Optimal sizing of 1D vibrating columns accounting for axial compression and self-weight
Federico Ferrari








dtu]Federico Ferraricor1
 feferr@dtu.dk





[cor1]Corresponding author
[dtu]Department of Civil and Mechanical Engineering, Technical University of Denmark, Koppels Allé, 2800 Lyngby, Denmark




We investigate the effect of axial compression on the optimal design of columns, for the maximization of the fundamental vibration frequency. The compression may be due to a force at the columns' tip or to a load distributed along its axis, which may act either independently or simultaneously. We discuss the influence of these contributions on the optimality conditions, and show how the optimal beam design, and the corresponding frequency gain drastically change with the level of compression. We also discuss the indirect effect of frequency optimization on the critical load factors for the tip (λ_P) and distributed (λ_Q) loads. Finally, we provide some quantitative results for the optimal design problem parametrized by the triple (λ_P, λ_Q, Ω^2) of buckling and dynamic eigenvalues.




 Optimal design, Eigenvalue optimization, Vibration, Buckling















§ INTRODUCTION

 


Sizing of beams is a classical topic within optimal structural design, and has been one of the driving applications in the early developments of the field<cit.>. Despite the simplicity of beam models, closed-form solutions to their optimal design are achievable only in a few cases, involving compliance, volume, or displacement objective or constraints, and linear elastic beams<cit.>, plastic design<cit.>, or design against buckling of uniformly compressed, statically determined columns<cit.>. In more advanced instances, such as optimization of vibration frequencies, or buckling loads for general boundary conditions and loads, the beams' optimal size can only be computed by numerical methods. 

Nevertheless, important insights about the qualitative behavior of the solution can still be obtained through local asymptotic analysis<cit.>, and bounds on the optimal frequencies or buckling loads can be given by isoperimetric inequalities<cit.>.

For instance, Niordson<cit.> studied the optimal design of a simply supported beam for maximum fundamental frequency of vibration, showing an optimal improvement of 6.6%. The analysis was extended to the cantilever beam by Karihaloo and Niordson<cit.>, giving a much higher frequency improvement: 425-678%, depending on the relationship between stiffness and inertia, and then to the optimization of higher frequencies by Olhoff<cit.>. These studies highlighted some delicate matters, which have been the focus of many later works, both dealing with vibration and buckling optimization.

First, at points where the beam's internal actions vanish, the optimal beam size will also go to zero, causing a singularity in one of the generalized displacement or strain components<cit.>. In particular, at locations where only the bending moment vanishes a hinge-like point appears (Type I singularity<cit.>), whereas at locations where both the bending moment and shear force vanish a complete separation of the beam (Type II singularity<cit.>) can occur. This may compromise the well-position of the Sturm-Liouville operator underlying the optimal design problem, <cit.>, and several works have focused on finding the correct setting, and admissible class of optimal size distributions to avoid this<cit.>.

Another delicate point concerns the very existence of non-trivial optimal solutions, depending on the relationship between stiffness and mass, the introduction of design-independent inertia terms, and the particular choice of boundary conditions<cit.>. These matters have been the focus of many mathematical studies, such as those by Brach<cit.>, Holmåker <cit.>, and Vepa<cit.>.


The literature on optimal beam design against buckling is equally vast <cit.>. Limiting the discussion to recent times, Keller<cit.> started it off by giving a closed-form solution for the optimal design of a simply supported column with maximum buckling load, under a uniform tip compression. Then, Tadjbakhsh and Keller<cit.> extended to other boundary conditions, including the clamped beam, and this latter case was then reassessed by Olhoff and Rasmussen<cit.>, accounting for the bimodal character of the optimal solution. The design of a heavy column, where the compression varies along the beam's axis according to self-weight, was addressed by Keller and Niordson<cit.>, using a combination of asymptotic and numerical methods. In this case, the same technical caveats encountered for frequency optimization are raised, as discussed by Cox and McCarthy<cit.>, or Barnes<cit.>, who introduced minimum requirements on the optimal beam size to ensure well position of the problem. From a more practical stance, the regularization of the problem by introduction of a minimum size had already been proposed by Adali <cit.>.

Buckling design under the combined effect of self-weight and a tip, design-independent compression, was studied by Atanackovic, <cit.>, also considering the bimodal optimization of the clamped column <cit.>. Other recent works on the topic are from Sadiku <cit.>, Seyranian and Privalova<cit.>, and Egorov <cit.>.


In the author's knowledge, no research work has yet systematically addressed the interplay between buckling and frequency optimization, while considering self-weight. The work by Rammerstorffer<cit.> is the only one we are aware of discussing the influence of a pre-stress on the beam frequency optimization problem. However, the analysis was limited to uniform pre-stress, simply supported configuration, and a linear control, since the design variable was the Young's modulus.

As the optimal size of frequency optimized beams shows large variations, and the tendency to vanish at some locations, a question about the beams' stability naturally arises. The designs in <ref> suggest that the buckling strength of beams optimized purely for vibrations may be severely compromised, to the extent that even self-weight may trigger instability.

