As for the formulations you have listed, I know you intend to use Euler beam theory. Though I don't derive the dynamic equation, I want to point out that the formulation 'u=-z*w(x,t) ' should be 'u=-z* dw(x,t)/dx '.
It is better to use a variational statement equivalent to the Navier equation. Then you can directly substitute your assumptions of the displacement field into the variational statement. however, you do have to note that to derive beam theories this way, one also need to invoke plane stress assumption. A better way to derive beam equations from 3D elasticity theory is using the variational asymptotic method, which is numerically implemented in a general-purpose beam modeling tool, VABS .
But I want to derive Timoshenko beam equation directly from Navier equation and it was not working untill now. I want to know what assumption in addition to displacement field, must be applied?
I had found variational approach, but I am searching for Navier approch.
Yours SincerelyMin Yi
As for the formulations you have listed, I know you intend to use Euler beam theory. Though I don't derive the dynamic equation, I want to point out that the formulation 'u=-z*w(x,t) ' should be 'u=-z* dw(x,t)/dx '.
Yours Sincerely Min Yi
In reply to Yours SincerelyMin Yi by Min Yi
It is better to use a
It is better to use a variational statement equivalent to the Navier equation. Then you can directly substitute your assumptions of the displacement field into the variational statement. however, you do have to note that to derive beam theories this way, one also need to invoke plane stress assumption. A better way to derive beam equations from 3D elasticity theory is using the variational asymptotic method, which is numerically implemented in a general-purpose beam modeling tool, VABS .
Timoshenko beam eq from Navier
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But I want to derive Timoshenko beam equation directly from Navier equation and it was not working untill now. I want to know what assumption in addition to displacement field, must be applied?
I had found variational approach, but I am searching for Navier approch.
Thank you all.