1.23 Why stresses jump

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This seems a trivial question. Stresses jump, because the displacements are

only C0, so the derivatives are discontinuous at interelement boundaries. But

perhaps it is worthwhile to study the phenomenon from the perspective of

influence functions. To keep things simple we consider a bar (Fig. 1.69). The

influence function for the jump

σR

x

− σL

x (1.298)

in the stress at the center node is the influence function for σR

x minus the

influence function for σL

x , and because this compound influence function is

not zero, the stress jumps.

At an interior point of an element the two influence functions are identical,

and therefore they cancel. In other words jumps in the stresses do not occur

at any interior point.

But this is no surprise; rather, it smacks of a circular argument. Because

the equivalent nodal forces for the influence functions of the stress σx are, up

to the factor E, the first derivative of the shape functions, the jump in the

stresses will always be zero if the first derivative is the same on both sides of

the point, i.e., if the stress is continuous ...

But there are two interesting points to make looking at Fig. 1.69. Obviously

the maximum jump occurs if the load is applied directly at either side of the

node and the jump will be zero if the load alternates, +p in the first element

and −p in the second element. This is probably also true in 2-D and 3-D

problems. Checkerboard loads leave few traces in Vh, i.e., the equivalent nodal

forces fi are relatively small.

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