A Critical Study of Howell et al.'s Nonlinear Beam Theory
aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n A Critical Study of Howell et al. ’s Nonlinear BeamTheory
Kavinda Jayawardana ∗ Abstract
In our analysis, we show that Howell et al. ’s nonlinear beam theory [1] doesnot depict a representation of the Euler-Bernoulli beam equation, nonlinearor otherwise. The authors’ nonlinear beam theory implies that one can benda beam in to a constant radius of deformation and maintain that constantradius of deformation with zero force. Thus, the model is disproven by show-ing that it is invalid when the curvature of deformation is constant, whileeven the linear Euler-Bernoulli beam equation stays perfectly valid undersuch deformations. To conclude, we derive a nonlinear beam equation byusing Ciarlet’s nonlinear plate equations [2] and show that our model is validfor constant radius of deformations.
Keywords:
Beam Theory, Finite Deformations, Plate Theory,Mathematical Elasticity
1. Introduction A beam is a structural element with one planar dimension is larger incomparison to its thickness and its other planar dimension. Thus, beamtheories are derived from the three-dimensional elastic theory by makingsuitable assumptions concerning the kinematics of deformation or the statestress through the thickness, thereby reducing three-dimensional elasticityproblem into a one-dimensional problem. Such theories provide methods ofcalculating the load-carrying and deflection characteristics of beams, wherethe most notable beam theory is the Euler-Bernoulli beam theory [3]: for the ∗ Corresponding author
Email address: [email protected] (Kavinda Jayawardana)
January 20, 2021 ero-Poisson’s ratio case, it has a governing equation of the following form,
EI ∂ w ( x ) ∂x − q ( x ) = 0 , (1)where w ( x ) ∈ R is the deflection (i.e. the displacement or the deformationform the z = 0 line, with respect to the 3D Cartesian coordinate system), E is the Young’s modulus, I is the second moment of inertia of the beam and q ( x ) is a transverse load acting on the beam.Euler-Bernoulli beam theory is a good approximation for the case of smalldeflections of a beam that are subjected to lateral loads. However, if one alsotakes into account the shear deformation and the rotational inertia effects,i.e. making it suitable for describing the behaviour of thick beams, then themost acceptable beam theory is considered to be is the Timoshenko beamtheory [4]. In fact, Euler-Bernoulli beam theory is a special case of Tim-oshenko beam theory. Note that the governing equations for Timoshenkobeam theory can be derived by eliminating the y dependency from Mindlin-Reissner plate theory.Another approach to study the nonlinear behaviour of beams was at-tempted by Howell, Kozyreff and Ockendon [1] in their publication AppliedSolid Mechanics , where the authors claim that their formulation is an alter-native representation of the nonlinear Euler-Bernoulli beam theory (see page190 of Howell et al. [1]). In our reading, we show that the authors’ model isflawed and it does not depict a representation of a beam equation, nonlinearor otherwise, as it is invalid for constant radius of deformations. To concludewe derive a nonlinear beam equation by the use of Ciarlet’s nonlinear plateequations [2] (see section B of Ciarlet [2]) with mathematical regiour, andshow that our model is valid for constant radius of deformations.
2. Howell et al. ’s Nonlinear Beam Theory
In section 4.3. of Howell et al. [1], the authors present a method forstudying the behaviour of large deflections of beams. With methods thatare inconsistent with the finite-deformation theory and differential geometry, https://books.google.co.uk/books?isbn=0521671094 R(a) (c)(b)
