Euler-Bernoulli beam
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![{\displaystyle {\begin{aligned}u_{1}&=u_{0}(x)-z{\cfrac {dw_{0}}{dx}}\\u_{2}&=0\\u_{3}&=w_{0}(x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f30a6d76195c2f6549af5d8c87bd769b34ea63)
![{\displaystyle \varepsilon _{11}=\varepsilon _{xx}=\varepsilon _{xx}^{0}+z\varepsilon _{xx}^{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e0e689a73c90868782223f5a98c05e4549865d)
![{\displaystyle {\begin{aligned}\varepsilon _{xx}^{0}&={\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\\\varepsilon _{xx}^{1}&=-{\cfrac {d^{2}w_{0}}{dx^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2042587e5cc1a5f28d03937a41f04b3b101f2a7)
![{\displaystyle \varepsilon _{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)+{\frac {1}{2}}\left({\frac {\partial u_{m}}{\partial x_{i}}}{\frac {\partial u_{m}}{\partial x_{j}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ae6078c9130e35d800467e0122c604c17fbdc4)
![{\displaystyle {\begin{aligned}\varepsilon _{11}&={\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{1}}}+{\frac {\partial u_{1}}{\partial x_{1}}}\right)+{\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{1}}}{\frac {\partial u_{1}}{\partial x_{1}}}+{\frac {\partial u_{2}}{\partial x_{1}}}{\frac {\partial u_{2}}{\partial x_{1}}}+{\frac {\partial u_{3}}{\partial x_{1}}}{\frac {\partial u_{3}}{\partial x_{1}}}\right)\\\varepsilon _{22}&={\frac {1}{2}}\left({\frac {\partial u_{2}}{\partial x_{2}}}+{\frac {\partial u_{2}}{\partial x_{2}}}\right)+{\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{2}}}{\frac {\partial u_{1}}{\partial x_{2}}}+{\frac {\partial u_{2}}{\partial x_{2}}}{\frac {\partial u_{2}}{\partial x_{2}}}+{\frac {\partial u_{3}}{\partial x_{2}}}{\frac {\partial u_{3}}{\partial x_{2}}}\right)\\\varepsilon _{33}&={\frac {1}{2}}\left({\frac {\partial u_{3}}{\partial x_{3}}}+{\frac {\partial u_{3}}{\partial x_{3}}}\right)+{\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{3}}}{\frac {\partial u_{1}}{\partial x_{3}}}+{\frac {\partial u_{2}}{\partial x_{3}}}{\frac {\partial u_{2}}{\partial x_{3}}}+{\frac {\partial u_{3}}{\partial x_{3}}}{\frac {\partial u_{3}}{\partial x_{3}}}\right)\\\varepsilon _{23}&={\frac {1}{2}}\left({\frac {\partial u_{2}}{\partial x_{3}}}+{\frac {\partial u_{3}}{\partial x_{2}}}\right)+{\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{2}}}{\frac {\partial u_{1}}{\partial x_{3}}}+{\frac {\partial u_{2}}{\partial x_{2}}}{\frac {\partial u_{2}}{\partial x_{3}}}+{\frac {\partial u_{3}}{\partial x_{2}}}{\frac {\partial u_{3}}{\partial x_{3}}}\right)\\\varepsilon _{31}&={\frac {1}{2}}\left({\frac {\partial u_{3}}{\partial x_{1}}}+{\frac {\partial u_{1}}{\partial x_{3}}}\right)+{\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{3}}}{\frac {\partial u_{1}}{\partial x_{1}}}+{\frac {\partial u_{2}}{\partial x_{3}}}{\frac {\partial