Timoshenko beam.
u
1
=
u
0
(
x
)
+
z
φ
x
u
2
=
0
u
3
=
w
0
(
x
)
{\displaystyle {\begin{aligned}u_{1}&=u_{0}(x)+z\varphi _{x}\\u_{2}&=0\\u_{3}&=w_{0}(x)\end{aligned}}}
ε
x
x
=
ε
x
x
0
+
z
ε
x
x
1
γ
x
z
=
γ
x
z
0
{\displaystyle {\begin{aligned}\varepsilon _{xx}&=\varepsilon _{xx}^{0}+z\varepsilon _{xx}^{1}\\\gamma _{xz}&=\gamma _{xz}^{0}\end{aligned}}}
ε
x
x
0
=
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
ε
x
x
1
=
d
φ
x
d
x
γ
x
z
0
=
φ
x
+
d
w
0
d
x
{\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\varphi _{x}}{dx}}\\\gamma _{xz}^{0}&=\varphi _{x}+{\cfrac {dw_{0}}{dx}}\end{aligned}}}
δ
W
int
=
δ
W
ext
{\displaystyle \delta W_{\text{int}}=\delta W_{\text{ext}}}
where
δ
W
int
=
∫
x
a
x
b
∫
A
(
σ
x
x
δ
ε
x
x
+
σ
x
z
δ
γ
x
z
)
d
A
d
x
=
∫
x
a
x
b
(
N
x
x
δ
ε
x
x
0
+
M
x
x
δ
ε
x
x
1
+
Q
x
δ
γ
x
z
0
)
d
x
δ
W
ext
=
∫
x
a
x
b
q
δ
w
0
d
x
+
∫
x
a
x
b
f
δ
u
0
d
x
{\displaystyle {\begin{aligned}\delta W_{\text{int}}&=\int _{x_{a}}^{x_{b}}\int _{A}(\sigma _{xx}\delta \varepsilon _{xx}+\sigma _{xz}\delta \gamma _{xz})dA~dx\\&=\int _{x_{a}}^{x_{b}}(N_{xx}\delta \varepsilon _{xx}^{0}+M_{xx}\delta \varepsilon _{xx}^{1}+Q_{x}\delta \gamma _{xz}^{0})~dx\\\delta W_{\text{ext}}&=\int _{x_{a}}^{x_{b}}q\delta w_{0}~dx+\int _{x_{a}}^{x_{b}}f\delta u_{0}~dx\end{aligned}}}
Q
x
=
K
s
∫
A
σ
x
z
d
A
{\displaystyle Q_{x}={K_{s}}\int _{A}\sigma _{xz}~dA}
K
s
{\displaystyle K_{s}}
ε
x
x
0
=
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
{\displaystyle \varepsilon _{xx}^{0}={\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}}
Take variation
δ
ε
x
x
0
=
d
(
δ
u
0
)
d
x
+
1
2
(
2
d
w
0
d
x
)
(
d
(
δ
w
0
)
d
x
)
=
d
(
δ
u
0
)
d
x
+
d
w
0
d
x
d
(
δ
w
0
)
d
x
.
{\displaystyle \delta \varepsilon _{xx}^{0}={\cfrac {d(\delta u_{0})}{dx}}+{\frac {1}{2}}\left(2{\cfrac {dw_{0}}{dx}}\right)\left({\cfrac {d(\delta w_{0})}{dx}}\right)={\cfrac {d(\delta u_{0})}{dx}}+{\cfrac {dw_{0}}{dx}}{\cfrac {d(\delta w_{0})}{dx}}~.}
ε
x
x
1
=
d
φ
x
d
x
{\displaystyle \varepsilon _{xx}^{1}={\cfrac {d\varphi _{x}}{dx}}}
Take variation
δ
ε
x
x
1
=
d
δ
φ
x
d
x
{\displaystyle \delta \varepsilon _{xx}^{1}={\cfrac {d\delta \varphi _{x}}{dx}}}
γ
x
z
0
=
φ
x
+
d
w
0
d
x
{\displaystyle \gamma _{xz}^{0}=\varphi _{x}+{\cfrac {dw_{0}}{dx}}}
Take variation
δ
γ
x
z
0
=
δ
φ
x
+
d
(
δ
w
0
)
d
x
{\displaystyle \delta \gamma _{xz}^{0}=\delta \varphi _{x}+{\cfrac {d(\delta w_{0})}{dx}}}
