani as U.S. drops arms to defenders | 6:58pm ED<br><br> |
| <math> \Delta w := \frac{\partial^2 w}{\partial x_\alpha \partial x_\alpha} = \frac{\partial^2w}{\partial x_1^2} + \frac{\partial^2w}{\partial x_2^2} </math></ref> | | Obama makes rare campaign trail appearance, some leave early 5:13am ED<br><br> |
| :<math>
| | Nigeria declared Ebola-free, holds lessons for others | 3:59pm ED<br><br> |
| \Delta^2 w := \frac{\partial^2}{\partial x_\alpha \partial x_\alpha}\left[\frac{\partial^2 w}{\partial x_\beta \partial x_\beta}\right]
| | U.S. stocks end higher despite drag from IBM 5:08pm ED<br> |
| = \frac{\partial^4 w}{\partial x_1^4} + \frac{\partial^4 w}{\partial x_2^4} + 2\frac{\partial^4 w}{\partial x_1^2 \partial x_2^2} \,.
| | Follow Reuters Faceboo<br> |
| </math>
| | Twitte<br> |
| Equation (1) above can be derived from [[kinematic]] assumptions and the [[constitutive relation]]s for the plate. Equations (2) are the two equations for the conservation of linear momentum in two dimensions where it is assumed that the out–of–plane stresses ({{math|''σ''<sub>33</sub>,''σ''<sub>13</sub>,''σ''<sub>23</sub>}}) are zero.
| | RS<br> |
| | | YouTub<br><br> |
| == Validity of the Föppl–von Kármán equations ==
| | Edition: U.S<br> |
| While the Föppl–von Kármán equations are of interest from a purely mathematical point of view, the physical validity of these equations is questionable.<ref>[http://imechanica.org/node/6618 von Karman plate equations http://imechanica.org/node/6618 Accessed Tue July 30 2013 14:20.]</ref> Ciarlet<ref name=Ciarlet>{{Citation|last=Ciarlet|first=P. G.|year=1990|title=Plates and Junctions in Elastic Multi-Structures|publisher=Springer-Verlag.}}</ref> states: ''The two-dimensional von Karman equations for plates, originally proposed by von Karman [1910], play a mythical role in applied mathematics. While they have been abundantly, and satisfactorily, studied from the mathematical standpoint, as regards notably various questions of existence, regularity, and bifurcation, of their solutions, their physical soundness has been often seriously questioned.'' Reasons include the facts that
| | Arabi<br><br> |
| # the theory depends on an approximate geometry which is not clearly defined
| | Argentin<br><br> |
| # a given variation of stress over a cross-section is assumed arbitrarily
| | Brazi<br><br> |
| # a linear constitutive relation is used that does not correspond to a known relation between well defined measures of stress and strain
| | Canad<br><br> |
| # some components of strain are arbitrarily ignored
| | Chin<br><br> |
| # there is a confusion between reference and deformed configurations which makes the theory inapplicable to the large deformations for which it was apparently devised.
| | Franc<br><br> |
| Conditions under which these equations are actually applicable and will give reasonable results when solved are discussed in Ciarlet.<ref name=Ciarlet/><ref>{{Citation|last= Ciarlet|first= Philippe G. | title=A justification of the von Kármán equations|journal= Archive for Rational Mechanics and Analysis |volume=73|year=1980|pages= 349–389.|issue=4|bibcode= 1980ArRMA..73..349C|doi= 10.1007/BF00247674}}</ref>
| | German<br><br> |
| | | Indi<br><br> |
| == Equations in terms of Airy stress function ==
| | Ital<br><br> |
| The three Föppl–von Kármán equations can be reduced to two by introducing the [[Stress functions#Airy stress function|Airy stress function]] <math>\varphi</math> where
| | Japa<br><br> |
| :<math>
| | Latin Americ<br><br> |
| \sigma_{11} = \frac{\partial^2 \varphi}{\partial x_2^2} ~,~~
| | Mexic<br><br> |
| \sigma_{22} = \frac{\partial^2 \varphi}{\partial x_1^2} ~,~~
| | Russi<br><br> |
| \sigma_{12} = - \frac{\partial^2 \varphi}{\partial x_1 \partial x_2} \,.
