Question

A rigid bar AB is hinged at A and supported by a bronze rod 2 m long and a steel rod 1 m long. A load of 80 tonnes is applied at the end B as shown in following figure. Using the data in the following table, calculate the stress in each rod and reaction at A.

	Area(mm2)	E(N/mm2)
Steel	900	2.05 x 105
Bronze	600	0.82 x 105

Answer 1

There are three unknowns, i.e. reaction R_A at A, force P₁ in the steel rod and force P₂ in the bronze rod. To solve the problem, the two equations of static equilibrium(i.e. the force equation and moment equation) must be used in combination with the additional equations obtained from the geometric relations between the elastic deformations.

Let suffix s stand for the steel rod and b for the bronze rod. 

Taking moments about A, we get

\(\sum\) M_A = 0 = (P₁ x 1) + (P₂ x 3) - (80000 x 4)

P₁ +3P₂ = 320000 (1)

If Δ₁ is the deformation of the steel rod and Δ₂ that of bronze rod we have from similar triangles,

from which P₁ = 2.5 P₂ .....(2) 

Substituting the value of P₁ in (1), we get 

2.5P₂ + 3P₂ = 320000 

From which, P₂ = 58182 

N P₁ = 145455 N 

For finding the reactions at A, use the force equation(i.e. \(\sum\) P = 0). Assuming R_A to act downward, 

P₁ + P₂ - R_A - 80000 = 0 

\(\therefore\) R_A = 145455 + 58182 - 80000 = 123637 N 

(The positive sign to R_A shows that the assumed direction of R_A is correct.)