Find out horizontal and vertical components of hinge force:

The frame shown in Figure is supported by a hinge at E and a roller support at D. Find out horizontal and vertical components of hinge force at C as it acts upon member BD.

Solution

 (i)        The roller support at D is replaced by a vertical reaction V_D and hinge support at E is replaced by the vertical and horizontal components of reaction V_E and V_H respectively as illustrated in Figure.

 (ii)       The free body diagram of the three members is illustrated in Figure.

          Figure 11.6

(iii)       For Member AB

              H _A = H _B ------ (a)

              V_A + V_B = 2400------- (b)

              M _A = 2400 × 1.2 - V_B × 2.2 = 0-------- (c)

           For Member DB

               H _B = H _C -------- (d)

               V_D + V_C = V_B           ---------- (e)

               DC cos 45^o = 1.5 and BC cos 45^o = 1.1

                DC = 2.12 m; and BC = 1.55 m

                BD cos 45^o = 1.5 + 1.1 = 2.6

                 or    BD = 2.6 /cos 45^o

                                 = 3.67 m

                M _D = V_C × 1.5 + H _C × 1.5 - V_B × (1.5 + 1.1) - H _B (1.5 + 1.1) = 0

or                V_C  + H _C  - V_B  × 1.73 - H _B  × 1.73 = 0            --------- (f)

              For Member AE

                    V_E - V_C - V_A = 0 -------- (g)

                  - H _E + H _C - H _A = 0----------- (h)

           M _E = V_C × 1.5 - H _C × 1.5 + H _A × 2.6 + V_A × 2.6 = 0

or         V_C  - H _C  + H _A  × 1.73 + V_A  × 1.73 = 0          ---------- (i)

 (iv)      For the three members there are nine equations. The unknowns are H_A, V_A, V_B, H_B, H_C, V_C, V_D, H_E and V_E also nine in numbers. Therefore, frame is statically determinate.

 (v)       From Eq. (c)

        V _B   =( 2400 × 1.2 )/2.2 = 1309 N

  Therefore, from Eq. (b),

             V_A = 2400 - V_B

or         V_A  = 2400 - 1309 = 1091 N

From Eqs. (f) and (i)

      V_C + H _C = 1309 × 1.73 + H _B × 1.73 ------- (j)

       V_C - H _C = - 1091 × 1.73 - H _A × 1.73-------- (k)

or         VC  - H C  = - 1091 × 1.73 - H B  × 1.73  ---------- (l)

Adding Eqs (j) and (l) :

                2V_C = 218 × 1.73

or         V_C  = 188.57 N         (Up side)

By Substituting for VC and VA in Eq. (i)

- H _C + 1.73 H _A = - 1091 × 1.73 - 188.57 = - 2076            --------- (m)

Since

H _A = H _B = H _C

∴ 0.73 H _C = - 2076

Or   H _C = 2843.8 N   (Right side)

         V_C = 188.57 N   (Up side )           Ans.