Question

a square prism of wood 50*50 mm in cross section and 300 mm long is subjected to a tensile stress of 4 t/cm2 along it's longitudinal axis and lateral compressive to a tensile stress of 2 t/cm2 along one pair of sides and a lateral tensile stress of 1-0 t/cm2 acting along the other pair of sides.If the value of E of the material is 1.5*10 to the power 5 kg/cm2,calculate it's changed dimensions.poisson's ratio for wood=.4

Answer 1

A square prism of wood $50*50 \, \text{mm}$ in cross section and $300 \, \text{mm}$ long is subjected to a tensile stress of 4 t/cm² along its longitudinal axis and lateral compressive to a tensile stress of 2 t/cm² along one pair of sides and a lateral tensile stress of 1-0 t/cm² acting along the other pair of sides. If the value of $\varepsilon$ of the material is $1.5*10$ to the power $5 \, \text{kg/cm}^2$, calculate its changed dimensions. Poisson's ratio for wood = .4

Let

$$\left\{ \begin{array}{c} l _ {x} = l _ {y} = 5 0 m m, l _ {z} = 3 0 0 m m \\ \sigma_ {x} = 2 0 0 0 \frac {k g}{c m ^ {2}}, \sigma_ {y} = 1 0 0 0 \frac {k g}{c m ^ {2}}, \sigma_ {z} = 4 0 0 0 \frac {k g}{c m ^ {2}} \\ E = 1. 5 \cdot 1 0 ^ {5} \frac {k g}{c m ^ {2}} \end{array} \right.$$ $$\sigma = E \frac {\Delta l}{l} \rightarrow \Delta l = \frac {\sigma l}{E}$$ $$\left\{ \begin{array}{r l} & {\rightarrow \Delta l _ {x} = \frac {\sigma_ {x} l}{E} = 2 0 0 0 \cdot \frac {0 . 5}{1 . 5 \cdot 1 0 ^ {5}} = 6 6 6 \cdot 1 0 ^ {- 5} c m} \\ & {\rightarrow \Delta l _ {y} = \frac {\sigma_ {y} l}{E} = 1 0 0 0 \cdot \frac {0 . 5}{1 . 5 \cdot 1 0 ^ {5}} = 3 3 3 \cdot 1 0 ^ {- 5} c m} \\ & {\rightarrow \Delta l _ {z} = \frac {\sigma_ {z} l}{E} = 4 0 0 0 \cdot \frac {3 0}{1 . 5 \cdot 1 0 ^ {5}} = 6 6 6 \cdot 1 0 ^ {- 5} = 0. 8 c m} \end{array} \right.$$ $$\left\{ \begin{array}{l} \rightarrow L _ {x} = l _ {x} + \Delta l _ {x} = 0. 5 + 6 6 6 \cdot 1 0 ^ {- 5} c m \\ \rightarrow L _ {y} = l _ {y} + \Delta l _ {y} = 0. 5 + 3 3 3 \cdot 1 0 ^ {- 5} c m \\ \rightarrow L _ {z} = l _ {z} + \Delta l _ {z} = 3. 8 c m \end{array} \right.$$