Monday, October 22, 2007

Column design and analysis


In a design situation, the expected load on a column would be known, along with the length required by the application. The designer would then specify the following:
1. The manner of attaching the ends to the structure that affects the end fixity.
2. The general shape of the column cross section (for example, round, square, rectangular, and hollow tube).
3. The material for the column.
4. The design factor, considering the application.
5. The final dimensions for the column.
It may be desirable to propose and analyze several different designs to approach an optimum for the application, so software such as this facilitates the process.
It is assumed that the designer for any given trial specifies items 1 through 4. For some simple shapes, such as the solid round or square section, the final dimensions are computed from the appropriate formula: the Euler formula, or the J. B. Johnson formula. If an algebraic solution is not possible, iteration can be done.
In a design situation, the unknown cross-sectional dimensions make computing the radius of gyration and therefore the slenderness ratio, KL/r, impossible. Without the slenderness ratio, we cannot determine whether the column is long (Euler) or short (Johnson). Thus, the proper formula to use is not known.

We overcome this difficulty by making an assumption that the column is either long or short and proceeding with the corresponding formula. Then, after the dimensions are determined for the cross section, the actual value of KL/r will be computed and compared with Cc. This will show whether or not the correct formula has been used. If so, the computed answer is correct. If not, the alternate formula must be used and the computation repeated to determine new dimensions.
Design factor
Under typical industrial conditions, the design factor of N = 3 is recommended. If the application is very smooth, a value as low as N = 2 may be justified. Under conditions of shock or impact, N = 4 or higher should be used, and careful testing is advised.
Column analysis

The procedure for analyzing straight, centrally loaded columns:
1. For the given column, compute its actual slenderness ratio.
2. Compute the value of Cc.
3. Compare Cc with KL/r. Because Cc represents the value of the slenderness ratio that separates a long column from a short one, the result of the comparison indicates which type of analysis should be used.
4. If the actual KL/r is greater than Cc the column is long. Use Euler's equation:
The equation gives the critical load, Pcr, at which the column would begin to buckle.
An alternative form of the Euler formula is often desirable. Note that:
But, from the definition of the radius of gyration, r,
Then
This form of the Euler equation aids in a design problem in which the objective is to specify a size and a shape of a column cross section to carry a certain load.

Notice that the buckling load is dependent only on the geometry (length and cross section) of the column and the stiffness of the material represented by the modulus of elasticity. The strength of the material is not involved at all. For these reasons, it is often of no benefit to specify a high-strength material in a long column application. A lower-strength material having the same stiffness, E, would perform as well.

5. If KL/r is less than Cc, the column is short. Use the J. B. Johnson formula:

Use of the Euler formula in this range would predict a critical load greater than it really is. The J. B. Johnson formula is written as follows:

The critical load for a short column is affected by the strength of the material in addition to its stiffness, E. As shown in the preceding section, strength is not a factor for a long column when the Euler formula is used.

Eccentrically Loaded Columns


An eccentric load is one that is applied away from the centroidal axis of the cross section of the column, as shown in the graphic help entitled “Eccentric column”. Such a load exerts bending in addition to the column action that results in the deflected shape shown in the figure. The maximum stress in the deflected column occurs in the outermost fibers of the cross section at the midlength of the column where the maximum deflection, ymax occurs. Let's denote the stress at this point as sL/2. Then, for any applied load, P,

Note that this stress is not directly proportional to the load. When evaluating the secant in this formula, note that its argument in the parentheses is in radians. Also, because most calculators do not have the secant function, recall that the secant is equal to 1/cosine.

For design purposes, we would like to specify a design factor, N, that can be applied to the failure load similar to that defined for straight, centrally loaded columns. However, in this case, failure is predicted when the maximum stress in the column exceeds the yield strength of the material. Let's now define a new term, Py, to be the load applied to the eccentrically loaded column when the maximum stress is equal to the yield strength. The equation then becomes

Now, if we define the allowable load to be

or

this equation becomes

Required

This equation cannot be solved for either A or Pa, so an iterative solution is required.
Another critical factor may be the amount of deflection of the axis of the column due to the eccentric load:

Find more details about column design and download FREE excel program to help you quickly calculate the critical buckling load.

More information with excel spreadsheet for download.





Stresses and Deformations in Beams

Reactions are the forces and/or couples acting at the supports and holding the beam and holding the beam in place. In some cases the user should enter a distributed load to account for the weight of the beam.

The shear V effective on a section is the algebraic sum of all forces acting parallel to and on one side of the section,


The bending moment is the algebraic sum of the moments due to applied loads and other applied moments to one side of the section of interest. Using value V bending moment can be calculated
where
x = position on the beam measured along its length
M0 = constant of integration evaluated from the boundary conditions.
A bending moment that bends a beam convex downward (tensile stress on bottom fiber) is considered positive, while convex upward (compressive on bottom fiber) is negative.

Moment and shear diagram constructed by plotting to scale the particular entity as the ordinate for each section of the beam. Such diagrams show in continuous form the variation among the length of the beam.

