AML 883 Properties and selection of engineering materials LECTURE 2. Materials choices for stiffnesslimited design Density and Modulus M P Gururajan Email: guru.courses@gmail.com Room No. MS 207/A 3 Phone: 1340 Course homepage: http://aml883.wikidot.com/
Structural materials and their properties Stiffness, strength and toughness at reasonable densities Materials for fan blades in windtunnels: touches on almost all the above topics Starting point Case study on choice of materials for fan blades in windtunnels
Reference S E Richardson, Design and manufacture of wood blades for windtunnel fans, NASA report (NASA/CR 1998 196708), 1998. Available at http://hdl.handle.net/2060/19980055204 All figures and data (unless noted otherwise)
Windtunnel A research tool to study effect of air moving around solid objects Air is blown or sucked through a duct using a series of fans Image courtesy: wiki
Windtunnel fan A closer look at the fan Fans may be powered by turbofans a type of jet engine What is the material of the fan blades? Why? Image courtesy: wiki
Sizes of windtunnels Based on fan diameter and power consumption Small less than 6 feet Medium 8 to 24 feet Large more than 24 feet
A zoomed out look! NASA Ames Research Center (world's largest windtunnel?) during construction 1940 1944 With wooden fans 1978 1986 Rebuilt
A closer look at the fan
A closer look at the blades
Loading of the blades (i) Air Bending Moments (ABM) the bending moments due to the aerodynamic forces on the blade sections and (ii) centrifugal loading due to rotations By suitable orientation of the main axis of the blade, centrifugal loading can be made to result in Centrifugal Force Bending Moment (CFBM) that is opposite to ABM in sign The Net Bending Moment (NBM) is reduced; hence, lower stresses in the blade
How to get the required CFBM CFBM: can be designed to have the required value by choosing the correct Track and Sweep Track offset of the C. of G. of blade sections from the rotation plane Sweep offset of the C. of G. Of blade sections from the true radial axis Note: usually, CFBM = (½) ABM. If not, what happens at the shutdown when ABM is zero, suddenly?
Track schematic
Sweep schematic
The first requirement! CFBM can be calculated from track and sweep alone, only to first approximation Provided there is no deflection of blade under load, i.e., the blades should be stiff! Stiffness the first desired property that we shall study (along with density)! Stiffness (S): For a given load (F), how much is the deflection ( ) S = F/
Questions! What is stiffness limited design? Which material properties are important if the design limitation is stiffness? (Density and modulus) How does density and modulus of materials come about from an atomistic point of view? Can density and modulus of materials be manipulated if not at the atomic level, at least at the microstructural level?
A matter of our approach Mathematical treatments tend to be relatively simple, and are used only where genuinely relevant and useful. Nevertheless, quantification is integral to the book, and this is one of its great strengths. T W Clyne, reviewing Materials by Ashby, Shercliff and Cebon for Materials Today, November 2007. Less mathematics (derivations); however, not less numeracy (calculations)
Structural components Ties Columns Beams Shafts Shells
Tie tensile loading
Column compressive loading
Beam Bending
Shaft Twisting
Shell Pressure difference
Elastic extension or compression Consider a rod of uniform cross sectional area A, Young's modulus E, and length L 0 subjected to a compressive or tensile load of F The stiffness, S = F/ S=AE / L 0
Optimization: light, stiff rods Cast balconies supported by cylindrical tie rods Its length is specified ( ) L 0 It must carry a tensile force F without extending elastically by more than Image courtesy: buildingscience site
Optimization: light, stiff rods Stiffness must at least be S* = F/ Of course, we also want it to be strong and tough Objective: how light can it be? The cross sectional area is the free parameter
Optimization: light, stiff rods Aim: minimize mass Free parameter: cross sectional area Constraint: given minimal stiffness and length of the rod How do we do this, preferably, mathematically?
The objective function Seek an equation that describes the quantity to be maximized or minimized; this equation is called the objective function Question: what is it that we wish to minimize?
The objective function We wish to minimize the mass, m What is the objective function in our case?
The objective function m=a L 0 How can we reduce m?
The objective function m=a L 0 Reduce m by reducing A. How much do we want to reduce?
The cosntraint equation Only so much as not to reduce the stiffness below the required minimum of S* = A E / L 0 Material with low E, large A is needed (or vice versa)
Optimization with constraint m= A L 0 S* = A E/L 0 m = S* L 0 2 / E
Material index of the problem m = S* Thus, lightest tie rod is that which will have the lowest L 0 2 /E / E ratio Typical to report in terms of maximum: M t =E / Specific stiffness: M t =E /
The density modulus combination Specific stiffness, M t =E / Materials with high value of specific stiffness are the best candidates (provided they meet other constraints) Modulus Density combination Shows you why the Modulus Density property chart is a good idea!
Optimization: summary Identify the quantity to be minimised or maximised Write the objective function in terms of the free variable Find the constraint equation Eliminate the free variable in the objective function using the constraint You obtain the material index for the problem in hand
Other optimizations Material cost Weight a light, stiff panel Weight a light, stiff beam The book describes several other case studies! Take a look. (Problem sheet 1: Available at the homepage!)
Summary Many structural applications require light and stiff components Light density Stiff Elastic modulus Question: What gives rise to the specific values of density and elastic modulus for the materials? Can we manipulate them? How?