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Stability of Building Structural Engineering Based on Fractional Differential Equations

Accepté: 17 Apr 2022
Détails du magazine
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Introduction

At present, the research on the stability of the combined compression rod in academia mostly focuses on the static force, and the research on dynamic stability is relatively rare. The combined pressure rod can be divided into strip type and strip type [1]. We use the energy method and the Hamilton principle to establish the Mathieu-type parametric vibration equations of the strip and plate composite compression bars under cyclic longitudinal loads. We solve for the dynamically unstable region surrounded by its periodic solution. At the same time, the paper discusses the dynamic stability of the parametric resonance of the two combined compression bars.

The combined compression rod comprises a split limb bearing axial load and a splicing piece connecting the split limb [2]. The shear deformation is much larger than that of a solid web column. Additional deflection effects due to shear forces cannot be ignored. Therefore, we must consider the effect of shear deformation in the stability study of the combined compression rod. The cross-section of the composite strut after the structure is bent is not perpendicular to the deflection curve υ. We use φ to represent the rotation angle of the section, then the bending moment at any point on the axis of the combined compression rod is: $M=−EIφ′$ M = - EI{\varphi^{'}}

The sheer force is: $FQ=Kv(dυ/dx−φ)$ {F_Q} = {K_v}\left({d\upsilon /dx - \varphi} \right)

EI is the bending stiffness of the combined compression rod. I is the moment of inertia of the cross-section concerning the centroid axis. Kv is the shear stiffness of the section. Using the Timoshenko beam theory, we can deduce the expression of the total potential energy of the composite compression rod considering the effect of shear deformation. In this way, a numerical method or energy method can be used to study the elastic stability performance of composite compression rods under the influence of shear deformation [3]. The total potential energy of the combined compression rod with shear deformation under the action of the axial pressure P is expressed as follows: $Π=12∫0l[EI(dφdx)2+Kv(dυdx−φ)2−P(dυdx)2]dx$ \Pi = {1 \over 2}\int_0^l {\left[{EI{{\left({{{d\varphi} \over {dx}}} \right)}^2} + {K_v}{{\left({{{d\upsilon} \over {dx}} - \varphi} \right)}^2} - P{{\left({{{d\upsilon} \over {dx}}} \right)}^2}} \right]dx}

Stable power of the strip-type combined pressure rod

The strip-type composite compression rod (Figure 1) can be calculated as a truss. The connection nodes between the column legs and the strips are hinge nodes. The rods in the truss cause only additional axial forces when the stability is lost. The axial force FN of the composite strut strip is approximated by the truss [4]. We can get the following expression: $FN"=±FQ/cos α$ {F_N}^{''} = \pm {F_Q}/\cos \,\alpha

In the formula, α is the angle between the strip and the horizontal axis.

Then the lateral shear deformation caused by unit shear force is: $Δ=1EA1cos2θ1bcos θ1+1EA2 cos2 θ2bcos θ2$ \Delta = {1 \over {E{A_1}{{\cos}^2}{\theta_1}}}{b \over {\cos \,{\theta_1}}} + {1 \over {E{A_2}\,{{\cos}^2}\,{\theta_2}}}{b \over {\cos \,{\theta_2}}}

A1, A2, is the area of the upper and lower inclined strips, respectively. θ1, θ2 is the angle between the upper and lower inclined strips and the horizontal axis, respectively. b is the width of the bar limb. The distance d = b tan θ1 + b tan θ2 between two nodes on the same column limb. Therefore, the shear stiffness of the combined compression bar can be expressed as: $1Kv=Δd=1tan θ1+tan θ2(1EA1 cos3 θ1+1EA2 cos3 θ2)$ {1 \over {{K_v}}} = {\Delta \over d} = {1 \over {\tan \,{\theta_1} + \tan \,{\theta_2}}}\left({{1 \over {E{A_1}\,{{\cos}^3}\,{\theta_1}}} + {1 \over {E{A_2}\,{{\cos}^3}\,{\theta_2}}}} \right)

The load potential energy is: $UP=−(P0+Pt cos θt)∫0l12(υ′)2 dx$ {U_P} = - \left({{P_0} + {P_t}\,\cos \,\theta t} \right)\int_0^l {{1 \over 2}{{\left({{\upsilon^{'}}} \right)}^2}\,dx}

Substitute into equation (3) to get the expression of the total potential energy. The kinetic energy is: $T=12∫0lm(υ˙)2 dx$ T = {1 \over 2}\int_0^l {m{{\left({\dot \upsilon} \right)}^2}\,dx}

