At a point in a plate subjected to plane stress loading along its surface, the cartesian stress components acting on the x-face of an element oriented along the (x,y) frame are σx=40MPa and τxy=70MPa,
The maximum allowable shear stress, to prevent yielding of the material, is τo=74MPa.
What is the permissible range (minimum and maximum values in MPa) for the normal stress on the y-face, σy, to ensure that the material will not yield?
σminy=
σmaxy=
To find the permissible range for the normal stress on the y-face (σy), we need to consider the yield criteria for the material. In this case, we'll use the maximum shear stress theory.
According to the maximum shear stress theory, the yield condition is satisfied when the maximum shear stress (τmax) is less than or equal to the permissible shear stress (τo). The maximum shear stress can be calculated using the given stress components as:
τmax = sqrt((σx - σy)^2 + 4τxy^2)
Given:
σx = 40 MPa
τxy = 70 MPa
τo = 74 MPa
We need to find the minimum and maximum values of σy that satisfy the yield condition. Let's start with the minimum value (σminy).
To find σminy, we assume that the maximum shear stress occurs with σy at its minimum value. Setting τmax to be equal to τo, we have:
τo = sqrt((σx - σminy)^2 + 4τxy^2)
Rearranging the equation, we can solve for σminy:
(σx - σminy)^2 + 4τxy^2 = τo^2
(40 - σminy)^2 + 4(70)^2 = (74)^2
Solving this equation will give us the minimum value of σy (σminy) that satisfies the yield condition.
Similarly, to find the maximum value of σy (σmaxy), we assume that the maximum shear stress occurs with σy at its maximum value. Setting τmax to be equal to τo, we have:
τo = sqrt((σx - σmaxy)^2 + 4τxy^2)
Rearranging the equation, we can solve for σmaxy:
(σx - σmaxy)^2 + 4τxy^2 = τo^2
(40 - σmaxy)^2 + 4(70)^2 = (74)^2
Solving this equation will give us the maximum value of σy (σmaxy) that satisfies the yield condition.
Now, you can solve these two equations to find σminy and σmaxy by substituting the given values of σx, τxy, and τo.
σminy = [solve the equation (40 - σminy)^2 + 4(70)^2 = (74)^2 for σminy]
σmaxy = [solve the equation (40 - σmaxy)^2 + 4(70)^2 = (74)^2 for σmaxy]
To ensure that the material will not yield, we need to consider the maximum allowable shear stress and the stress transformation equations. The stress transformation equations in plane stress state are:
σx = σx'
τxy = τxy'
τyx = τyx'
σy = σy'
Where σx' and σy' are the normal stress components on the x' and y' faces, and τxy' and τyx' are the shear stress components on the x'y' and y'x' faces, respectively.
Using these equations, we can determine the range of σy values:
From the given data, we have:
σx = 40 MPa
τxy = 70 MPa
τo = 74 MPa
To find the maximum σy, we can first calculate the maximum shear stress (τmax) using the maximum normal stress (σmax) on the x' face:
τmax = τo = σmax / 2
σmax = 2 * τo
Substituting the given value of τo:
σmax = 2 * 74 = 148 MPa
Next, we can determine the minimum normal stress (σmin) on the x' face using the maximum shear stress:
σmin = σmax - τmax = σmax - τo
Substituting the values:
σmin = 148 - 74 = 74 MPa
Therefore, the permissible range for the normal stress on the y-face is:
σminy = 74 MPa
σmaxy = 148 MPa