Load-Induced Stiffness Matrix of Plates

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Load-induced stiffness matrix of plates Shi-Jun Zhou

Abstract: In this paper, a rectangular plate element for the finite-element method, which takes into consideration the stiffening effect of dead loads, is proposed. The element stiffness matrices that include the effect of dead loads are derived. The effect of dead loads on dynamic behaviors of plates is analyzed using the finite-element method. It is shown that the stiffness of plates increases when the effect of dead loads is included in the calculation and that the effect is more significant for plates with a smaller stiffness. The validity of the proposed procedure is confirmed by numerical examples. Although the finite-element results obtained are in agreement with the approximate closed-form solutions, the proposed method based on a finite-element formulation is more easily applied to practical structures under various support conditions and various types of dead loads. Key words: load-induced stiffness matrix of plate, stiffening effect of dead loads. Résumé : Dans cet article, il est proposé d’étudier un élément de plaque rectangulaire par la méthode des éléments finis, en tenant compte de l’effet de raideur provenant des charges mortes. Les matrices de rigidité de cet élément sont dérivées. L’effet des charges mortes sur le comportement dynamique des plaques est analysé par la méthode des éléments finis. Il est montré que la rigidité des plaques augmente lorsque l’effet des charges mortes est pris en compte par les calculs, et que cet effet est d’autant plus significatif sur les plaques dont la rigidité est moindre. La validité de la procédure proposée est confirmée par les exemples numériques. Bien que les résultats obtenus par la méthode des éléments finis soient en accord avec ceux obtenus par celle des solutions fermées approchées, la méthode proposée, qui est formulée en terme d’éléments finis, peut être plus facilement appliquée dans le cadre de structures réelles soumises à différentes conditions de support et à différents types de charges mortes. Mots clés : matrice de rigidité d’une plaque liée au chargement, effet de raideur dû aux charges mortes. [Traduit par la Rédaction]

Note

184

Introduction Plates or shells, like structures in general, are always subjected to dead loads. When plates are subjected to live loads in addition to dead loads, the plates deflect from a reference state caused by the initial dead loads. The deflection should include the effect of the conservative initial bending stresses. This effect of dead loads is more significant for plates and shells than for beams because of the smaller stiffness of the plates; nevertheless, it was ignored in most previous studies. A better understanding of the dead-load effect will lead to a more accurate estimate of the effects of live loads; thus the safety factors for heavyweight structures and lightweight structures will be equalized, and truly safe structural designs will be possible. Hibbitt (1979) considered the load stiffness associated with pressure loading, and in ABAQUS the pressure load stiffness is implemented as a symmetric form. Takabatake (1991) studied the effect of dead loads on the natural frequencies of beam and proposed a governing equation of beam including the effect of dead loads. Zhou and Received 2 October 2000. Revised manuscript accepted 26 September 2001. Published on the NRC Research Press Web site at http://cjce.nrc.ca on 14 December 2001. S.-J. Zhou. Department of Civil Engineering, Lanzhou Railway Institute, Lanzhou, 730070, P.R. China. Written discussion of this note is welcomed and will be received by the Editor until 30 June 2002. Can. J. Civ. Eng. 29: 181–184 (2002)

I:\cjce\cjce_29\cjce-01\L01-064.vp Thursday, February 07, 2002 1:17:57 PM

Zhu (1996) developed a conception of load-induced stiffness matrix and a beam element for the finite-element method. Takabatake (1992) extended the elementary plate theory and analyzed the effects of dead loads in dynamic plates. However, it is difficult to use Takabatake’s approaches for complex plate or shell structures. In this paper, the finite-element formulation for plates, which takes into consideration the stiffening effect of dead loads, is proposed. The element stiffness matrices including the effect of dead loads are derived. The validity of the proposed procedure is confirmed by the numerical examples.

Finite-element formulations Zhou and Zhu (1996) presented the following dynamic equation for beam elements, which takes into consideration the stiffening effect of dead loads: [1] {F} = [M ]{&&δ} + ([K ] + [K ] − [K ]){δ} − {F } c

g

eq

in which {δ} is the element nodal displacement vector, {F} is the element nodal force vector, [M] is the consistent mass matrix, [Ke] is the elastic stiffness matrix, [Kg] is the geometric stiffness matrix, [K] is the load-induced stiffness matrix, and {Feq} is the element equivalent concentrated force vector. The dynamic equation for plate elements is of the same form as eq. [1] for beam elements; they differ only in the content and volume of matrices connected with the equation.

