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L11 — Plate Bending（板弯曲）🔴 样卷未考

Q77 🔴 板和梁的区别

EN: A plate is the 2D extension of a beam. Key differences:

  • A beam resists bending about one axis; a plate resists bending about two perpendicular axes AND twisting
  • A beam uses 1D theory (v(x)v(x)); a plate uses 2D theory (w(x,y)w(x,y))
  • Beam governing equation: EI d4v/dx4=qEI\,d^4v/dx^4 = q; Plate: D∇4w=qD\nabla^4 w = q
  • A plate is initially flat; if initially curved, it is called a shell

中文： 板是梁的二维推广。梁抵抗单向弯曲，板抵抗两个方向的弯曲+扭转。梁方程 EI w′′′′=qEI\,w''''=q；板方程 D∇4w=qD\nabla^4 w=q。初始为平面的是板，初始为曲面的是壳。

Q78 🔴 Kirchhoff薄板假设

EN: The Kirchhoff (thin) plate theory is based on three key assumptions:

  1. Normals remain normal — lines normal to the mid-surface remain straight and normal after deformation. This implies zero transverse shear strains: γxz=γyz=0\gamma_{xz} = \gamma_{yz} = 0.

  2. Normal stress is negligible — the transverse normal stress is zero: σz=0\sigma_z = 0. The plate is in a state of plane stress parallel to the mid-surface. Also, thickness change is neglected: εz=0\varepsilon_z = 0.

  3. Membrane forces are neglected — in-plane (membrane) displacements of the mid-surface are zero: u(x,y,0)=v(x,y,0)=0u(x,y,0) = v(x,y,0) = 0. This is valid when the transverse deflection is small compared to thickness (w≪tw \ll t).

中文： Kirchhoff薄板假设——(1)法线保持法线：γxz=γyz=0\gamma_{xz}=\gamma_{yz}=0（横向剪切为零）；(2)法向应力忽略：σz≈0\sigma_z\approx0（平面应力状态），εz≈0\varepsilon_z\approx0；(3)中面无面内位移：u0=v0=0u_0=v_0=0（小挠度 w≪tw\ll t）。这些假设类似于梁的Euler-Bernoulli假设。

Q79 🔴 板的弯曲刚度D

EN: The bending rigidity (flexural rigidity) of a plate is:

D=Et312(1−ν2)\boxed{D = \frac{Et^3}{12(1-\nu^2)}}

Analogous to EIEI for beams. The factor 1/(1−ν2)1/(1-\nu^2) accounts for the biaxial stress state in plates. DD has units of force × length (e.g., N·m). Note the t3t^3 dependence — doubling the thickness increases bending stiffness by a factor of 8.

中文： 板的弯曲刚度 D=Et3/[12(1−ν2)]D=Et^3/[12(1-\nu^2)]，类似梁的 EIEI。1/(1−ν2)1/(1-\nu^2) 因子来自板的双轴应力状态。DD 正比于 t3t^3——厚度翻倍，刚度增加8倍。这是必背公式。

Q80 🔴 板的控制方程

EN: The governing differential equation for Kirchhoff plate bending:

D(∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4)=q\boxed{D\left(\frac{\partial^4 w}{\partial x^4} + 2\frac{\partial^4 w}{\partial x^2 \partial y^2} + \frac{\partial^4 w}{\partial y^4}\right) = q}

or compactly: D∇4w=qD \nabla^4 w = q, where ∇4=∇2∇2\nabla^4 = \nabla^2 \nabla^2 is the biharmonic operator.

Compare with beam: EI d4w/dx4=qEI\,d^4w/dx^4 = q. The plate equation is the 2D biharmonic generalization.

