INTERIM REPORT - atwebpages.comstressanalysis.atwebpages.com/Nailbuckling.pdf1. Short compression nails or columns 2. Intermediate nails or columns 3. Long slender nails or columns

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INTERIM REPORT

Date:10/26/07 D Number: IR #:07-07 Name: John Stearns Subject: The mathematical theory of nail buckling and its applications. Abstract: Nails will occasionally bend or buckle when they are inserted into seal hubs during collector assembly. Nail buckling will usually occur when the seals are dry, increasing the force to insert them into the hub. Nail buckling is dependant on: nail diameter, length and material type. Tin plating does not contribute to the ability to resist buckling. This IR develops mathematical relationship for buckling for all nails in production. Conclusions:

1. EBC nails can be classified as intermediate type columns, but are very close to being long slender type columns for 3A, 2A and D size.

2. The 2A/3A nails are the easiest to bend at 31lbf, and the F nail requires the highest force to bend at 221lbf

Recommendations:

1. Test mathematical predictions to experimental results by bending nails in the laboratory for all sizes.

2. Determine force to insert nails for the EBC product line Discussion: Nails will occasionally bend or buckle when they are inserted into seal hubs during collector assembly. Nail buckling will usually occur when the seals are dry, increasing the force to insert them into the hub. The Important determinants of nail strength for a given brass material are as follows:

1. Length of nail 2. Diameter of nail 3. Moment of inertia of nail

From the diameter, the moment of inertia can be determined. This means that both of these maybe grouped together as one factor. The single factor used in column design which is




related to to both the nail cross sectional area and the moment of inertia is called the radius of gyration (r), so the list of determinates for nail strength reduces to:

1. Length of nail 2. Radius of gyration

To determine the radius of gyration for a nail, we compute the area (a) and moment of inertia (I) as

4*,

64* 24 dadI ππ

==

Then

4d

aIr ==

Both of these terms are measured in the same units, there ratio is called the slenderness ratio, and is given by:

rlratiosSlendernes =

Nails are essentially compression members when they are inserted into a seal hub, and can be subdivided into 3 categories

1. Short compression nails or columns 2. Intermediate nails or columns 3. Long slender nails or columns

The subdivisions can be visualized from the experimental curve taken from reference 1 shown in Figure 1, Where F/A is force per unit area.

Figure 1: Plot of F/A (at failure) vs. Slenderness ratio (l/r) for compression members




Figure 1 shows a experimental curve plotting force/area=stress vs. l/r. This curve has three distinct portions corresponding to the three categories of nails. For low slenderness ratios the experimental curve is a straight line. This means the stress is constant in this range regardless of the slenderness ratio. Nails in this range are considered short compression members and will not buckle under load and the failure of the nail is determined from its ultimate strength of the nail material. For brass, the ultimate strength is around 24,000 psi. As an example, consider a very short 2A nail, with diameter of 0.0455, area is 0.00162 in2 , so force to failure is 0.00162*24,000= 38.8 pounds to failure. None of the EBC nails are considered in this category. From Figure 1 it is clear that nail buckling does not become significant until a certain l/r value is exceeded. For members with large values of l/r the experimental curve in Figure 1 shows that a very small stress value may cause failure. Such members are categorized as slender columns. Experimental curves for slender columns fit an equation proposed by Euler,

( )rl

EaF 2π=

Most of EBC nails fail in the Intermediate column range. Intermediate columns are members whose l/r values lie in the range between short and slender columns. The actual shape of the curve describing intermediate columns is difficult to determine, a straight line form and parabolic fit form are most popular. See reference #1 for more detailed descriptions. The MIT software calculates several forms for immediate column types. End conditions: The strength of the nail is dependant upon the ways in which the nail is held, in addition to the slenderness ratio. The various end conditions maybe classified according to Figure 2. The effect of the end condition, will essentially, lengthen or shorten the nail, dependant on the condition for which it is held.