Figure 1: ‘A beam (a) before bending and (b) after bending; (c) a close-upof the displacement field’ [1], where h is the thickness of the beam and R isthe radius of the curvature of deformation.the authors derive their own version of the nonlinear Euler-Bernoulli beamequation, where the authors define the governing equation to this model asfollows, EI d θ ( s ) ds + N cos θ − T sin θ = 0 , (2)where θ ( s ) and s are respectively the angle and the arc length between thecentre-line of the beam (see figure 1), and T and N are the forces applied atthe boundary which are parallel to the x and y axis respectively (note thatthe authors’ y dimension is the z or the transverse-dimension in terms of thestandard beam theory terminology). The authors assume that the centre-lineis virtually unchanged , and thus, large in-plane strains of the centre-line areignored in the derivation of the equation (2). However, what the authorsput forward is flawed and it does not depict a representation of the Euler-Bernoulli beam equation, nonlinear or otherwise. For a comprehensive studyof the nonlinear Euler-Bernoulli beam equation please, consult Reddy [5] orHodges et al. [6].To examine why equation (2) is flawed, consider a case where one is3ending a beam into a shape with constant radius of curvature R (i.e. θ ( s ) = R − s ), where R ≫ I , by applying appropriate boundary forces N and T .Then equation (2) reduces to EI d ds (cid:16) sR (cid:17) + N cos θ − T sin θ = 0 . As R is a constant, one finds that N cos θ − T sin θ = 0, ∀ θ , and thus, N = 0and T = 0. This implies that it takes zero force to bend the beam with aconstant radius of deformation, regardless of the magnitude of the deforma-tion, which is not physically viable. Now, the reader can see that when theradius of curvature of deformation is constant, equation (2) is no longer valid.However, those who are familiar with beam theories may argue that onecannot bend a beam so that the radius of the deformation is constant withoutan external forcing. Thus, if one considers equation 4.9.3 of Howell et al. [1],then one can re-express equation (2) as follows, EI d θ ( s ) ds + N ( θ ) = 0 . (3)where N ( θ ) is an normal force acting on the beam. Again, bend the beamin to a constant radius of curvature R by applying a normal force of N ( θ ) toobtain EI d ds ( R − s ) + N ( θ ) = 0, which implies that N ( θ ) = 0, ∀ θ . As thereader can see that the authors’ nonlinear beam equation is, again, no longervalid, regardless of the magnitude of the deformation, i.e. regardless of thelimits of θ or the magnitude of the constant R .On another note, equation (2) implies that one can bend a beam in toa constant radius of deformation and maintain that constant radius of de-formation with zero force. To illustrate this, consider the deformation of adiving board as a thought experiment (to stay in consistent with the numer-ical modelling of section 4.9.2 of Howell et al. [1]). If one bends this divingboard to either semicircular shape with radius R or over a barrel with a ra-dius R respectively, then equation (2) or equation (3) imply that N , T = 0(zero boundary forces) or N ( θ ) = 0 (zero transverse load), ∀ θ , respectively,as now EI d θds ≡
0. Can the reader see why this is not physically realistic?In contrast, consider that if one were to bend a beam into a shape witha constant radius R by a transverse load q ( x ) below the x -axis (i.e. below4 = 0 line) with respect the linear Euler-Bernoulli beam equation (1), thenone finds that EI (cid:20) u ( x ) + R ) + 18 x ( u ( x ) + R ) + 15 x ( u ( x ) + R ) (cid:21) + q ( x ) = 0 , (4)where u ( x ) = √ R − x − R and x ≪ R . As the reader can see that Euler-Bernoulli equation stays perfectly valid under such deformations. Equation(4) also shows that (i.e. Euler-Bernoulli equation implies that) it takes anon-zero transverse load (i.e. q ( x ) = 0) to bend a beam in to a constantradius of curvature, which directly contradicts Howell et al. ’s model [1]. Canthe reader see now why Howell et al. ’s beam theory [1] is flawed?The flaws of equation (2) has arisen from the derivation of the equa-tion (see section 4.9.1 