8 The ernoulli- Euler eam 8 1
Chapter 8: THE ERNOULLI- EULER E TLE OF CONTENTS Page 8.1. Introduction 8 3 8.2. The eam odel 8 3 8.2.1. Field Equations................. 8 3 8.2.2. oundary Conditions.............. 8 4 8.2.3. The SF Tonti Diagram.............. 8 4 8.3. The TPE (Primal) Functional 8 5 8.4. The TCPE (Dual) Functional 8 7 8.5. The Hellinger-Reissner Functional 8 9 8. Exercises...................... 8 11 8 2
8.2 THE E ODEL z, (x) q(x) (a) L EI(x) x eam and applied load (b) + Internal forces. Note sign conventions (c) Prescribed transverse displacement and rotation θ at left end. Note sign conventions +θ + +V + +V Prescribed bending moment and transverse shear V at right end. Note sign conventions 8.1. Introduction Figure 8.1. The ernoulli-euler beam model: (a) beam and transverse load; (b) positive convention for moment and shear; (c) boundary conditions. From the Poisson equation e move to elasticity and structural mechanics. Rather than tackling the full 3D problem first this Chapter illustrates, in a tutorial style, the derivation of Variational Forms for a one-dimensional model: the ernoulli-euler beam. Despite the restriction to 1D, the mathematics offers a ne and challenging ingredient: the handling of functionals ith second space derivatives of displacements. Physically these are curvatures of deflected shapes. In structural mechanics curvatures appear in problems involving beams, plates and shells. In fluid mechanics second derivatives appear in slo viscous flos. 8.2. The eam odel The beam under consideration extends from x = tox = L and has a bending rigidity EI, hich may be a function of x. See Figure 8.1(a). The transverse load q(x) is given in units of force per length. The unknon fields are the transverse displacement (x), rotation θ(x), curvature (x), bending moment (x) and transverse shear V (x). Positive sign conventions for and V are illustrated in Figure 8.1(b). oundary conditions are applied only at and. For definiteness the end conditions shon in Figure 8.1(c) ill be used. 8.2.1. Field Equations In hat follos a prime denotes derivative ith respect to x. The field equations over x L are as follos. 8 3
Chapter 8: THE ERNOULLI- EULER E (KE) Kinematic equations: θ = d dx =, = d2 dx 2 = = θ, (8.1) here θ is the rotation of a cross section and the curvature of the deflected longitudinal axis. Relations (8.1) express the kinematics of an ernoulli-euler beam: plane sections remain plane and normal to the deflected neutral axis. (CE) Constitutive equation: = EI, (8.2) here I is the second moment of inertia of the cross section ith respect to y (the neutral axis). This moment-curvature relation is a consequence of assuming a linear distribution of strains and stresses across the cross section. It is derived in elementary courses of echanics of aterials. (E) alance (equilibrium) equations: V = d dx =, dv dx q = V q = q =. (8.3) Equations (8.3) are established by elementary means in echanics of aterials courses. 1 8.2.2. oundary Conditions For the sake of specificity, the boundary conditions assumed for the example beam are of primaryvariable (PC) type on the left and of flux type (FC) on the right: (PC) at (x = ) : = ŵ, θ = ˆθ, (FC) at (x = L) : = ˆ, V = ˆV, (8.4) in hich ŵ, ˆθ, ˆ and ˆV are prescribed. If ŵ = ˆθ =, these conditions physically correspond to a cantilever (fixed-free) beam. We ill let ŵ and ˆθ be arbitrary, hoever, to further illuminate their role in the functionals. Remark 8.1. Note that a positive ˆV acts donard (in the z direction) as can be seen from Figure 8.1(b), so it disagrees ith the positive deflection +. On the other hand a positive ˆ acts counterclockise, hich agrees ith the positive rotation +θ. 1 In those courses, hoever, +q(x) is sometimes taken to act donard, leading to V + q = and + q =. 8 4