∫
x
a
x
b
N
x
x
δ
ε
x
x
0
d
x
=
∫
x
a
x
b
N
x
x
[
d
(
δ
u
0
)
d
x
+
d
w
0
d
x
d
(
δ
w
0
)
d
x
]
d
x
∫
x
a
x
b
M
x
x
δ
ε
x
x
1
d
x
=
∫
x
a
x
b
M
x
x
[
d
δ
φ
x
d
x
]
d
x
∫
x
a
x
b
Q
x
δ
γ
x
z
0
d
x
=
∫
x
a
x
b
Q
x
[
δ
φ
x
+
d
(
δ
w
0
)
d
x
]
d
x
{\displaystyle {\begin{aligned}\int _{x_{a}}^{x_{b}}N_{xx}\delta \varepsilon _{xx}^{0}~dx&=\int _{x_{a}}^{x_{b}}N_{xx}\left[{\cfrac {d(\delta u_{0})}{dx}}+{\cfrac {dw_{0}}{dx}}{\cfrac {d(\delta w_{0})}{dx}}\right]~dx\\\int _{x_{a}}^{x_{b}}M_{xx}\delta \varepsilon _{xx}^{1}~dx&=\int _{x_{a}}^{x_{b}}M_{xx}\left[{\cfrac {d\delta \varphi _{x}}{dx}}\right]~dx\\\int _{x_{a}}^{x_{b}}Q_{x}\delta \gamma _{xz}^{0}~dx&=\int _{x_{a}}^{x_{b}}Q_{x}\left[\delta \varphi _{x}+{\cfrac {d(\delta w_{0})}{dx}}\right]~dx\end{aligned}}}
δ
W
int
=
∫
x
a
x
b
{
N
x
x
[
d
(
δ
u
0
)
d
x
+
d
w
0
d
x
d
(
δ
w
0
)
d
x
]
+
M
x
x
[
d
δ
φ
x
d
x
]
+
Q
x
[
δ
φ
x
+
d
(
δ
w
0
)
d
x
]
}
d
x
{\displaystyle {\begin{aligned}\delta W_{\text{int}}=\int _{x_{a}}^{x_{b}}&\left\{N_{xx}\left[{\cfrac {d(\delta u_{0})}{dx}}+{\cfrac {dw_{0}}{dx}}{\cfrac {d(\delta w_{0})}{dx}}\right]\right.+\\&M_{xx}\left[{\cfrac {d\delta \varphi _{x}}{dx}}\right]+\left.Q_{x}\left[\delta \varphi _{x}+{\cfrac {d(\delta w_{0})}{dx}}\right]\right\}~dx\end{aligned}}}
Get rid of derivatives of the variations.
∫
x
a
x
b
N
x
x
[
d
(
δ
u
0
)
d
x
+
d
w
0
d
x
d
(
δ
w
0
)
d
x
]
d
x
=
[
N
x
x
δ
u
0
]
x
a
x
b
−
∫
x
a
x
b
d
N
x
x
d
x
δ
u
0
d
x
+
[
N
x
x
d
w
0
d
x
δ
w
0
]
x
a
x
b
−
∫
x
a
x
b
d
d
x
(
N
x
x
d
w
0
d
x
)
δ
w
0
d
x
∫
x
a
x
b
M
x
x
[
d
(
δ
φ
x
)
d
x
]
d
x
=
[
M
x
x
δ
φ
x
]
x
a
x
b
−
∫
x
a
x
b
d
M
x
x
d
x
δ
φ
x
d
x
∫
x
a
x
b
Q
x
[
δ
φ
x
+
d
(
δ
w
0
)
d
x
]
d
x
=
∫
x
a
x
b
Q
x
δ
φ
x
d
x
+
[
Q
x
δ
w
0
]
x
a
x
b
−
∫
x
a
x
b
d
Q
x
d
x
δ
w
0
d
x
{\displaystyle {\begin{aligned}\int _{x_{a}}^{x_{b}}N_{xx}&\left[{\cfrac {d(\delta u_{0})}{dx}}+{\cfrac {dw_{0}}{dx}}{\cfrac {d(\delta w_{0})}{dx}}\right]~dx=\left[N_{xx}\delta u_{0}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}{\cfrac {dN_{xx}}{dx}}\delta u_{0}~dx+\\&\qquad \qquad \qquad \left[N_{xx}{\cfrac {dw_{0}}{dx}}\delta w_{0}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}{\cfrac {d}{dx}}\left(N_{xx}{\cfrac {dw_{0}}{dx}}\right)\delta w_{0}~dx\\\\\int _{x_{a}}^{x_{b}}M_{xx}&\left[{\cfrac {d(\delta \varphi _{x})}{dx}}\right]~dx=\left[M_{xx}\delta \varphi _{x}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}{\cfrac {dM_{xx}}{dx}}\delta \varphi _{x}~dx\\\\\int _{x_{a}}^{x_{b}}Q_{x}&\left[\delta \varphi _{x}+{\cfrac {d(\delta w_{0})}{dx}}\right]~dx=\int _{x_{a}}^{x_{b}}Q_{x}\delta \varphi _{x}~dx+\left[Q_{x}\delta w_{0}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}{\cfrac {dQ_{x}}{dx}}\delta w_{0}~dx\end{aligned}}}
