Investigations on the Incompletely Developed Plane Diagonal-Tension Field Page: 3 of 26
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THE INCOMPLETELY DEVELOPED PLANE DIAGONAL-TENSION FIELd)
Some compressive stress continues to exist, and this
diagonal compression combines with part of the diag-
onal tension into a shear stress. As the shear force
increases to higher and higher values, the relative im-
portance of the compressive stress decreases and the
condition of pure diagonal tension is approached more
and more closely but is never quite reached in an actual
beam. The finite strength of the material will permit
a failure before the diagonal-tension field is fully devel-
oped; in other words, practical design has to deal with
webs constituting an incompletely developed diagonal-
tension field somewhere between the limiting stages of
pure shear and pure diagonal tension.
A number of authors have adopted an assumption
sometimes used in the analysis of trusses with double
diagonals, namely, that the compressive stress remains
constant after buckling and equal to the stress at
budding. Under this assumption, only the excess
stress over the building shear stress is converted into
diagonal tension, while a shear stress equal in magni-
tude to the building stress is carried by the sheet as
true shear. This assumption can be written(7)
where r is the purely nominal applied shear stress Slht,
prD is the part of the applied shear stress that is carried
as diagonal tension, and 7,, is the critical stress that
continues to exist as a true shear stress. The second
form of writing the expression indicates that the diag-
onal tension is a function of the ratio 7-,/r, which is the
inverse of the loading ratio r/r,, that is, the ratio of the
applied stress to the buckling stress or of the applied
load to the budding load.
Wagner himself proposed (reference 7) to apply the
assumption expressed by equation (7) to the analysis of
curved diagonal-tension fields. In such fields, the load-
ing ratio 7r/r, at the design load probably never exceeds
10 and is generally below 5 because the folds become
objectionably deep soon after budding occurs and
permanent set and failure soon follow.
In plane diagonal-tension fields, the loading ratio is,
as a rule, much higher than in curved fields and equa-
tion (7) was experimentally found to be inadequate to
deal with such cases. Realizing this weakness of the
theory, Wagner conducted experiments (reference 5)
that were intended to give empirical relations for the
stresses in incomplete diagonal-tension fields. Ques-
tions have been raised, however, about the direct ap-
plicability of the test results to practical design.
Schapitz (references 8 and 9) developed a theory of
the incomplete diagonal-tension field, including at the
same time the effect of superposed normal stresses on
the panel. He began with an assumption on the stress
distribution within the panel, leaving one basic param-
eter (two for curved panels) free to be adjusted so
that the results would fit experiments. Like all other
authors preceding him, he assumed the diagonal com-
pression to be constant after buckling. Attempts to
407300A-41-26correlate this theory with the N. A. C. A. tests have
been unsuccessful.
PROPOSED NEW THEORY OF INCOMPLETE DIAGONAL-TENSION
FIELD
Test observations have shown that the behavior of
a shear web working in partial diagonal tension is often
quite irregular and is apparently influenced by a number
of factors which cannot be evaluated. This observation
is, of course, merely a repetition of similar experiences
in the study of built-up structures of thin sheet metal,
but it again emphasizes the fact that too much accuracy
should not be expected from a "rigorous" and perhaps
mathematically very elaborate theory if the physical
action of such complicated structures depends on en-
tirely too many unknown and uncontrollable factors.
In view of the fact that the ultimate aim of any engi-
neering theory of stresses is application to practical de-
sign work, it seems rational under such circumstances to
develop a theory with an eye toward ease of appli-
cation.
A modification of the theory given in reference 10
for curved diagonal-tension fields was found to describe
the experimental facts with an accuracy compatible with
the scatter of the test points. The basic assumption of
this theory is that the total shear force in the web can
be divided into a shear force carried by shear in the
sheet and a shear force carried by diagonal tension;
this assumption may be written as
S=SS+SDT
orSDr= kS and Ss= (1-k)S
(8)
where k is the fraction of the total shear that is carried
as diagonal tension.
The diagonal-tension fraction k is assumed to be
given by the expressionk= (1-r7,/c)n
(9)
The form of this expression was suggested by formula
(7). According to Wagner (reference 5), the degree of
development of the diagonal tension is a function of
au/, where au is the compressive stress in the upright.
Experimental values of na derived from the N. A. 0. A.
tests to be described in part II were therefore plotted
against u/-r; the relation between n and au/7- could be
expressed byn= I + 57/r
(10'
For beams with very heavy uprights (a-u->O) the
assumption expressed by equations (8), (9), and (10)
reduces to that given by equation (7); that is, the true
shear in the sheet remains constant afte the buckling
load is passed. For finite values of ou, however, the
true shear continues to increase according to equations
(8), (9), and (10) after the buckling load has been
passed. This distinction is the main difference between
the proposed new theory and the existing theories.389
TDr= T -r e,= (1i- Cr7fT)
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Kuhn, Paul. Investigations on the Incompletely Developed Plane Diagonal-Tension Field, report, March 28, 1940; (https://digital.library.unt.edu/ark:/67531/metadc66357/m1/3/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.