Find out horizontal and vertical components of hinge force, Mechanical Engineering

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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 VD and hinge support at E is replaced by the vertical and horizontal components of reaction VE and VH 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)

              VA + VB = 2400------- (b)

              M A = 2400 × 1.2 - VB × 2.2 = 0-------- (c)

           For Member DB

               H B = H C -------- (d)

               VD + VC = VB           ---------- (e)

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

                DC = 2.12 m; and BC = 1.55 m

                BD cos 45o = 1.5 + 1.1 = 2.6

                 or    BD = 2.6 /cos 45o

                                 = 3.67 m

                M D = VC × 1.5 + H C × 1.5 - VB × (1.5 + 1.1) - H B (1.5 + 1.1) = 0

or                VC  + H C  - VB  × 1.73 - H B  × 1.73 = 0            --------- (f)

              For Member AE

                    VE - VC - VA = 0 -------- (g)

 

                  - H E + H C - H A = 0----------- (h)

           M E = VC × 1.5 - H C × 1.5 + H A × 2.6 + VA × 2.6 = 0

or         V- H C  + H A  × 1.73 + VA  × 1.73 = 0          ---------- (i)

 (iv)      For the three members there are nine equations. The unknowns are HA, VA, VB, HB, HC, VC, VD, HE and VE 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),

             VA = 2400 - VB

or         VA  = 2400 - 1309 = 1091 N

From Eqs. (f) and (i)

      VC + H C = 1309 × 1.73 + H B × 1.73 ------- (j)

       VC - 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) :

                2VC = 218 × 1.73

or         VC  = 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)

         VC = 188.57 N   (Up side )           Ans.


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