[
N
x
x
δ
u
0
]
x
a
x
b
−
∫
x
a
x
b
d
N
x
x
d
x
δ
u
0
d
x
+
[
N
x
x
d
w
0
d
x
δ
w
0
+
Q
x
δ
w
0
]
x
a
x
b
−
∫
x
a
x
b
[
d
d
x
(
N
x
x
d
w
0
d
x
)
+
d
Q
x
d
x
]
δ
w
0
d
x
+
[
M
x
x
δ
φ
x
]
x
a
x
b
−
∫
x
a
x
b
(
d
M
x
x
d
x
−
Q
x
)
δ
φ
x
d
x
=
∫
x
a
x
b
q
δ
w
0
d
x
+
∫
x
a
x
b
f
δ
u
0
d
x
{\displaystyle {\begin{aligned}&\left[N_{xx}{\delta u_{0}}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}{\cfrac {dN_{xx}}{dx}}{\delta u_{0}}~dx+\\&\left[N_{xx}{\cfrac {dw_{0}}{dx}}\delta w_{0}+Q_{x}\delta w_{0}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}\left[{\cfrac {d}{dx}}\left(N_{xx}{\cfrac {dw_{0}}{dx}}\right)+{\cfrac {dQ_{x}}{dx}}\right]\delta w_{0}~dx+\\&\left[M_{xx}\delta \varphi _{x}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}\left({\cfrac {dM_{xx}}{dx}}-Q_{x}\right)\delta \varphi _{x}~dx\\&=\int _{x_{a}}^{x_{b}}q~\delta w_{0}~dx+\int _{x_{a}}^{x_{b}}f{\delta u_{0}}~dx\end{aligned}}}
∫
x
a
x
b
(
d
N
x
x
d
x
+
f
)
δ
u
0
d
x
=
[
N
x
x
δ
u
0
]
x
a
x
b
∫
x
a
x
b
[
d
d
x
(
N
x
x
d
w
0
d
x
)
+
d
Q
x
d
x
+
q
]
δ
w
0
d
x
=
[
N
x
x
d
w
0
d
x
δ
w
0
+
Q
x
δ
w
0
]
x
a
x
b
∫
x
a
x
b
(
d
M
x
x
d
x
−
Q
x
)
δ
φ
x
d
x
=
[
M
x
x
δ
φ
x
]
x
a
x
b
{\displaystyle {\begin{aligned}\int _{x_{a}}^{x_{b}}\left({\cfrac {dN_{xx}}{dx}}+f\right){\delta u_{0}}~dx&=\left[N_{xx}{\delta u_{0}}\right]_{x_{a}}^{x_{b}}\\\int _{x_{a}}^{x_{b}}\left[{\cfrac {d}{dx}}\left(N_{xx}{\cfrac {dw_{0}}{dx}}\right)+{\cfrac {dQ_{x}}{dx}}+q\right]~\delta w_{0}~dx&=\left[N_{xx}{\cfrac {dw_{0}}{dx}}~\delta w_{0}+Q_{x}~\delta w_{0}\right]_{x_{a}}^{x_{b}}\\\int _{x_{a}}^{x_{b}}\left({\cfrac {dM_{xx}}{dx}}-Q_{x}\right)~\delta \varphi _{x}~dx&=\left[M_{xx}~\delta \varphi _{x}\right]_{x_{a}}^{x_{b}}\end{aligned}}}
d
N
x
x
d
x
+
f
=
0
d
d
x
(
N
x
x
d
w
0
d
x
)
+
d
Q
x
d
x
+
q
=
0
d
M
x
x
d
x
−
Q
x
=
0
{\displaystyle {\begin{aligned}{\cfrac {dN_{xx}}{dx}}+f&=0\\{\cfrac {d}{dx}}\left(N_{xx}{\cfrac {dw_{0}}{dx}}\right)+{\cfrac {dQ_{x}}{dx}}+q&=0\\{\cfrac {dM_{xx}}{dx}}-Q_{x}&=0\end{aligned}}}
σ
x
x
=
E
ε
x
x
;
σ
x
z
=