u_{2}}{\partial x_{1}}}+{\frac {\partial u_{3}}{\partial x_{3}}}{\frac {\partial u_{3}}{\partial x_{1}}}\right)\\\varepsilon _{12}&={\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{2}}}+{\frac {\partial u_{2}}{\partial x_{1}}}\right)+{\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{1}}}{\frac {\partial u_{1}}{\partial x_{2}}}+{\frac {\partial u_{2}}{\partial x_{1}}}{\frac {\partial u_{2}}{\partial x_{2}}}+{\frac {\partial u_{3}}{\partial x_{1}}}{\frac {\partial u_{3}}{\partial x_{2}}}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43601a5cfda455119cb3f0fecb9cb4c02815d990)
The displacements
![{\displaystyle u_{1}=u_{0}(x_{1})-x_{3}{\cfrac {dw_{0}}{dx_{1}}}~;~~u_{2}=0~;~~u_{3}=w_{0}(x_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeda272373a940dbeb8bb9f9293c2aef77c5dae2)
The derivatives
![{\displaystyle {\begin{aligned}{\frac {\partial u_{1}}{\partial x_{1}}}&={\cfrac {du_{0}}{dx_{1}}}-x_{3}{\cfrac {d^{2}w_{0}}{dx_{1}^{2}}}~;~~&{\frac {\partial u_{1}}{\partial x_{2}}}&=0~;~~&{\frac {\partial u_{1}}{\partial x_{3}}}&=-{\cfrac {dw_{0}}{dx_{1}}}\\{\frac {\partial u_{2}}{\partial x_{1}}}&=0~;~~&{\frac {\partial u_{2}}{\partial x_{2}}}&=0~;~~&{\frac {\partial u_{2}}{\partial x_{3}}}&=0\\{\frac {\partial u_{3}}{\partial x_{1}}}&={\cfrac {dw_{0}}{dx_{1}}}~;~~&{\frac {\partial u_{3}}{\partial x_{2}}}&=0~;~~&{\frac {\partial u_{3}}{\partial x_{3}}}&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af6317b2f53d12cd46d74046fb5159c91ac2ed77)
The von Karman strains
![{\displaystyle {\begin{aligned}\varepsilon _{11}&={\cfrac {du_{0}}{dx_{1}}}-x_{3}{\cfrac {d^{2}w_{0}}{dx_{1}^{2}}}+{\frac {1}{2}}\left[\left({\cfrac {du_{0}}{dx_{1}}}-x_{3}{\cfrac {d^{2}w_{0}}{dx_{1}^{2}}}\right)^{2}+\left({\cfrac {dw_{0}}{dx_{1}}}\right)^{2}\right]\\\varepsilon _{22}&=0\\\varepsilon _{33}&={\frac {1}{2}}\left({\cfrac {dw_{0}}{dx_{1}}}\right)^{2}\\\varepsilon _{23}&=0\\\varepsilon _{31}&={\frac {1}{2}}\left({\cfrac {dw_{0}}{dx_{1}}}-{\cfrac {dw_{0}}{dx_{1}}}\right)-{\frac {1}{2}}\left[\left({\cfrac {du_{0}}{dx_{1}}}-x_{3}{\cfrac {d^{2}w_{0}}{dx_{1}^{2}}}\right)\left({\cfrac {dw_{0}}{dx_{1}}}\right)\right]\\\varepsilon _{12}&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a21b537e2ea75e84391480ff4da1f5a3f5aaf967)
![{\displaystyle {\begin{aligned}{\cfrac {dN_{xx}}{dx}}+f(x)&=0\\{\cfrac {d^{2}M_{xx}}{dx^{2}}}+q(x)+{\cfrac {d}{dx}}\left(N_{xx}{\cfrac {dw_{0}}{dx}}\right)&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6da79553cecfa57fe7ea76ac1b62dbc4010ab3d)
![{\displaystyle {\begin{aligned}N_{xx}&=\int _{A}\sigma _{xx}~dA\\M_{xx}&=\int _{A}z\sigma _{xx}~dA\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d81d34e7e8c9ed452e8441107d3ea48420fd7e1b)
![{\displaystyle \sigma _{xx}=E\varepsilon _{xx}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e19dd435d5ac3be56bdbb0511a31669e5f47d)