中文： 板控制方程：D(wxxxx+2wxxyy+wyyyy)=qD(w_{xxxx}+2w_{xxyy}+w_{yyyy})=q（双调和方程）。类比梁：EI w′′′′=qEI\,w''''=q。板方程多了混合导数项 2wxxyy2w_{xxyy}（来自两个方向的弯曲耦合和扭转）。

Q81 🔴 板位移场

EN: Under Kirchhoff assumptions, the 3D displacement components are expressed in terms of the mid-surface deflection w(x,y)w(x,y):

u(x,y,z)=−z∂w∂x,v(x,y,z)=−z∂w∂y,w(x,y,z)=w(x,y)u(x,y,z) = -z\frac{\partial w}{\partial x}, \quad v(x,y,z) = -z\frac{\partial w}{\partial y}, \quad w(x,y,z) = w(x,y)

The in-plane displacements uu and vv vary linearly through the thickness and are zero at the mid-surface (z=0z=0). The transverse displacement ww is constant through the thickness.

中文： 板内位移场：u=−z∂w/∂xu=-z\partial w/\partial x, v=−z∂w/∂yv=-z\partial w/\partial y, w=w(x,y)w=w(x,y)。面内位移 u,vu,v 沿厚度线性变化（中面为零），挠度 ww 沿厚度不变。这完全类似于梁的 u=−y dv/dxu=-y\,dv/dx。

Q82 🔴 曲率与应变的关系

EN: Plate curvatures are defined as:

κx=−∂2w∂x2,κy=−∂2w∂y2,κxy=−2∂2w∂x∂y\kappa_x = -\frac{\partial^2 w}{\partial x^2}, \quad \kappa_y = -\frac{\partial^2 w}{\partial y^2}, \quad \kappa_{xy} = -2\frac{\partial^2 w}{\partial x \partial y}

The in-plane strains at any distance zz from the mid-surface are:

εx=zκx,εy=zκy,γxy=zκxy\varepsilon_x = z\kappa_x, \quad \varepsilon_y = z\kappa_y, \quad \gamma_{xy} = z\kappa_{xy}

The strain varies linearly through the thickness (zero at mid-surface, maximum at surfaces z=±t/2z = \pm t/2).

中文： 曲率 κx=−w,xx\kappa_x=-w_{,xx}, κy=−w,yy\kappa_y=-w_{,yy}, κxy=−2w,xy\kappa_{xy}=-2w_{,xy}。应变 = z × 曲率，沿厚度线性分布（类似梁的 ε=−yv′′\varepsilon=-yv''）。注意负号约定。

Q83 🔴 弯矩-曲率关系

EN: The bending and twisting moments per unit length are:

{MxMyMxy}=D[1ν0ν10001−ν2]{κxκyκxy}\begin{Bmatrix} M_x \\ M_y \\ M_{xy} \end{Bmatrix} = D\begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1-\nu}{2} \end{bmatrix} \begin{Bmatrix} \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{Bmatrix}

where Mx=∫−t/2t/2σxz dzM_x = \int_{-t/2}^{t/2} \sigma_x z\,dz, My=∫−t/2t/2σyz dzM_y = \int_{-t/2}^{t/2} \sigma_y z\,dz, Mxy=∫−t/2t/2τxyz dzM_{xy} = \int_{-t/2}^{t/2} \tau_{xy} z\,dz.

The moment-curvature material matrix has the same form as the plane stress D-matrix but multiplied by DD instead of E/(1−ν2)E/(1-\nu^2).

中文： 板的弯矩 = 曲率 × D矩阵（形式同平面应力D矩阵，但因子为 D=Et3/[12(1−ν2)]D=Et^3/[12(1-\nu^2)]）。Mx,MyM_x, M_y 是单位长度的弯矩，MxyM_{xy} 是单位长度的扭矩。

Q84 🔴 板单元每节点自由度

EN: A plate bending element has 3 DOFs per node:

{di}={wiθxiθyi}\{d_i\} = \begin{Bmatrix} w_i \\ \theta_{xi} \\ \theta_{yi} \end{Bmatrix}

where:

  • wiw_i — transverse deflection
  • θxi=+∂w/∂y\theta_{xi} = +\partial w/\partial y — rotation about the x-axis
  • θyi=−∂w/∂x\theta_{yi} = -\partial w/\partial x — rotation about the y-axis

A 4-node rectangular plate element has 12 DOFs total (4×3=124 \times 3 = 12).

