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Shear Buckling of Unstiffened Plates Formula and Calculator

Shear Buckling of Unstiffened Plates Formula and Calculator

Plate girder webs are usually very slender and fail by shear buckling rather than yielding. To control this, intermediate web stiffeners are used to raise the shear strength, as illustrated in Figure 1.

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Eq. 1

${\displaystyle {\mathrm{\tau <!--\; t\; -->}}_{cr}=k\frac{{\mathrm{\pi <!--\; p\; -->}}^{2}E}{12(1-<!--\; -\; -->{\mathrm{\nu <!--\; ?\; -->}}^2)}{\left(\frac{t}{b}\right)}^2}$

For steel, Poisson’s ratio ν = 0.3 and Young’s modulus E = 210,000 N/mm² and Equation 1 becomes

Eq. 2

${\displaystyle {\mathrm{\tau <!--\; t\; -->}}_{cr}=k\times <!--\; \times \; -->190000\times <!--\; \times \; -->{\left(\frac{t}{b}\right)}^2}$

For the most common situation of plates simply supported on all four edges, the buckling coefficient is

Eq. 3
(SeeCompression Buckling in Unstiffened Plates Formula and Calculator webpage Figure 2, for K values)

${\displaystyle k=5.34{\left(\frac{b}{a}\right)}^2+4ifa\le <!--\; =\; -->b}$

Figure 1 Illustration of shear buckling for a plate girder web. (a) No intermediate web stiffeners and (b) with intermediate stiffeners.

where a is the length and b is the width as illustrated in Figure 1. As stated previously, plates bounded on four sides by flanges and web stiffeners have a post-buckling reserve of strength due to a phenomenon called tension field action. This means that the elastic critical shear stress generally provides a conservative estimate of strength, despite imperfections, such as weld shrinkage stresses and lack of straightness. For this reason, the design stress can be taken as either the elastic critical shear stress or the yield shear stress, whichever is less. The shear stress, which will cause first yielding, is

Eq. 4

${\displaystyle {\mathrm{\tau <!--\; t\; -->}}_y=\frac{{\mathrm{\sigma <!--\; s\; -->}}_y}{\sqrt{3}}}$

The design value of shear stress (τ_Rd) is the lesser of either τ_cr or τ_y, i.e.,

Eq. 5

${\displaystyle {\mathrm{\tau <!--\; t\; -->}}_{Rd}={\mathrm{\tau <!--\; t\; -->}}_{cr}or\frac{{\mathrm{\sigma <!--\; s\; -->}}_y}{\sqrt{3}}(whicheverisless)}$

and the design shear strength (V_Rd) is approximately given by

Eq. 6

${\displaystyle V_{Rd}={\mathrm{\tau <!--\; t\; -->}}_{Rd}\times <!--\; \times \; -->platearea}$

where

τ_cr = is the elastic critical buckling shear stress.
ν is Poisson’s ratio.
E is Young’s modulus.
t is the plate thickness.
a is the plate length.
E is the Modulus of elasticity
k is the buckling coefficient.
b is the plate width (height) (see Figure 2).
t is the plate thickness.

Source:

• Structural Design First Principles, Michael Byfield

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