1.6.1
GENERAL — A theoretical treatment of columns can be found in standard texts on the strength of materials. Some of the problems which are not well defined by theory are discussed in this section. Actual strengths of columns of various materials are provided in subsequent chapters.
1.6.2
PRIMARY INSTABILITY FAILURES — A column can fail through primary instability by bending laterally (stable sections) or by twisting about some axis parallel to its own axis. This latter type of primary failure is particularly common to columns having unsymmetrical open sections. The twisting failure of a closed section column is precluded by its inherently high torsional rigidity. Since the amount of available information is limited, it is advisable to conduct tests on all columns subject to this type of failure.
1.6.2.1
Columns with Stable Sections — The Euler formula for columns which fail by lateral bending is given by Equation 1.3.8.2. A conservative approach in using this equation is to replace the elastic modulus (E) by the tangent modulus (Et) given by Equation 1.3.8.1. Values for the restraint coefficient (c) depend on degrees of ends and lateral fixities. End fixities tend to modify the effective column length as indicated in Equation 1.3.8.1. For a pin-ended column having no end restraint, c = 1.0 and L' = L. A fixity coefficient of c = 2 corresponds to an effective column length of L' = 0.707 times the total length.
The tangent modulus equation takes into account plasticity of a material and is valid when the following conditions are met:
- (a) The column adjusts itself to forcible shortening only by bending and not by twisting.
- (b) No buckling of any portion of the cross section occurs.
- (c) Loading is applied concentrically along the longitudinal axis of the column.
- (d) The cross section of the column is constant along its entire length.
MIL-HDBK-5 provides typical stress versus tangent modulus diagrams for many materials, forms, and grain directions. These information are not intended for design purposes. Methodology is contained in Chapter 9 for the development of allowable tangent modulus curves.
1.6.2.2
Column Stress ( ) — The upper limit of column stress for primary failure is designated as . By definition, this term should not exceed the compression ultimate strength, regardless of how the latter term is defined.
1.6.2.3
Other Considerations — Methods of analysis by which column failure stresses can be computed, accounting for fixities, torsional instability, load eccentricity, combined lateral loads, or varying column sections are contained in References 1.6.2.3(a) through (d).
1.6.3
LOCAL INSTABILITY FAILURES — Columns are subject to failure by local collapse of walls at stresses below the primary failure strength. The buckling analysis of a column subject to local instability requires consideration of the shape of the column cross section and can be quite complex. Local buckling, which can combine with primary buckling, leads to an instability failure commonly identified as crippling.
1.6.3.1
Crushing or Crippling Stress ( ) — The upper limit of column stress for local failure is defined by either its crushing or crippling stress. The strengths of round tubes have been
MIL-HDBK-5J
1.6.4
CORRECTION OF COLUMN TEST RESULTS — In the case of columns having unconventional cross sections which are subject to local instability, it is necessary to establish curves of transition from local to primary failure. In determining these column curves, sufficient tests should be made to cover the following points.
1.6.4.1
Nature of "Short Column Curve" — Test specimens should cover a range of L'/p values. When columns are to be attached eccentrically in structural application, tests should be designed to cover such conditions. This is important particularly in the case of open sections, as maximum load carrying capabilities are affected by locations of load and reaction points.
1.6.4.2
Local Failure — When local failure occurs, the crushing or crippling stress can be determined by extending the short column curve to a point corresponding to a zero value for L'/p. When a family of columns of the same general cross section is used, it is often possible to determine a relationship between crushing or crippling stress and some geometric factor. Examples are wall thickness, width, diameter, or some combination of these dimensions. Extrapolation of such data to conditions beyond test geometry extremes should be avoided.
1.6.4.3
Reduction of Column Test Results on Aluminum and Magnesium Alloys to Standard Material — The use of correction factors provided in Figures 1.6.4.3(a) through (i) is acceptable to the Air Force, the Navy, the Army, and the Federal Aviation Administration for use in reducing aluminum and magnesium alloys column test data into allowables. (Note that an alternate method is provided in Section 1.6.4.4). In using Figures 1.6.4.3(a) through (i), the correction of column test results to standard material is made by multiplying the stress obtained from testing a column specimen by the factor K. This factor may be considered applicable regardless of the type of failure involved, i.e., column crushing, crippling or twisting. Note that not all the information provided in these figures pertains to allowable stresses, as explained below.
The following terms are used in reducing column test results into allowable column stress:
- is the design compression yield stress of the material in question, applicable to the gage, temper and grain direction along the longitudinal axis of a test column.
- is the maximum test column stress achieved in test. Note that a letter (F) is used rather the customary lower case (f). This value can be an individual test result.
- is the compressive yield strength of the column material. Note that a letter (F) is used rather than the customary lower case (f). This value can be an individual test result using a standard compression test specimen.
Using the ratio of (F c' / F cy'), enter the appropriate diagram along the abscissa and extend a line upwards to the intersection of a curve with a value of (F cy' / F cy). Linear interpolation between curves is permissible. At this location, extend a horizontal line to the ordinate and read the corresponding K-factor. This factor is then used as a multiplier on the measured column strength to obtain the allowable. The basis for this allowable is the same as that noted for the compression yield stress allowable obtained from the room temperature allowables table.
MIL-HDBK-5J
1.6.4.4
Reduction of Column Test Results to Standard Material-Alternate Method — For materials that are not covered by Figures 1.6.4.4(a) through (i), the following method is acceptable for all materials to the Air Force, the Navy, the Army, and the Federal Aviation Administration.
- (1) Obtain the column material compression properties: .
- (2) Determine the test material column stress (fc') from one or more column tests.
- (3) Determine the test material compression yield stress (f_{\mathrm{cy}}^{\prime}) from one or more tests.
- (4) Assume and from (1) apply directly to the column material. They should be the same material.
- (5) Assume that geometry of the test column is the same as that intended for design. This means that a critical slenderness ratio value of (L'/p) applies to both cases.
- (6) Using the conservative form of the basic column formula provided in Equation 1.3.8.1, this enables an equality to be written between column test properties and allowables. If
[1.6.4.4(a)]
Then
- (7) Tangent modulus is defined as:
[1.6.4.4(c)]
- (8) Total strain (e) is defined as the sum of elastic and plastic strains, and throughout the Handbook is used as:
[1.6.4.4(d)]
or,
[1.6.4.4(e)]