Determination of Ideal Spring Stiffness for Lateral Supports of Buckling Members
Technical Article
If a member is laterally supported to prevent buckling due to a compressive axial force, it must be ensured that the lateral support is actually able to prevent buckling. Therefore, the aim of this article is to determine the ideal spring stiffness of a lateral support using the Winter model.
According to George Winter, the ideal spring stiffness is that which is at least necessary to completely prevent the lateral buckling of the main member with regard to its critical buckling load and to act accordingly as a full support. Winter speaks of "full bracing". According to this, the zero crossing of the buckling curve should be located at this support spring, so that the buckling curve itself has two or more waves instead of one.
In the Winter model, an ideally straight compression member with pinned ends on both sides is considered, which is restrained in the middle by a support spring. To determine the ideal spring stiffness, Winter developed the idealized model shown in Image 01.
The notional release is based on the assumption of an inflection point in the flexural buckling curve, if the span lengths are the same. If the critical buckling load P_{e} is applied as compression axial force and the member is displaced by the dimension w in the support spring's region, we obtain the ideal spring stiffness C_{ideal}, after clearing the zone around the notional release by imaginary cuts and setting up conditions for the moment equilibrium.
${\mathrm{C}}_{\mathrm{ideal}}=\frac{2\cdot {\mathrm{P}}_{\mathrm{e}}}{\mathrm{L}}$
C_{ideal} | Ideal spring stiffness |
P_{e} | Critical load |
L | Span length between support and support spring |
This correlation between spring stiffness and critical buckling load results in the function shown in Image 02. Thus, a buckling shape with lateral displacement in the support spring's region occurs for spring stiffnesses smaller than C_{ideal}.
The critical load P_{e} can be determined with the add-on modules RSBUCK and RF-STABILITY, or manually, as follows.
${\mathrm{P}}_{\mathrm{e}}=\frac{\mathrm{\pi}\xb2\cdot \mathrm{E}\cdot \mathrm{I}}{\mathrm{L}\xb2}$
P_{e} | Critical load |
E | Modulus of elasticity |
I | Second moment of area |
L | Span length between support and support spring |
Determination of the ideal spring stiffness described by the example
In the model (Image 03), a compression member (IPE 400) with pinned ends and the parameters E = 21,000 kN/cm^{2}, I_{z} = 1,318 cm^{4} and L = 5 m is restrained in the middle by a support spring.
This results in a critical load P_{e} of 1,089 kN, which results in an ideal spring stiffness C_{ideal} for the support spring defined in the member's center of 436 kN/m.
Determination of the stabilizing force in the support spring using the example of a buckling member with an imperfection
After performing ultimate load tests on buckling columns in addition to the theoretical considerations mentioned above, we ascertained that the theoretically ideal spring stiffness is insufficient for columns with geometric imperfections.
Accordingly, deformation w from Image 01 is supplemented by predeformation w_{0} to w_{tot}.
w_{tot} = w + w_{0}
After setting up the moment equilibrium around the notional hinge (Image 01), the result is:
P ⋅ (w + w_{0}) = C ⋅ w ⋅ L / 2
This results in:
${\mathrm{w}}_{\mathrm{tot}}=\frac{{\mathrm{w}}_{0}}{1-{\displaystyle \frac{2*\mathrm{P}}{\mathrm{C}*\mathrm{L}}}}$
w_{tot} | Total deformation from buckling deflection and precamber |
w_{0} | Pre-deformation from precamber due to geometric imperfection |
P | Existing compression axial force in buckling member |
C | Spring stiffness of lateral support spring |
L | Span between support and support spring |
And for C_{ideal} = 2 ⋅ P_{e} / L:
${\mathrm{w}}_{\mathrm{tot}}=\frac{{\mathrm{w}}_{0}}{1-{\displaystyle \frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{e}}}}}$
w_{tot} | Total deformation from buckling deflection and precamber |
w_{0} | Pre-deformation from precamber due to geometric imperfection |
P | Existing compression axial force in buckling member |
P_{e} | Critical load in buckling member |
Based on these equations, the stabilizing force F_{c} results in:
${\mathrm{F}}_{\mathrm{c}}=\mathrm{C}\xb7\mathrm{w}=\frac{2\xb7\mathrm{P}}{\mathrm{L}}\xb7\frac{{\mathrm{w}}_{0}}{1-{\displaystyle \frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{e}}}}}$
F_{c} | Lateral stabilizing force |
C | Spring stiffness of lateral support |
w | Lateral deflection of buckling member in center |
P | Compression axial force in buckling member |
L | Span between support and support spring |
w_{0} | Pre-deformation from precamber due to geometric imperfection |
P_{e} | Critical load of buckling member |
The stabilizing force F_{c} can thus be determined from the following parameters:
Existing compressive force P = 500 kN
Span between support and support spring L = 5 m
Precamber from imperfection w_{0} = L_{total} / 300 = 10 / 300 = 0.0333 m
Critical load P_{e} = 1,089 kN
This results in a stabilizing load of F_{c} = 12.3 kN. RFEM determines 11.7 kN.
Conclusion
To check the correctness of the determined spring stiffness, you can look at the results from RF-STABILITY. The first mode shape is a double-wave buckling curve with zero crossing at the support spring level, while the second shape is a single-wave buckling curve supported by the support spring (Image 04). Both have approximately the same critical buckling load.
Author
Bastian Ackermann, M.Sc.
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Mr. Ackermann is the contact person for sales inquiries.
Keywords
Ideal Spring stiffness Buckling member Lateral Support
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