Where Fn is the force in the nth member and An and Ln are its cross - sectional area and length.
Let W be the total load, the total load carried will be the sum of all loads for all the members.
Therefore, each member carries a portion of the total load W proportional of EA / L value. The above expression may be writen as
if the length of each individual member in same then, we may write Thus, the stress in member '1' may be determined as s1 = F1 / A1
hus, the stress in member '1' may be determined as s1 = F1 / A1 Determination of common extension of compound bars: In order to determine the common extension of a compound bar it is convenient to consider it as a single bar of an imaginary material with an equivalent or combined modulus Ec. Assumption: Here it is necessary to assume that both the extension and original lengths of the individual members of the compound bar are the same, the strains in all members will than be equal. Total load on compound bar = F1 + F2+ F3 +………+ Fn
where F1 , F 2 ,….,etc are the loads in members 1,2 etc
But force = stress . area,therefore
s (A 1 + A 2 + ……+ A n ) = s1 A1 + s2 A2 + ........+sn An Where s is the stress in the equivalent single bar
Dividing throughout by the common strain Î .
Compound bars subjected to Temp. Change : Ordinary materials expand when heated and contract when cooled, hence , an increase in temperature produce a positive thermal strain. Thermal strains usually are reversible in a sense that the member returns to its original shape when the temperature return to its original value. However, there here are some materials which do not behave in this manner. These metals differs from ordinary materials in a sence that the strains are related non linearly to temperature and some times are irreversible .when a material is subjected to a change in temp. is a length will change by an amount. dt = a .L.t
or Ît= a .L.t or s t= E .a.t
a = coefficient of linear expansoin for the material
L = original Length
t = temp. change
Thus an increase in temperature produces an increase in length and a decrease in temperature results in a decrease in length except in very special cases of materials with zero or negative coefficients of expansion which need not to be considered here. If however, the free expansion of the material is prevented by some external force, then a stress is set up in the material. They stress is equal in magnitude to that which would be produced in the bar by initially allowing the bar to its free length and then applying sufficient force to return the bar to its original length. Change in Length = a L t
Therefore, strain = a L t / L
= a t
Therefore ,the stress generated in the material by the application of sufficient force to remove this strain = strain x E
or Stress = E a t
Consider now a compound bar constructed from two different materials rigidly joined together, for simplicity. Let us consider that the materials in this case are steel and brass.
If we have both applied stresses and a temp. change, thermal strains may be added to those given by generalized hook's law equation –e.g.