| | Spai<br><br> |
| </math>
| | United Kingdo<br><br> |
| Then the above equations become<ref name="ld"/>
| | Back to to<br> |
| :<math>
| | Reuters.com Busines<br> |
| \frac{Eh^3}{12(1-\nu^2)}\Delta^2 w-h\left(\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2 w}{\partial x_1^2}+\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2 w}{\partial x_2^2}-2\frac{\partial^2\varphi}{\partial x_1 \, \partial x_2}\frac{\partial^2 w}{\partial x_1 \, \partial x_2}\right)=P
| | Market<br> |
| </math> | | Worl<br> |
| :<math>
| | Politic<br> |
| \Delta^2\varphi+E\left\{\frac{\partial^2 w}{\partial x_1^2}\frac{\partial^2 w}{\partial x_2^2}-\left(\frac{\partial^2 w}{\partial x_1 \, \partial x_2}\right)^2\right\}=0 \,.
| | Technolog<br> |
| </math> | | Opinio<br> |
| | | Mone<br> |
| == Pure bending==
| | Picture<br> |
| For the [[pure bending]] of thin plates the equation of equilibrium is <math>D\Delta^2\ w=P</math>, where
| | Video<br> |
| :<math>
| | Site Inde<br> |
| D :=\frac{Eh^3}{12(1-\nu^2)}
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| </math> | | Brand Attribution Guideline<br> |
| is called [[flexural rigidity|flexural]] or ''cylindrical rigidity'' of the plate.<ref name="ld"/>
| | Delivery Option<br> |
| | | Support & Contact Suppor<br> |
| == Kinematic assumptions (Kirchhoff hypothesis) ==
| | Correction<br> |
| In the derivation of the Föppl–von Kármán equations the main kinematic assumption (also known as the '''Kirchhoff hypothesis''') is that [[surface normal]]s to the plane of the plate remain perpendicular to the plate after deformation. It is also assumed that the in-plane (membrane) displacements and the change in thickness of the plate are negligible. These assumptions imply that the displacement field {{math|'''u'''}} in the plate can be expressed as
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| :<math>
| | Facebook <br> |
| u_1(x_1,x_2,x_3) = -x_3\,\frac{\partial w}{\partial x_1} ~,~~
| | LinkedIn <br> |
| u_2(x_1,x_2,x_3) = -x_3\,\frac{\partial w}{\partial x_2} ~,~~
| | RSS <br> |
| u_3(x_1, x_2, x_3) = w(x_1,x_2)
| | Podcast <br> |
| </math>
| | Newsletters <br> |
| This form of the displacement field implicitly assumes that the amount of rotation of the plate is small.
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| | | About Privacy Polic<br> |
| == Strain-displacement relations (von Kármán strains) ==
| | Terms of Us<br> |
| The components of the three-dimensional Lagrangian [[Finite strain theory|Green strain tensor]] are defined as
| | Advertise With U<br> |
| :<math>
| | AdChoice<br> |
| E_{ij} := \frac{1}{2}\left[\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}
| | Copyrigh<br> |
| + \frac{\partial u_k}{\partial x_i}\,\frac{\partial u_k}{\partial x_j}\right] \,.
