Professor Terje Haukaas University of British Columbia, Vancouver The M4 Element. Figure 1: Bilinear Mindlin element.

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1 Professor Terje Hakaas University of British Colmbia, ancover The M Element variety of plate elements exist, some being characterized as Kirchhoff elements, i.e., for thin plates, and others as Mindlin elements, i.e., for thick plates. The early developments focsed on trianglar elements bt a qadrilateral Mindlin element with bilinear shape fnctions is shown in Figre 1 and derived below q z 3 8 y z x Figre 1: Bilinear Mindlin element. s a starting point, consider the principle of virtal displacements: δ ε T σ d q z δ wd (1) #" #" where, in the Mindlin formlation, the stress and strain vectors contain everything except the zz-components: δw ext ε {ε xx,ε yy,γ # ",γ xy "$ yz,γ } () zx σ {σ xx,σ yy,τ #",τ xy "$ yz,τ } (3) zx σ b The three first components of the stress and strain vectors are associated with bending; hence, they are collected in vectors with the sbscript b. The relationship between those stress and strain components are governed by the plain stress material law, which is appropriate when the plate is not nreasonably thick: σ x σ y τ xy E 1 ν 1 ν ν 1 1 ν σ s ε x ε y γ xy σ b () The last two stress and strain components, labelled with the sbscript s, are related to shear deformation and are independent of the other components, related by the shear modls GE/((1+ν)): The M Element Updated December, 1 Page 1

2 Professor Terje Hakaas University of British Colmbia, ancover τ yz τ zx G G γ yz γ zx σ s ε s (5) Sbstittion of Eqs. () and (5) into Eq. (1) yields the following two-part expression for the internal virtal work δ ε T b σ b d + δ ε T s σ s d δ ε T b d + δ ε T s ε s d Next, the kinematic eqations from Mindlin plate theory are written in matrix form, maintaining the separation between the bending-related and shear-related parts: ε x ε y γ xy,x z θ x,y θ x,x θ y,y γ yz γ zx θ x + w,y θ y + w,x (6) z (7) ε s (8) where and are loosely referred to as crvatre vectors for bending and shear, respectively. Sbstittion of Eqs. (7) and (8) into Eq. (1) yields z δ κ T b d + δ κ T s d (9) Those volme integrals are now transformed into area integrals by carrying ot the integration with respect to z from h/ to h/. Becase all the components of the crvatre vectors are constant with respect to z, the reslt is: h3 1 δ κ T b d + h δ κ T s d (1) This can also be expressed in terms of modified D-matrices that inclde the plate thickness: where T δ Db T d + δ Ds d (11) Eh D 3 b 1(1 ν ) 1 ν ν 1 1 ν (1) The M Element Updated December, 1 Page

3 Professor Terje Hakaas University of British Colmbia, ancover and Gh Gh (13) Having sbstitted material law and kinematics into the internal virtal work in Eq. (1), the discretization of the problem by shape fnctions is addressed. For Mindlin elements the displacement field, w, as well as the two rotation fields, θ x and, are discretized. For the for-node element shown Figre 1, the following bilinear shape fnctions are sed: N 1 (ξ,η) 1 (1 ξ)(1 η) N (ξ,η) 1 (1+ ξ)(1 η) N 3 (ξ,η) 1 (1+ ξ)(1+ η) (1) N (ξ,η) 1 (1 ξ)(1+ η) With reference to the degrees of freedom in Figre 1, the complete discretization reads: w θ x N 1 N N 1 N N 1 N In short, the discretization reads It can also be spelled ot as (15) N (16) w N 1 + N (17) θ x N 1 + N (18) N 1 + N (19) The M Element Updated December, 1 Page 3

4 Professor Terje Hakaas University of British Colmbia, ancover This discretization is sbstitted into the weak form of the bondary vale problem in Eq. (11) via the crvatre vectors, which in trn are expressed in terms of the shape fnctions by the B-matrices:,x θ x,y θ x,x θ y,y N 1,x N,x N 3,x N,x N 1,y,y,y + N,y N 1,x,x,x + N,x N 1,y N,y N 3,y N,y N 1,x N,x N 3,x N,x N 1,y,y,y N,y N 1,x N 1,y N,x N,y N 3,x N 3,y N,x N,y B b () θ x + w,y + w,x N 1 N N 3 N + N 1,y,y,y + N,y N 1 + N + N 1,x,x,x + N,x 1 N 1,y N 1 N,y N N 3,y N 3 N,y N N 1,x N 1 N,x N N 3,x N 3 N,x N B s Sbstittion of Eqs. () and (1) into Eq. (11), which is now combined with Eq. (1) yields where (1) (B b δ ) T Db (B b )d + (B s δ ) T Ds (B s )d q z (N w δ )d () Rearranging Eq. () yields N w N 1 N δ T T T B b B b d + B s B s d (3) q z N T w d () The M Element Updated December, 1 Page

5 Professor Terje Hakaas University of British Colmbia, ancover For arbitrary virtal displacements δ the parenthesis mst be zero, hence T B b Db T B b d + B s Ds B s d q z N T w d " $$$$$ $$$$ % " $ % where the stiffness matrix and load vector have been identified. K F (5) The M Element Updated December, 1 Page 5