Chapter 8 Columns 255 Chapter 9 Thin Plates 294
9.6 Failure stress in plates and stiffened panels
9.6 Failure stress in plates and stiffened panels
The previous discussion on plates and stiffened panels investigated the prediction of buckling stresses. However, as we have seen, plates retain some of their capacity to carry load even though a portion of the plate has buckled. In fact, the ultimate load is not reached until the stress in the majority of the plate exceeds the elastic limit. The theoretical calculation of the ultimate stress is difficult since non-linearity results from both large deflections and the inelastic stress–strain relationship.
Gerard1 proposes a semi-empirical solution for flat plates supported on all four edges. After elastic buckling occurs theory and experiment indicate that the average compressive stress,σ¯a, in the plate and the unloaded edge stress,σe, are related by the following expression
¯ σa σCR =α1
σe σCR
n
(9.8) where
σCR= kπ2E 12(1−ν2)
t b
2
and α1 is some unknown constant. Theoretical work by Stowell7 and Mayers and Budiansky8 shows that failure occurs when the stress along the unloaded edge is approximately equal to the compressive yield strength, σcy, of the material. Hence substitutingσcyforσein Eq. (9.8) and rearranging gives
¯ σf σcy =α1
σCR σcy
1−n
(9.9) where the average compressive stress in the plate has become the average stress at failureσ¯f. Substituting forσCRin Eq. (9.9) and putting
α1π2(1−n)
[12(1−ν2)]1−n =α yields
¯ σf
σcy =αk1−n
t b
E σcy
1
2
2(1−n)
(9.10)
or, in a simplified form
¯ σf σcy =β
t b
E σcy
12m
(9.11) whereβ=αkm/2. The constantsβand m are determined by the best fit of Eq. (9.11) to test data.
Experiments on simply supported flat plates and square tubes of various aluminium and magnesium alloys and steel show thatβ=1.42 and m=0.85 fit the results within
±10 per cent up to the yield strength. Corresponding values for long clamped flat plates areβ=1.80, m=0.85.
Gerard9–12extended the above method to the prediction of local failure stresses for the plate elements of thin-walled columns. Equation (9.11) becomes
¯ σf σcy =βg
gt2 A
E σcy
12m
(9.12) where A is the cross-sectional area of the column,βg and m are empirical constants and g is the number of cuts required to reduce the cross-section to a series of flanged sections plus the number of flanges that would exist after the cuts are made. Examples of the determination of g are shown in Fig. 9.6.
The local failure stress in longitudinally stiffened panels was determined by Gerard10,12using a slightly modified form of Eqs (9.11) and (9.12). Thus, for a section of the panel consisting of a stiffener and a width of skin equal to the stiffener spacing
¯ σf
σcy =βg
gtsktst A
E
¯ σcy
1
2
m
(9.13)
Fig. 9.6Determination of empirical constantg.
9.6 Failure stress in plates and stiffened panels 305 where tskand tst are the skin and stiffener thicknesses, respectively. A weighted yield stressσ¯cyis used for a panel in which the material of the skin and stiffener have different yield stresses, thus
¯
σcy = σcy+σcy,sk[(t/tst)−1]
t/tst
where¯t is the average or equivalent skin thickness previously defined. The parameter g is obtained in a similar manner to that for a thin-walled column, except that the number of cuts in the skin and the number of equivalent flanges of the skin are included. A cut to the left of a stiffener is not counted since it is regarded as belonging to the stiffener to the left of that cut. The calculation of g for two types of skin/stiffener combination is illustrated in Fig. 9.7. Equation (9.13) is applicable to either monolithic or built up panels when, in the latter case, interrivet buckling and wrinkling stresses are greater than the local failure stress.
The values of failure stress given by Eqs (9.11), (9.12) and (9.13) are asso- ciated with local or secondary instability modes. Consequently, they apply when le/r≤20. In the intermediate range between the local and primary modes, failure occurs through a combination of both. At the moment there is no theory that predicts satisfactorily failure in this range and we rely on test data and empirical methods.
The NACA (now NASA) have produced direct reading charts for the failure of ‘top hat’, Z- and Y-section stiffened panels; a bibliography of the results is given by Gerard.10
It must be remembered that research into methods of predicting the instabil- ity and post-buckling strength of the thin-walled types of structure associated with aircraft construction is a continuous process. Modern developments include the use of the computer-based finite element technique (see Chapter 6) and the study of the sensitivity of thin-walled structures to imperfections produced during fab- rication; much useful information and an extensive bibliography is contained in Murray.2
Fig. 9.7Determination ofgfor two types of stiffener/skin combination.
Fig. 9.8Diagonal tension field beam.