8.3 THE TPE (PRIL) FUNCTIONL θ PC: θ = ' q = θ = θ at KE: = '' E: '' q = = EI V = ' FC: = V = V at V Figure 8.2. Tonti diagram of Strong Form of ernoulli-euler beam model. 8.2.3. The SF Tonti Diagram The Strong Form Tonti diagram for the ernoulli-euler model of Figure 8.1 is dran in Figure 8.2. The diagram lists the three field equations (KE, CE, E) and the boundary conditions (PC, FC). The latter are chosen in the very specific manner indicated above to simplify the boundary terms. 2 Note than in this beam model the rotation θ = and the transverse shear V = play the role of auxiliary variables that are not constitutively related. The only constitutive equation is the moment-curvature equation = EI. The reason for the presence of such auxiliary variables is their direct appearance in boundary conditions. 3 8.3. The TPE (Primal) Functional Select as only master field. Weaken the E and FC connections to get the Weak Form (WF) diagram of Figure 8.4 as departure point. Choose the eighting functions on the eak links E, FC on and FC on V to be δ, δθ and δ, respectively. The eak links are combined as follos: [ ] δ = ( ) q δ dx + ( ˆ)δθ (V ˆV )δ =. (8.5) Why the different signs for the moment and shear boundary terms? If confused, read Remark 4.1. Next, integrate ( ) δ dx tice by parts: [ ] ( ) δ dx = ( ) δ dx + ( ) δ [ ] [ = δ dx + ( ) δ δ ] (8.6) = δ dx + V δ δθ. 2 In fact 2 4 = 16 boundary condition combinations are mathematically possible. Some of these correspond to physically realizable support conditions, for example simply supports, hereas others do not. 3 In the Timoshenko beam model, hich accounts for transverse shear energy, θ appears in the constitutive equations. 8 5
Chapter 8: THE ERNOULLI- EULER E θ PC: = θ = θ at θ = ' aster q E: ('' q) δ dx = KE: = '' = EI V = ( )' FC: V ( ) δθ = (V V ) δ = at Figure 8.3. Weak Form used as departure point to derive the TPE functional for the ernoulli-euler beam The disappearance of boundary terms at in the last equation results from δ =, δ = δθ = on account of the strong PC connection at x =. Inserting (8.6) into (8.5) gives δ = ( δ q δ) dx ˆ δθ + ˆV δ (8.7) = (EI δ q δ) dx ˆ δ + ˆV δ. This is the first variation of the functional TPE [] = 1 2 EI( ) 2 dx q dx ˆ + ˆV. (8.8) This is called the Total Potential Energy (TPE) functional of the ernoulli-euler beam. It as used in the Introduction to Finite Element ethods course to derive the ell knon Hermitian beam element in Chapter 12. For many developments it is customarily split into to terms in hich U[] = 1 2 [] = U[] W [], (8.9) EI( ) 2 dx, W [] = q dx + ˆ ˆV. (8.1) Here U is the internal energy (strain energy) of the beam due to bending deformations (bending moments orking on curvatures), hereas W gathers the other terms that collectively represent the external ork of the applied loads. 4 4 Recall that ork and energy have opposite signs, since energy is the capacity to produce ork. It is customary to rite = U W instead of the equivalent = U + V, here V = W is the external ork potential. This notational device also frees the symbol V to be used for transverse shear in beams and voltage in electromagnetics. 8 6
8.4THE TCPE (DUL) FUNCTIONL KE: θ PC: = θ = θ at ( ) δ dx = θ = ' Ignorable = /(EI) aster q V = E: '' q = FC: = V = V at V Figure 8.4. Weak Form used as departure point to derive the TCPE functional for the ernoulli-euler beam Remark 8.2. Using integration by parts one can sho that if δ =, U = 1 W. (8.11) 2 In other ords: at equilibrium the internal energy is half the external ork. This property is valid for any linear elastic continuum. (It is called Clapeyron s theorem in the literature of Structural echanics.) It has a simple geometric interpretation for structures ith finite number of degrees of freedom. 8.4. The TCPE (Dual) Functional The Total Complementary Potential Energy (TCPE) functional is mathematically the dual of the primal (TPE) functional. Select the bending moment as the only master field. ake KE eak to get the Weak Form diagram displayed in Figure 8.4. Choose the eighting function on the eakened KE to be δ. The only contribution to the variation of the functional is δ = ( )δdx= ( )δdx=. (8.12) Integrate δdxby parts tice: [ ] δdx= δ dx + δ [ ] [ = δ dx + δ δ ] = δ dx ˆθ δ +ŵδv. (8.13) The disappearance of the boundary terms at results from enforcing strongly the free-end boundary conditions = ˆ and V = ˆV, hence the variations δ =, δv = δ =. ecause of 8 7