G
γ
x
z
{\displaystyle \sigma _{xx}=E\varepsilon _{xx}~;\qquad \sigma _{xz}=G\gamma _{xz}}
Then,
N
x
x
=
A
x
x
ε
x
x
0
+
B
x
x
ε
x
x
1
M
x
x
=
B
x
x
ε
x
x
0
+
D
x
x
ε
x
x
1
Q
x
=
S
x
x
γ
x
z
0
{\displaystyle {\begin{aligned}N_{xx}&=A_{xx}~\varepsilon _{xx}^{0}+B_{xx}~\varepsilon _{xx}^{1}\\\\M_{xx}&=B_{xx}~\varepsilon _{xx}^{0}+D_{xx}~\varepsilon _{xx}^{1}\\\\Q_{x}&=S_{xx}~\gamma _{xz}^{0}\end{aligned}}}
where
S
x
x
=
K
s
∫
A
G
d
A
←
shear stiffness
{\displaystyle S_{xx}=K_{s}\int _{A}G~dA\qquad \leftarrow \qquad {\text{shear stiffness}}}
d
d
x
{
A
x
x
[
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
]
}
+
f
=
0
d
d
x
{
S
x
x
(
d
w
0
d
x
+
φ
x
)
+
A
x
x
d
w
0
d
x
[
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
]
}
+
q
=
0
d
d
x
(
D
x
x
d
φ
x
d
x
)
+
S
x
x
(
d
w
0
d
x
+
φ
x
)
=
0
{\displaystyle {\begin{aligned}{\cfrac {d}{dx}}\left\{A_{xx}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]\right\}+f&=0\\{\cfrac {d}{dx}}\left\{S_{xx}\left({\cfrac {dw_{0}}{dx}}+\varphi _{x}\right)+A_{xx}{\cfrac {dw_{0}}{dx}}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]\right\}+q&=0\\{\cfrac {d}{dx}}\left(D_{xx}{\cfrac {d\varphi _{x}}{dx}}\right)+S_{xx}\left({\cfrac {dw_{0}}{dx}}+\varphi _{x}\right)&=0\end{aligned}}}
∫
x
a
x
b
A
x
x
d
(
δ
u
0
)
d
x
[
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
]
d
x
=
∫
x
a
x
b
f
δ
u
0
d
x
+
[
N
x
x
δ
u
0
]
x
a
x
b
∫
x
a
x
b
A
x
x
d
(
δ
w
0
)
d
x
d
w
0
d
x
[
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
]
d
x
+
∫
x
a
x
b
S
x
x
d
(
δ
w
0
)
d
x
(
d
w
0
d
x
+
φ
x
)
d
x
=
∫
x
a
x
b
q
δ
u
0
d
x
+
[
(
N
x
x
d
w
0
d
x
+
Q
x
)
δ
w
0
]
x
a
x
b
−
∫
x
a
x
b
S
x
x
δ
φ
x
(
d
w
0
d
x
+
φ
x
)
d
x
+
∫
x
a
x
b
D
x
x
d
(
δ
φ
x
)
d
x
d
φ
x
d
x
d
x
=
[
M
x
x
δ
φ
x
]
x
a
x
b
{\displaystyle {\begin{aligned}\int _{x_{a}}^{x_{b}}A_{xx}{\cfrac {d(\delta u_{0})}{dx}}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]~dx&=\int _{x_{a}}^{x_{b}}f\delta u_{0}~dx+\left[N_{xx}\delta u_{0}\right]_{x_{a}}^{x_{b}}\\\\\int _{x_{a}}^{x_{b}}A_{xx}{\cfrac {d(\delta w_{0})}{dx}}{\cfrac {dw_{0}}{dx}}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]~dx&+\int _{x_{a}}^{x_{b}}S_{xx}{\cfrac {d(\delta w_{0})}{dx}}\left({\cfrac {dw_{0}}{dx}}+\varphi _{x}\right)~dx=\\&\int _{x_{a}}^{x_{b}}q\delta u_{0}~dx+\left[\left(N_{xx}{\cfrac {dw_{0}}{dx}}+Q_{x}\right)\delta w_{0}\right]_{x_{a}}^{x_{b}}\\\\-\int _{x_{a}}^{x_{b}}S_{xx}\delta \varphi _{x}\left({\cfrac {dw_{0}}{dx}}+\varphi _{x}\right)~dx&+\int _{x_{a}}^{x_{b}}D_{xx}{\cfrac {d(\delta \varphi _{x})}{dx}}{\cfrac {d\varphi _{x}}{dx}}~dx=\left[M_{xx}\delta \varphi _{x}\right]_{x_{a}}^{x_{b}}\end{aligned}}}