![{\displaystyle {\begin{aligned}N_{xx}&=A_{xx}\varepsilon _{xx}^{0}+B_{xx}\varepsilon _{xx}^{1}=A_{xx}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]-B_{xx}{\cfrac {d^{2}w_{0}}{dx^{2}}}\\M_{xx}&=B_{xx}\varepsilon _{xx}^{0}+D_{xx}\varepsilon _{xx}^{1}=B_{xx}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]-D_{xx}{\cfrac {d^{2}w_{0}}{dx^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09e134796671c8d4bf39a1a7ed5e4fac40e0d4ff)
![{\displaystyle {\begin{aligned}A_{xx}&=\int _{A}E~dA\qquad \leftarrow \qquad {\text{extensional stiffness}}\\B_{xx}&=\int _{A}zE~dA\qquad \leftarrow \qquad {\text{extensional-bending stiffness}}\\D_{xx}&=\int _{A}z^{2}E~dA\qquad \leftarrow \qquad {\text{bending stiffness}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f843ac688f0028bad978469996e2de7528ca0fc1)
If
is constant, and
-axis passes through centroid
![{\displaystyle {\begin{aligned}A_{xx}&=E\int _{A}~dA=EA\\B_{xx}&=E\int _{A}z~dA=0\\D_{xx}&=E\int _{A}z^{2}~dA=EI\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84d3aa2f55b94a65321b123956c03baeec4b4d34)
![{\displaystyle {\begin{aligned}\int _{x_{a}}^{x_{b}}{\cfrac {d(\delta u_{0})}{dx}}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]A_{xx}~dx&=\int _{x_{a}}^{x_{b}}(\delta u_{0})f~dx+\\&\delta u_{0}(x_{a})Q_{1}+\delta u_{0}(x_{b})Q_{4}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e6cb90b2907d59be6976f3f4a0b2e67ca7944d2)
where
![{\displaystyle {\begin{aligned}\delta u_{0}&:=v_{1}\\Q_{1}&:=-N_{xx}(x_{a})\\Q_{4}&:=N_{xx}(x_{b})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9224aba5e3cabfb7f50a5cf2d94d7767382052)
![{\displaystyle {\begin{aligned}\int _{x_{a}}^{x_{b}}\left\{{\cfrac {d(\delta w_{0})}{dx}}\right.&\left[{\cfrac {du_{0}}{dx}}+{\cfrac {1}{2}}~\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]{\cfrac {dw_{0}}{dx}}A_{xx}+\left.{\cfrac {d^{2}(\delta w_{0})}{dx^{2}}}\left({\cfrac {d^{2}w_{0}}{dx^{2}}}\right)D_{xx}\right\}~dx=\\&\int _{x_{a}}^{x_{b}}(\delta w_{0})q~dx+\delta w_{0}(x_{a})Q_{2}+\delta w_{0}(x_{b})Q_{5}+\delta \theta (x_{a})Q_{3}+\delta \theta (x_{b})Q_{6}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13315a5aae21e0bdb3fc1eedd4d3f2fe689b09e4)
where
![{\displaystyle {\begin{aligned}\delta w_{0}&:=v_{2}&\delta \theta &:={\cfrac {dv_{2}}{dx}}\\Q_{2}&:=-\left[{\cfrac {dM_{xx}}{dx}}+N_{xx}{\cfrac {dw_{0}}{dx}}\right]_{x_{a}}&Q_{5}&:=\left[{\cfrac {dM_{xx}}{dx}}+N_{xx}{\cfrac {dw_{0}}{dx}}\right]_{x_{b}}\\Q_{3}&:=-M_{xx}(x_{a})&Q_{6}&:=M_{xx}(x_{b})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c10c5324568f9e62a1216ac98ddbe5ebf7438531)
Finite element model for Euler Bernoulli beam
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![{\displaystyle {\begin{aligned}u_{0}(x)&=u_{1}\psi _{1}(x)+u_{2}\psi _{2}(x)\\w_{0}(x)&=w_{1}\phi _{1}(x)+\theta _{1}\phi _{2}(x)+w_{2}\phi _{3}(x)+\theta _{2}\phi _{4}(x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8fd8af18a3243d5f53aa59142b6b55e50816a51)
where
.