中文： 板单元每节点3个DOF：挠度 ww、绕x轴转角 θx=+∂w/∂y\theta_x=+\partial w/\partial y、绕y轴转角 θy=−∂w/∂x\theta_y=-\partial w/\partial x。4节点矩形板单元共12个DOF。注意 θy\theta_y 的负号。

Q85 🔴 板的12项位移多项式

EN: For a 4-node rectangular plate element, the transverse displacement is interpolated with a 12-term polynomial:

w=a1+a2x+a3y+a4x2+a5xy+a6y2+a7x3+a8x2y+a9xy2+a10y3+a11x3y+a12xy3\begin{aligned} w &= a_1 + a_2 x + a_3 y + a_4 x^2 + a_5 xy + a_6 y^2 \\ &+ a_7 x^3 + a_8 x^2 y + a_9 x y^2 + a_{10} y^3 \\ &+ a_{11} x^3 y + a_{12} x y^3 \end{aligned}

The 12 coefficients are determined by the 12 nodal DOFs (4 nodes × 3 DOFs each). Note that the polynomial is not complete — the x2y2x^2y^2 term is missing (a complete quartic would have 15 terms). This incompleteness means the element may not satisfy inter-element slope continuity (C¹ compatibility) on all edges.

中文： 板单元位移多项式含12项（对应12个DOF）。注意缺少 x2y2x^2y^2 项——12项多项式不是完全四次（完全四次需15项）。这导致该单元在某些边上不满足C¹连续性（斜率不连续），是板单元的经典问题。

Q86 🟡 板弯曲的等效节点力

EN: For a plate under distributed transverse load q(x,y)q(x,y), the equivalent nodal forces are:

{Fs}=∫A[Ns]Tq dx dy\{\mathbf{F}_s\} = \int_A [\mathbf{N}_s]^T q \, dx\,dy

where [Ns][\mathbf{N}_s] contains the shape functions corresponding to the transverse displacement DOFs. For a uniform load, the equivalent nodal loads include both nodal forces AND nodal moments (analogous to the fixed-end moments in beam elements under uniform load).

中文： 板受均布载荷时，等效节点力包含节点力和节点力矩（类比梁的固端力）。通过 Fs=∫NsTq dA\mathbf{F}_s=\int\mathbf{N}_s^T q\,dA 积分得到。

Q87 🟡 薄板判据

EN: A plate is considered “thin” (Kirchhoff theory applicable) when:

tL<110ort≪b,  t≪c\frac{t}{L} < \frac{1}{10} \quad \text{or} \quad t \ll b, \; t \ll c

where tt is the thickness and LL (or b,cb, c) is the characteristic in-plane dimension. If the thickness is larger, transverse shear deformation becomes significant, and Mindlin-Reissner (thick plate) theory should be used instead (which includes γxz\gamma_{xz} and γyz\gamma_{yz}).

中文： 薄板条件：t<L/10t < L/10（厚度远小于面内尺寸）。若厚度较大，横向剪切变形不可忽略，需用Mindlin厚板理论（考虑 γxz,γyz\gamma_{xz}, \gamma_{yz}）。

Q88 🟡 板的应力-应变关系

EN: Under the Kirchhoff plate assumption (σz=0\sigma_z = 0), the in-plane stresses are:

σx=E1−ν2(εx+νεy),σy=E1−ν2(εy+νεx),τxy=Gγxy\sigma_x = \frac{E}{1-\nu^2}(\varepsilon_x + \nu\varepsilon_y), \quad \sigma_y = \frac{E}{1-\nu^2}(\varepsilon_y + \nu\varepsilon_x), \quad \tau_{xy} = G\gamma_{xy}

where G=E/[2(1+ν)]G = E/[2(1+\nu)]. This is identical to the plane stress constitutive law, applied to each layer through the thickness.

中文： 板的面内应力-应变关系 = 平面应力本构（因为 σz≈0\sigma_z\approx0）。在每个厚度层上应用 σx=E/(1−ν2)(εx+νεy)\sigma_x = E/(1-\nu^2)(\varepsilon_x+\nu\varepsilon_y) 等。最大应力出现在板的上下表面（z=±t/2z=\pm t/2）。

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