 # SEM212 2016: Sagging of Beams under their Own Weight & Sandwich panels - Engineering Assessment Answers

November 02, 2018
##### Author : Julia Miles

Solution Code: 1EJE

## Question: Engineering

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### Engineering Assignment

Sagging of beams under their own weight

The aim of this demonstration is to provide opportunity to develop familiarity with beam to introduce the concept of a material index.

Governing equations:

I. Deflection at the mid?point of a beam under its own weight, supported at its ends: Questions:

1. Rearrange Equation 1 so that it provides an expression for E

2. Calculate the 2nd moment area of inertia I and then fill out the table below for the three materials provided in the demonstration class and back calculate E using Equation 1. The first few rows provide data for calculating the density. Present all data in a table format

3. Estimate a range of uncertainty for the data on the table. Then calculate an uncertainty error range for the modulus of elasticity E

4. For your three materials, calculate the critical height for the column buckling under its own weight.

5. From inspection of the above two equations, determine the material indexes that must be used to

a) Minimise the deflection of a beam under its own weight and;

b) Maximise the critical buckling height.

6. Using the CES software, or a chart of E vs density, identify two materials that show optimum values of the material index for free column buckling under self?weight.

7. Give an example of a free (or nearly free) standing column from nature or from man?made structures and roughly estimate how far (in terms of percentage %) the column is from its maximum allowable height. Provide a justification for this percentage Experimental Data table

Data Tables   Sandwich panels

The aim of this demonstration is to ‘meet’ some sandwich panels and to become familiar with the basis of their superior performance in bending for lightness combined with stiffness. In particular, we are interested to know if sandwich panels can enable us to enter the desired hole in property space shown below: Figure 1. Density – stiffness plot showing the ‘desirable’ hole in the top left corner.

Governing equations:

1. Basic beam deflection equation: 2. Sandwich panel ‘stiffness’ in bending:

Despite the complexity of the cores of sandwich panels, it is convenient to compare sandwich panels with solid materials of the same outer dimensions. This helps show how good sandwich panels can be. To do this we can define an effective stiffness, E~ , which we simply use in the panel deflection  equation: It is also convenient to define a similar parameter for the effective or apparent density, p ~ , based on the panel mass and its outer dimensions

Questions:

1. You have in front of you some examples of sandwich panels. Measure and record the masses of the panels and their outer dimensions to establish their effective densities. 2. Estimate the effective stiffness E~ of one of the panels by supporting its ends, applying ‘narrow’ weights to the mid span and measuring the deflection. Record measurements and calculations below. Plot the result on Figure 1 or a version of it and comment on whether or not the material is in the desirable ‘hole’. Discuss what you think of the potential of the material to achieve desirable properties.

3. Most sandwich panels are made up of two face sheets and a core. The core is often light, weak and contributes very little to the stiffness of the panel. This enables a very rough analysis to be made by assuming that the core is just air and that the cross?section of the panel is equivalent to a rectangular crosssection tube. Using this assumption derive an equation for E~ in terms of the outer dimensions of the panel (b>>h) and the value of E for the face material. Hint: equate equations 1 and 2 for the same values of span, deflection, force and C1 and note that because b>>h the rectangular tube second moment of area can be given as I ~ h2tb / 2 . Note that you are not calculating a value here, just finding an expression in terms of the aforementioned variables. 4. Use the assumption in the previous point to derive an approximate equation for the effective density, p ~ , of the panel. Hint: start by assuming the panel is b x b x h in size and calculate the volume of material in the two face sheets of thickness t (you can ignore the side strips). Your equation will be in terms of t/h.

5. Use your equations to estimate the effective stiffness of a wood (parallel direction) face?sheet sandwich panel (with a negligible weak light foam core). Assume the panel apparent density is 50 kg/m3. Comment on whether the result falls in the ‘desired’ region of property space in Figure 1.

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