Chapter 8: THE ERNOULLI- EULER E θ KE: ( ) δ dx = = /(EI) aster V = ' Figure 8.5. The collapsed WF diagram for the TCPE functional, shoing only leftovers boxes. the strong E connection, q =, δ vanishes identically in x L. Consequently δdx= ˆθ δ +ŵδv. (8.14) Replacing into (8.12) yields δ = This is the first variation of the functional [] = 1 2 and since = /EI,efinally get δdx+ ˆθ δ ŵδv =. (8.15) TCPE [] = 1 2 dx+ ˆθ V ŵ, (8.16) 2 EI dx + ˆθ V ŵ. (8.17) This is the TCPE functional for the ernoulli-euler beam model. s in the case of the TPE, it is customarily split as [] = U [] W [], (8.18) here U = 1 2 2 EI dx, W = ˆθ + V ŵ. (8.19) Here U is the internal complementary energy stored in the beam by virtue of its deformation, and W is the external complementary energy that collects the ork of the prescribed end displacements and rotations. Note that only (and its slaves), ˆθ and ŵ remain in this functional. The transverse displacement (x) is gone and consequently is labeled as ignorable in Figure 8.5. Through the integration by parts process the WF diagram of Figure 8.4 collapses to the one sketched in Figure 8.5. The reduction may be obtained by invoking the folloing to rules: 8 8
8.5 THE HELLINGER-REISSNER FUNCTIONL (1) The ignorable box, θ of Figure 8.4 may be replaced by the data box ŵ, ˆθ because only the boundary values of those quantities survive. (2) The data boxes q and ˆ, ˆV of Figure 8.4 may be removed because they are strongly connected to the varied field. The collapsed WF diagram of Figure 8.5 displays the five quantities (, V,, ŵ, ˆθ ) that survive in the TCPE functional. Remark 8.3. One can easily sho that for the actual solution of the beam problem, U = U, a property valid for any linear elastic continuum. Furthermore U = 1 2 W. 8.5. The Hellinger-Reissner Functional The TPE and TCPE functionals are single-field, because there is only one master field that is varied: displacements in the former and moments in the latter. We next illustrate the derivation of a tofield mixed functional, identified as the Hellinger-Reissner (HR) functional HR [, ], for the ernoulli-euler beam. Here both displacements and moments are picked as master fields and thus are independently varied. The point of departure is the WF diagram of Figure 8.6. s illustrated, three links: KE, E and FC, have been eakened. The master (varied) fields are and. It is necessary to distinguish beteen displacement-derived curvatures = and moment-derived curvatures = /EI, as shon in the figure. The to curvature boxes are eakly connected, expressing that the equality = /EI is not enforced strongly. The mathematical expression of the WF, having chosen eights δ, δ, δ = δθ and δ for the eak connections KE, E, moment in FC and shear V in FC, respectively, is δ [, ] = ( )δdx+ ( q)δdx + ( ˆ)δθ (V ˆV )δ. (8.2) Integrating δ tice by parts as in the TPE derivation, inserting in (8.2), and enforcing the strong PCs at, yields [ δ [, ] = ( )δ + δ ] dx q δ dx + ˆV δ ˆ δθ, (8.21) This is the first variation of the Hellinger-Reissner (HR) functional ) HR [, ] = ( 2 2EI q dx + ˆV ˆ θ. (8.22) gain this can be split as HR = U W, in hich ) U[, ] = ( 2 dx, W [] = 2EI represent internal energy and external ork, respectively. q dx ˆV + ˆ θ, (8.23) 8 9
