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.

 

[Mene]

I can confirm that your understanding of strains and change in temperature is correct. The strain on the length axis of a beam, assuming no external forces, can be calculated using the equation ε=αΔΤ, where α is the coefficient of thermal expansion and ΔΤ is the change in temperature. This holds true for isotropic materials, where the strain is equal in all directions and can be represented by εxx=εyy=εzz=αΔΤ.

In the case of a statically indeterminate beam that is supported at both ends and cannot change its length, the strain on the length axis (εxx) will be 0, as you mentioned. However, the strains on the y and z axes (εyy and εzz) will not be equal to αΔΤ, as there is a constraint on the length of the beam. Instead, these strains can be calculated using Hooke's law, which takes into account the stress (σxx), Poisson's ratio (ν), and Young's modulus (E). Therefore, the strains on the y and z axes would be εyy=εzz=αΔΤ - σxxν/E.

To answer your question about the movement of a circle drawn on a beam after deformation, it would depend on the direction of the applied force and the constraints on the beam. If the beam is free to deform in all directions, the circle would likely become an ellipsoid. However, if there are constraints on certain axes, the circle may not deform equally in all directions.

In conclusion, your understanding of strains and change in temperature is correct, and your reasoning for the strains on the y and z axes is also accurate. Keep questioning and exploring different scenarios to deepen your understanding of this concept.