User:Eas4200c.f08.blue.a/Lecture 37

Continuation of problem using Eulet cut principle Stringer 3: Refer to the image below



$$\sum{F_{x}}=0$$

$$\int_{A_{3}}^{}{\left[\sigma_{xx}(x+dx)-\sigma_{xx}(x) \right]dA_{3}}+\left[\tilde{-q_{23}}-\tilde{q_{43}}+\tilde{q_{31}} \right]dx$$

Taylor series: $$\frac{d\sigma_{xx}dx}{dx} + h.o.t.$$

$$\tilde{q_{31}}=\tilde{q_{23}}+\tilde{q_{43}}+\tilde{q}^{(3)}$$

$$q^{3}\Rightarrow$$ stringer 3 contribution $$-\int_{A_{3}}^{}{\frac{d\sigma_{xx}}{dx}}dA_{3}$$

Recall $$v_{y}=\frac{\partial Mz}{\partial x}$$, $$v_{z}=\frac{\partial My}{\partial x}$$

$$q^{(3)}=-\left(k_{y}v_{y}-k_{yz}v_{z}\right)Q_{z}^{(3)}-\left(k_{z}v_{z}-k_{yz}v_{y} \right)Q_{y}^{(3)}$$

$$Q_{z}^{(3)}=\int_{A_{3}}^{}{y}dA_{3}$$ and  $$Q_{y}^{(3)}=\int_{A_{3}}^{}{z}dA_{3}$$

String 2

$$\tilde{q_{23}}=\tilde{q_{12}}+\tilde{q_{24}}+\tilde{q}^{(2)}$$ but because of the way you cut the cells $$\tilde{q_{12}}$$ and $$\tilde{q_{24}}$$ equal zero.

$$q^{(2)}$$ is computed the same as $$q^{(3)}$$ except $$A_{3}$$ becomes $$A_{2}$$

$$Q_{z}^{(2)}=y_{2}A_{2}$$ and $$Q_{y}^{(2)}=z_{2}A_{2}$$

Stringer 4 is $$\tilde{q}_{43}=\tilde{q}_{24}-\tilde{q}_{41}+q^{(4)}$$ but $$\tilde{q}_{24}$$ and $$\tilde{q}_{41}$$ become zero because of the cut of the cells.$$q^{(4)}$$ can be computed as above for $$q^{(2)}$$

This marks the end of problem 2

Now returning to the superposition principle

$$q_{ij}=\tilde{q}_{ij}+q_{k}$$

$$q_{12}=\tilde{q}_{12}+q_{1}$$

$$q_{23}=\tilde{q}_{23}+q_{1}$$

All of the $$\tilde{q}$$ are known

Continuing this process for $$q_{24}$$, $$q_{43}$$, and $$q_{41}$$

$$q_{24}=\tilde{q}_{24}+q_{1}$$

$$q_{43}=\tilde{q}_{43}+q_{1}$$

$$q_{41}=\tilde{q}_{41}+q_{1}$$

To solve problems, such as problem 2, you'll have a certain number of unknown shear stresses and you'll get the same amount of equations to solve for the unknowns from,

1)The moment equation: Take the moment of $$v_{y}$$, $$v_{z}$$, and $$\left\{q_{12}, ..., q_{41} \right\}$$ about any convenient point (usually where lines of action of $$v_{y}$$, $$v_{z}$$ cross)

2)$$\theta _{1}=\theta _{2}$$

3)$$\theta _{2}=\theta _{3}$$