Appendix - Indeterminate assemblies with multiple components

Compatibility         The assemblage is shown deformed as a result of the external tensile load F₀.
The extensions of succesive segments of A between the connectors are   δ^A₁, δ^A₂ . . . etc, and of B   δ^B₁, δ^B₂. . . etc.
The deflections of successive connectors are  δ^C₁, δ^C₂. . . and so on as shown. From the diagram, compatibility requires :

( i)         δ^C_n+1   +   δ^A_n   =   δ^C_n   +   δ^B_n       ;       1 ≤ n ≤ N = 5
 
Equilibrium         Let the forces in the inter-connector segments of A be   F₁,   F₂ . . . etc, while for equilibrium (of the RHS free body eg.) the corresponding forces in B must be   F₀ - F₁,   F₀ - F₂ . . . and so on, as sketched. A free body of the typical n'th connector illustrates that the A/B force which is transferred through the connector is   F_n-1 - F_n.

Constitutive laws         For A and B in tension :

( ii)         δ^A_n   =   c_A F_n       and       δ^B_n   =   c_B ( F₀ - F_n )       ;       1 ≤ n ≤ N = 5

and for the connectors :

( iii)       δ^C_n   =   c_C ( F_n-1 - F_n )       ;       1 ≤ n ≤ N+1 = 6       ;       F₆   =   0

Solution         Inserting the 3N+1 constitutive laws into the N compatibility equations leads to the final N equations from which the N unknown forces may be found : The equations for N other than 5 may evidently be written down immediately.
The fraction of the total load which passes through each of the N+1 =6 connectors, as found from solution of ( iv), is shown in the distributions below for B rigid and for various ratios c_A/c_C.
In the leftmost illustration both A and B are rigid therefore all connectors experience the same deformation - a fact that can be seen qualitatively by imagining A and B to be made of steel and the connectors to take the form of rubber blocks. Since all connectors deform in an identical manner they must transmit the same force, 1/6 or 16.7% of the external load on the assemblage, F₀. The compliance (flexibility) of A increases in the three other distributions illustrated above. The area under the distributions remains the same (F₀), but it is apparent that the connectors at the right hand end of the assembly (where the unconnected A meets the connection assemblage) progressively take more of the load, until, as c_A/c_C tends to infinity, the first connector at the right hand end takes all 100% of the external load. This is the case argued qualitatively earlier when a flexible rubber bolt in a steel nut with steel connecting threads, stretches such that all the load is taken by the first engaged thread adjacent to the pulled bolt head.

Some further results from ( iv), in which B becomes progressively more flexible with respect to A, are also presented. As may be seen, the effect of B's increasing flexibility is to place more load on the connectors at the left hand end of the connection assembly adjacent to the unconnected B, until, when the compliances of A and B are equal as in the rightmost graph, the load distribution is symmetric. If B's compliance were to further increase with respect to that of A, then the distribution would become skewed with the larger peak at the left hand end.

Although this simple model is somewhat unrealistic - for example in its assuming A and B are in tension only - the resulting load distributions are certainly realistic and confirm the general conclusion that the load distribution between components of a statically indeterminate assembly depends markedly upon the relative stiffnesses of the components.

A similar situation arises when the connector is continuous as in the case of a continuous weld C between A and B. The non-uniform load transfer is often referred to as shear lagand will be examined in the chapter on Welded Joints.