Figure 2: End Conditions

In the case of nails pushed into seal hubs, Condition B is most applicable, The K factor for condition B is 0.8, meaning the length is effectively reduced by 20% or K*l/r. A great variety of




column design characteristics has been presented thus far. The American Institute for steel construction (AISC) has provided guidelines for which formula to used. Table 1 below summarizes AISCs recommended practice. Table 1:Design Equations for Axially loaded metal columns Short Compression Members Intermediate Columns Long Slender Columns

NySy

aF=max

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛

−=E

rKlSy

NySy

aF

2

2

*41

π

2

2

⎟⎠⎞

⎜⎝⎛

=

rKlNy

EaF π

40<rl M

rl<≤40 M

rl>

The slenderness ratio defines the categories of columns, if,

l/r<40: Short column 40<l/r<M: Intermediate column

l/r>M: Slender column The M factor will be defined shortly, l/r factor for nails are given in Table 2 below Table 2: Column Parameters for EBC nails 4A 3A 2A C D F r (radius of gyration)(in)

0.0113 0.0113 0.0113 0.018 0.018 0.022

l (length)(in) 0.788 1.25 1.25 1.6 1.6 1.75 Diameter .0455 .0455 .0455 .072 .072 .091 a (area) (in^2)

0.00162 0.00162 0.00162 0.0040 0.0040 0.0065

l/r 69.275 109.89 109.89 88.88 88.88 76.92 To determine the M factor, we need to consider how the nail is held when it buckles. When nail buckling occurs its when nail is fist pushed into the seal. In this condition, one end is fixed while the other is pinned. The end condition (K) factor according to ref 1 is K=0.8 The M factor is based on the nails material properties and end condition (K) and can be defined as

2**2

KSEM

y

π=




Where E= Youngs modulus for brass (1.5e7 psi), Sy= Yield stress (50,000 psi). We can now determine the category of columns for various nails, this is shown in table 2 below Table 3: Categorizing EBC nails 4A 3A 2A C D F M 96.19 96.19 96.19 96.19 96.19 96.19 l/r 69.275 109.89 109.89 88.88 88.88 76.92 Category Intermediate Slender Slender Intermediate Slender Intermediate The final factor in AISC column calculations is the determination of factor of safety (Ny)

Factor of safety vs l/r

y = -6E-08x3 + 4E-07x2 + 0.0029x + 1.6699R2 = 0.9997

1.65

1.7

1.75

1.8

1.85

1.9

1.95

0 50 100 150

l/r (slenderness ratio)

Ny

(fact

or o

f saf

ety)

NyPoly. (Ny)

Figure 3: Factor of safety for AISC columns

Now taking all of these factors for nail buckling, the following predictions are made and compared to MIT’s Column calculations spread sheet as a reference. The MIT column calculator spreadsheet is a write protected spreadsheet filled with macros and look-up tables to facilitate ease of use. Based on geometric static values and material values, it will calculate effective column length and force to buckle the column. The results are plotted for the euler, linear, parabolic and secant solutions for intermediate and long slender columns. Examples can be seen in Figures 4 through 7. These values can then be compared to experimental values obtained in the lab. Justin Begg did this experiment and bent 8 AA NGS nails, with an average force of 22.8 lbf. Nail head geometry and nail point is not accounted for, but should not change any of the resulting calculations.




Table 4: Predicted buckling forces for nails 4A 3A 2A C D F ASIC value 60.2 28.2 28.2 110 110 221 MIT (parbolic)

60 31 31 124 124 221

MIT (linear) 52 31 31 111 111 195 ODE Calc 50.29 19.9 19.9 88.2 88.2 162.7 Authors calculations are in good agreement with the MIT spreadsheet and differs slightly from the experimental results for the 2A nail. Since most nails fall in the intermediate range, the nails tend to fail due to inelastic buckling. This occurs then the stress in the nail exceed its proportional limit before reaching the critical load. If the nails are considered slender, elastic buckling will generally cause failure. The derivation of the critical buckling force

L Θ

y Figure 4: Buckling of a long column

The failure of a long nail is predominantly one of buckling. A first mode shape buckling is shown in Figure 4. The ends are pinned and are free to rotate. The load P is increased until the column bows outward as shown. The bending load can be computed. The instant the load, P is large enough, buckling occurs and a bending moment, M is induced. The moment caused by buckling is negative and is given by M=-PΘ . The standard beam deflection ordinary differential equation (ODE) can be used to solve the buckling problem. This equation is:

