I have a simple question which is bothering me :
If I take a clamped-clamped beam submitted to a uniformely distributed force, I can calculate its shape for a given force per unit length, as long as one stays with small deformation. Because the deflection at any position along the beam is proportional to the applied charge, I can calculate a rigidity (k) for the beam as q*l/Ymax
(q*l) the total force and Ymax the deflection of the center of the beam. I can then calculate the Energy needed to defrom the beam to reach a given deflection at its center using U=0.5*k*Ymax^2
But I can also calculate the bending energy needed to reach that deformation using U=Integrate[M^2/(2EI),{x,0,l}] With M the bending moment obtained from the second derivative of the deformation function.
My problem is that I do not find the same result with the two methods. With the first one I get U= 192EI*Ymax/l^3 and with the second, I get U=(512/5)EI*Ymax/l^3
Can someone explain me why I don't get the same result?
Thanks in advance.
If I take a clamped-clamped beam submitted to a uniformely distributed force, I can calculate its shape for a given force per unit length, as long as one stays with small deformation. Because the deflection at any position along the beam is proportional to the applied charge, I can calculate a rigidity (k) for the beam as q*l/Ymax
(q*l) the total force and Ymax the deflection of the center of the beam. I can then calculate the Energy needed to defrom the beam to reach a given deflection at its center using U=0.5*k*Ymax^2
But I can also calculate the bending energy needed to reach that deformation using U=Integrate[M^2/(2EI),{x,0,l}] With M the bending moment obtained from the second derivative of the deformation function.
My problem is that I do not find the same result with the two methods. With the first one I get U= 192EI*Ymax/l^3 and with the second, I get U=(512/5)EI*Ymax/l^3
Can someone explain me why I don't get the same result?
Thanks in advance.