At time t0, t1 we can obtain the variational equation according to the Hamilton principle: $δ∫t0t1(T−Π)dt=0$ \delta \int_{{t_0}}^{{t_1}} {\left({T - \Pi} \right)dt = 0}

We vary υ, φ separately to obtain the dynamic differential equation of the combined pressure rod, which is expressed as follows $mυ−Kv(υ′−φ)′+(P0+Pt cos θt)υ"=0EIφ"+Kv(υ′−φ)=0$ \matrix{{m\upsilon - {K_v}{{\left({{\upsilon^{'}} - \varphi} \right)}^{'}} + \left({{P_0} + {P_t}\,\cos \,\theta t} \right){\upsilon^{''}} = 0} \hfill \cr {EI{\varphi^{''}} + {K_v}\left({{\upsilon^{'}} - \varphi} \right) = 0} \hfill \cr}

Assume that the lateral rod displacement $υ(x,t)=∑n=1∞fn(t) sin(nπx/l)$ \upsilon \left({x,t} \right) = \sum\limits_{n = 1}^\infty {{f_n}\left(t \right)\,\sin \left({n\pi x/l} \right)} satisfies the rod boundary conditions. We substitute it into formula (10) to eliminate variable φ. We utilize the Galerkin method for discretization [5]. Then the equation can be transformed into the following formula: $mf"(t)+EIπ4l4[1+π2Il2(tan θ1+tan θ2)×(1A1 cos3 θ1+1A2 cos3 θ2)]−1gf(t)−(P0+Ptcos θt)π2l2f(t)=0$ \matrix{{m{f^{''}}\left(t \right) + {{EI{\pi^4}} \over {{l^4}}}{{\left[{1 + {{{\pi^2}I} \over {{l^2}\left({\tan \,{\theta_1} + \tan \,{\theta_2}} \right)}} \times \left({{1 \over {{A_1}\,{{\cos}^3}\,{\theta_1}}} + {1 \over {{A_2}\,{{\cos}^3}\,{\theta_2}}}} \right)} \right]}^{- 1}}{\rm{g}}} \hfill \cr {f\left(t \right) - \left({{P_0} + {P_t}\cos \,\theta t} \right){{{\pi^2}} \over {{l^2}}}f\left(t \right) = 0} \hfill \cr}

Equation (10) can be further simplified as: $f"(t)+Ω2(1−2 μ cos θt)f(t)=0$ {f^{''}}\left(t \right) + {\Omega^2}\left({1 - 2\,\mu \,\cos \,\theta t} \right)f\left(t \right) = 0

Equation (12) is the Mathieu-type parametric vibration equation of the strip-type composite compression bar under the action of longitudinal periodic loads [6]. The elastic critical load of the strip-type composite compression rod considering the shear deformation is as follows: $P=π2EIl2[1+π2Il2(tan θ1+tan θ2)(1A1 cos3 θ1+1A2 cos3 θ2)]−1$ P = {{{\pi^2}EI} \over {{l^2}}}{\left[{1 + {{{\pi^2}I} \over {{l^2}\left({\tan \,{\theta_1} + \tan \,{\theta_2}} \right)}}\left({{1 \over {{A_1}\,{{\cos}^3}\,{\theta_1}}} + {1 \over {{A_2}\,{{\cos}^3}\,{\theta_2}}}} \right)} \right]^{- 1}}

In: $ω={π2EIml4[1+π2Il2(tan θ1+tan θ2)(1A1 cos3 θ1+1A2 cos3 θ2)]−1}12Ω=ω1−P0/Pμ=Pt2(P−P0)$ \matrix{{\omega = {{\left\{{{{{\pi^2}EI} \over {m{l^4}}}{{\left[{1 + {{{\pi^2}I} \over {{l^2}\left({\tan \,{\theta_1} + \tan \,{\theta_2}} \right)}}\left({{1 \over {{A_1}\,{{\cos}^3}\,{\theta_1}}} + {1 \over {{A_2}\,{{\cos}^3}\,{\theta_2}}}} \right)} \right]}^{- 1}}} \right\}}^{{1 \over 2}}}} \hfill \cr {\Omega = \omega \sqrt {1 - {P_0}/P}} \hfill \cr {\mu = {{{P_t}} \over {2\left({P - {P_0}} \right)}}} \hfill \cr}

