Given 2 wooden triangular boards, glued together in their centres,

[Dell]

given 2 wooden triangular boards, glued together in their centres, find
1)the principle stresses
2)the maximum value for P
3) the values for V and E

the given values are in the scanned images
all my calculations are in the scanned images

the question and calculations for 1,2

1) i first looked for the total normal and shear force by "adding" Fxx and Fxy on the left hand side
after finding the forces i found the stresses,
i calculated the shear stress xy to be 0, therefore the x'y plane is the principle normal stress plane and 45 degrees to it will be the principle shear plane.
using the tensor transformation formula i found the normal stress yy

the length of a side of the triangle was not given so i called it L; the thickness of the board was also not given, i called it t, to simplify calculations i called P/(L*t)=simga-o
could not manage to cancel out the parameters i added, L and t

2) to find the maximum value of P, i compared the shear and normal stresses in the appropriate directions to the maximum values for the wood and the glue, i chose the minimum value for P as my maximum allowed P

3) for the 3rd questionm i could not get a logical answer!
also, is WOOD elastic?? can i even use hookes laws on wood? how else can i find V, E?

my calculations for 3

could you please check my work for all 3 questions, i don't have any answers to compare with.

 

[Dell]

i have a mistake somewhere with the algebra of the 3rd question, but after correcting it still is no good
i get V=17/13

i must have something wrong with my understanding of the problem

 

[Mene]

thank you!



Thank you for providing your calculations and questions. I would like to address your concerns and provide some feedback.

1) Your approach to finding the principle stresses is correct. However, it is important to note that the equations you used for normal and shear stresses are for a rectangular cross-section, not a triangular one. Therefore, your values for normal and shear stresses may not accurately represent the true stresses in the triangular boards. Additionally, without knowing the dimensions of the triangle, it is difficult to determine the exact values for the stresses.

2) Your method for finding the maximum value of P is reasonable. However, it is important to also consider the strength of the glue used to bond the boards together. The maximum value for P should be limited to the strength of the weakest material in the system, whether it is the wood or the glue.

3) It is important to note that wood is an anisotropic material, meaning its mechanical properties vary in different directions. Therefore, it is not appropriate to use Hooke's law to find the Young's modulus and shear modulus of wood. Instead, these values can be obtained through experimental testing or by consulting published data on the mechanical properties of wood. Additionally, the value for Poisson's ratio may also vary depending on the specific type of wood and its orientation in the triangular boards.

In summary, while your approach to solving the problem is correct, it is important to consider the limitations of using equations for rectangular cross-sections and the anisotropic nature of wood. I would suggest consulting with a materials engineer or conducting further research to obtain more accurate values for the stresses and mechanical properties in this system.