Chapter 8: THE ERNOULLI- EULER E θ PC: = θ = θ at θ = ' KE: = '' aster fields ( and ) q E: ('' q) δ dx = ( ) δ dx = fields = /(EI) V = ' FC: V ( ) δθ = (V V ) δ = at Figure 8.6. The WF Tonti diagram used as departure point for deriving the Hellinger- Reissner (HR) functional. Remark 8.4. If the primal boundary conditions (PC) at are eakened, the functional (8.22) gains to extra boundary terms. Remark 8.5. The = term in (8.23) may be transformed by applying integration by parts once: dx = L dx = ˆ θ ˆθ dx (8.24) to get an alternative form of the HR equation ith balanced derivatives in and. Such transformations are common in the finite element applications of mixed functionals. The objective is to exert control over interelement continuity conditions. 8 1
Exercises Homeork Exercises for Chapter 8 The ernoulli- Euler eam EXERCISE 8.1 [:25] n assumed-curvature mixed functional. The WF diagram of a to-field displacementcurvature functional (, ) for the ernoulli-euler beam is shon in Figure E8.1. Starting from this form, derive the functional [, ] = ( 1 2 )dx q dx ˆ θ ˆV. (E8.1) Which extra term appears if link PC is eakened? Hints: The integral ( δ + δ ) dx is the variation of dx (ork it out, it is a bit tricky) hereas the integral δ dx is the variation of 1 L dx. 2 θ PC: = θ = θ θ = ' aster at KE: = '' = E I γ q E: ('' q) δ dx = aster : ( ) δ dx = = EI V = ( )' FC: V ( ) δθ = (V V ) δ = at Figure E8.1. The WF Tonti diagram used as departure point for deriving the curvature-displacement mixed functional. EXERCISE 8.2 [:15] Prove the property stated in Remark 8.2. 8 11
Chapter 8: THE ERNOULLI- EULER E θ PC: = θ = θ at θ = ' aster KE: = '' q E: ('' q) δ dx = CC: ( ) δ dx = aster = EI γ : ( ) δ dx = aster V = ' FC: V ( ) δθ = (V V ) δ = at Figure E8.2. Starting WF diagram to derive the three-master-field Veubeke-Hu- Washizu mixed functional, hich is the topic of Exercise 8.3. EXERCISE 8.3 [:3] The most general mixed functional in elasticity is called the Veubeke-Hu-Washizu or VHW functional. The three internal fields: displacements, curvatures and moments, are selected as masters and independently varied to get [,, ] = [ 1 + ( ) ] dx q dx ˆθ ˆV 2. (E8.2) Derive this functional starting from the WF diagram shon in Figure E8.2. EXERCISE 8.4 [:35] (dvanced, research paper level) Suppose that on the beam of Figure 8.1, loaded by q(x), one applies an additional concentrated load P at an arbitrary cross section x = x P. The additional transverse displacement under that load is P. The additional deflection elsehere is P φ(x), here φ(x) is called an influence function, hose value at x = x P is 1. For simplicity assume that the end forces at vanish: ˆ = ˆV =. The TPE functional can be vieed as function of to arguments: [, P ] = 1 2 EI( + P φ ) 2 dx [ q + Pδ(x P ) ] ( + P φ)dx, (E8.3) here = (x) denotes here the deflection for P = and (x P ) is Dirac s delta function. 5 Sho that if the 5 This is denoted by (.) instead of the usual δ(.) to avoid confusion ith the variation symbol. 8 12
Exercises beam is in equilibrium (that is, δ = ): P = U[, P] P. (E8.4) This is called Castigliano s theorem on forces (also Castigliano s first theorem). 6 In ords: the partial derivative of the internal (strain) energy expressed in terms of the beam deflections ith respect to the displacement under a concentrated force gives the value of that force. Note: This energy theorem can be generalized to arbitrary elastic bodies (not just beams) but requires fancy mathematics. It also applies to concentrated couples by replacing displacement of the load by rotation of the couple. This result is often used in Structural echanics to calculate reaction forces at supports. Castigliano s energy theorem on deflections (also called Castigliano s second theorem), hich is Q = U / Q in hich U is the internal complementary energy, is the one normally taught in undergraduate courses for Structures. Textbooks normally prove these theorems only for systems ith finite number of degrees of freedom. Proofs for arbitrary continua are usually faulty because singular integrals are not properly handled. 6 Some mathematical facility ith integration by parts and delta functions is needed to prove this, but it is an excellent exercise for advanced math exams. 8 13