u
0
(
x
)
=
∑
j
=
1
m
u
j
ψ
j
(
1
)
;
δ
u
0
=
ψ
i
(
1
)
w
0
(
x
)
=
∑
j
=
1
n
w
j
ψ
j
(
2
)
;
δ
w
0
=
ψ
i
(
2
)
φ
x
(
x
)
=
∑
j
=
1
p
s
j
ψ
j
(
3
)
;
δ
φ
x
=
ψ
i
(
3
)
{\displaystyle {\begin{aligned}u_{0}(x)&=\sum _{j=1}^{m}u_{j}~\psi _{j}^{(1)}~;\qquad \qquad \delta u_{0}=\psi _{i}^{(1)}\\\\w_{0}(x)&=\sum _{j=1}^{n}w_{j}~\psi _{j}^{(2)}~;\qquad \qquad \delta w_{0}=\psi _{i}^{(2)}\\\\\varphi _{x}(x)&=\sum _{j=1}^{p}s_{j}~\psi _{j}^{(3)}~;\qquad \qquad \delta \varphi _{x}=\psi _{i}^{(3)}\end{aligned}}}
[
K
11
⋮
K
12
⋮
K
13
⋮
⋮
K
21
⋮
K
22
⋮
K
23
⋮
⋮
K
31
⋮
K
32
⋮
K
33
]
[
u
w
s
]
=
[
F
1
F
2
F
3
]
{\displaystyle {\begin{bmatrix}\mathbf {K} ^{11}&\vdots &\mathbf {K} ^{12}&\vdots &\mathbf {K} ^{13}\\&\vdots &&\vdots &\\\mathbf {K} ^{21}&\vdots &\mathbf {K} ^{22}&\vdots &\mathbf {K} ^{23}\\&\vdots &&\vdots &\\\mathbf {K} ^{31}&\vdots &\mathbf {K} ^{32}&\vdots &\mathbf {K} ^{33}\\\end{bmatrix}}{\begin{bmatrix}\mathbf {u} \\\\\mathbf {w} \\\\\mathbf {s} \end{bmatrix}}={\begin{bmatrix}\mathbf {F} ^{1}\\\\\mathbf {F} ^{2}\\\\\mathbf {F} ^{3}\end{bmatrix}}}
ψ
(
1
)
{\displaystyle \psi ^{(1)}}
m
=
2
{\displaystyle m=2}
ψ
(
2
)
{\displaystyle \psi ^{(2)}}
n
=
2
{\displaystyle n=2}
ψ
(
3
)
{\displaystyle \psi ^{(3)}}
p
=
2
{\displaystyle p=2}
Nearly singular stiffness matrix (
6
×
6
{\displaystyle 6\times 6}
ψ
(
1
)
{\displaystyle \psi ^{(1)}}
m
=
2
{\displaystyle m=2}
ψ
(
2
)
{\displaystyle \psi ^{(2)}}
n
=
3
{\displaystyle n=3}
ψ
(
3
)
{\displaystyle \psi ^{(3)}}
p
=
2
{\displaystyle p=2}
The stiffness matrix is (
7
×
7
{\displaystyle 7\times 7}
6
×
6
{\displaystyle 6\times 6}
ψ
(
1
)
{\displaystyle \psi ^{(1)}}
m
=
2
{\displaystyle m=2}
ψ
(
2
)
{\displaystyle \psi ^{(2)}}
n
=
4
{\displaystyle n=4}
ψ
(
3
)
{\displaystyle \psi ^{(3)}}
p
=
3
{\displaystyle p=3}
The stiffness matrix is (
9
×
9
{\displaystyle 9\times 9}
6
×
6
{\displaystyle 6\times 6}
Linear
u
0
{\displaystyle u_{0}}
w
0
{\displaystyle w_{0}}
φ
x
{\displaystyle \varphi _{x}}
d
w
0
d
x
=
constant
.
{\displaystyle {\cfrac {dw_{0}}{dx}}=~~{\text{constant}}.}
But, for thin beams,
d
w
0
d
x
=
slope
=
−
φ
x
←
(
linear!