Hermite shape functions for beam
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![{\displaystyle {\begin{bmatrix}\mathbf {K} ^{11}&\vdots &\mathbf {K} ^{12}\\&\vdots &\\\mathbf {K} ^{21}&\vdots &\mathbf {K} ^{22}\end{bmatrix}}{\begin{bmatrix}\mathbf {u} \\\\\mathbf {d} \end{bmatrix}}={\begin{bmatrix}\mathbf {F} ^{1}\\\\\mathbf {F} ^{2}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b78b4f753208bc930b4e0d6b66e5093aa43a747)
where
![{\displaystyle {\begin{aligned}\mathbf {u} &=[u_{1}\quad u_{2}]^{T}\\\mathbf {d} &=[w_{1}\quad \theta _{1}\quad w_{2}\quad \theta _{2}]^{T}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/411d9c55d69d78c56b388005af2fcc0c2f7860c2)
![{\displaystyle {\begin{aligned}\mathbf {K} ^{11}&=2\times 2;\qquad &\mathbf {K} ^{12}&=2\times 4\\\mathbf {K} ^{21}&=4\times 2;\qquad &\mathbf {K} ^{22}&=4\times 4\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f56b31e8af34069be913e598a2c4230cfa048430)
![{\displaystyle {\begin{aligned}K_{ij}^{11}&=\int _{x_{a}}^{x_{b}}A_{xx}{\cfrac {d\psi _{i}}{dx}}{\cfrac {d\psi _{j}}{dx}}~dx\\K_{ij}^{12}&={\frac {1}{2}}\int _{x_{a}}^{x_{b}}\left(A_{xx}{\cfrac {dw_{0}}{dx}}\right){\cfrac {d\psi _{i}}{dx}}{\cfrac {d\phi _{j}}{dx}}~dx\\K_{ij}^{21}&={\frac {1}{2}}\int _{x_{a}}^{x_{b}}\left(A_{xx}{\cfrac {dw_{0}}{dx}}\right){\cfrac {d\phi _{i}}{dx}}{\cfrac {d\psi _{j}}{dx}}~dx\\K_{ij}^{22}&=\int _{x_{a}}^{x_{b}}\left\{{\frac {1}{2}}A_{xx}\left[{\cfrac {du_{0}}{dx}}+\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]{\cfrac {d\phi _{i}}{dx}}{\cfrac {d\phi _{j}}{dx}}+D_{xx}{\cfrac {d^{2}\phi _{i}}{dx^{2}}}{\cfrac {d^{2}\phi _{j}}{dx^{2}}}\right\}~dx\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b56a59bc9b53f9a98d0d75c41d6f10d49cf2df4)
![{\displaystyle {\begin{aligned}F_{i}^{1}&=\int _{x_{a}}^{x_{b}}\psi _{i}f~dx+\psi _{i}(x_{a})Q_{1}+\psi _{i}(x_{b})Q_{4}\\F_{i}^{2}&=\int _{x_{a}}^{x_{b}}\phi _{i}q~dx+\phi _{i}(x_{a})Q_{2}+\phi _{i}(x_{b})Q_{5}+{\cfrac {d\phi _{i}}{dx}}(x_{a})Q_{3}+{\cfrac {d\phi _{i}}{dx}}(x_{b})Q_{6}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51483114b4d96de6686856f8591249d20daf650f)
![{\displaystyle \mathbf {K} (\mathbf {U} )\mathbf {U} =\mathbf {F} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b04cd3302321183f57d7cb74ee215cb2a229d3da)
where
![{\displaystyle {\begin{aligned}U_{1}&=u_{1},~U_{2}=u_{2},~U_{3}=d_{1},~U_{4}=d_{2},~U_{5}=d_{3},~U_{6}=d_{4}\\F_{1}&=F_{1}^{1},~F_{2}=F_{2}^{1},~F_{3}=F_{1}^{2},~F_{4}=F_{2}^{2},~F_{5}=F_{3}^{2},~F_{6}=F_{4}^{2}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e6114e6c8c58f295414b7edefdf57fa016ff10)
The residual is
![{\displaystyle \mathbf {R} =\mathbf {K} \mathbf {U} -\mathbf {F} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9734256826e3a99ab5eff07d7823a0974d2294f)
For Newton iterations, we use the algorithm
![{\displaystyle \mathbf {U} ^{r+1}=\mathbf {U} ^{r}-(\mathbf {T} ^{r})^{-1}\mathbf {R} ^{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7b7cd04f14144b0d13fb724674ae9d547d930e)
where the tangent stiffness matrix is given by
![{\displaystyle \mathbf {T} ^{r}={\frac {\partial \mathbf {R} ^{r}}{\partial \mathbf {U} }};\quad {\text{or}}\quad T_{ij}={\frac {\partial R_{i}}{\partial U_{j}}},\qquad i=1\dots 6,j=1\dots 6~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37e7ea6a6000ec2ef540a59355a600028cdbe0a9)