of Howell et al. [1]) as the authors did not considerany valid mathematical techniques in the study of thin objects subjected tofinite-strains and mathematical techniques in study of coordinate transforms.The authors make the fundamental mistake of assuming that an arbitrarycoordinate transform is the same as a deformation of an elastic body (seeequations 4.9.1 of of Howell et al. [1]), resulting in a nonsensical model thatis inconsistent with the behaviour of beams.To conclude our analysis, we derive a nonlinear beam equation by usingCiarlet’s nonlinear plate equations [2] (see section B of Ciarlet [2]). For theboundary conditions, as one needs to consider the pure-traction case andthe displacement-traction case separately, we only consider the pure-tractioncase as this appears to be the only case that is consistent Howell et al. ’sderivation [1]. Note that Einstein’s summation notation is assumed through-out, we regard the indices i, j, k, l ∈ { , , } and α, β, γ, δ ∈ { , } , and thecoordinates ( x , x , x ) = ( x, y, z ), unless it is strictly states otherwise.Recall Ciarlet’s nonlinear plate equations [2], and thus, one can express5he energy functional of a plate for the pure-traction case as follows, J ( u ) = h Z Ω " A αβγδ (cid:0) ∂ α ¯ R i ( u ) ∂ β ¯ R i ( u ) − δ αβ (cid:1) (cid:0) ∂ γ ¯ R k ( u ) ∂ δ ¯ R k ( u ) − δ γδ (cid:1) + 124 h A αβγδ ∂ αβ ¯ R i ( u ) (cid:0) ∂ x ¯ R ( u ) × ∂ y ¯ R ( u ) (cid:1) i || ∂ x ¯ R ( u ) × ∂ y ¯ R ( u ) || ! ∂ γδ ¯ R k ( u ) (cid:0) ∂ x ¯ R ( u ) × ∂ y ¯ R ( u ) (cid:1) k || ∂ x ¯ R ( u ) × ∂ y ¯ R ( u ) || ! − f i u i dxdy , − h Z ∂ Ω " τ i u i + 112 h η iα ∂ α u i d ( ∂ Ω) (5) u ∈{ v | ∂ α ¯ R i ( v ) ∂ β ¯ R i ( v ) = , ∀ ( x, y ) ∈ Ω } , where ¯ R ( u ) = ( x, y, ) + ( u ( x, y ) , u ( x, y ) , u ( x, y ) ) , and where u is the displacement field, f is an external force density field, τ is an external traction (i.e. applied stress) field, η is an external change inmoments density (i.e. applied change in torque density) field, A αβγδ = 2 µλ ( λ + 2 µ ) δ αβ δ γδ + µ (cid:0) δ αγ δ βδ + δ αδ δ βγ (cid:1) is the elasticity tensor for a plate, ∂ i is the partial derivate with respect to thecoordinate x i , δ ji is the Kronecker delta, × is the Euclidean cross-product, || · || is the standard Euclidean norm, Ω ⊂ R is a 2D-plane that describesthe unstrained mid-plane of the plate. Note that λ = νE (1 + ν )(1 − ν ) and µ = E ν )are the first and the second Lam´e’s parameters of the plate respectively, and ν is the Poisson ratio and E Young’s modulus of the plate.6ow, assume that one is considering a plate with infinite length in y -dimension, and, also assume that the plate is independent of any displace-ments in the x -dimension, and thus, we may assume that u = 0 and u = 0.Thus, the energy functional of a plate per unit y can be expressed as follows, J ( u ) = h Z ω " A (cid:0) ∂ ¯ R i ( u ) ∂ ¯ R i ( u ) − δ (cid:1) (cid:0) ∂ ¯ R k ( u ) ∂ ¯ R k ( u ) − δ (cid:1) + 124 h A ∂ ¯ R i ( u ) (cid:0) ∂ x ¯ R ( u ) × ∂ y ¯ R ( u ) (cid:1) i || ∂ x ¯ R ( u ) × ∂ y ¯ R ( u ) || ! ∂ ¯ R k ( u ) (cid:0) ∂ x ¯ R ( u ) × ∂ y ¯ R ( u ) (cid:1) k || ∂ x ¯ R ( u ) × ∂ y ¯ R ( u ) || ! − f u dx − h " τ u + 112 h η ∂ u | ∂ω u ∈{ v | ∂ ¯ R i ( v ) ∂ ¯ R i ( v ) = , ∀ x ∈ ω } , where ω ⊂ R describes the unstrained centre-line of the beam. Now, to stayconsistent with the standard beam theory notations, we redefine u = w , f = ˜ q (= q/h ). We also let τ = τ and η = η for convenience, were τ isan external normal traction and η is an external change in normal momentsdensity (note that τ and η are not constants as they may attain differentvalues on the boundary of the set ω , i.e. in ∂ω ). Thus, one can express anenergy functional of a nonlinear beam as follows, J ( w ) = h Z ω "
18 Λ( ∂ x w ) + 124 h Λ ( ∂ xx w ) ∂ x w ) − ˜ qw dx − h " τ w + 112 h η ∂ x w | ∂ω , (6) w ∈{ v | ∂ x v = 0 , ∀ x ∈ ω } , where Λ = 4 µ ( λ + µ )( λ + 2 µ )= E (1 + ν )(1 − ν ) .
