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Buckling length calculation

In order to calculate buckling lengths:

  • Check if the default bar groups make sence for your structure. If not, correct them.
    The calculation of the buckling lengths is worth nothing when the group definition is faulty.
  • Click on or go to the menu ‘Analysis – Calculate buckling lengths’.
    The following dialogue appears:
  • Flexural buckling can occur around two axes: the local y'(u) and z'(v)-axis
    • Check around which axes Diamonds should calculate the buckling lengths.
    • Select the analysis strategy.
      You can choose between: displaceable, non-displaceable and semi-displaceable nodes.
      Especially for steel structures, the analysis strategy for the bucklings lengths, depends on the horizontal stiffness of the structure (more info).
      The background of the 3 strategies is given at the bottom of this article.
    • Group illogically ungrouped bars
      If this option is checked, bars lying on the same axis, without any support, nor other bars connected and defined as ‘ungrouped’ will be grouped.
      This option is default checked, because it’s a safe approach.
  • Indicate if Diamonds should calculate the buckling lengths for all bars or only the selected bars.
  • Combination for determination of non-linearities
    This option will be visible when the the structure contains non-linearities (tie rods, supports that only work under compression, stiffnes diagrams, …).
    Let’s say a structure contains 20 tie rods. In load combination 1 a few tie rods work under tension. In load combination 2 a different set of tie rods work. The distribution of which tie rods work and which don’t, determines the stiffness of the structure. A stiffness that differs depending on the selected load combination. Since the buckling length calculation depends on the stiffness of the structure, you’ll have to choose a reference combination (usually ULS FC 1). It’s with this stiffness distribution that Diamonds will calculate the buckling lengths.

Background information on the analysis strategies

In order to explain the different analysis strategies, we need to explain how the buckling length calculation in Diamonds works first:

  • Step 1: the bar for which you want to calculate the buckling length
  • Step 2: Diamonds applies a uniformly distributed load on the bar, perpendicularly to the axis for which you want to determine the buckling length.
    When two (or more bars) are grouped because they can physically buckle as a whole, the uniformly distributed load applies simultaneously on all the grouped bars, and the boundary conditions to be taken into account in Euler’s differential equation will correspond to the supports related to the ends of the grouped bars.
  • Step 3: Due to the uniform load, internal forces M, V will appear and the bar will deform u and φ.
  • Step 4: The ratios V/u and M/φ determine the transverse and angular stiffness of equivalent springs at each of the ends.
  • Step 5: For a bar held in the two ends by a transverse an angular spring, we can apply Euler’s differential equation:

        \[E \cdot I \cdot u" + P \cdot u = 0 \]


    This equation has the following general solution:

        \[u=A\cdot sin(\alpha x)+B\cdot cos(\alpha x)+C\cdot x+D\]


    with \alpha=\sqrt{\frac{P}{E \cdot I}}
    By expressing the 4 conditions at the supports according to the global solution, we obtain a system of 4 equations with 4 unknowns A, B, C and D.
    This system includes a non-trivial solution when the determinant equals zero. The determinant can only equal zero for some values of α.
    The smallest of the α values corresponds to the critical load Pk which makes the bar buckle.
  • Step 6: from the buckling load Pk the corresponding buckling length lk can be deducted.

        \[P_k=\frac{\pi^2 \cdot E \cdot I}{l_k^2}\]

  • Step 7: Diamonds repeats steps 1-6 in the other direction.

In the buckling length calculation discussed above, the rigidity of the adjacent structure is in the ratios (V/u) and (M/φ). However, Eurocode doesn’t contain this scenario. Eurocode provides two buckling lengths: one for displaceable nodes and one for non-displaceable ones. So in Diamonds:

  • when you selected “non-displaceable nodes“
    The rigitiy of the adjacent structure is assumed to be infinitely rigid. For Diamonds this means that the ratio V/u is set to ininity.
    This method is to be used in the case of a 2nd order calculation or during a 1st order analysis of a braced frame.
    This method results in buckling lengths smaller or equal to the system length.
  • when you selected “displaceable nodes“
    The rigitiy of the adjacent structure is assumed to be zero. For Diamonds this means that the smallest V/u ratio is set to zero.
    In this method neglects the rigidity of the adjacent structure. This simulates the situation in which all compressed bars simultaneously lose their buckling stability.
    This method is to be used with a 1st order calculation of an unbraced frame.
    This method results in buckling lengths larger or equal to the system length. This is the most conservative option.
  • when you selected “semi-displaceable nodes“
    The rigidity of the adjacent structure is taken into account.
    This method is to be used in the case of analysis of the 1st order of a braced frame.
    Results in buckling lengths between the non-displaceable and displaceable method.
Limitations of the buckling length calculation in Diamonds

You can do a lot with the buckling length calculation in Diamonds. But it can and will not handle every situation well. Some alertness is advised.

Example 1:

  • Problem: what is the buckling lenght of column a, assuming you use semi-displaceable nodes?
  • Answer: You expect the buckling lenght to be twice the height of the column, 2*L. But Diamonds calculates a buckling length that is smaller.
  • Reason: if column a wants to buckle, columns b until e have to follow. Since the unit load is only place on column a, columns b until e don’t have the tendency to buckling. Columns b until e help carry the unit load on column a. Therefor the buckling length calculated by Diamonds, is smaller than what you expect.
  • Solution: an internal solution would be to apply the unit load to all columns when the buckling length of column a is calculated. But that would imply that Diamonds has some form of intelligence to detect this issue… Which it has not. So the only alternative is that you impose the buckling lengths manually.

Example 2:

  • Problem: what is the buckling length of column a (assume semi-displaceable buckling lengths)
  • Answer: you expect that the buckling length equal the height of one floor because the buckling form is a sinus wave. But Diamonds calculates a buckling length that is smaller.
  • Reason: if column a wants to buckle, columns b and c have to follow. Because the unit load is only placed on column a, columns b and c don’t buckle together with column a. Columns b and c help carry the unit load on column a. Therefor the buckling length calculated by Diamonds, is smaller than what you expect.
  • Solution: an internal solution would be to apply the unit load alternating to the columns when the buckling length of column a is calculated. But that would imply that Diamonds has some form of intelligence to detect this issue… Which it has not. So the only alternative is that you impose the buckling lengths manually.

On this page

  • Background information on the analysis strategies