Partially Fixed Cantilever
Partially Fixed Cantilever
(OP)
I need to aproximate the deflection stiffness of an "L" shape frame (one with a vertical section supporting a shorter horizontal section - see attached *.png). The existing steel supporting the vertical member (not shown in picture) is assumed to provide complete fixity.
I am going to assume small rotations and deformations to keep it all in the elastic range. For discussion purposes, assume the point load at the end of the horizontal member. I'm using Stadd Pro to validate my equations and I'm clearly missing something; I'm assuming its the partial fixity of the cantilevered horizontal member.
Where:
Member S1 has length = L1 and IZ = I1
Member S2 has length = L2 and IZ = I2
Then the vertical deflection at point P2 can be approximated as:
[Part 1] [Part 2]
Delta = (P*L2^3 / (3*E*I2)) + (2 * L2 * sin(P * L2 * L1 / (E * I1)))
Part 1 is simply the fixed cantilever deflection w/ a point load at the end.
Part 2 is the deflection caused by rotation at P1 from the moment applied at P1. The 2 at the front of part 2 is my assumption to account for the partially fixity of member S1.
Where am I going wrong?
I am going to assume small rotations and deformations to keep it all in the elastic range. For discussion purposes, assume the point load at the end of the horizontal member. I'm using Stadd Pro to validate my equations and I'm clearly missing something; I'm assuming its the partial fixity of the cantilevered horizontal member.
Where:
Member S1 has length = L1 and IZ = I1
Member S2 has length = L2 and IZ = I2
Then the vertical deflection at point P2 can be approximated as:
[Part 1] [Part 2]
Delta = (P*L2^3 / (3*E*I2)) + (2 * L2 * sin(P * L2 * L1 / (E * I1)))
Part 1 is simply the fixed cantilever deflection w/ a point load at the end.
Part 2 is the deflection caused by rotation at P1 from the moment applied at P1. The 2 at the front of part 2 is my assumption to account for the partially fixity of member S1.
Where am I going wrong?






RE: Partially Fixed Cantilever
The first part is right, just a cantilever beam call it Delta1.
The second part should be Delta2 = Theta (L2) where Theta = (Moment (L1))/EI1
And the Moment at P1 is just P(L2)
Then Delta2 = (P(L2)(L2)(L1))/EI1
Total vertical deflection a P2 is Delta1 + Delta2
RE: Partially Fixed Cantilever
RE: Partially Fixed Cantilever
Unless you have deflections which are large compared to the beam lengths then sine Theta = Theta, and Theta is in radians is a good assumption.
If you really think about it, all of our BASIC assumptions of beam deflection theory are that loading remains orthogonal to beam centerlines.
RE: Partially Fixed Cantilever
But my calculations and Staad still don't agree. P1 actually deflects horizontally a small amount. This would add an additional rigid body rotation to the P2 deflection so I'm going to add that in and see what happens.
RE: Partially Fixed Cantilever
This should read:
Delta = (P*L2^3 / (3*E*I2)) + (L2 * sin(P * L2 * L1 / (E * I1)))
which for a small angle is approximately equal to:
Delta = (P*L2^3 / (3*E*I2)) + (L2 * P * L2 * L1 / (E * I1))
You had a misplaced factor of 2 in the second expression.
BA
RE: Partially Fixed Cantilever
RE: Partially Fixed Cantilever
P1 deflects horizontally by M1*L1^2/2E*I1 where M1 = P*L2. The horizontal deflection of P1 does not add a "rigid body rotation to the P2 deflection" as you stated earlier. It adds a horizontal rigid body translation to point P2 but has no effect on the vertical deflection of point P2.
BA
RE: Partially Fixed Cantilever
Delta = P*L2^3 / (3*E*I2) + L2*P*L2*L1/(E*I1) + P*L1/(A1 *E)
Now I'm within 5% of Staad. Why I can't get closer I don't know.
RE: Partially Fixed Cantilever
BA
RE: Partially Fixed Cantilever
RE: Partially Fixed Cantilever