Here we aim at providing a broad analysis of the influence of compression on the optimization outcomes. We address the classical problem of maximizing the fundamental frequency of vibration, introducing both constant (size-independent) and variable (size-dependent) compression sources. First, we discuss how these enter the optimally conditions of the problem, also considering the particular case of self-weight. Then, we provide numerical results considering the two compression sources acting either separately or simultaneously. We will show that both substantially change the optimized design, and, generally, also the frequency improvement which can be obtained. Depending on the beam's boundary conditions, and for a compression which is close to the critical one, the natural frequency increases between 7 to 14.2 times, when the applied compression is uniform, and up to 15 times when it is size-dependent. These improvements are much higher than those achieved when optimizing an unloaded beam<cit.>, making frequency optimization even more appealing for higher compressed beams<cit.>.

We also discuss the indirect effect that the optimal design process has on the buckling capacity of the beam. Again, this heavily depends on the amount of compression considered in the optimization, and it is not obvious that the optimized design is stronger than the uniform one as soon as we account for compression in the optimization process.

Finally, we point out that here we consider the technical Bernoulli beam model, following the majority of the research works in the field. Relatively few works have dealt with the Timoshenko beam<cit.>, or with geometrically nonlinear models<cit.>, or again with non-conservative loads<cit.>.











§ GOVERNING EQUATIONS AND STATEMENT OF THE OPTIMIZATION PROBLEM


We consider the initially slender, straight, non-uniform column depicted in <ref>(a), whose axis occupies the interval [0,L]⊂ℝ. The column is subjected to the tip force of magnitude P, and to the distributed axial load with intensity q̃(s), s∈ (0,L), both inducing compression.

Assuming the moderately nonlinear kinematic Bernoulli theory, the Lagrangian density associated with the beam's small amplitude transversal vibrations is


    ℓ[ s, ṽ( s, t ) ] = 
       1/2[ ρ A ṽ^2_t +
       ( P + ∫^L_sq̃( ζ)   dζ)ṽ^2_s - EJ ṽ^2_ss]

where s and t are the space and time coordinates, ṽ(s,t) is the lateral deflection, A = A(s), ρ = ρ(s), J = J(s) and E = E(s) are the beam area, density, modulus of inertia and Young's modulus, respectively. In <ref>, and later on, subscript lowercase letters will denote differentiation with respect to the corresponding variable.

Taking one characteristic dimension of the cross section as variable over the beam length, according to the function ỹ(s) ∈𝒴 := { y ∈ C^0(0,L)  | ỹ_ min≤ỹ(s) ≤ỹ_ max}<cit.>, the area becomes A(s) = y_0ỹ(s), where y_0 is a constant length <cit.>. The modulus of inertia can be expressed as


    J(s,p) = c_1 A(s)^p
     = α y^2(2-p)_0[ y_0ỹ(s)]^p = α  y^(4-p)_0ỹ^p(s)

where c_1 = α y^2(2-p)_0, α∈ℝ, and p ∈ℕ_+ (usually, p = {1, 2, 3}). Choosing the y_0 and p, (<ref>) can be particularized to the case of some commonly used cross sections shapes<cit.>.

Introducing the non-dimensional variables x = s/L, τ = t/T, v(x,τ) = ṽ(s,t)/L, y(x) = y_0Lỹ(s)/V, and q(x) = Lq̃(s)/Q, where V is the beam's volume and Q = ∫^L_0q̃(s)   ds the magnitude of the distributed load, the substitution of (<ref>) in (<ref>) gives


    ℓ[ x, v(x,τ) ] = 
       1/2[ ϖ y v^2_τ + 
       ( λ_P + 
       λ_Q∫^1_x q(ζ)
         dζ)v^2_x - 
       y^p v^2_xx]

where we have introduced the non-dimensional parameters


    λ_P = P/Vβ  ,   λ_Q = Q/Vβ  ,   ϖ = ρ L/T^2β

and β = Eα V^p-1y^4-2p_0/L^p+2.

For λ_P < λ^( Cr)_P and λ_Q < λ^( Cr)_Q, where λ^( Cr)_P and λ^( Cr)_Q are the critical multipliers for the tip force and distributed load, respectively, the beam will undergo small amplitude harmonic oscillations. Therefore, we assume v(x, τ) = v(x)e^-iωτ and v^2_τ = - ω^2 v^2(x) (ω > 0), and imposing stationarity of (<ref>) with respect to δ v ∈𝒱 := H^2_0(0,1) gives the elasto-dynamic linear field equation governing the beam's motion


    (y^pv_xx)_xx + ( λ_P + 
     λ_Q∫^1_x q(ζ)   dζ)v_xx
     - λ_Q q(x) v_x - Ω^2 y v = 0
          x ∈ (0,1)

where Ω^2 := ϖω^2 is the dynamic eigenvalue, proportional to the square of the angular frequency ω. The boundary terms arising from the integration by parts of (<ref>) can be collected in the linear operator


    B[v,δ v] := .
     [ (y^pv_xx)_x + ( λ_P + 
     λ_Q∫^1_x q(ζ)   dζ)v_x]δ v - 
     y^p v_xxδ v_x|_x={0,1}

corresponding to one of the following pairs of boundary conditions at x={0,1}


    either   v = 0         or   (y^pv_xx)_x + ( λ_P +
      λ_Q∫^1_x q(ζ)   dζ)v_x = 0
      
     either   v_x = 0         or   y^p v_xx = 0


When the distributed axial load is proportional to self-weight, we have q̃(s) = ρ g A(s) = ρ g y_0ỹ(s) and (<ref>) and (<ref>) hold with q(ζ) = y(ζ) and λ_Q = ρ g β^-1.





