Engineering Cross Section of Beams?

Tom, you are describing a classic cantilever beam. The cross section does not change when a load is applied and the beam bends.

The top of the beam is in tension (experiencing tensile stress) and the bottom of the beam is in compression (experiencing compressive stress). Hooke's Law will apply that is E = stress/strain. There will be an extension of the fibres of the material in the top of the beam, there will be contraction of the fibres of the material in the bottom of the beam. The stress varies from maximum tensile in the top to zero at the 'neutral axis' to maximum compressive at the bottom. I acknowledge the sense of your suggestion that the cross section deforms, which it of course does under these stresses. Such deformation is not taken into account in beam design. It may become evident in the case of failure. If you stress a mild steel bar to failure in tension you will see a very marked reduction in its cross section prior to failure. Have a look at this test on a mild steel bar and watch how it narrows. http://www.youtube.com/watch?v=YpmYMj92NZU Also this one: http://www.youtube.com/watch?v=5QaqwmZ7Sic&feature=related

when we design, we usually stay within the elastic range. that means the deflected shape will bounce back to it's original shape after the load is removed. the governing law is s=Ex/L, so u can see stress has a linear relationship with displacement. and the beam's deflection is small. and the cross section remains the same cross section. however, when we design a building under earthquake, we design the beams and columns into the plastic range to save money and materials because the earthquake load is hugh and vast materials is involved. by plastic range, it means the beams is stress beyond it's elastic limit and it won't be able to recover it's original shape.

when we design, we usually stay within the elastic range. that means the deflected shape will bounce back to it's original shape after the load is removed. the governing law is s=Ex/L, so u can see stress has a linear relationship with displacement. and the beam's deflection is small. and the cross section remains the same cross section. however, when we design a building under earthquake, we design the beams and columns into the plastic range to save money and materials because the earthquake load is hugh and vast materials is involved. by plastic range, it means the beams is stress beyond it's elastic limit and it won't be able to recover it's original shape.

The top of the beam is in tension (experiencing tensile stress) and the bottom of the beam is in compression (experiencing compressive stress). Hooke's Law will apply that is E = stress/strain. There will be an extension of the fibres of the material in the top of the beam, there will be contraction of the fibres of the material in the bottom of the beam. The stress varies from maximum tensile in the top to zero at the 'neutral axis' to maximum compressive at the bottom. I acknowledge the sense of your suggestion that the cross section deforms, which it of course does under these stresses. Such deformation is not taken into account in beam design. It may become evident in the case of failure. If you stress a mild steel bar to failure in tension you will see a very marked reduction in its cross section prior to failure. Have a look at this test on a mild steel bar and watch how it narrows. http://www.youtube.com/watch?v=YpmYMj92NZU Also this one: http://www.youtube.com/watch?v=5QaqwmZ7Sic&feature=related