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| </math>
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| Substitution of the expressions for the displacement field into the above gives
| | Investor Relation<br> |
| :<math>
| | Career<br> |
| \begin{align}
| | Contact U<br> |
| E_{11} & = \frac{\partial u_1}{\partial x_1}
| | Thomson Reuters is the world's largest international multimedia news agency, providing investing news, world news, business news, technology news, headline news, small business news, news alerts, personal finance, stock market, and mutual funds information available [http://tinyurl.com/ku6vjks uggs on sale] Reuters.com, video, mobile, and interactive television platforms. Thomson Reuters journalists are subject to an Editorial Handbook which [http://tinyurl.com/ku6vjks uggs on sale] requires fair presentation and disclosure of relevant interests<br> |
| + \frac{1}{2}\left[\left(\frac{\partial u_1}{\partial x_1}\right)^2
| | NYSE and AMEX quotes delayed by at least 20 minutes. Nasdaq delayed by at least 15 minutes. For a complete list of exchanges and delays, please click here. |
| + \left(\frac{\partial u_2}{\partial x_1}\right)^2
| |
| + \left(\frac{\partial u_3}{\partial x_1}\right)^2\right]\\
| |
| &= -x_3\,\frac{\partial^2 w}{\partial x_1^2}
| |
| + \frac{1}{2}\left[x_3^2\left(\frac{\partial^2 w}{\partial x_1^2}\right)^2
| |
| + x_3^2\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)^2
| |
| + \left(\frac{\partial w}{\partial x_1}\right)^2\right]\\
| |
| E_{22} & = \frac{\partial u_2}{\partial x_2}
| |
| + \frac{1}{2}\left[\left(\frac{\partial u_1}{\partial x_2}\right)^2
| |
| + \left(\frac{\partial u_2}{\partial x_2}\right)^2
| |
| + \left(\frac{\partial u_3}{\partial x_2}\right)^2\right]\\
| |
| &= -x_3\,\frac{\partial^2 w}{\partial x_2^2}
| |
| + \frac{1}{2}\left[x_3^2\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)^2
| |
| + x_3^2\left(\frac{\partial^2 w}{\partial x_2^2}\right)^2
| |
| + \left(\frac{\partial w}{\partial x_2}\right)^2\right]\\
| |
| E_{33} & = \frac{\partial u_3}{\partial x_3}
| |
| + \frac{1}{2}\left[\left(\frac{\partial u_1}{\partial x_3}\right)^2
| |
| + \left(\frac{\partial u_2}{\partial x_3}\right)^2
| |
| + \left(\frac{\partial u_3}{\partial x_3}\right)^2\right]\\
| |
| &= \frac{1}{2}\left[\left(\frac{\partial w}{\partial x_1}\right)^2
| |
| + \left(\frac{\partial w}{\partial x_2}\right)^2
| |
| \right]\\
| |
| E_{12} & = \frac{1}{2}\left[\frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1}
| |
| + \frac{\partial u_1}{\partial x_1}\,\frac{\partial u_1}{\partial x_2}
| |
| + \frac{\partial u_2}{\partial x_1}\,\frac{\partial u_2}{\partial x_2}
| |
| + \frac{\partial u_3}{\partial x_1}\,\frac{\partial u_3}{\partial x_2}\right]\\
| |
| & = -x_3\frac{\partial^2 w}{\partial x_1 \partial x_2}
| |
| + \frac{1}{2}\left[x_3^2\left(\frac{\partial^2 w}{\partial x_1^2}\right)\left(\frac{\partial^2 w}{\partial x_1\partial x_2}\right)
| |
| + x_3^2\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)\left(\frac{\partial^2 w}{\partial x_2^2}\right)
| |
| + \frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}\right]\\
| |
| E_{23} & = \frac{1}{2}\left[\frac{\partial u_2}{\partial x_3} + \frac{\partial u_3}{\partial x_2}
| |
| + \frac{\partial u_1}{\partial x_2}\,\frac{\partial u_1}{\partial x_3}
| |
| + \frac{\partial u_2}{\partial x_2}\,\frac{\partial u_2}{\partial x_3}
| |
| + \frac{\partial u_3}{\partial x_2}\,\frac{\partial u_3}{\partial x_3}\right]\\
| |
