1-1.  The force acting on the truss is  as shown in Fig. 1.1.1. If the allowable normal stress of each of the bar is  , what are the probabilities of failure of these three bars? Assume that P and S_a are independent.

Solution

Internal Loadings: The force developed in each member of the truss can be determined by using the method of joints. First, consider the equilibrium of joint C, Fig. 1.1.2,

;       ;    

;   ; 

Subsequently, the equilibrium of joint B, Fig. 1.1.3, is considered

;       ;  

The cross-sectional area of each of the bars is

.

For bar BC, the probability of failure   is

 (1)

Since ,, and P and S_a are independent, Y also follows a normal distribution, .

Equation (1) can be written as

                                 Ans.

Similarly, for bar AC, the probability of failure   is

   (2)

Equation (2) can be written as

                         Ans.

For bar AB, the probability of failure   is

          (3)

Equation (3) can be written as

                          Ans.