![{\displaystyle {\begin{aligned}i=1\dots 2;~j=1\dots 2&:\\&{T_{ij}^{11}=K_{ij}^{11}}\\\\i=1\dots 2;~j=1\dots 4&:\\&{T_{ij}^{12}=2K_{ij}^{12}}\\\\i=1\dots 4;~j=1\dots 2&:\\&{T_{ij}^{21}=2K_{ij}^{21}}\\\\i=1\dots 4;~j=1\dots 4&:\\&{T_{ij}^{22}=K_{ij}^{22}+{\frac {1}{2}}\int _{x_{a}}^{x_{b}}A_{xx}\left[{\cfrac {du_{0}}{dx}}+2\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]{\cfrac {d\phi _{i}}{dx}}{\cfrac {d\phi _{j}}{dx}}~dx}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b3ce0ac52051a90d8fed30a8e9fea5e3b348da)
Recall
![{\displaystyle N_{xx}=A_{xx}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]-{B_{xx}}~~{\cfrac {d^{2}w_{0}}{dx^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9237fbff37b9f40a5b1a43e292ea2d5a8b8835)
- Divide load into small increments.
![{\displaystyle F=\sum _{i=1}^{N}\Delta F_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a5353301a680909155fcb10c9220ae02fbb4df)
- Compute
and
for first load step,
![{\displaystyle \mathbf {K} (\mathbf {U} _{0})\mathbf {U} _{1}=\Delta F_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa92dc1d01c22ea93237d5ee6e5fcc749e902c6)
Stiffness of Euler-Bernoulli beam.
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- Compute
and
for second load step,
![{\displaystyle \mathbf {K} (\mathbf {U} _{1})\mathbf {U} _{2}=\Delta F_{1}+\Delta F_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c237606f046abcafede3b8ddbabe0b563388007f)
- Continue until F is reached.
Recall
![{\displaystyle {\begin{bmatrix}\mathbf {K} ^{11}&\vdots &\mathbf {K} ^{12}\\&\vdots &\\\mathbf {K} ^{21}&\vdots &\mathbf {K} ^{22}\end{bmatrix}}{\begin{bmatrix}\mathbf {u} \\\\\mathbf {d} \end{bmatrix}}={\begin{bmatrix}\mathbf {F} ^{1}\\\\\mathbf {F} ^{2}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b78b4f753208bc930b4e0d6b66e5093aa43a747)
where
![{\displaystyle {\begin{aligned}K_{ij}^{11}&=\int _{x_{a}}^{x_{b}}A_{xx}{\cfrac {d\psi _{i}}{dx}}{\cfrac {d\psi _{j}}{dx}}~dx\\K_{ij}^{12}&={\frac {1}{2}}\int _{x_{a}}^{x_{b}}\left(A_{xx}{\cfrac {dw_{0}}{dx}}\right){\cfrac {d\psi _{i}}{dx}}{\cfrac {d\phi _{j}}{dx}}~dx\\K_{ij}^{21}&={\frac {1}{2}}\int _{x_{a}}^{x_{b}}\left(A_{xx}{\cfrac {dw_{0}}{dx}}\right){\cfrac {d\phi _{i}}{dx}}{\cfrac {d\psi _{j}}{dx}}~dx\\K_{ij}^{22}&=\int _{x_{a}}^{x_{b}}\left\{{\frac {1}{2}}A_{xx}\left[{\cfrac {du_{0}}{dx}}+\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]{\cfrac {d\phi _{i}}{dx}}{\cfrac {d\phi _{j}}{dx}}+D_{xx}{\cfrac {d^{2}\phi _{i}}{dx^{2}}}{\cfrac {d^{2}\phi _{j}}{dx^{2}}}\right\}~dx\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b56a59bc9b53f9a98d0d75c41d6f10d49cf2df4)
Mebrane locking in Euler-Bernoulli beam
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Membrane strain:
![{\displaystyle \varepsilon _{xx}^{0}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/129e9829dbac7cf0223404d836cd6a57ab2195bb)
or
![{\displaystyle {\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1448804aeb19bfcc882a321a7547f14031dd5e0a)
Hence, shape functions should be such that
![{\displaystyle {\cfrac {du_{0}}{dx}}\approx \left({\cfrac {dw_{0}}{dx}}\right)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e005b1ea2b961ea41ecc89aa436014fc4dfbf71f)
linear,
cubic
Element Locks!