| & = \frac{1}{2}\left[x_3\left(\frac{\partial^2 w}{\partial x_1\partial x_2}\right)\left(\frac{\partial w}{\partial x_1}\right)
| |
| + x_3\left(\frac{\partial^2 w}{\partial x_2^2}\right)\left(\frac{\partial w}{\partial x_2}\right)
| |
| \right]\\
| |
| E_{31} & = \frac{1}{2}\left[\frac{\partial u_3}{\partial x_1} + \frac{\partial u_1}{\partial x_3}
| |
| + \frac{\partial u_1}{\partial x_3}\,\frac{\partial u_1}{\partial x_1}
| |
| + \frac{\partial u_2}{\partial x_3}\,\frac{\partial u_2}{\partial x_1}
| |
| + \frac{\partial u_3}{\partial x_3}\,\frac{\partial u_3}{\partial x_1}\right] \\
| |
| & = \frac{1}{2}\left[x_3\left(\frac{\partial w}{\partial x_1}\right)\left(\frac{\partial^2 w}{\partial x_1^2}\right)
| |
| + x_3\left(\frac{\partial w}{\partial x_2}\right)\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)
| |
| \right]
| |
| \end{align}
| |
| </math>
| |
| For small strains but '''moderate rotations''', the higher order terms that cannot be neglected are
| |
| :<math>
| |
| \left(\frac{\partial w}{\partial x_1}\right)^2 ~,~~ \left(\frac{\partial w}{\partial x_2}\right)^2 ~,~~
| |
| \frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2} \,.
| |
| </math>
| |
| Neglecting all other higher order terms, and enforcing the requirement that the plate does not change its thickness, the strain tensor components reduce to the '''von Kármán strains'''
| |
| :<math>
| |
| \begin{align}
| |
| E_{11} & = -x_3\,\frac{\partial^2 w}{\partial x_1^2}
| |
| + \frac{1}{2}\left(\frac{\partial w}{\partial x_1}\right)^2 \\
| |
| E_{22} & = -x_3\,\frac{\partial^2 w}{\partial x_2^2}
| |
| + \frac{1}{2}\left(\frac{\partial w}{\partial x_2}\right)^2 \\
| |
| E_{12} & = -x_3\frac{\partial^2 w}{\partial x_1 \partial x_2}
| |
| + \frac{1}{2}\,\frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}\\
| |
| E_{33} & = 0 ~,~~ E_{23} = 0 ~,~~ E_{31} = 0 \,.
| |
| \end{align}
| |
| </math>
| |
| | |
| == Stress-strain relations ==
| |
| If we assume that the [[Cauchy stress tensor]] components are linearly related to the von Kármán strains by [[Hooke's law]], the plate is isotropic and homogeneous, and that the plate in under a [[plane stress]] condition,<ref>Typically, an assumption of '''zero out-of-plane stress''' is made at this point.</ref> we have {{math|''σ''<sub>33</sub>}} = {{math|''σ''<sub>13</sub>}} = {{math|''σ''<sub>23</sub>}} = 0 and
| |
| :<math>
| |
| \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix}
| |
| = \cfrac{E}{(1-\nu^2)}
| |
| \begin{bmatrix} 1 & \nu & 0 \\
| |
| \nu & 1 & 0 \\
| |
| 0 & 0 & 1-\nu \end{bmatrix}
| |
| \begin{bmatrix} E_{11} \\ E_{22} \\ E_{12} \end{bmatrix}
| |
| </math>
| |
| Expanding the terms, the three non-zero stresses are
| |
| :<math>
| |
| \begin{align}
| |
| \sigma_{11} &= \cfrac{E}{(1-\nu^2)}\left[\left(-x_3\,\frac{\partial^2 w}{\partial x_1^2}
| |
| + \frac{1}{2}\left(\frac{\partial w}{\partial x_1}\right)^2 \right) +
| |
| \nu\left(-x_3\,\frac{\partial^2 w}{\partial x_2^2}
| |
| + \frac{1}{2}\left(\frac{\partial w}{\partial x_2}\right)^2 \right) \right] \\
| |
| \sigma_{22} &= \cfrac{E}{(1-\nu^2)}\left[\nu\left(-x_3\,\frac{\partial^2 w}{\partial x_1^2}
| |
| + \frac{1}{2}\left(\frac{\partial w}{\partial x_1}\right)^2 \right) +
| |
| \left(-x_3\,\frac{\partial^2 w}{\partial x_2^2}
| |
| + \frac{1}{2}\left(\frac{\partial w}{\partial x_2}\right)^2 \right) \right] \\
| |
| \sigma_{12} &= \cfrac{E}{(1+\nu)}\left[-x_3\frac{\partial^2 w}{\partial x_1 \partial x_2}
| |
| + \frac{1}{2}\,\frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}\right] \,.
