Published on January 23, 2016
1. Lecture 6 Energy principles Energy methods and variational principles Print version Lecture on Theory of Elasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACEG 6.1 Contents 1 Work and energy 1 2 Strain energy 2 3 Principle of Minimum Potential Energy 3 4 Ritz method 5 6.2 1 Work and energy Work and energy Work • A material particle is moved from point A to point B by a force F • The inﬁnitesimal distance along the path from A to B is a displacement du • The work dW performed by the force F is deﬁned as dW = F·du • The work done is the product of the displacement and the force in the direction of the displacement • The total work is W = B A F·du 6.3 Work and energy Work 1
2. • Work = Force × Displacement 6.4 Work and energy Energy • The energy is the capacity to do work • It is a measure of the capacity of all forces on a body to do work • Work is performed on a body trough a change in energy. 6.5 2 Strain energy Strain energy Internal work • The work done by forces on an elastic solid is stored inside the body in the form of a strain energy • Consider the uniaxial tension test where we can assume that the stresses increases slowly from zero to σ 6.6 Strain energy Internal work • The strain energy stored is equal to the work done on the differential element dU = σ 0 σd u+ ∂u ∂x dx dydz− σ 0 σdudydz = σ2 2E dxdydz 6.7 Strain energy Internal work • The strain energy density is the strain energy per unite volume U = dU dxdydz = 1 2 σε 6.8 2
3. Strain energy Internal work • The strain energy density is the shaded area under the stress-strain curve U = 1 2 σε 6.9 Strain energy Internal work • General expression for the strain energy density U = 1 2 (σxxεxx +σyyεyy +σzzεzz +2σxyεxy +2σyzεyz +2σzxεzx) = 1 2 σijεij = 1 2 σ : ε • The total strain energy is Utot = 1 2 V σ : εdV • The potential of the applied forces is W = − V f·udV + S t·udS 6.10 3 Principle of Minimum Potential Energy Principle of Minimum Potential Energy TPE functional • Total potential energy of the body ΠTPE = Utot +W = 1 2 V σ : εdV − V f·udV − S t·udS 6.11 Principle of Minimum Potential Energy TPE functional • The principle states that: The body is in equilibrium if there is an admissible displacement ﬁeld u that makes the total potential energy a minimum δΠTPE(u) = 0 where δΠTPE is a variation of the functional ΠTPE(u) • The admissible displacement ﬁeld is the one that satisfy the displacement BCs. Note The variational operator δ is much like the differential operator d except that it operates with respect to the dependent variable u rather than the independent variable x 6.12 3
4. Principle of Minimum Potential Energy 0.5 0.5 1.0 1.5 2.0 u, u 500 500 1000 W, U, TPE functional • Local minimum of the TPE functional 6.13 Principle of Minimum Potential Energy A gentle touch to the variational analysis • The condition for minimum of a functional I(u) is δI = ∂I ∂u δu = 0 • Almost the same as the condition for a minimum of a function u(x) du = ∂u ∂x dx = 0 6.14 Principle of Minimum Potential Energy A gentle touch to the variational analysis • Consider a functional I(u) = b a F x,u(x),u (x) dx • The minimum condition is δI(u) = b a ∂F ∂x δx+ ∂F ∂u δu+ ∂F ∂u δu dx = 0 • The integral can be manipulated to get the expression for the variation of u (integration by parts of the 3-rd addend) 6.15 Principle of Minimum Potential Energy A gentle touch to the variational analysis • The variation of the functional is δI(u) = b a ∂F ∂u − ∂ ∂x ∂F ∂u δudx • The non-trivial solution gives ∂F ∂u − ∂ ∂x ∂F ∂u = 0 • The above expression is called Euler equation 6.16 4
5. Principle of Minimum Potential Energy A gentle touch to the variational analysis • A more general 2D case is given by I(u,v) = A F (x,y,u,v,u,x,v,x,...,v,yy) • The Euler equations are ∂F ∂u − ∂ ∂x ∂F ∂u,x − ∂ ∂y ∂F ∂u,y + ∂2 ∂x2 ∂F ∂u,xx + ∂2 ∂x∂y ∂F ∂u,xy + ∂2 ∂y2 ∂F ∂u,yy = 0 ∂F ∂u − ∂ ∂x ∂F ∂u,x − ∂ ∂y ∂F ∂u,y + ∂2 ∂x2 ∂F ∂u,xx + ∂2 ∂x∂y ∂F ∂u,xy + ∂2 ∂y2 ∂F ∂u,yy = 0 6.17 Principle of Minimum Potential Energy Example • Consider a cantilever beam with length of and subjected to uniform load q. Using the principle of a minimum of TPE work out the equilibrium equation 6.18 4 Ritz method Ritz method Approximate solution • A lot of problems in elasticity there is no analytical solution of the ﬁeld equations • For such cases approximate solution schemes have been developed based on the varia- tional formulation of the problem (i.e. the principle of minimum potential energy) • The Ritz method is based on the idea of constructing a series of trial approximating func- tions that satisfy the essential (displacement) BCs but not differential equations exactly Note • Since the TPE functional includes the force BCs, it is require the trial solution satisﬁes only the displacement BCs! 6.19 Ritz method Approximate solution • Walter Ritz (1878-1909) 6.20 5
6. Ritz method Approximate solution • The original paper 6.21 Ritz method Approximate solution • The displacement can be expressed as u = u0 +a1u1 +a2u2 +...+anun = u0 + n ∑ i=1 aiui v = v0 +b1v1 +b2v2 +...+bnvn = v0 + n ∑ i=1 bivi w = w0 +c1w1 +c2w2 +...+cnwn = w0 + n ∑ i=1 ciwi • The terms of u0, v0 and w0 are chosen to satisfy any non-homogeneous displacement BCs and ui, vi and wi satisfy the corresponding homogeneous BCs • These forms are not required to satisfy the stress BCs 6.22 Ritz method Approximate solution • These trial functions are chosen from the some combinations of elementary functions (polynomials, trigonometric or hyperbolic forms) • The unknown coefﬁcients ai, bi and ci are to be determined so as to minimize the TPE functional of the problem • Thus we approximately satisfy the variational formulation of the problem • Using this approximation the TPE functional will be a function of these unknown coefﬁ- cients ΠTPE = ΠTPE(ai,bi,ci) 6.23 Ritz method Approximate solution • The minimizing condition ca be expressed as a series of ∂ΠTPE ∂ai = 0, ∂ΠTPE ∂bi = 0, ∂ΠTPE ∂ci = 0 • This set forms a system of 3n equations which gives ai, bi and ci 6
7. • Under suitable conditions on the choice of trial functions (completeness) the approxima- tion will improve as the number of included terms is increased • When the approximate displacement solution is obtained the strains and stresses can be calculated from the appropriate ﬁeld equations Note The method is suitable to apply at problems involving one or two displacements (bars, beams, plates and shells) 6.24 Ritz method Example • Consider a bending of simply supported beam of length carrying a uniform load q 6.25 Ritz method The End • Any questions, opinions, discussions? 6.26 7
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