 §.§ Optimization problem and solution strategy

 

We aim at finding the optimal size distribution y_∗∈𝒴 maximizing the beam's fundamental frequency Ω_1 = Ω_1(λ_P,λ_Q), for a prescribed total volume fraction V and compression level, given by the pair (λ_P,λ_Q). This can be stated as


    max_y∈𝒴Ω^2_1(λ_P, λ_Q) :=    {min_v∈𝒱-0⟨ y^pv_xx, v_xx⟩_[0,1] - ⟨ (λ_P + λ_Q I[x,y])
       v_x, v_x⟩_[0,1]/⟨ y v, v ⟩_[0,1]}
     s.t.    ⟨ y, 1 ⟩_[0,1] - 1 ≤ 0

where we have introduced the shorthand notation ⟨ u,v ⟩_[a,b] = ∫^b_a u v   dx, and set I[x,y] = ⟨ q(ξ,y(ξ)), 1 ⟩_[x,1]. In this latter term we account for a general dependence on both x and y, allowing for the case of self-weight.

The objective of (<ref>) is the squared frequency, reduced by both the tip and axially distributed compression, and expressed through the Rayleigh quotient<cit.>. This can be related to the one of the unloaded beam Ω̅^2_1 by


    Ω^2_1(λ_P, λ_Q)
      = Ω̅^2_1
      - ⟨ (λ_P + λ_Q I[x,y]) v_x,
      v_x⟩_[0,1]

where, without loss of generality, we assumed ⟨ y v, v ⟩_[0,1] = 1. <ref> expresses the well-known inverse proportionality between the squared natural frequency and axial compression <cit.>, and we remark that Ω^2_1 = 0 when either λ_P = λ^ (Cr)_P or λ_Q = λ^ (Cr)_Q.

Taking the variation of the Lagrangian associated with (<ref>), with respect to δ y ∈𝒴, we obtain the adjoint equation


    p y^p-1 v^2_xx - λ_Q
      I_y[x,y]
      v^2_x - Ω^2_1 v^2
      - μ = 0

where I_y[x,y] = δ I[x,δ y] and μ∈ℝ_+ is the Lagrange multiplier associated with the volume constraint. Assuming that such constraint is active at the optimum <cit.>, we have


    μ_∗ = ( p - 1 ) Ω̅^2_1 + ⟨[λ_P + λ_Q(I[x,y] - yI_y[x,y])] v_x, v_x⟩_[0,1]


The optimal beam size distribution is then found by solving (<ref>) with respect to y


    y_∗(x) = [
     μ + λ_QI_y[x,y]v^2_x+Ω^2_1v^2/p v_xx]^1-p

and once this is known, the optimal displacement v_∗ is computed from (<ref>), paired with the appropriate set of boundary conditions from (<ref>), and the eigenvalue Ω^2_1∗ is given by the condition ⟨ y, 1 ⟩_[0,1] = 1.

The set of nonlinear field equations governing the state, adjoint and optimal size is summarized here


    0     = (y^pv_xx)_xx + γ_1v_xx - γ_0v_x - Ω̅^2_1y v 
    
      0     = p y^p-1 v^2_xx - γ_2v^2_x - 
      (Ω̅^2_1 - ⟨γ_1 v_x, v_x⟩_[0,1]) v^2 - 
      (p-1)Ω̅^2_1 - ⟨ (γ_1 - yγ_2) v_x, v_x⟩_[0,1]
    
      y     = [
      γ_2v^2_x + ( Ω̅^2_1 - ⟨γ_1 v_x, v_x⟩_[0,1]) v^2 +
      (p-1)Ω̅^2_1 + ⟨ (γ_1 - yγ_2) v_x, v_x⟩_[0,1]/p v_xx]^1-p

where we have set γ_0 := λ_Qq(x), γ_1 := λ_P + λ_QI[x,y], and γ_2 := λ_QI_y[x,y].

<ref> is a system of non-linear, coupled, integro-differential equations expressing the optimality conditions for the optimization problem of (<ref>). We notice that the design variable y enters non-linearly in both the state and adjoint equations, whereas the state variable v appears only linearly in the state equation, as the coefficients γ_i do not depend on it. Also, we remark that in (<ref>) we have written everything in terms of the eigenvalue of the unloaded beam Ω̅^2_1, and explicitly substituted for the Lagrange multipler expression (<ref>), to fully show the influence of each term, in the most general case. However, we will see in the following example that (<ref>) can be considerably simplified, when tackling specific cases.













§ NUMERICAL RESULTS

 

We consider a slender beam with aspect ratio L/y_0 = 12, where y_0 = 0.5 is the constant cross section dimension, and also the upper bound ỹ_ max for the size distribution. The maximum allowed volume is V = 0.25 y_0 L, and the Young modulus and density are E = 2.5· 10^7 MPa and ρ = 2.4 kg/m^3. We assume similar cross sections<cit.>, setting p = 2 and α = 1/12 in (<ref>). Four cases of boundary conditions, also called beam configurations in the following, are considered: cantilever (CF), simply supported (SS), doubly clamped (CC) and clamped-pinned (CP).

