- #1

roldy

- 237

- 2

## Homework Statement

The plane strain solution for the stresses in the rectangular block 0 < x < a, 0 < y < b, 0 < z < c under certain loading conditions is given by.

[itex] \sigma_{xx} = \frac{3Fxy}{2b^3}[/itex]

[itex] \sigma_{xy} = \frac{3F(b^2-y^2)}{4b^3}[/itex]

[itex] \sigma_{yy} = 0[/itex]

[itex] \sigma_{zz} = \frac{3\nu Fxy}{2b^3}[/itex]

Find the tractions on all the six surfaces of the block. Determine the resultants (Forces and Moments) on the surface x = a.

## Homework Equations

[itex]F = \int\int \sigma dA[/itex]

[itex]M_x = Fx[/itex]

[itex]M_y = Fy[/itex]

[itex]M_z = Fz[/itex]

## The Attempt at a Solution

For the traction on each of the surfaces, I just plug the min and max values of the dimensions into the stress given.

x faces:

x = 0,

[itex]\sigma_{xx} = 0[/itex]

[itex]\sigma_{xy} = \frac{3F(b^2-y^2)}{4b^3}[/itex]

[itex]\sigma_{xz} = 0[/itex]

x = a,

[itex]\sigma_{xx} = \frac{3Fay}{2b^3}[/itex]

[itex]\sigma_{xy} = \frac{3F(b^2-y^2)}{4b^3}[/itex]

[itex]\sigma_{xz} = 0[/itex]

y faces:

y = 0,

[itex]\sigma_{yy} = 0[/itex]

[itex]\sigma_{yx} = \frac{3F}{4b}[/itex]

[itex]\sigma_{yz} = 0[/itex]

y = b,

[itex]\sigma_{xx} = 0[/itex]

[itex]\sigma_{xy} = 0[/itex]

[itex]\sigma_{xz} = 0[/itex]

z faces:

z = 0,

[itex]\sigma_{zz} = \frac{3 \nu Fxy}{2b^3}[/itex]

[itex]\sigma_{zy} = 0[/itex]

[itex]\sigma_{xz} = 0[/itex]

z = c,

[itex]\sigma_{zz} = \frac{3 \nu Fxy}{2b^3}[/itex]

[itex]\sigma_{zy} = 0[/itex]

[itex]\sigma_{xz} = 0[/itex]

The part I am confused about is finding the resultant forces on the face x = a.

I believe that all I do is use the tractions that I found for x = a. Thus,

[itex]F_x = \int \limits_0^b \int \limits_0^c \sigma_{xx}dzdy = \int \limits_0^b \int \limits_0^c \frac{3Fay}{2b^3}dzdy[/itex]

[itex]F_y = \int \limits_0^b \int \limits_0^c \sigma_{xy}dzdy = \int \limits_0^b \int \limits_0^c \frac{3F(b^2-y^2)}{4b^3}dzdy[/itex]

[itex]F_z = \int \limits_0^b \int \limits_0^c \sigma_{xz}dzdy = \int \limits_0^b \int \limits_0^c (0)dzdy[/itex]

Then for the moments I just take these forces and multiply them by the distance to the x axis.

Is my work correct so far or am I missing something.