0=+⎟⎠⎞

⎜⎝⎛ θθ P

dxdEI

dxd

Equation 1 Or

P x




02

2

=+ θθ PEIdxd

Equation 2 To obtain a solution to the differential equation for beam bending (Equation 2), lets divide Equation 2 by EI (E= Youngs modulus and I is moment of inertia, the term EI is often called the bending stiffness) Yielding Equation 3

02

2

=+EIP

dxd θθ

Equation 3

, let EIPk =2

Then Equation 3 becomes

022

2

=+ θθ kdxd

Equation 4 This is a linear homogenous ordinary differential equation with constant coefficients. From any differential text book, the solution of this ODE has the form

mxCe=θ Equation 5

Substituting Equation 5 into equation 4 by knowing the derivatives are

mxnCedxd

=θ and mxCem

dxd 2

2

2

=θ

Substituting into Equation 4, takes on the form

022

12 =+ mxmx eCkeCm

Equation 6 The generic C, has been replaced by C1 and C2 to donate that they are two different constants. Note that mxe can never be zero (0), so that forces

022 =+ km or kim ±= , where i is the imaginary number. Substituting this back into Equation 5 yields:

kixCe=θ . Using this new expression into Equation 6, it now becomes

021 =+ −ikxikx eCeCθ Equation 7




The expression in Equation 7 with e, i can be transformed using Euler’s formula. Euler’s formula states that:

)cos()sin( kxBkxA +=θ Equation 8

Before we substitute Euler’s formula into Equation 7 we need to establish boundary conditions to determine constants A & B. Refer back to Figure 4. The top and bottom of the column cannot bow and is fixed so the side deflection at these points are zero, ie 0 and 00 == (L))( θθ , Substituting into Equation 8 and also knowing sin(0)=0 and cos(0)=1, yields B)( == 00θ and

)sin(0 kLA(L) ==θ Equation 9

The only way Equation 9 can equal 0 is if, and only if kL=π, 2π, 3π or kL=n π, n=1,2,3…. or

Lnk π

=

Equation 10

But we also stated that EIPk =2 , so that now

2

222

Lnk π

= , and 2

22

Ln

EIP π

= , solving for critical load P

2

22

LEInP π

=

Equation 11 Using equation 11, the critical force can now be calculated. The results are shown in Table 4. Note equation 11, as n is increased from 1 to 2,3 etc, the predicted buckling force increase by n2. Typically, the first mode shape buckle (n=1), the nail bows in the middle and buckles. Higher mode shape buckling (n=2 or more) requires more energy to buckle the same nail and the resulting buckled nail shape is also different. These higher order buckling shapes rarely occur in most metal columns, hence first mode buckling is the most important. In Summary, Theoretical calculations compare reasonably well to experimental results in the lab. AISC column calculations adjust the length of the column by considering the method in which it is held. The MIT calculation spreadsheet also predicts similar results. Figure 6 shows the spreadsheet for the C/D nail. An error exists on the nail length used (1.5 inches), but the numbers in Table 4 are correct for the real length of 1.6 inches. This MIT calculator had a one month usage so, the adjustment of the error could not be printed because the software has expired. The software can be purchased for $14 from MIT.




Figure 4: MIT Column calculator for 2A & 3A nails




Figure 5: MIT column calculator for 4A nail




Figure 6: MIT column calculator for C & D nail




Figure 7: MIT column calculator for F nail




Written by: Date: (signature) Read and understood by: Date: (signature) Read and understood by: Jeremy Paul (typed) References:

1. “Statics and Strength of Materials” Bassin, Bodsky and Wolkoff, 1979. 2. Slender strut (column) buckling ver 1.13 (26.06.2006), MIT. http://www.mitcalc.com 3. IR01-10, Janmey, R.M. “Finite element analysis- Nail insertion force of the 2A next generation

and E91/X91 – Comparison of tin plated versus Burnished Brass nails”

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INTERIM REPORT - atwebpages.comstressanalysis.atwebpages.com/Nailbuckling.pdf1. Short compression nails or columns 2. Intermediate nails or columns 3. Long slender nails or columns