The solution strategy for (<ref>) is based on iterating between the state and adjoint equations (<ref>), until both conditions y_(k) - y_(k-1)_∞≤ 10^-3 and |Ω_(k) - Ω_(k - 1)| ≤ 10^-4 are fulfilled. This is carried out repeatedly, starting by setting the lower bound ỹ_ min = 10^-3, then reducing it until ỹ_ min = 10^-10. In each of these steps, we use the final size distribution from the previous one as initial guess. We will take that a given design shows a Type I, or Type II singularity<cit.> if its final size distribution y_∗ hits ỹ_ min at an isolated point, or at multiple adjacent points, respectively. The state variable v_h is discretized by C^1 elements <cit.>, ensuring nodal continuity of both v and v_x, whereas the design variable y_h is only continuous at nodes. 

In the following, the tip compression is scaled by λ_P/λ^ Cr_P(0)∈ [0, 1), where λ^ Cr_P(0) = 
L^-2_cπ^2EJ_0 is the critical load of the uniform beam, and L_c is the characteristic length depending on the boundary conditions <cit.>. Likewise, the distributed axial load is scaled by λ_Q/λ^ Cr_Q(0)∈ [0,1), where λ^ Cr_Q(0) is the critical load corresponding to a constant load distribution. To ease the notation, we will denote these multipliers by the symbols λ_P and λ_Q. As we focus on the fundamental vibration frequency, we drop the subscript “1”, and we denote the frequencies of the uniform and optimized size distributions by Ω_0 and Ω_∗, respectively. Also, we will call “reduced frequencies” those reduced by the axial compression through (<ref>).





 §.§ Influence of the tip force alone (λ_P≥ 0, λ_Q = 0)

 

In this case γ_0 = γ_2 = 0, γ_1 = λ_P, and the state and adjoint equations in (<ref>) simplify to


    0     = (y^pv_xx)_xx + λ_Pv_xx - Ω̅^2 y v 
    
       0     = p y^p-1 v^2_xx - (Ω̅^2 - λ_P⟨ v_x, v_x⟩_[0,1]) v^2 - (p-1)Ω̅^2 - λ_P⟨ v_x, v_x⟩_[0,1]

and the design-independent compression influences the solution of (<ref>) through the state equation and the Lagrange multiplier μ.

The compression, which is constant during the optimization, is applied in the range λ_P∈ [0, 0.995], with Δλ_P = 1/200. We refer to the smallest level of non-zero compression (λ_P = 0.005) as λ_P≈ 0, and to the last multiplier as λ_P≈ 1. The BLF of the beam, say λ^ Cr_P, will be indirectly modified during the optimization, and we must check that λ_P≤λ^ Cr_P at each step (see <ref>); otherwise the beam will undergo divergence, voiding the assumption of harmonic motion. 

<ref> shows the influence of the compression level on the frequency of the uniform beam (Ω_0, blue dashed line), and of the optimized one (Ω_∗, blue continuous line). While the first follows the well-known (nearly) inverse proportionality between Ω^2_0 and λ_P <cit.>, the frequency attained in the optimization decreases less than proportionally with the compression level. The relative frequency gain corresponding to each compression level is given by the ratio r_Ω := Ω_∗/Ω_0 (see red line in <ref>). For λ_P = 0 (optimization of the unloaded beam), r_Ω attains values very close to the theoretical bounds recalled in <ref> (b). We clearly have r_Ω→∞ as λ_P→ 1; however, we recall that we bound λ_P to be close but slightly below one. The largest absolute r_Ω is reached by the CF configuration, which for λ_P≈ 1 shows a frequency gain of over 14 times, whereas the SS configuration is the one showing the largest relative improvement between r_Ω(0) (unloaded beam) and r_Ω(≈ 1) (highly compressed beam).

For all the configurations but the SS, the relationship between λ_P and r_Ω is non-monotonic. For the CF, CS and CC beams, as soon as λ_P > 0 the frequency attained by the optimized beam (Ω_∗) decreases much more rapidly than the frequency of the uniform one, and r_Ω drops. After reaching a minimum at λ_P = l_1 (white dot in <ref>), the frequency gain r_Ω equals the one for the unloaded beam at λ_P = l_2 (black dot in <ref>), then increases further as λ_P→ 1.

This non-monotonic behaviour can be explained by looking at how the optimized size distribution y_∗(x) changes with the increase of the compression (<ref>). For the SS beam, the optimized size shows minor changes and only gets thicker near the beam center, and more flat near the pinned ends, as λ_P increases. For all the other beam configurations, the optimized size shows an abrupt change as soon as λ_P > 0, and immediately adapts to sustain the design-independent tip load. The size distribution is prevented from vanishing according to a Type II singularity, whereas we still have points resembling a Type I singularity, where the bending moment vanishes (see <ref>). For the CS and CC configurations, these points appear within the beam domain as soon as λ_P > 0 and, for λ_P≈ 1, they are at x≈ 1/3 and x ≈ 1/6, respectively<cit.>. However, we stress that these points do not strictly correspond to singularities, since we always have y_∗ > y_ min, as soon as λ_P > 0. 

