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.