Problem 10.7

Determine the displacements and rotation at the free end of the cantilevered quarter ring using Castigliano's theorem II.  Use the complementary strain energy of curved beams as established in the class notes, and include the shear term V²/2GA_s.  Let A be the area of the cross-section, and Z be the shape factor. 

Let

u = displacement in direction of P.

v = radial outward displacement.

j = rotation in direction of M_o.

To determine v by Castigliano II, we need to apply a force in the direction of v.  Let 

V_o = radial outward force applied at the free end. 

By Castigliano's theorem II,

(u, v, j) = (∂U*/∂P, (∂U*/∂V_o, ∂U*/∂M_o)

where

V_o is set to zero after taking the partial derivatives of U*.  

From the free-body diagram,

N = Pcosq + V_osinq

V = Psinq - V_ocosq

M = M_o + PR(1-cosq) - V_oRsinq

Substitute for N, V, and M into the integrand above, and integrate.  The integrals of (sinq, cosq, sin²q, cos²q, sinqcosq)dq in the interval (0, p/2) are equal to (1, 1, p/4, p/4, 1/2).  Find

U* = (R/2EA)(P²p/4 + V_o²p/4 + PV_o)

        +(1/EA)(PM_o + P²R(1-p/4) - PV_oR/2 + V_oM_o + PV_oR/2 - V_o²Rp/4)

        +[R(1+Z)/2EI'](M_o²p/2 + P²R²(3p/4 - 2) + V_o²R²p/4 + 2M_oPR(p/2 - 1) - 2M_oV_oR - PV_oR²)

        +(R/2GA_s)(P²p/4 + V_o²p/4 - PV_o)

or, after simplification,

U* = (R/2EA)( P²(2-p/4) - V_o²p/4 + PV_o )

        +(1/EA)(PM_o + V_oM_o )

        +[R(1+Z)/2EI'][M_o²p/2 + P²R²(3p/4 - 2) + V_o²R²p/4 + 2M_oPR(p/2 - 1) - 2M_oV_oR - PV_oR²]

        +(R/2GA_s)(P²p/4 + V_o²p/4 - PV_o)

Further simplifications occur by combining EI' and EA terms using the relation AR²Z = I'.  We obtain

U* = P²(R³/2EI')(3p/4 - 2 + Zp/2) + P²(R/2GA_s)(p/4)

        + V_o²(R³/2EI')(p/4) +  V_o²(R/2GAs)(p/4)

        + M_o²(R/2EI')(1+Z)p/2

        + PV_o(R³/EI')(-1/2) +PV_o(R/GA_s)(-1/2)  + PM_o(R²/EI')(p/2 - 1 + Zp/2) + V_oM_o(R²/EI')(-1)

Taking the partial derivatives of U*, and putting the result in matrix form, we obtain

The answer to the problem is obtained by setting V_o = 0.