The optimized design shown as a red dashed line in <ref> corresponds to the compression level λ_P = l_2, where we match the frequency gain of the unloaded beam. We also plot, as green dashed line, the one corresponding to λ_P = l_3); this is the first designs meeting the same buckling strength of the uniform beam, as will be further discussed in <ref>.

The above discussion shows that for the CF, CS, and CC beams the magnitude of λ_P has a huge effect on the frequency optimization. Even the slightest design-independent compression prevents the optimal beam size from vanishing according to a Type II singularity, and this drastically lowers the frequency gain. Then, as the compression level increases, the reduction in the initial frequency is so severe that the frequency gain becomes even higher than the one reached when optimizing the unloaded beam. For the SS beam, the frequency gain is always higher for the compressed beam, compared to the unloaded one.







 §.§ Influence of the self-weight (λ_P≥ 0, q(x) = y(x), λ_Q = 1)

 

We repeat the analysis of the previous section, now introducing the self-weight contribution. This amounts to the full system of equations (<ref>), where q(x) = y(x), I[x,y] = ⟨ y, 1 ⟩_[x,1], and I_y[x,y] = (1-x). Thus, we can simplify γ_0 = λ_Qy, γ_1 = λ_P + λ_Q⟨ y, 1 ⟩_[x,1], and γ_2 = λ_Q(1-x).

We remark that, due to the self-weight, buckling may also happen with λ_P = 0. This will be discussed in the next section, and we now fix λ_Q = 1. In the following, we will refer to the beam loaded only by the tip force as the “light beam”, whereas the “heavy beam” also accounts for the self-weight.

We refer to <ref> where, against the left axis, we plot Δ^ (SW)_Ω := Ω/Ω^ (SW) - 1, which is the relative difference between the reduced frequency for the light beam (Ω, same as in the previous section), and that of the heavy beam (Ω^ (SW)). For the uniform design, this quantity is clearly larger than one, as the self-weight introduces an additional compression, further reducing the vibration frequency. Also, it is monotonically increasing with λ_P for all the beam configurations, and this comes from the coupling of λ_Q in the v_x and v_xx terms of the state equation.

Let us focus the discussion on the CF configuration first. For the uniform beam (blue dashed curve), the self-weight alone slightly reduces the frequency (Δ^ (SW)_Ω≈ 0.125% for λ_P = 0). Then, its effect increases with increasing λ_P, and for λ_P≈ 1 we have Δ^ (SW)_Ω≈ 38%. The trend is the opposite when looking at the relative difference in the frequency attained by the optimized beam (blue continuous curve). Now the largest difference happens for λ_P = 0 (Δ^ (SW)_Ω≈ 28%), as the self-weight is the first source of compression perturbing the optimal design from the one corresponding to the unloaded beam. Then, as λ_P increases and the size-independent compression dominates the self-weight, Δ^ (SW)_Ω quickly drops to a few . However, for the optimized frequency, the ratio Δ^ (SW)_Ω is non-monotonous and after reaching the minimum at λ_P = l_4 = 0.385, the influence of self-weight (slightly) increases again, giving Δ^ (SW)_Ω≈ 0.415% for λ_P≈ 1.

This trend is also reflected by the relative difference between the frequency gains for the light and heavy beams, Δ^ (SW)_r_Ω := r_Ω/r^ (SW)_Ω - 1 (see red curves in <ref>). For low magnitudes of the tip compression we have Δ^ (SW)_r_Ω > 0, and the optimization gives a higher frequency gain for the light beam, compared to the heavy one. For λ_P = l_5≈ 0.455 we have Δ^ (SW)_r_Ω = 0, and the frequency gains for the heavy and light beams are exactly the same. Then, for higher values of the tip compression we have Δ^ (SW)_r_Ω < 0 and, for λ_P≈ 1 the heavy beam benefits from the optimization about 27.5% more, compared to the light one. However, we observe that apart form the two regions where λ_P is close to zero or one, the self-weight has a very little influence on the optimization outcomes. For instance, when 0.175≤λ_P≤ 0.655 we have |Δ^ (SW)_r_Ω| ≤ 10^-3, and therefore the self-weight has a negligible impact on the frequency gain (see horizontal lines in <ref>).



The CS and CC configurations share the same qualitative behaviour of the CF, tough we have different values for the ratios Δ^ (SW)_Ω, Δ^ (SW)_r_Ω, and location of points l_4 and l_5 (see also <ref>). Also, both configurations show a much wider range where |Δ^ (SW)_r_Ω| ≤ 10^-3, and therefore the self-weight has negligible influence on the optimization outcomes (λ_P∈ [0.01,0.88] and λ_P∈ [0.01,0.86] for the CS and CC beams, respectively). This can be clearly seen by the trend of the red curves in <ref>, for the CF and CS beams. For the SS configuration, both Δ^ (SW)_Ω and Δ^ (SW)_r_Ω are monotonically increasing and decreasing, respectively. Therefore, as λ_P increases, we gain systematically more when optimizing the heavy beam, compared to the light one.