The article computes unstable regions surrounded by periodic solutions. Assuming that equation (12) has periodicity, the periodic solution of 4π / θ is expressed as follows: $f(t)=∑n=1,3,5…∞(ansinnθt2+bn cosnθt2)$ f\left(t \right) = \sum\limits_{n = 1,3,5 \ldots}^\infty {\left({{a_n}\sin {{n\theta t} \over 2} + {b_n}\,\cos {{n\theta t} \over 2}} \right)} . We substitute it into Eq. (12) to sort and merge similar items. We obtain the critical frequency equation according to the periodic solution's condition [7]. At the same time, assuming the first-order row and column 1 ± μθ2 / (4Ω2) = 0, we can obtain the approximate formula $θ=2Ω1±μ$ \theta = 2\Omega \sqrt {1 \pm \mu} of the critical frequency by solving. Similarly, Equation (12) has periodicity, then the periodic solution of 2π / θ is expressed as $f(t)=∑n=2,4,6…∞(a0+ansinnθt2+bn cosnθt2)$ f\left(t \right) = \sum\limits_{n = 2,4,6 \ldots}^\infty {\left({{a_0} + {a_n}\sin {{n\theta t} \over 2} + {b_n}\,\cos {{n\theta t} \over 2}} \right)} .

We substitute into equation (12) and take the second-order determinant of the critical frequency equation [8]. The critical frequency approximation formula can be obtained by solving in this way: $θ1=Ω1+μ2/3θ2=Ω1−2μ2$ \matrix{{{\theta_1} = \Omega \sqrt {1 + {\mu^2}/3}} \hfill \cr {{\theta_2} = \Omega \sqrt {1 - 2{\mu^2}}} \hfill \cr}

Stable power of the plate-mounted combined pressure rod

A rigid frame can be taken as the plate-type combined pressing rod (Fig. 2). Its deformation state can be decomposed into two parts: the overall deformation as a rod and the local bending deformation between the nodes as a rigid frame [9]. The latter can be seen as additional bending moments due to internode shear forces.

Under the action of unit shear force, the linear stiffness of the siding is generally greater than that of the column leg [10]. We can use graph multiplication to obtain the lateral shear deformation: $Δ=4EI′d2×4d223d4+2EI"d2×2b223d2=d324EI′+bd212EI"$ \Delta = {4 \over {E{I^{'}}}}{d \over {2 \times 4}}{d \over 2}{2 \over 3}{d \over 4} + {2 \over {E{I^{''}}}}{d \over {2 \times 2}}{b \over 2}{2 \over 3}{d \over 2} = {{{d^3}} \over {24E{I^{'}}}} + {{b{d^2}} \over {12E{I^{''}}}}

Where I, I is the moment of inertia of the column leg and the siding, respectively. $1kv=Δd=d224EI′+bd12EI"$ {1 \over {{k_v}}} = {\Delta \over d} = {{{d^2}} \over {24E{I^{'}}}} + {{bd} \over {12E{I^{''}}}}

The load potential energy is: $UP=−(P0+Pt cos θt)∫0l12(υ′)2 dx$ {U_P} = - \left({{P_0} + {P_t}\,\cos \,\theta t} \right)\int_0^l {{1 \over 2}{{\left({{\upsilon^{'}}} \right)}^2}\,dx}

The kinetic energy is: $V=12∫0lm(υ˙)2 dx$ V = {1 \over 2}\int_0^l {m{{\left({\dot \upsilon} \right)}^2}\,dx}

According to Hamilton's principle, we can obtain the dynamic differential equation of the combined pressure rod: $mυ−Kv(υ′−φ)′+(P0+Pt cos θt)υ"=0EIφ"+Kv(υ′−φ)=0$ \matrix{{m\upsilon - {K_v}{{\left({{\upsilon^{'}} - \varphi} \right)}^{'}} + \left({{P_0} + {P_t}\,\cos \,\theta t} \right){\upsilon^{''}} = 0} \hfill \cr {EI{\varphi^{''}} + {K_v}\left({{\upsilon^{'}} - \varphi} \right) = 0} \hfill \cr}

We discretize the equation using the Galerkin method. We assume that the lateral rod displacement $υ(x,t)=∑n=1∞fn(t) sin(nπx/l)$ \upsilon \left({x,t} \right) = \sum\limits_{n = 1}^\infty {{f_n}\left(t \right)\,\sin \left({n\pi x/l} \right)} satisfies the rod boundary conditions [11]. We substitute it into equation (18) to get the dynamic differential equation of the combined pressure rod $mf"(t)+EIπ4l4[1+π2EIl2(d224 EI′ +bd12 EI")]−1f(t)−(P0+Ptcos θt)π2l2f(t)=0$ m{f^{''}}\left(t \right) + {{EI{\pi^4}} \over {{l^4}}}{\left[{1 + {{{\pi^2}EI} \over {{l^2}}}\left({{{{d^2}} \over {24\,E{I^{'}}}} + {{bd} \over {12\,E{I^{''}}}}} \right)} \right]^{- 1}}f\left(t \right) - \left({{P_0} + {P_t}\cos \,\theta t} \right){{{\pi^2}} \over {{l^2}}}f\left(t \right) = 0