| |
| \end{align}
| |
| </math>
| |
| | |
| == Stress resultants ==
| |
| The [[stress resultants]] in the plate are defined as
| |
| :<math>
| |
| N_{\alpha\beta} := \int_{-h/2}^{h/2} \sigma_{\alpha\beta}\, d x_3 ~,~~
| |
| M_{\alpha\beta} := \int_{-h/2}^{h/2} x_3\,\sigma_{\alpha\beta}\, d x_3 \,.
| |
| </math>
| |
| Therefore,
| |
| :<math>
| |
| \begin{align}
| |
| N_{11} &= \cfrac{Eh}{2(1-\nu^2)}\left[\left(\frac{\partial w}{\partial x_1}\right)^2
| |
| + \nu\left(\frac{\partial w}{\partial x_2}\right)^2 \right] \\
| |
| N_{22} &= \cfrac{Eh}{2(1-\nu^2)}\left[\nu\left(\frac{\partial w}{\partial x_1}\right)^2
| |
| + \left(\frac{\partial w}{\partial x_2}\right)^2 \right] \\
| |
| N_{12} &= \cfrac{Eh}{2(1+\nu)}\,\frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}
| |
| \end{align}
| |
| </math>
| |
| and | |
| :<math> | |
| \begin{align}
| |
| M_{11} &= -\cfrac{Eh^3}{12(1-\nu^2)}\left[\frac{\partial^2 w}{\partial x_1^2} +\nu \,\frac{\partial^2 w}{\partial x_2^2} \right] \\
| |
| M_{22} &= -\cfrac{Eh^3}{12(1-\nu^2)}\left[\nu \,\frac{\partial^2 w}{\partial x_1^2} +\frac{\partial^2 w}{\partial x_2^2} \right] \\
| |
| M_{12} &= -\cfrac{Eh^3}{12(1+\nu)}\,\frac{\partial^2 w}{\partial x_1 \partial x_2} \,.
| |
| \end{align}
| |
| </math>
| |
| Solutions are easier to find when the governing equations are expressed in terms of stress resultants rather than the in-plane stresses.
| |
| | |
| == Föppl–von_Kármán equations in terms of stress resultants ==
| |
| The Föppl–von_Kármán equations are typically derived with an energy approach by considering [[variational calculus|variation]]s of internal energy and the work done by external forces. A similar approach can be used to write these equations in terms of stress resultants. The resulting governing equations are
| |
| :<math>
| |
| \begin{align}
| |
| &\frac{\partial^2 M_{11}}{\partial x_1^2} + \frac{\partial^2 M_{22}}{\partial x_2^2} + 2\frac{\partial^2 M_{12}}{\partial x_1\partial x_2} +
| |
| \frac{\partial}{\partial x_1}\left(N_{11}\,\frac{\partial w}{\partial x_1} + N_{12}\,\frac{\partial w}{\partial x_2}\right) +
| |
| \frac{\partial}{\partial x_2}\left(N_{12}\,\frac{\partial w}{\partial x_1} + N_{22}\,\frac{\partial w}{\partial x_2}\right) = P \\
| |
| & \frac{\partial N_{\alpha\beta}}{\partial x_\beta} = 0 \,.
| |
| \end{align}
| |
| </math>
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| == See also ==
| |
| * [[Plate theory]]
| |
| | |
| {{DEFAULTSORT:Foppl-Von Karman Equations}}
| |
| [[Category:Partial differential equations]]
| |
| [[Category:Continuum mechanics]]
| |
The S&P 500 closed at a record on Thursday and ended in positive territory for the year after Federal Reserve Chair Janet Yellen said harsh weather seems to be to behind recent U.S. economic softness.