DOI: 10.1139/L01-064

© 2002 NRC Canada


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Can. J. Civ. Eng. Vol. 29, 2002

For a plate element, it is assumed that the external forces are transverse loads only and the axial forces are neglected. Deflections w are produced by the dead loads p of the plates. The deformed state by dead loads p is considered as the reference state. The dynamic deflections w by live loads p are measured from the reference state. The element strain energy U* can be expressed as [2]

U* = U + U

where U is the strain energy produced by live loads p and U is the additional strain energy resulting from the conservative initial bending stresses produced by dead loads p. Different types of strain–displacement relation for U and U are used to include the effect of dead loads. For the strain energy U, the linear strain–displacement relations are used; for the strain energy U, the nonlinear strain–displacement relations are used. Thus, the strain energies U and U can be written as D [3] U = ∫ ∫ {(w , xx + w , yy ) 2 + 2(1 − ν)[(w , xy ) 2 − w , xxw , yy ]} dx dy 2 [4]

U = D∫ ∫ [w , xxw , xx + w , yyw , yy + 2w , xyw , xy + ν(w , yyw , xx + w , xxw , yy − 2w , xyw , xy )] dx dy +

Eh {(w , x ) 2 (w , x ) 2 + (w , y ) 2 (w , y ) 2 + 2w , xw , yw , xw , y + ν[(w , y ) 2 (w , x ) 2 + (w , x ) 2 (w , y ) 2 − 2w , xw , yw , xw , y ]} dx dy 4(1 − ν2 ) ∫ ∫

in which h is the thickness of the plate, D is the bending rigidity of the plate, and ν is Poisson’s ratio (Takabatake 1992). The strain energy U is a function of both the unknown displacement w due to live loads and the known displacements w due to dead loads. Equations [3] and [4] can be expressed as 1 [5] U = {δ}T[Ke ]{δ} 2 1 [6] U = {δ}T[Ke ]{δ} + {δ}T[K ]{δ} 2 where [7] [8]

1 T D {δ} [Ke ]{δ} = ∫ ∫ {(w , xx + w , yy ) 2 + 2(1 − ν)[(w , xy ) 2 − w , xxw , yy ]} dx dy 2 2 1 T Eh {δ} [K ]{δ} = {(w , x ) 2 (w , x ) 2 + (w , y ) 2 (w , y ) 2 + 2w , xw , yw , xw , y 2 4(1 − ν2 ) ∫ ∫ + ν[(w , y ) 2 (w , x ) 2 + (w , x ) 2 (w , y ) 2 − 2w , xw , yw , xw , y ]} dx dy

in which [Ke] is the element elastic stiffness matrix of the plate element and [K] is the load-induced stiffness matrix of the plate element. Suppose the element displacements are expressed as follows: [9a]

w = [N ]{δ}

[9b]

w = [N ]{δ}

where [N] is a shape function. The element elastic stiffness matrix can be written as [10]

[Ke ] = ∫ ∫ [B]T[D][B] dx dy

in which [B] is the element strain matrix and [D] is the elastic matrix. Therefore, the load-induced stiffness matrix is Eh [11] [K ] = {({δ}T[N , x ]T[N , x ]{δ} + ν{δ}T[N , y ]T[N , y ]{δ})[N , x ]T[N , x ] + ({δ}T[N , y ]T[N , y ]{δ} 2(1 − ν2 ) ∫ ∫ + ν{δ}T[N , x ]T[N , x ]{δ})[N , y ]T[N , y ] + (1 − ν){δ}T[N , x ]T[N , y ]{δ}([N , x ]T[N , y ] + [N , y ]T[N , x ])} dx dy From eq. [11], we can see that it is somewhat complicated to calculate the element load-induced stiffness matrix [K], although a simple shape function can be selected. For convenience, a simple expression for the transverse displacement w is used to calculate [K]. Figure 1 shows a 4-node plate element. The element nodal displacement vector {δ} and element nodal force vector {F} can be defined as [12]

{δ} = [w1, θx1, θ y1, w 2 , θx 2 , θ y 2 , w 3, θx 3, θ y 3, w 4, θx 4, θ y 4 ]T

[13]

{F} = [Q1, M x1, M y1, Q2 , M x 2 , M y 2 , Q3, M x 3, M y 3, Q4, M x 4, M y 4 ]T © 2002 NRC Canada



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183

Fig. 1. A 4-node plate element.

stiffness matrix [Ke] can easily be obtained. For convenience, in calculating the element load-induced stiffness matrix [K ], the dead load displacement function is simply selected as [18]

w = c0 + c1x + c2 y + c3xy

Applying the boundary conditions, we have

[19] Suppose [14]

w = [H ]{α}

where [15]

[H ] = [1, x, y, x 2 , xy, y 2 , x 3, x 2 y, xy 2 , y 3, x 3y, xy 3 ]

[16]

{α} = [α1, α 2 , K , α12 ]T

Applying the boundary conditions, the constant {α} can be expressed as [17]

−1

{α} = [C ] {δ}

Thus the element shape function [N] and the element elastic

[22]

From eq. [18], [20]

and

c0 = w1 w − w1 c1 = 2 a w 4 − w1 c2 = b w1 + w 3 − w 2 − w 4 c3 = ab w , x = c1 + c3y w y = c2 + c3x

Equation [11] can simply be expressed as [21]

[K ] =

Eh ([C ]−1) T[ S ][C ]−1 2(1 − ν2 )

in which

[ S ] = ∫ ∫ {[(c1 + c3y) 2 + ν(c2 + c3x) 2 ][H, x ]T[H, x ] + [(c2 + c3x) 2 + ν(c1 + c3y) 2 ][H, y ]T[H, y ] + (1 − ν)(c1 + c3y)(c2 + c3x)([H, x ]T[H, y ] + [H, y ]T[H, x ])} dx dy