Magnitude of the direct stresses (tension and compressive) can be calculated from the direct stress formula
where
F = tensile/compressive force
A = cross sectional shape area

At the point of maximum bending stress, the flexure formula gives the stress

where
M = magnitude of the bending moment in the section
I = moment of inertia of the cross section with respect to its neutral axis
c = distance from the neutral axis to the outermost fiber of the beam cross section
A beam carrying loads transverse to its axis will experience shearing force. The resulting shearing stress can be computed from
where
I = rectangular moment of inertia of the cross section of the beam
t = thickness of the section at the place where the shearing stress is to be computed
Q = First moment of the area to the outside of the axis of interest with respect to the overall centroidal axis of the cross section of interest
V =shearing force

When a torque is applied to a member, it tends to deform by twisting, causing a rotation of one part of the member relative to another. The value of the maximum torsion shear stress can be computed

where
T = moment due to torque
c = distance from the neutral axis to the outermost fiber of the beam cross section
J = polar moment of inertia

When the beam is subjected to bending, the fibers on one side elongate, while the fibers on the other side shorten. These changes in length cause the beam to deflect. All points on the beam expect those directly over the support fall below their original position. The fundamental equation from which the elastic curve of a bent beam can be developed and the deflection of any beam obtained is,

If the loads are applied in vertical and horizontal plane or the forces are applied at angle to planes, in this case it is necessary to use a principle of superposition. Therefore result deflection at specific locations is computed from
where
fh = deflection in horizontal plane
fv = deflection in vertical plane

Using value of bending moment, slope can be calculated


where C0 = constant of integration evaluated from the boundary conditions.

The shortening/prolongation due to a direct axial tensile/compressive load is computed from

where
= total deformation of the member carrying the axial load
F = direct axial load
L = original total length of the member
E = modulus of elasticity of the material
A = cross-sectional area of the member

The following formula computes the relative elongation due to a direct axial tensile (compressive) load

When a shaft is subjected to a torque, it undergoes a twisting in which one cross-section is rotated relative to the other cross section in the shaft.

The relative twisting angle is defined


The angle of twist is computed from




where
= angle of twist (radians)
L = length of the shaft over which the angle of twist is being computed
G = modulus of elasticity of the shaft material in shear


Note:
All applied loads, forces and moments are considered to pass through the center of gravity of the cross-section.

The mass of the beam is not considered. If the consideration of the beam's mass is necessary, then represent it with a distributed load.


Type of beam support

There are two very common beam arrangements. A beam that is supported at one fixed end is commonly referred to as a cantilever. The other is simply supported at two points along the length of the beam. Both of these arrangements are statically defined and are referred to as “determinate.”
The beam that is supported at one end can experience shearing force, longitudinal reaction, twisting, and bending moments.
The beam supported at two points can result in shear forces and reactions due to axial and shear forces.





Distance to considered cross-section


Distance to the desired cross-section, where the calculation of the force factors and deformations is needed, is defined with “Distance to considered cross-section” variable.

Type of cross-sectional shape

For standard structural shapes, click on the table icon and a menu of standard shapes will appear and the software will automatically input the geometric values into the input page. According to the selected cross-section type the following geometrical data must be entered:

Circle:
D diameter mm

Rectangle:
b width mm
h height mm

Shapes:
w depth mm
b flange width mm
t web thickness mm
t1 flange thickness mm
A area mm^2
IX-X moment of inertia mm^4
IY-Y moment of inertia mm^4
SX-X section modulus mm^3
SY-Y section modulus mm^3

Hollow round tube:
d inner diameter mm
D outer diameter mm

Modulus of elasticity in tension

For the part of the stress-strain diagram that is straight, stress is proportional to strain, and the value of E is the constant of proportionality. That is,

This is the slope of the straight-line portion of the diagram. The modulus of elasticity indicates the stiffness of the material, or the resistance to deformation.

Modulus of elasticity in shear

The modulus of elasticity in shear (G) is the ratio of shearing stress to shearing strain. This property indicates a material’s stiffness under shear loading, that is, the resistance to shear deformation. Generally this material property is available from published data. It is commonly provided in this software by clicking on the table icon and selecting the appropriate material.

There is a simple relationship between E, G and Poisson’s ratio:

where
E = modulus of elasticity in tension
= Poisson’s ratio

This equation is valid within the elastic range of the material.

Forces and distributed loads

The shearing forces input list “Forces and distributed loads” has the following parameters: distance from the beam’s beginning to the application point “Distance”; length of a distributed load “Length”, (should be 0 for a concentrated force); force/load value; angle of projection’s inclination of the force’s action line on the OYZ plane to Z-axis “a ”. The “Value” sign defines the direction of the force’s action. Force projections on coordinate axis are found using the method of superposition.
Note: Distributed load has unit “lb/in” or "N/mm".

Bending moments

The list of bending moments has the following parameters: distance from the beam’s beginning to the application point “Distance”; value of the moment “Value” and angle of inclination of the moment action plane to Z-axis. The “Value” sign defines the direction of the moment’s action. Moment projections on OXY and OXZ planes are found using the method of superposition.



Twisting moments

“Twisting moments” list is used for twisting moments input. The list has the following parameters:
distance from the beam’s beginning to the application point, “Distance”, in,
value of the moment, “Value”, lb.in. or N.mm The sign defines the direction of application. The default is shown in the corresponding graphic help.

Monday, October 1, 2007

Fasteners

A set of n bolts is to be used to provide a clamping force of F between two components. The load is shared equally among the bolts. Specify suitable bolts, including the grade of the material, if each is to be stressed to K % of its proof strength. The variable K is called the demand factor.

The load on each screw is to be
P = F /n

Specify a bolt made from an SAE grade steel, having a proof strength s psi. Then the allowable stress is
sa = K.[s]

The required tensile stress area for the bolt is then
At = P / sa

From a table find the required tensile stress area for the thread. The required tightening torque will be
T = k1 x D x P

where
D - nominal outside diameter of threads
P - clamping load
k1- constant dependent on the lubrication present


Constant dependent on the lubrication present
For average commercial conditions, use k1 = 0.15 if any lubrication at all is present. Even cutting fluids or other residual deposits on the threads will produce conditions consistent with k1 = 0.15. If the threads are well cleaned and dried, k1 = 0.20 is better. Of course, these values are approximate, and variations among seemingly identical assemblies should be expected. Testing and statistical analysis of the results are recommended.

Related article: Bolts, Nuts, Screws, Studs