What Is the Maximum Load P That Glued Boards Can Withstand?

[Dell]

in the below question

2 identical boards are glued together at their centers, the strength of the glue is 10Mpa shearing and 20Mpa normal, the strength of the boards is 10Mpa shearing and 30Mpa normal


i found the stress on the system and found

$$\sigma$$xx= 5P
$$\sigma$$yy= P
$$\sigma$$xy= 0

this meaning that $$\theta$$=0 is the principal plane and that
$$\sigma$$s max=+-2P


now the question asks what the maximum value of P can be,
i clearly need to compare the values of the principal stresses to the strengths of the wood and glue,
my question is whether i need to compare the principal normal stress to the glues strength even though it is at 45 degrees to the plane where the glue lies??

meaning: the glue lies on the x-axis but the maximum stress is at 45 degrees so do i need to compare the normal strength at 90 degrees (P) or the maximum stress (5P)??
same goes for the shear stress?

at first i thought i just need to compare it to the stress perpindicular to the x axis, and the shear stress parallel to it, but someone told me that since the "stress cube" is tiny and the glue does have some volume, i need to compare the maximum stresses to the glues strength


therefore i would get P<=4

whereas had i not done that ii would have P<=5

 

[Valkarie]

so which answer is correct?? and why??The correct answer is P<=4, since the glue has some volume, the maximum stresses need to be compared to the strength of the glue. The shear stress of 10Mpa should be compared to the shear stress of 5P, and the normal stress of 20Mpa should be compared to the normal stress of P.

 

[Mene]

I would like to clarify a few things regarding the maximum stress in this scenario. First, it is important to note that the maximum stress in a material is the highest stress that the material can withstand before it fails. In this case, the maximum stress will depend on the type and strength of the material being used.

In order to determine the maximum value of P, we need to compare the maximum stress (5P) to the strength of the wood and glue. Since the strength of the glue is given as 10Mpa shearing and 20Mpa normal, we need to compare the maximum shear stress (5P) to 10Mpa and the maximum normal stress (2P) to 20Mpa. This means that P should not exceed 2Mpa in order for the glued boards to remain intact.

It is also important to note that stress is a vector quantity and has both magnitude and direction. In this case, the maximum normal stress (2P) is at 45 degrees to the x-axis and the maximum shear stress (5P) is parallel to the x-axis. Therefore, we need to compare the maximum normal stress at 45 degrees to the glue's normal strength, and the maximum shear stress parallel to the glue's shear strength.

In summary, the maximum value of P should be P<=2Mpa in order for the glued boards to remain intact, based on the strengths of the glue and wood provided in the question.