* Low Aspect Ratio Plates ( Topic 7: Additional Notes)
Advanced Local Structural Design & Analysis of Marine Structures
OST
Since 2015
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OST
Since 2015
All rights reserved by DK
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[Part I] Plastic Design of Structures
– Plastic theory of bending (Topic 1) – Ultimate loads on beams (Topic 2)
– Collapse of frames and grillage structures (Topic 3)
[Part II] Elastic Plate Theory under Pressure
– Basic (Topic 4)
– Simply supported plates under Sinusoidal Loading (Topic 5) – Long clamped plates (Topic 6)
– Short clamped plates (Topic 7)
– Low aspect ratio plates, strength & permanent set (Topic 7A)
[Part III] Buckling of Stiffened Panels
– Failure modes (Topic 8)
[Theory of Plates and Grillages]
Adv. Local Structural Design & Analysis of Marine Structures (Overview)
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Introduction
In this lecture we will:
1. Define a normalised capacity for plates, and examine how to extend the capacity description to low aspect ratio plates (i.e. not long plates)
2. Examine the Clarkson method for designing plates for a specified permanent set.
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• We define anormalised load Q:
𝑄 = 𝑝𝐸 𝜎𝑌2
• From previously, for long plates with Poisson’s effect 𝑝𝑌 = 2
1 − 𝜈 + 𝜈2 𝑡 𝑏
2
𝜎𝑌
• For the yield condition in bending we have:
𝑄 = 𝑝𝑌𝐸
𝜎2 = 2 𝑡
𝑏
2
𝜎𝑌 𝐸 𝜎2
Low Aspect Ratio Plates (1/16)
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• We define 𝛽 = 𝑏
𝑡
𝜎𝑌
𝐸 as the slenderness ratio, which lets us write:
𝑄 =
21−𝜈+𝜈2 1 𝛽2
• This is for long plates, a >> b (say a > 3b):
Low Aspect Ratio Plates (2/16)
𝑄 = 𝑝𝑦𝐸 𝜎𝑌2
= 2
1 − 𝜈 + 𝜈2 𝑡 𝑏
2
𝜎𝑌 𝐸 𝜎𝑌2
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Low Aspect Ratio Plates (3/16)
• For low aspect ratio plates (a < 3b), we can use the equation:
𝑄 = 2
1 − 𝜈 + 𝜈2 1
𝛽2 1 + 0.6 𝑏 𝑎
4
• The extra term results in an increase of the load capacity as demonstrated in the following table:
• This is for long plates, a >> b (say a > 3b):
𝑄 =
21−𝜈+𝜈2 1 𝛽2
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Low Aspect Ratio Plates (4/16)
• The above equation refers to the yield condition. When just brought to yield, there will be no permanent deformation (set).
• For loads above Qy there will be increasing levels of permanent set.
• So far we have discussed load-deflection curves. We have approached plastic bending strength from the point of view of load capacity.
• Another approach to plastic design is to allow some level of permanent set, as a % of the plate thickness, or as a % of plate span.
For low aspect ratio plates (a < 3b),
𝑄 = 2
1 − 𝜈 + 𝜈2 1
𝛽2 1 + 0.6 𝑏 𝑎
4
𝑄 =
21−𝜈+𝜈2 1 𝛽2
For long plates, ( a > 3b):
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Low Aspect Ratio Plates (5/16)
• Refer to Hughes and Paik (2010, Chapter 9). Hughes describes the experimental and analytical work of Clarkson
• Hughes develops a set of ‘design’ plots. Using the plots, you can determine capacity of scantlings to achieve a given level of permanent set (i.e. given level of denting for the design load).
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Since 2015
Low Aspect Ratio Plates (6/16)
• Figure 1 shows plate capacity Q, vs. the slenderness ratio β, for various aspect ratios (a/b) of plate. This particular plot gives values for a level of permanent set of:
𝑤𝑝
𝛽𝑡 = 1.0
• This is equivalent to:
𝑤𝑝
𝑏 = 1.0 𝐸 𝜎𝑌
• For a typical shipbuilding steel
(σY = 250 ~ 400), this is approximately 𝑤𝑝
= 0.039
Figure 1
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Low Aspect Ratio Plates (7/16)
EXAMPLE:
• For a plate with the following parameters, what is the load capacity and permanent deflection according to Figure 1?
a = 1000 mm b = 500 mm
σY = 300 MPa t = 15 mm E=207000MPa
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Low Aspect Ratio Plates (8/16)
SOLUTION:
• Step 1: findβ 𝛽 = 500
15
300
207000 = 1.59
• Step 2: lookupQ on the a/b = 2 curve.
Q = 4.1
• Step 3: find p
𝑝 = 4.1×3002
207000 = 1.78 𝑀𝑃𝑎
• Step 4: find wp
a = 1000 mm b = 500 mm t = 15 mm σY = 300 MPa E = 207000MPa
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Low Aspect Ratio Plates (9/16)
• Note that pY = 2.25 σY (t/b)2 = 0.39 MPa
• The p at 19 mm deflection is not only above yield. It is above the 3 hinge load pc.
• The plots in Figure 2 show four additional cases, for a range of permanent set. These plots can be used just like Figure 1.
• We will explore avariety of ways to use the plots.
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Since 2015
OST
Since 2015
[Part I] Plastic Design of Structures
– Plastic theory of bending (Topic 1) – Ultimate loads on beams (Topic 2)
– Collapse of frames and grillage structures (Topic 3)
[Part II] Elastic Plate Theory
– Basic (Topic 4)
– Simply supported plates under Sinusoidal Loading (Topic 5) – Long clamped plates (Topic 6)
– Short Clamped plates (Topic 7)
– Additional (Low aspect ratio plates, strength & permanent set)
[Part III] Buckling of Stiffened Panels
– Failure modes (Topic 8) – Tripping (Topic 9)
[Theory of Plates and Grillages]
Adv. Marine Structures / Adv. Structural Design & Analysis (Next class)
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