The effect of self-weight on the optimized size distributions is shown in <ref>. Here, the SS beam is not shown, as for this case we have only minor design changes. The plots in the top row compare the relative size distribution of the light (dashed line) and of the heavy beams (continuous line), for three notable tip compression values: λ_P = 0, λ_P≈ 1, and λ_P = l_5. For all the beam configurations, we can distinguish between y^ (SW)_∗ and y_∗ only for λ_P = 0, as the self-weight has the effect of reducing the size at the clamped ends, while increasing it at the points where the unloaded beam would develop a Type II singularity. For the compression levels λ_P = l_5 and λ_P≈ 1, y^ (SW)_∗ and y_∗ are almost indistinguishable; however, we can refer to the plots of the bottom row, showing a logarithmic plot of their difference, and recognize a similar trend. For λ_P = 0, the CF beam's sizes show the largest differences due to self-weight. At x = 0 the size is reduced of about 30%, whereas the maximum increase of about 13.5% occurs at x ≈ 0.5. For the CS beam, the size's maximum decrease and increase are lower (4.3% and 5.5%), and both occur within the domain, at about x≈ 0.75. For the CC beam, the size is reduced of about 24% at the clamped ends, and the maximum increase (10%) occurs, interestingly, at the symmetrical locations x≈ 0.25 and x≈ 0.75, and not at the mid-point. In all cases, for the higher compression levels the size relative differences quickly drops to very small values. However, it is interesting to note that for the CS configuration we still have size differences up to ± 2%, corresponding to λ_P = l_5, without affecting the frequency gain.

Summarizing, the self-weight contribution sensibly impacts the frequency optimization both when considering zero or high values of the design-independent compression. In particular, frequency optimization becomes highly beneficial for heavy and highly compressed columns. Otherwise, the influence of self-weight on the optimization outcome is very small. 









 §.§ Influence of the distributed axial load alone (λ_P = 0, q(x) = y(x), λ_Q≥ 0)

 

We now remove the size-independent compression, while keeping the distributed axial load proportional to self-weight; thus the coefficients in  (<ref>) become γ_0 = λ_Qy, γ_1 = λ_Q⟨ y, 1 ⟩_[x,1], and γ_2 = λ_Q(1-x).

We increase the axial load multiplier up to λ_Q = 0.995, with Δλ_Q = 1/200. Again, we have an inverse relationship between Ω^2_0 and λ_Q and, for the uniform design, the multiplier causing buckling under self-weight (and thus Ω_0 = 0) is λ^ Cr_Q = {942.97, 397.52, 2664.6, 3781.8}, for the SS, CF, CS and CC configurations, respectively <cit.>.

We point out that scaling λ_Q, when considering self-weight, can still be given a meaning as a “tallest column” design problem<cit.>. Indeed, from (<ref>) we have λ_Q∝ L^4 and ϖ/λ_Q∝ L. Therefore, we may alternatively interpret this as the optimal design of a the tallest column with the same frequency and buckling strength of the initial one. However, the main difference with the classical tallest column problem is that now λ_Q is set as a parameter.

<ref> (a,b) show the trend of the optimized frequency Ω_∗ = Ω_∗(λ_Q) and frequency gain r_Ω = r_Ω(λ_Q), for the four beam configurations. For comparison, we also display, as dashed lines, the same curves corresponding to the case of tip compression, identical to those in <ref>. For all the beam configurations, we highlight the points λ_Q = l_6, where r_Ω(l_6) = r_Ω(0) (see also <ref>). As for the case of tip compression, we have l_6 = 0 for the SS beam, and the frequency gain is monotonically increasing with the load parameter.

For the SS and CF beams, the optimized frequency Ω_∗ corresponding to a given λ_Q is always larger than that obtained for the same value of λ_P; also, we have r_Ω(λ_Q) > r_Ω(λ_P). Therefore, for these two configurations frequency optimization leads to larger gains when the beam is loaded by its own self-weight, compared to the tip force compression. The opposite is true for the CS and CC beams, as we now have Ω_∗(λ_Q) < Ω_∗(λ_P) and r_Ω(λ_Q) < r_Ω(λ_P), for all values of the load multiplier.

These trends are clearly visible from <ref>(c) showing the ratio r_Ω(λ_Q)/r_Ω(λ_P), which is always above one for the SS and CF beams, and always below one for the CS and CC. For the SS configuration the ratio only shows a mild variation, whereas, for the CF beam we have a sudden increase, as the slightest amount of tip compression reduces Ω_∗ much more than the slightest amount of axial load. Then we reach the maximum ratio r_Ω(λ_Q)/r_Ω(λ_P) ≈ 1.15 for λ_Q = 0.495, and a slight decrease after. For the CS and CC configurations the decrease of this ratio is monotonic, and very strong for the CC. For the case λ_Q≈ 1 we have frequency gains 1.1% and 6.2% larger, for the SS and CF configurations, and 22% and 36% smaller, for the CS and CC configurations, compared to the case λ_P≈ 1.

<ref> displays the comparison between the optimized size distribution corresponding to the axially distributed and tip load. The designs are compared for nearly critical compression values (λ_Q≈ 1 and λ_P≈ 1), where the two show the largest differences. We see that the optimal size distributions change from those in <ref>, to a design closer to the tallest column one <cit.>, clearly showing a magnification of the self-weight effect. We also show the designs corresponding to the two notable compression level λ_Q = l_6, where the frequency gain matches the one for the unloaded beam, and λ_Q = l_7, where the critical load matches that of the initial design (see discussion in the next section).