7o derive the governing equations, we apply principle of virtual displacements (see section 2.2.2 of Reddy [7]) to equation (6) to obtain the following, J ′ ( w )( δw ) = h Z ω "
12 Λ( ∂ x w ) ∂ x δw + 112 h Λ h ∂ xx w ∂ x w ) ∂ xx δw − ( ∂ xx w ) ∂ x w (1 + ( ∂ x w ) ) ∂ x δw i − ˜ qδw dx − h " τ δw + 112 h η ∂ x δw | ∂ω ,w ∈{ v | ∂ x v = 0 , ∀ x ∈ ω } . Now, applying integration by parts and collecting all the δw terms to obtaina nonlinear beam equation for a traverse loading q ( x ) of the following form,112 h Λ " ∂ xx (cid:18) ∂ xx w ∂ x w ) (cid:19) + ∂ x (cid:18) ( ∂ xx w ) ∂ x w (1 + ( ∂ x w ) ) (cid:19) −
12 Λ ∂ x ( ∂ x w ) − ˜ q ( x ) = 0 , (7)where the boundary conditions can be expressed as follows, "
12 Λ( ∂ x w ) − h Λ h ∂ x (cid:18) ∂ xx w ∂ x w ) (cid:19) + ( ∂ xx w ) ∂ x w (1 + ( ∂ x w ) ) i | ∂ω = τ , Λ " ∂ xx w ∂ x w ) | ∂ω = η . To emphasise to the reader that equation (7) does, indeed, represents thegoverning equations of a beam, should one linearise equation (7), one getsthe following, 112 h Λ ∂ w ( x ) ∂x − ˜ q ( x ) = 0 , which is the Euler-Bernoulli beam equation for a non-zero Poisson’s ratio,where the linearised boundary conditions can be expressed as follows, − h Λ ∂ w ( x ) ∂x | ∂ω = τ , Λ ∂ w ( x ) ∂x | ∂ω = η . et al. ’s equations 4.9.1 [1] (see page 118 of Howell etal. [1]), i.e. ∂x ( s ) ∂s = cos( θ ( s )) and (8) ∂w ( s ) ∂s = sin( θ ( s )) . (9)Note that in our analysis, we do not treat equations (8) and (9) as coordi-nate transforms or as deformations, as equations (8) and (9) satisfies neitherdefinitions. We merely treat these equations as change of variables. Thus, bythe changing variables in equation (7) with equations (8) and (9), one findsthe following,112 h Λ sec ( θ ) " ∂ θ ( s ) ∂s + 6 tan( θ ) ∂θ ( s ) ∂s ∂ θ ( s ) ∂s +4 sec ( θ ) (cid:18) ∂θ ( s ) ∂s (cid:19) + 2 (cid:18) ∂θ ( s ) ∂s (cid:19) −
32 Λ sec ( θ ) tan ( θ ) ∂θ ( s ) ∂s − ˜ q ( s ) = 0 , (10)where the boundary conditions can be expressed as follows, "
12 Λ tan ( θ ) − h Λ sec ( θ ) h ∂ θ ( s ) ∂s + 2 tan( θ ) (cid:18) ∂θ ( s ) ∂s (cid:19) i | ∂ω = τ , h Λ sec( θ ) ∂θ ( s ) ∂s i | ∂ω = η . Recall that what Howell et al. [1] present as the nonlinear beam equationfails to be valid under constant curvature of deformation. Thus, consider thecase where one bends a beam in to a constant radius of curvature, where theradius is r , i.e. θ ( s ) = s/r . Thus, equation (10) implies a transverse load ofthe form, ˜ q ( s ) = 16 (cid:18) hr (cid:19) Λ r sec ( s/r ) h ( s/r ) + 1 i −
32 Λ r sec ( s/r ) tan ( s/r ) , (11)9nd where the boundary conditions can be expressed as follows, "
12 Λ tan ( s/r ) − (cid:18) hr (cid:19) Λ sec ( s/r ) tan( s/r ) | ∂ω = τ , " Λ r sec( s/r ) | ∂ω = η . From equation (11), one can see that a non-zero transverse load is requiredto deform a beam in to a constant radius of curvature, and the magnitude(i.e. the strength of the force required) is a function of the curvature ofdeformation (i.e. 1 /r ) and the limits of the arc s , which far more realisticthan what Howell et al. [1] present.
3. Conclusions
With a counter example, we showed that Howell et al. ’s nonlinear beamtheory [1] is flawed. The model was disproven by showing that it is in-valid when the curvature of deformation is constant, while even the linearEuler-Bernoulli beam equation stays perfectly valid under such deformations.Should the reader examine section 4.9.1 of Howell et al. [1], it becomes evi-dent that the authors assumes an arbitrary coordinate transform is the sameas a deformation of an elastic body. To understand the distinction betweena coordinate transform and a vector displacement, we refer the reader tosection 1 and section 2 of Morassi and Paroni [8], and for a comprehensivemathematical study of thin bodies (plates and shells) subjected to finite-strains, we refer the reader to sections Bs of Ciarlet [2] and Ciarlet [9].With our analysis, we also showed that that Howell et al. ’s nonlinearbeam theory [1] implies that one can bend a beam in to a constant radiusof deformation and maintain that constant radius of deformation with zeroforce, which is both physically unrealistic and mathematically disprovable.To conclude, we derived a nonlinear beam equation by using Ciarlet’snonlinear plate equations [2] (see section B of Ciarlet [2]) with mathematicalregiour, and show that our model is valid for constant radius of deformations.10 eferenceseferences