)
{\displaystyle {\cfrac {dw_{0}}{dx}}=~~{\text{slope}}~~=-\varphi _{x}~~\leftarrow ~~({\text{linear!}})}
If constant
φ
x
{\displaystyle \varphi _{x}}
d
φ
x
d
x
=
0
{\displaystyle {\cfrac {d\varphi _{x}}{dx}}=0}
Also
Q
x
=
S
x
x
φ
x
≠
0
⟹
{\displaystyle Q_{x}=S_{xx}\varphi _{x}\neq 0\implies }
M
x
x
=
D
x
x
d
φ
x
d
x
=
0
⟹
{\displaystyle M_{xx}=D_{xx}{\cfrac {d\varphi _{x}}{dx}}=0\implies }
Result : Zero displacements and rotations
⟹
{\displaystyle \implies }
Recall
d
d
x
{
A
x
x
[
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
]
}
+
f
=
0
{\displaystyle {\cfrac {d}{dx}}\left\{A_{xx}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]\right\}+f=0}
or,
d
d
x
{
A
x
x
ε
x
x
0
}
+
f
=
0
{\displaystyle {\cfrac {d}{dx}}\left\{A_{xx}\varepsilon _{xx}^{0}\right\}+f=0}
If
f
=
0
{\displaystyle f=0}
A
x
x
=
{\displaystyle A_{xx}=}
A
x
x
d
d
x
(
ε
x
x
0
)
=
0
⟹
ε
x
x
0
=
constant
.
{\displaystyle A_{xx}{\cfrac {d}{dx}}(\varepsilon _{xx}^{0})=0\qquad \implies \qquad \varepsilon _{xx}^{0}=~{\text{constant}}.}
If there is only bending but no stretching,
ε
x
x
0
=
0
=
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
{\displaystyle \varepsilon _{xx}^{0}=0={\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}}
Hence,
d
u
0
d
x
≈
(
d
w
0
d
x
)
2
{\displaystyle {\cfrac {du_{0}}{dx}}\approx \left({\cfrac {dw_{0}}{dx}}\right)^{2}}
Also recall:
d
d
x
{
S
x
x
(
d
w
0
d
x
+
φ
x
)
+
A
x
x
d
w
0
d
x
[
d
u
0
d
x
+
1
2
(
d
w
0
d
x
)
2
]
}
+
q
=
0
{\displaystyle {\cfrac {d}{dx}}\left\{S_{xx}\left({\cfrac {dw_{0}}{dx}}+\varphi _{x}\right)+A_{xx}{\cfrac {dw_{0}}{dx}}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]\right\}+q=0}
or,
d
d
x
{
S
x
x
γ
x
z
0
+
A
x
x
d
w
0
d
x
ε
x
x
0
}
+
q
=
0
{\displaystyle {\cfrac {d}{dx}}\left\{S_{xx}\gamma _{xz}^{0}+A_{xx}{\cfrac {dw_{0}}{dx}}\varepsilon _{xx}^{0}\right\}+q=0}
If
q
=
0
{\displaystyle q=0}
S
x
x
=
{\displaystyle S_{xx}=}
S
x
x
d
d
x
(
γ
x
z
0
)
=
0
⟹
γ
x
z
=
constant
=
d
w
0
d
x
+
φ
x
{\displaystyle S_{xx}{\cfrac {d}{dx}}(\gamma _{xz}^{0})=0\qquad \implies \qquad {\gamma _{xz}=~{\text{constant}}}~={\cfrac {dw_{0}}{dx}}+\varphi _{x}}
Hence,
φ
x
≈
d
w
0
d
x
{\displaystyle \varphi _{x}\approx {\cfrac {dw_{0}}{dx}}}
Shape functions need to satisfy:
d
u
0
d
x
≈
(
d
w
0
d
x
)
2
;
and
φ
x
≈
d
w
0
d
x
{\displaystyle {\cfrac {du_{0}}{dx}}\approx \left({\cfrac {dw_{0}}{dx}}\right)^{2}~;\qquad {\text{and}}\qquad \varphi _{x}\approx {\cfrac {dw_{0}}{dx}}}
Linear
u
0
{\displaystyle u_{0}}
w
0
{\displaystyle w_{0}}
φ
x
{\displaystyle \varphi _{x}}
First condition
⟹
{\displaystyle \implies }
=
{\displaystyle =}
Second condition
⟹
{\displaystyle \implies }
=
{\displaystyle =}
Linear
u
0
{\displaystyle u_{0}}
w
0
{\displaystyle w_{0}}
φ
x
{\displaystyle \varphi _{x}}
First condition
⟹
{\displaystyle \implies }
=
{\displaystyle =}
Second condition
⟹
{\displaystyle \implies }
=
{\displaystyle =}
Quadratic
u
0
{\displaystyle u_{0}}
w
0
{\displaystyle w_{0}}
φ
x
{\displaystyle \varphi _{x}}
First condition
⟹
{\displaystyle \implies }
=
{\displaystyle =}
Second condition
⟹
{\displaystyle \implies }
=
{\displaystyle =}
Cubic
u
0
{\displaystyle u_{0}}
w
0
{\displaystyle w_{0}}
φ
x
{\displaystyle \varphi _{x}}
First condition
⟹
{\displaystyle \implies }
=
{\displaystyle =}
Second condition
⟹
{\displaystyle \implies }
=
{\displaystyle =}
Linear
u
0
{\displaystyle u_{0}}
w
0
{\displaystyle w_{0}}
φ
x
{\displaystyle \varphi _{x}}
Equal interpolation for both
w
0
{\displaystyle w_{0}}
ϕ
x
{\displaystyle \phi _{x}}
Reduced integration for terms containing
ϕ
x
{\displaystyle \phi _{x}}
Cubic
u
0
{\displaystyle u_{0}}
w
0
{\displaystyle w_{0}}
φ
x
{\displaystyle \varphi _{x}}
Stiffness matrix is
9
×
9
{\displaystyle 9\times 9}
Hard to implement.