We point out that, in none of these cases the design develops a Type II singularity and even for λ_Q≈ 0 we have y_∗ = y_ min only at isolated points. This, which may look counter-intuitive, is due to the fact that the multiplier λ_Q is here a constant parameter, and is not optimized for as in the classical tallest column formulation<cit.>.







 §.§ Indirect effect on the buckling load factor

 

Finally we investigate the indirect effect of maximizing Ω_1 at a given compression level, on the beams' critical load. This latter, which is not directly optimized, may either increase or decrease during the optimization. <ref> shows the ratios r_λ_P := λ^ Cr_P(∗)/λ^ Cr_P(0) (viz. r_λ_Q := λ^ Cr_Q(0)/λ^ Cr_Q(∗)), between the critical load of the optimized beam and that of the uniform one, depending on the magnitude of the tip compression λ_P, or axial distributed load λ_Q. The critical load multiplier of the compressed, uniform beam clearly follows the relationship λ^ Cr_P(0) = 1/λ_P (viz., λ^ Cr_Q(0) = 1/λ_Q), plotted as a black dashed line in <ref>. Since both r_λ_P and r_λ_Q are always above this line, our assumption of harmonic motion is valid.

For the SS beam, r_λ_P > 1 and r_λ_Q > 1 for all compression levels, so for this configuration the maximization of the vibration frequency always gives, as a side effect, the increase of the critical load. Also, the range of variation of the two ratios is very narrow for the SS beam, especially when considering the tip compression, for which 1.328 ≤ r_λ_P≤ 1.333. This is expected, as in this case the optimized beam size is only minimally modified (see <ref>(a)). On the other hand, we observe a larger range of variation when considering the axially distributed load, for which 1.23 ≤ r_λ_Q≤ 1.406, and this is consistent with the more marked modification of the beam's profile in this case (see <ref>(a)).

For the CF, CS, and CC configurations the buckling capacity of the optimized design heavily depends on the compression level considered in the optimization. Up to the values λ_P = l_3 and λ_Q = l_6, respectively, we have r_λ_P < 1 and r_λ_Q < 1; therefore the optimized beam will show a lower buckling capacity than the uniform one. Then, for higher compression levels, the optimization also improves the beam's critical load. The designs corresponding to points l_3 and l_6, featuring the same critical load as the uniform beam, are marked by green dotted lines in <ref> and <ref>, respectively.

The first configuration to meet the condition r_λ_P = 1, at l_3 = 0.335, is the CS, whereas the last is the CC, at l_3 = 0.89. For the CF configuration, the value of r_λ_P is almost always between those of the CS and CC beams, and we have l_3 = 0.7. We also observe that r_λ_P has an approximately linear trend for the CF and CC configurations, whereas it shows a marked non-linear trend for the CS case. Moreover, for λ_P≈ 1 the CS beam attains a critical load which is slightly below (less than 1%) that of the SS beam, whereas the other two beam configurations attain lower values.

When considering the axially distributed load, we see some differences. First, for all beam configurations the same critical load of the uniform design is now reached for higher values of compression (thus l_6 > l_3). The trend of r_λ_Q for the CF and CC beams shows little deviations compared to the case of tip compression. On the contrary, the trend for the CS configuration is very different, and this configuration is now the last to reach r_λ_Q = 1, for a compression level close to the critial one (l_6 = 0.96).







§ CONCLUSIONS

 

We have investigated the effect of axial compression on the optimal sizing of beams for maximum fundamental vibration frequency, considering both a tip force generating a constant, control-independent compression, and a distributed axial load proportional to self-weight, giving a control-dependent compression. Some notable numerical results discussed throughout <ref> are summarized in <ref>.

Both sources of compression have an impact on the frequency gains attainable by the optimization and, for all beam configurations but the SS, also on the optimal size distribution. As soon as compression is introduced, the optimal design deviates from that obtained for the unloaded beam (pure vibrations), and is prevented from developing singularities anywhere in the beam domain. This determines a substantial reduction of the optimal frequency gain, compared to the case of an unloaded beam. However, when looking at high compression levels, frequency optimization becomes much more beneficial, compared to the unloaded case. When the compression is just below the critical one, the frequency gains relative to those for the unloaded beam range between 2 to 7.1 for the tip force, and between 11.34 to 7.2 times for the distributed load proportional to self-weight.

When paired to the design-independent compression, the self-weight has a very little contribution for most of the “intermediate” values of the first. However, it does have an influence when the compression is close to the critical value, where again, its account leads to higher frequency gains, especially for the cantilever and simply supported configuration.





 
 



§ ACKNOWLEDGEMENTS

The author gratefully acknowledges the support of the Villum Foundation, through the Villum Investigator project “InnoTop”.







§ REMARKS ON THE LOCAL BEHAVIOUR OF THE DYNAMIC SOLUTION

 

Here we summarize the classical local asymptotic analysis for the optimal design of a purely vibrating beam, recalling the definition of the two types of singularities occurring in this case<cit.>.