That gave some relief to investors who supported the view that heavy snowstorms and unusually cold weather - and not worsening fundamentals - were to blame for weak U.S. employment, retail sales and other data.
The advance lifted the S&P 500 above its 2013 year-end closing level of 1,848.36, which has served as resistance in recent sessions.
"The market was worried. She could have excluded weather and perhaps talked more about the soft patch," said Quincy Krosby, market strategist at Prudential Financial, which is based in Newark, New Jersey.
"I think she gave the market some comfort that she thought it was probably mostly due to weather-related issues."
Testifying before the Senate Banking Committee, Yellen also said the Fed would watch carefully to make sure weather was indeed behind the recent weakness. But she said it would take a "significant change" to the economy's prospects for the central bank to put plans to reduce its bond-buying program on hold.
Some retailers scored sharp gains for a third session, with the shares of J.C. Penney Co Inc (JCP.N) and others jumping after the companies posted strong results.
Mylan Inc (MYL.O) gave one of the biggest boosts to both the S&P 500 and Nasdaq. Mylan's shares shot up 9.4 percent to end at $56.27 after the U.S. generic drugmaker gave a 2014 forecast above Wall Street's estimates. Mylan also said it plans to make a "substantial" transaction this year that would add to future earnings.
The Dow Jones industrial average .DJI rose 74.24 points or 0.46 percent, to end at 16,272.65. The S&P 500 .SPX gained 9.13 points or 0.49 percent, to finish at 1,854.29, surpassing its previous record closing high set on January 15.
The Nasdaq Composite .IXIC added 26.869 points or 0.63 percent, to close at 4,318.933.
For the year, the S&P 500 index is now up 0.3 percent.
After the bell, Gap Inc GSP.N shares slid 1 percent to $43.24 after the clothing retailer reported results. Shares of Deckers Outdoor Corp (DECK.O) tumbled 12.5 percent to $74.05 after the company, whose brands include ugg boots sale ugg boots sale and Teva sandals, posted earnings.
During the regular session, J.C. Penney shares surged 25.3 percent to $7.47, a day after the U.S. department store chain forecast more improvement in its comparable sales and gross profit margin this fiscal year.
The S&P retail index .SPXRT has climbed 4.2 percent for the week so far, including its slim gain of 0.1 percent on Thursday.
Among other retailers, Best Buy Co Inc (BBY.N) reported a better-than-expected profit on Thursday. The stock rose as high as $28.19 before ending at $25.57, down 1 percent.
Sears Holding Corp (SHLD.O) reported a quarterly loss that narrowed from the year-ago period, sending its stock up 6.5 percent to $43.01. Kohl's Corp (KSS.N) said it expected modest sales gains in its new fiscal year and reported a lower fourth-quarter profit. Shares of Kohl's rose 2.4 percent to $55.74.
The day's economic data added to the positive tone, with orders for long-lasting U.S. manufactured goods excluding transportation, or durable goods excluding transportation, and a gauge of business spending unexpectedly rising in January.
About 6.5 billion shares changed hands on U.S. exchanges, below the 7 billion average so far this month, according to data from BATS Global Markets.
Advancers beat decliners on the New York Stock Exchange by a ratio of 2 to 1. On the Nasdaq, 17 stocks rose for every nine that fell.
(Editing by Jan Paschal)
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Feb 27, 2014 10:51am EST -- Report as abuse Simplerman wrote: JC Penny will now accept EBT?? What did I miss?
Feb 27, 2014 11:35am EST -- Report as abuse TRADEEDGE wrote: ugg boots usa Nice so now they will sleep over this again in the name of continuous monitoring till the its too late.
Feb 27, 2014 8:34pm EST -- Report as abuse This discussion is now closed. We welcome comments on our articles for a limited period after their publication. See All Comments �
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