Determination of the ideal spring stiffness for lateral supports of buckling members
Technical Article
If a member is laterally supported due to a compressive axial force to prevent buckling, 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 the critical load and to act accordingly as a full support. Winter speaks of "full bracing". Accordingly, the zero crossing of the buckling line should be located at the location of this support spring, so that the buckling line itself has two or more waves instead of one.
In the winter model, an ideally straight compression member hinged on both sides is considered, which is held in the middle by a support spring. To determine the ideal spring stiffness, Winter developed the idealized model shown in Figure 01.
The fictitious hinge is based on the assumption of an inflection point in the bending buckling line with the same span. Is as a normal force the buckling load P_{e} recognized and the rod in the region of the support spring displaced by the dimension w, obtained by cutting-free at the notional hinge and positioning of the moment equilibrium ideal spring stiffness C_{ideal.}
${\mathrm{C}}_{\mathrm{ideal}}=\frac{2\cdot {\mathrm{P}}_{\mathrm{e}}}{\mathrm{L}}$
C_{ideal} | Ideal spring stiffness |
P_{e} | Critical load |
L. | Span between support and support spring |
This correlation between spring stiffness and critical load results in the function shown in Figure 02. Accordingly occurs in spring stiffness less than_{ideal} C a buckling shape with lateral displacement in the region of the support spring.
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 | Moment of inertia |
L. | Span between support and support spring |
Determination of the ideal spring stiffness using an example
In the model (Figure 03), a compression member (IPE 400) hinged on both sides with the parameters E = 21,000 kN/cm², I_{z} = 1,318 cm ^{4} and L = 5 m is held 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 center of the member of 436 kN/m.
Determination of the stabilizing force in the support spring using the example of a buckling member with imperfection
After performing ultimate load tests on buckling columns in addition to the theoretical considerations mentioned above, it was found that the theoretically ideal spring stiffness is not sufficient for columns with geometric imperfections.
Accordingly, the deformation w from Figure 01 is supplemented by the pre -deformation w_{0} to w_{tot} .
w_{tot} = w + w_{0}
After setting up the moment equilibrium about the fictitious hinge (Figure 01), the result is:
P ⋅ (w + w_{0} ) = C ⋅ w ⋅ L/2
This results in:
${\mathrm{w}}_{\mathrm{ges}}=\frac{{\mathrm{w}}_{0}}{1-{\displaystyle \frac{2\xb7\mathrm{P}}{\mathrm{C}\xb7\mathrm{L}}}}$
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 the buckling member |
C | Spring stiffness of the lateral support spring |
L. | Span between support and support spring |
And for C_{ideal} = 2 ⋅ P_{e}/L:
${\mathrm{w}}_{\mathrm{ges}}=\frac{{\mathrm{w}}_{0}}{1-{\displaystyle \frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{e}}}}}$
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 the buckling member |
P_{e} | Critical load in the buckling member |
The stabilizing force F_{c} results from these equations:
${\mathrm{F}}_{\mathrm{c}}=\mathrm{C}\xb7\mathrm{w}=\frac{2\xb7\mathrm{P}}{\mathrm{L}}\xb7\frac{{\mathrm{w}}_{0}}{1-{\displaystyle \frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{e}}}}}$
F_{c} | Lateral stabilization force |
C | Spring stiffness of the lateral support |
w | Lateral deflection of the buckling member in the middle |
P. | Compression axial force in the buckling member |
L. | Span between support and support spring |
w_{0} | Pre -deformation from precamber due to geometric imperfection |
P_{e} | Critical load of the 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 = 5.00 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 stabilization load F_{c} = 12.3 kN. RFEM determines 11.7 kN.
Summary
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 line with zero crossing at the level of the support spring, while the second mode shape is a single -wave buckling line supported by the support spring (Figure 04). Both have approximately the same critical load.
Keywords
Ideal Spring stiffness Buckling member Lateral Support
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