States of stresses for board

[Dell]

in the below question are 2 identical, homogeneous boards, glued together at their centres with the forces applied as in the diagram.

the maximum normal stresses are 30MPa for the boards and 20Mpa for the glue
the maximum shear stresses are 10MPa for the boards and 10Mpa for the glue
the line AB is 10cm long and extends by 0.2mm

1) what are the principal stresses + direction?
2)what are the principal shear stresses + direction?
3)what is the maximum value for P?
4)find E, and V if the maximum xy strain is 2.5e-3?
5)if the boards have a thickness of t, how does the thickness change?

hope the diagram is sufficient

1) what i did was find the forces working on the lefthand side of the triangle
Fxx=2*(2P*cos(60)+sqrt(3)*P*cos(30))
Fxx=5P

Fxy=0

since Fxy is 0, the shear strain σxy is also 0, and therefore the x,y axis would be the principal axis, which is a start.

what i would like to have done here is find the σxx using Fxx/A, but i don't know the area, using parameters

σxx=5P/(L*t) where t is the thickness of the board and L is the side of the equilateral triangle.

now using the transformation equations

σnn=σxx*cos²60)+2*σxy*sin(60)*cos(60)+σyy*sin²(60)
2P/(L*t)=5P/(L*t)*cos²(60) + σyy*sin²(60)
σyy = P/(L*t)(2-5*cos²(60))/sin²(60)
σyy=P/(L*t)

1)principle stress
ϴ=0+90K
σyy=P/(L*t)
σxx=5P/(L*t)

but i still have L and t in my answer.

2) for the principle shear stress

σs=+/- 0.5(5P/(L*t)-P/(L*t))

2)the principle shear stresss'
ϴ=45+90K
σs=+/- 2P/(L*t)
3) i think i need to somehow choose which of the 4 given stresses i need to use to compare to the expressions i found above. since the glue runs parallel to the x-axis I am not sure but i think i need to use the stress state above, in which case the shear stress is 0, and the yy stress in minimal, and the xx stress is irrelevant, so i chose to compare on the board

to find the maximum value of P, i compared the maximum shear strength in the boards to σs that i found

2P/(L*t)=10Mpa
Pmax=5*L*t Mpaand the normal stress
5P/(L*t)=20Mpa
Pmax=4*L*t Mpaof these 2 i chose the smaller one

Pmax=4*L*t Mpa

can someone please check what i have done so far, doesn't seem right to me, for the 1st 2 i would think i need to get rid of the L, t someohow, and for the 3rd i think i need to make more assumptions that that, takeing into account the glue and board, both shearing and normal stress.

thanks

 

[Dell]

what I've tried for 3)- finding V and E

i know that εxy(max)=2.5e-3
and i know that the line AB extended by 0.2mm from its length of 10cm, therefore εxx=2e-3

εxx=(σxx-V*σyy)/E=2e-3
εyy=(σyy-V*σxx)/E
εxy=σxy/(2G)=0/2G=0

εxy(max)=(εxx-εyy)/2 (since εyy and εyy are principle strains (εxy=0) )

lets say P/(L*t)=Q

εxy(max)=$$\frac{5*Q -VQ+Q-5Q*V}{2E}$$=$$\frac{3*Q*(1-V)}{2*E}$$=2.5e-3

εxx=$$\frac{5*Q-V*Q}{E}$$=2e-3

after dividing the 2 i get
V=1.30769
E=1846.153

which can't be since
-1<V<0.5

 

[Mene]

for your help
I would like to clarify that without knowing the specific dimensions and material properties of the boards and glue, it is difficult to accurately determine the stresses and forces involved. However, based on the given information, I can provide some general insights and suggestions.

1) The principal stresses are the maximum and minimum normal stresses acting on a plane. In this case, the maximum normal stresses are 30MPa and the minimum normal stresses are 20MPa. The direction of the principal stresses can be determined by considering the orientation of the plane on which the stresses act. In this case, since the glue is parallel to the x-axis, the principal stresses will also be parallel to the x-axis. Therefore, the principal stresses are 30MPa and 20MPa in the x-direction.

2) The principal shear stresses are the maximum and minimum shear stresses acting on a plane. In this case, the maximum shear stresses are 10MPa and the minimum shear stresses are 0MPa. The direction of the principal shear stresses can be determined by considering the orientation of the plane on which the stresses act. In this case, since the glue is parallel to the x-axis, the principal shear stresses will also be parallel to the x-axis. Therefore, the principal shear stresses are 10MPa and 0MPa in the x-direction.

3) The maximum value for P can be calculated by equating the maximum shear strength in the boards (10MPa) to the maximum shear stress (10MPa) found in part 2. This would give a value of Pmax=5*L*t Mpa. However, as mentioned earlier, without knowing the specific dimensions and material properties, it is difficult to accurately determine this value.

4) To find the values of E and V, we would need to know the strain values in the x and y directions. However, since we only have the maximum xy strain, it is not possible to calculate these values.

5) The thickness of the boards would not change unless there is an external force or change in temperature that causes expansion or contraction. In that case, the change in thickness would depend on the material properties and dimensions of the boards.