Too stiff.
- Assume
is linear ;~~
is cubic.
- Then
is constant, and
is quadratic.
- Try to keep
constant.
![{\displaystyle {\begin{aligned}K_{ij}^{11}&=\int _{x_{a}}^{x_{b}}A_{xx}{\cfrac {d\psi _{i}}{dx}}{\cfrac {d\psi _{j}}{dx}}~dx\\K_{ij}^{12}&={\frac {1}{2}}\int _{x_{a}}^{x_{b}}\left(A_{xx}{\cfrac {dw_{0}}{dx}}\right){\cfrac {d\psi _{i}}{dx}}{\cfrac {d\phi _{j}}{dx}}~dx\\K_{ij}^{22}&=\int _{x_{a}}^{x_{b}}\left\{{\frac {1}{2}}A_{xx}\left[{\cfrac {du_{0}}{dx}}+\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]{\cfrac {d\phi _{i}}{dx}}{\cfrac {d\phi _{j}}{dx}}+D_{xx}{\cfrac {d^{2}\phi _{i}}{dx^{2}}}{\cfrac {d^{2}\phi _{j}}{dx^{2}}}\right\}~dx\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/497fa462a0e19f01af90bb151137dba2fb91ed03)
integrand is constant,
integrand is fourth-order ,
integrand is eighth-order
![{\displaystyle n_{\text{gauss pt}}={\text{int}}[(p+1)/2]+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05aed9f00bfa383b3fbb39b445b0de4c0ff86fa6)
Assume
= constant.
![{\displaystyle {\begin{aligned}K_{ij}^{11}&=A_{xx}\int _{x_{a}}^{x_{b}}{\cfrac {d\psi _{i}}{dx}}{\cfrac {d\psi _{j}}{dx}}~dx\\&=A_{xx}\int _{-1}^{1}\left(J^{-1}{\cfrac {d\psi _{i}(\xi )}{d\xi }}\right)\left(J^{-1}{\cfrac {d\psi _{j}(\xi )}{d\xi }}\right)J~d\xi =A_{xx}\int _{-1}^{1}F(\xi )~d\xi \\&\approx A_{xx}W_{1}F(\xi _{1})\leftarrow {\text{one-point integration}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44cd4370381e91a3b5175a0eb538e7cd06cdc515)
![{\displaystyle {\begin{aligned}K_{ij}^{12}&={\frac {1}{2}}\int _{x_{a}}^{x_{b}}\left(A_{xx}{\cfrac {dw_{0}}{dx}}\right){\cfrac {d\psi _{i}}{dx}}{\cfrac {d\phi _{j}}{dx}}~dx\\&={\cfrac {A_{xx}}{2}}\int _{-1}^{1}\left(\sum _{i=1}^{4}w_{i}J^{-1}{\cfrac {d\phi _{i}(\xi )}{d\xi }}\right)\left(J^{-1}{\cfrac {d\psi _{i}(\xi )}{d\xi }}\right)\left(J^{-1}{\cfrac {d\phi _{j}(\xi )}{d\xi }}\right)~dx\\&\approx A_{xx}\left[W_{1}F(\xi _{1})+W_{2}F(\xi _{2})+W_{3}F(\xi _{3})\right]\leftarrow {\text{full integration}}\\&\approx A_{xx}\left[W_{1}F(\xi _{1})+W_{2}F(\xi _{2})\right]\leftarrow {\text{reduced integration}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ebb006a71b2fd85988dad504cab84728908db86)