Strains and change in temperature

[Dell]

i know that the strains in a beam with no external forces on it, under a change in temperature will be

ε=αΔΤ

that i know is true for the strain on the axis of the length of the beam, but what about the height and width, if the length axis is x, what will the strains on the y and z axes be? are they 0?? common sense tells me that is the material is isotropic it must act the same in all directions therefore εxx=εyy=εzz=αΔΤ (presuming there is nothing limiting these changes in dimentions)

for the more specific case of a statically indetermined beam which is supported at both ends and cannot change its length, i know that now εxx=0 but what about yy and zz? are they now αΔΤ - σ_xxν/E? (according to hookes law)

if i draw a circle on a beam like this, where will it move to after deformation? on the x-axis will it stay the same point, and the y and z axis will it move or will it just get bigger (and become an ellipsoid)

 

[Mapes]

i know that the strains in a beam with no external forces on it, under a change in temperature will be

ε=αΔΤ

that i know is true for the strain on the axis of the length of the beam, but what about the height and width, if the length axis is x, what will the strains on the y and z axes be? are they 0?? common sense tells me that is the material is isotropic it must act the same in all directions therefore εxx=εyy=εzz=αΔΤ (presuming there is nothing limiting these changes in dimentions)

Agreed.

for the more specific case of a statically indetermined beam which is supported at both ends and cannot change its length, i know that now εxx=0 but what about yy and zz? are they now αΔΤ - σ_xxν/E? (according to hookes law)

Yes, exactly.

if i draw a circle on a beam like this, where will it move to after deformation? on the x-axis will it stay the same point, and the y and z axis will it move or will it just get bigger (and become an ellipsoid)

Yes, it will become an ellipse.