![{\displaystyle {\begin{aligned}\int _{x_{a}}^{x_{b}}N_{xx}\delta \varepsilon _{xx}^{0}~dx&=\int _{x_{a}}^{x_{b}}N_{xx}\left[{\cfrac {d(\delta u_{0})}{dx}}+{\cfrac {dw_{0}}{dx}}{\cfrac {d(\delta w_{0})}{dx}}\right]~dx\\\int _{x_{a}}^{x_{b}}M_{xx}\delta \varepsilon _{xx}^{1}~dx&=\int _{x_{a}}^{x_{b}}M_{xx}\left[{\cfrac {d\delta \varphi _{x}}{dx}}\right]~dx\\\int _{x_{a}}^{x_{b}}Q_{x}\delta \gamma _{xz}^{0}~dx&=\int _{x_{a}}^{x_{b}}Q_{x}\left[\delta \varphi _{x}+{\cfrac {d(\delta w_{0})}{dx}}\right]~dx\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82c8fd2ed2141e00bbc789e9799ce316e30c5763)
![{\displaystyle {\begin{aligned}\delta W_{\text{int}}=\int _{x_{a}}^{x_{b}}&\left\{N_{xx}\left[{\cfrac {d(\delta u_{0})}{dx}}+{\cfrac {dw_{0}}{dx}}{\cfrac {d(\delta w_{0})}{dx}}\right]\right.+\\&M_{xx}\left[{\cfrac {d\delta \varphi _{x}}{dx}}\right]+\left.Q_{x}\left[\delta \varphi _{x}+{\cfrac {d(\delta w_{0})}{dx}}\right]\right\}~dx\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45401fa97862eef9a90bbf9ee6a8746ef940fc19)
![{\displaystyle {\begin{aligned}\int _{x_{a}}^{x_{b}}N_{xx}&\left[{\cfrac {d(\delta u_{0})}{dx}}+{\cfrac {dw_{0}}{dx}}{\cfrac {d(\delta w_{0})}{dx}}\right]~dx=\left[N_{xx}\delta u_{0}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}{\cfrac {dN_{xx}}{dx}}\delta u_{0}~dx+\\&\qquad \qquad \qquad \left[N_{xx}{\cfrac {dw_{0}}{dx}}\delta w_{0}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}{\cfrac {d}{dx}}\left(N_{xx}{\cfrac {dw_{0}}{dx}}\right)\delta w_{0}~dx\\\\\int _{x_{a}}^{x_{b}}M_{xx}&\left[{\cfrac {d(\delta \varphi _{x})}{dx}}\right]~dx=\left[M_{xx}\delta \varphi _{x}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}{\cfrac {dM_{xx}}{dx}}\delta \varphi _{x}~dx\\\\\int _{x_{a}}^{x_{b}}Q_{x}&\left[\delta \varphi _{x}+{\cfrac {d(\delta w_{0})}{dx}}\right]~dx=\int _{x_{a}}^{x_{b}}Q_{x}\delta \varphi _{x}~dx+\left[Q_{x}\delta w_{0}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}{\cfrac {dQ_{x}}{dx}}\delta w_{0}~dx\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb0e4d4a972d9681f1bed27c722cf7cb9173bba)
![{\displaystyle {\begin{aligned}&\left[N_{xx}{\delta u_{0}}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}{\cfrac {dN_{xx}}{dx}}{\delta u_{0}}~dx+\\&\left[N_{xx}{\cfrac {dw_{0}}{dx}}\delta w_{0}+Q_{x}\delta w_{0}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}\left[{\cfrac {d}{dx}}\left(N_{xx}{\cfrac {dw_{0}}{dx}}\right)+{\cfrac {dQ_{x}}{dx}}\right]\delta w_{0}~dx+\\&\left[M_{xx}\delta \varphi _{x}\right]_{x_{a}}^{x_{b}}-\int _{x_{a}}^{x_{b}}\left({\cfrac {dM_{xx}}{dx}}-Q_{x}\right)\delta \varphi _{x}~dx\\&=\int _{x_{a}}^{x_{b}}q~\delta w_{0}~dx+\int _{x_{a}}^{x_{b}}f{\delta u_{0}}~dx\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58586c6e2caf2577dd9448f123bdbe10c9546069)