Equation (19) can be further simplified as: $f"(t)+Ω2(1−2 μ cos θt)f(t)=0$ {f^{''}}\left(t \right) + {\Omega^2}\left({1 - 2\,\mu \,\cos \,\theta t} \right)f\left(t \right) = 0

In: $ω={π2EIml2[1+π2EIl2(d224EI′+bd12EI")]−1}12P=π2EIl2[1+π2EIl2(d224EI′+bd12EI")]−1Ω=ω1−P0/P, μ=Pt/[2(P−P0)]$ \matrix{{\omega = {{\left\{{{{{\pi^2}EI} \over {m{l^2}}}{{\left[{1 + {{{\pi^2}EI} \over {{l^2}}}\left({{{{d^2}} \over {24E{I^{'}}}} + {{bd} \over {12E{I^{''}}}}} \right)} \right]}^{- 1}}} \right\}}^{{1 \over 2}}}} \hfill \cr {P = {{{\pi^2}EI} \over {{l^2}}}{{\left[{1 + {{{\pi^2}EI} \over {{l^2}}}\left({{{{d^2}} \over {24E{I^{'}}}} + {{bd} \over {12E{I^{''}}}}} \right)} \right]}^{- 1}}} \hfill \cr {\Omega = \omega \sqrt {1 - {P_0}/P},\,\mu = {P_t}/\left[{2\left({P - {P_0}} \right)} \right]} \hfill \cr}

In this way, we can determine the dynamic instability area in the same way as the tie-bar combination strut.

Example analysis

We make the following assumptions for the hinge support at both ends of the double-limb strip (plate) type combined pressure rod: $A′=2×10−3 m2, E=2×108kN/m2, I=2.5×10−4m4, b=d=0.5m.$ {A^{'}} = 2 \times {10^{- 3}}\,{m^2},\,E = 2 \times {10^8}kN/{m^2},\,I = 2.5 \times {10^{- 4}}{m^4},\,b = d = 0.5m.

We give the theoretical value of the elastic critical load of the combined compression bar without considering the shear deformation as a comparison: $Pcr=π2 EI/l2$ {P_{cr}} = {\pi^2}\,EI/{l^2}

Where I is the moment of inertia of the section composed of the limbs.

When the width of the compression bar is large, the elastic critical load without considering the shear deformation is much larger than the finite element calculation result [12]. This shows that the closer the length and width of the beam or column, the greater the effect of shear deformation (Figure 3). The elastic critical load value of the composite compression rod considering shear deformation in this paper is in good agreement with the finite element calculation results.

At this time, the dynamic instability area expands rapidly. This shows that the greater the slenderness ratio, the greater the possibility of parametric resonance [13]. Therefore, the slenderness ratio is an important factor in determining the dynamic instability region of the two combined struts (Fig. 4). With the increase in the area of the slatted combination strut, the stable critical bearing capacity will also increase. At this time, the frequency of the parametric resonance of the pressure rod increases, and the dynamic instability region narrows accordingly (Fig. 5). This shows that the larger the area of the slats, the less likely the rod is to suffer from instability failure caused by parametric excitation vibration. But the strip area has little effect on the dynamic instability region. With the increase of the rigidity of the affixed plate combined pressure rod, the stable critical bearing capacity of the pressure rod will increase [14]. At this time, the frequency of parametric resonance of the structure also increases, and the dynamic instability region also narrows (Fig. 6). This shows that the greater the stiffness of the slat line, the less likely the rod is to suffer from instability failure caused by parametric excitation vibration. However, the effect of panel line stiffness on the dynamic instability region is not obvious.

Conclusion

The critical load of the composite compression bar is always smaller than that of the solid web column with the same moment of inertia, so the influence of shear deformation must be considered in the stability study of the composite compression bar. With the increase of the slenderness ratio of the combined strut, the frequency of the parametric resonance of the structure decreases, and the dynamic instability region expands rapidly. Therefore, the slenderness ratio is an important factor in determining the dynamic instability region of the two lattice columns. With the increase of the strip area, the frequency of parametric resonance of the strip-type combined compression bar increases. At this time, the structure is less likely to be unstable. However, it has little effect on the dynamic instability region. Similarly, with the increase of the stiffness of the slatted line, the stable critical bearing capacity of the slatted composite compression bar increases, and the frequency of the parametric resonance of the structure also increases. At this time, the possibility of parametric vibration of the rod becomes smaller, but its influence on the dynamic instability region is not obvious.

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