By evaluating the integrals in eq. [22] and from eq. [21], we obtain [K ]. Numerical results show that [K ] computed from eq. [21] is sufficiently accurate. However, in practical applications, the expression for [K ] given by eq. [21] is somewhat complicated and the formulations need to be simplified. Let [23]

w , x = cx = constant w , y = c y = constant

and

[26]

1 (w 2 + w 3 − w1 − w 4) 2a 1 cy = (w 3 + w 4 − w1 − w 2 ) 2b cx =

[24]

then [25]

[K ] =

Eh ([C ]−1) T[T ][C ]−1 2(1 − ν2 )

in which

[T ] = ∫ ∫ {(cx2 + νc2y )[H, x ]T[H, x ] + (c2y + νcx2 )[H, y ]T[H, y ] + (1 − ν) cx c y ([H, x ]T[H, y ] + [H, y ]T [H, x ])} dx dy

Evaluating the integrals in eq. [26] gives all the elements of the symmetrical matrix [T].

Example analysis In the following analysis, the calculated values obtained from the proposed element are compared with the approximate closed-form solutions. The natural frequencies of a

simply supported plate with the following reference properties (Takabatake 1992) are calculated: lx = ly = 5 m, h = 0.05 m, E = 20.59 × 1010 N/m2, v = 0.3. A reference dead load p0 is assumed to be p0 = 3825 N/m2. In Figs. 2 and 3, ∆ = [(ω n – ω 0n)/ω 0n] × 100%, where ω n is the nth natural frequency including the effect of dead loads and ω 0n is the nth natural frequency excluding the effect of dead loads. Figure 2 shows the closed-form approximate solution © 2002 NRC Canada



184 Fig. 2. Comparison of calculated results (simply supported plate: 8 × 8 elements) (lx = ly = 5 m, h = 0.05 m, E = 20.59 × 1010 N/m2, v = 0.3, p0 = 3825 N/m2).

Fig. 3. Relationship between ∆ and P/P0 (simply supported plate: 8 × 8 elements) (lx = ly = 5 m, h = 0.05 m, E = 20.59 × 1010 N/m2, v = 0.3, p0 = 3825 N/m2, P0 = 5000 N).

Can. J. Civ. Eng. Vol. 29, 2002

require too many elements, and the number of elements used in the analysis depends only on the accuracy of the frequencies ω 0n, which also shows that the calculation of the loadinduced stiffness matrix is sufficiently accurate. Figure 3 shows the effect of the concentrated load P at the center of the plate on the natural frequencies of the simply supported plate. The reference concentrated load P0 is assumed to be P0 = 5000 N, while p0 = 3825 N/m2 = constant. It is shown that the effect of dead load is more obvious for the first natural frequency and that it can be negligible for frequencies higher than the first five natural frequencies. The larger the dead load is, the more apparent the effect will be. The improved analysis shows that the effects of dead loads on bending stiffness are determined by the following factors: dead loads, thickness of plate, span of plate, and boundary conditions. In other words, the load-stiffening effects are related to the dead loads and the stiffness of the structures. The load-stiffening effects are more apparent in plates than in beams (Zhou and Zhu 1996), since the bending stiffness of plates is smaller due to their thin thickness, and should be considered in the design of plate or shell structures for which the geometric nonlinearity is not negligible. To improve the accuracy of plate elements, we can select other types of plate elements, such as a serendipity 8-node isoparametric element. In this case, the proposed method for calculating the load-induced stiffness matrix is also useful.

Conclusions The load-induced stiffness matrix of a rectangular plate element has been developed for the finite-element method and the validity of the proposed procedure is confirmed by the numerical examples. Although the results obtained by the finite-element method are in good agreement with those given by Takabatake, the proposed method based on a finiteelement formulation is more easily applied to practical structures under various support conditions and various types of dead loads. The calculated results also show that the bending stiffness of plates is obviously increased when the effect of dead loads is considered. This effect of dead loads on bending stiffness should be included in the design of plate or shell structures for which the geometric nonlinearity is not negligible.

References

given by Takabatake(1992) for the three lowest frequencies and the finite-element solutions from eqs. [21] and [25]. From Fig. 2, we can see that the results obtained using the proposed method are in good agreement with those given by Takabatake. The simplified formulation (eq. [25]) is sufficiently accurate for the calculation of load-induced stiffness matrix K. In addition, the calculation of the ratio ∆ does not

Hibbitt, H.D. 1979. Some follower force and load stiffness. International Journal for Numerical Methods in Engineering, 14: 937–941. Takabatake, H. 1991. Effect of dead loads on natural frequencies of beams. ASCE Journal of Structural Engineering, 117(4): 1039– 1052. Takabatake, H. 1992. Effects of dead loads in dynamic plates. ASCE Journal of Structural Engineering, 118(1): 35–51. Zhou, S.J., and Zhu, X. 1996. Analysis of effect of dead loads on natural frequencies of beams using finite-element techniques. ASCE Journal of Structural Engineering, 122(5): 512–516.

© 2002 NRC Canada


Load-Induced Stiffness Matrix of Plates

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