For the optimal design of an uncompressed beam (λ_Q = λ_P = 0), for maximum frequency of vibration, the system governing the optimality (<ref>) reduces to


    (y^pv_xx)_xx - Ω̅^2 y v     = 0 
    
       p y^p-1 v^2_xx - Ω̅^2 v^2 - Ω̅^2(p-1)     = 0 
    
       B(y,v)     = 0 
    
        M v_x_x_k    = 0   ,      k = 1, … n_ I
    
        T v _x_k    = 0    ,      k = 1, … n_ II

where we have also introduced the boundary operator B(y, v), and we allowed for a finite number of jumps in the optimal energy distribution associated with the bending  moment M := y^pv_xx and shear force T := (y^pv_xx)_x. Continuity of these internal actions must hold everywhere within the beam domain, as we do not consider point loads<cit.>, but depending on the vanishing of either or both of them, we can identify two types of points where the optimal size y_∗→ 0, and some of the response components becomes singular.

The first set of points is defined as follows:


 Let x^ (I)_k∈ [0,1] be a point where M_∗(x^ (I)_k) = 0, and T_∗(x^ (I)_k) ≠ 0. Then, y_∗(ξ) = O(ξ^1/3), where ξ = |x - x^ (I)_k|. Also, v_∗ is continuous, its derivative v_∗, x has a finite jump, and the curvature v_∗, xx is singular at x^ (I)_k.


At such points, the continuity of the bending moment is automatically fulfilled, M_∗(x^ I_k) = 0, and therefore v_∗, x(x^ (I)_k) can have a finite jump, provided that y_∗(x^ I_k) = 0, such that the energy product y^p_∗v^2_∗, xx_x^ I_k remains finite. Mechanically, this point mimics the introduction of a hinge in the beam's domain.

The second set of points leading to a response singularity can be introduced as follows:


 Let x^ II_k∈[0,1] be a point where M_∗(x^ II_k) = T_∗(x^ II_k) = 0. Then, y_∗(ξ) = O(ξ^2), where ξ = |x - x^ II_k|. Also, v_∗ has a jump, and both v_∗, x and v_∗, xx are singular at x^ II_k.


At such points both internal actions are vanishing, and this allows for a jump on v_∗. Then, the singularity of v_∗,x(x^ II_k) and v_∗,xx(x^ II_k) requires y_∗((x^ II_k)) = 0, in order to have finite energy products y^p_∗v^2_∗,xx_x^ II_k and (y^p_∗v^2_∗,xx)_x v _x^ II_k. Mechanically, this second singularity represents a complete separation of the beam.




The behaviour of the response functions near these two types of points can be deduced by an asymptotic analysis on the following non-linear differential equation


    [( Ω̅^2(v^2_x + p - 1)/p v^2_xx)^p(1-p) v_xx]_xx -
      (Ω̅^2(v^2 + p - 1)/p v^2_xx)^1-p v = 0

obtained by eliminating the expression for y_∗ from the state equation in (<ref>). Then we represent the deflection v_∗(ξ) according to a regular, or a singular expansion <cit.>.

In the first case we assume v^ (I)(ξ) = a_0 + a_1ξ^r, where r∈ℚ and (a_0, a_1)∈ℝ, and substituting within (<ref>), we obtain the residual


    ℛ[v^ (I)] =
     Ω̅^2ξ^4-3r/t_0[
     (a^4_0+3(1+2a^2_0))t_1 +
     o(ξ) ] = 0

where t_0 = 4a_2r^3(r-1)^3 and t_1 = 3(r-2)(3r-5). Non-trivial solutions to (<ref>) require t_1 = 0, while t_0≠ 0. This correspond to the pair of solutions r_1 = 5/3 and r_2 = 2, which plugged in (<ref>) give


    lim_ξ→ 0^+ℛ{r_1,r_2} =
     {0, (a_0+a^3_0)Ω̅^2/8 a^2_2}

and therefore only the solution r = 5/3 is acceptable. The behaviour of the functions near a Type I points is (see <ref>)


    v_∗( ξ)     = a_0 + a_1ξ + ξ^ 5/3  ,   
       v_∗, x( ξ)   = a_1 + a_2ξ^ 2/3  ,   
      v_∗, xx( ξ)   = a_3ξ^-1/3
    
           y_∗( ξ)     = a_4ξ^1/3 + O(ξ)    ,   
           M_∗( ξ)   = ξ + O(ξ^2)          ,   
           T_∗( ξ)   = a_5 + O(ξ)

where the expression of the a_i coefficients can be found in <cit.>.

If we consider the singular expansion v^ (II)(ξ) = b_0ξ^-s, s∈ℕ_+, b_0∈ℝ the residual of (<ref>) becomes


    ℛ[v^ (II)] =
     Ω̅^2ξ^4-s/q_0[
     -b^4_1 q_1 + o(ξ^2 s) ] = 0

where q_0 = 4b^3_1s^3(s+1)^3 and q_1 = (s+15)(s-2). Now the only acceptable solution is s = 2, and the functions near a Type II point behave as follows (see <ref>)


    v_∗( ξ)     =   b_0ξ^-2  ,   
       v_∗, x( ξ)   = - b_1ξ^-3  ,   
      v_∗, xx( ξ)   =   b_2ξ^-4
    
           y_∗( ξ)     =   b_3ξ^2  ,   
           M_∗( ξ)   =   b_4ξ^4  ,   
           T_∗( ξ)   =   b_5ξ^3

where the expression of the b_i coefficients can be found in <cit.>.







 spmpsci 
 