![{\displaystyle {\begin{aligned}\int _{x_{a}}^{x_{b}}\left({\cfrac {dN_{xx}}{dx}}+f\right){\delta u_{0}}~dx&=\left[N_{xx}{\delta u_{0}}\right]_{x_{a}}^{x_{b}}\\\int _{x_{a}}^{x_{b}}\left[{\cfrac {d}{dx}}\left(N_{xx}{\cfrac {dw_{0}}{dx}}\right)+{\cfrac {dQ_{x}}{dx}}+q\right]~\delta w_{0}~dx&=\left[N_{xx}{\cfrac {dw_{0}}{dx}}~\delta w_{0}+Q_{x}~\delta w_{0}\right]_{x_{a}}^{x_{b}}\\\int _{x_{a}}^{x_{b}}\left({\cfrac {dM_{xx}}{dx}}-Q_{x}\right)~\delta \varphi _{x}~dx&=\left[M_{xx}~\delta \varphi _{x}\right]_{x_{a}}^{x_{b}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e03f89946175e9e1ca3b2d69e13c56eb4d31708b)
![{\displaystyle {\begin{aligned}{\cfrac {d}{dx}}\left\{A_{xx}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]\right\}+f&=0\\{\cfrac {d}{dx}}\left\{S_{xx}\left({\cfrac {dw_{0}}{dx}}+\varphi _{x}\right)+A_{xx}{\cfrac {dw_{0}}{dx}}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]\right\}+q&=0\\{\cfrac {d}{dx}}\left(D_{xx}{\cfrac {d\varphi _{x}}{dx}}\right)+S_{xx}\left({\cfrac {dw_{0}}{dx}}+\varphi _{x}\right)&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/089e0a9f830af368fa73fb7f0f50d7bd4969310e)
![{\displaystyle {\begin{aligned}\int _{x_{a}}^{x_{b}}A_{xx}{\cfrac {d(\delta u_{0})}{dx}}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]~dx&=\int _{x_{a}}^{x_{b}}f\delta u_{0}~dx+\left[N_{xx}\delta u_{0}\right]_{x_{a}}^{x_{b}}\\\\\int _{x_{a}}^{x_{b}}A_{xx}{\cfrac {d(\delta w_{0})}{dx}}{\cfrac {dw_{0}}{dx}}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]~dx&+\int _{x_{a}}^{x_{b}}S_{xx}{\cfrac {d(\delta w_{0})}{dx}}\left({\cfrac {dw_{0}}{dx}}+\varphi _{x}\right)~dx=\\&\int _{x_{a}}^{x_{b}}q\delta u_{0}~dx+\left[\left(N_{xx}{\cfrac {dw_{0}}{dx}}+Q_{x}\right)\delta w_{0}\right]_{x_{a}}^{x_{b}}\\\\-\int _{x_{a}}^{x_{b}}S_{xx}\delta \varphi _{x}\left({\cfrac {dw_{0}}{dx}}+\varphi _{x}\right)~dx&+\int _{x_{a}}^{x_{b}}D_{xx}{\cfrac {d(\delta \varphi _{x})}{dx}}{\cfrac {d\varphi _{x}}{dx}}~dx=\left[M_{xx}\delta \varphi _{x}\right]_{x_{a}}^{x_{b}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17c86f1575b49923c7998742922073904c5fb1f2)
![{\displaystyle {\cfrac {d}{dx}}\left\{A_{xx}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]\right\}+f=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d279b51f966e02a41f277bdb11d98ed73c6c79)
![{\displaystyle {\cfrac {d}{dx}}\left\{S_{xx}\left({\cfrac {dw_{0}}{dx}}+\varphi _{x}\right)+A_{xx}{\cfrac {dw_{0}}{dx}}\left[{\cfrac {du_{0}}{dx}}+{\frac {1}{2}}\left({\cfrac {dw_{0}}{dx}}\right)^{2}\right]\right\}+q=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d5d1c2c7bc5e313563a377c68115799e5dfccc)
