Structural-mode instability can lead to unexpected strut failure
By King-Yuen Chu, DrEng, PE,On-Board Corp.
Evaluating instability of short, linear-type pipe supports ensures conservative design
Linear-type pipe supports such as rigid struts and snubbers are normally checked for buckling strength using Euler`s buckling load, which is inversely proportional to the square of the effective column lengths. This check of compression members for buckling load is sufficient when they are supported rigidly at the two ends. However, when the end supports in the orthogonal directions are not quite rigid, there may exist another unstable condition for the compression members, whereby the critical load may be smaller than the Euler load.
This condition could occur for pipe supports attached to flexible piping systems, especially when the length of the compression member is very short. Therefore, when the design involves a very short strut, the piping stiffness at the strut location should be evaluated to assure that the specific allowable load provided by strut manufacturers is indeed conservative.
Strut attached to building structure
Figure 1(a) shows schematically a strut connected to a pipe through a pipe clamp, and attached to a building structure through a bracket. Figure 1(b) shows the structural model representing the strut. There are two unstable conditions which could lead to a strut failure. Figure 1(c) shows the conventional failure mode. Its critical load is the Euler load as follows:
where E1, I1 and l1 are the elastic modulus, moment of inertia and length of the strut respectively.
The other unstable failure mode is reached when the spring support representing the pipe stiffness starts to move laterally as shown in Figure 1(d). The critical load of this unstable condition is determined through the force equilibrium condition as follows:
where k1 is the spring constant representing the pipe stiffness at the orthogonal direction of the strut, and k2 is the spring constant representing the support stiffness of the building structure and attachments, i.gif. bracket and baseplate. Normally, k2 is much larger than k1. In that case, Equation 2 may be simplified as follows:
It is obvious that the shorter the strut is, the smaller the critical load P2 is. This is contrary to the conventional buckling load which is greater for a shorter strut. This failure mode can be more critical than the buckling failure mode when the strut is very short and the piping does not have many lateral supports. For example, the minimum strut length (pin to pin) of a standard pipe support, Part 2000-20, from Bergen-Paterson Pipe Support Corp.,1 is 18 inches (in.). Its allowable load (Level A & B) is 20,000 pounds (lbs). If the pipe is flexible in the direction orthogonal to the strut, the allowable load can be much smaller than 20,000 lbs. Assuming the pipe at this location has a stiffness of 1,000 lbs/in., then
Consistent with the requirements of ASME III2 and AISC specification3 the allowable load should be as follows:
This allowable load is only 47 percent of the support`s specified allowable load. The allowable load will reach the specified allowable only when k1 is 2,130 lbs/in or larger. Therefore, when the strut is very short, it is necessary to evaluate the piping stiffness to assure that the specified allowable load is indeed conservative.
For non-standard pipe supports, P2 will be critical and will control the design if the following condition exists:
Strut attached to a column
Figure 2(a) shows a typical design of a strut-column assembly. There are two critical loads similar to the first case, except the spring constant k2 involved in deriving P2 is quite complicated. It is actually not a constant value because the deflection of the column is not only a function of the load perpendicular to the column, but also includes the P effect.3 Therefore, it is more practical to derive the critical load in terms of the effective column length as follows:
where E2, I2 and K are theelastic modulus, moment of inertia and effective length factor of the column respectively.
Figure 2(b) shows the deflected shape of the strut-column assembly. The bending moment of the column which causes the deflection is:
where d1 and d2 are the translational displacements at the two ends of the strut as shown in Figure 2(b).
Equation 6 is transferred to the following differential equation:
The general solution is:
The two constants c1 and c2 are determined by their boundary conditions y = 0 and y` = 0 at x = 0.
The characteristic equation is determined by the condition of y” = 0 at x = l2 as follows:
Finally, the characteristic equation of the unstable condition is formulated as:
The solution of this equation requires first assuming a value for K, and then calculating the corresponding l1/l2. Figure 3 shows a series of curves plotted for the effective length factor K versus l1/l2 for various values of a. The curve for a = 0 reflects the case of rigid support at the pipe connection, i.gif. k1= . When the strut length is much longer than the column length, the value of K will approach 2, which is the effective length factor of a cantilever subjected to a load parallel to the undeformed column axis. As in the case of a strut attached to a building structure, the softer the piping is, the smaller the critical load. The following two examples (column size W 8 x 18) illustrate the use of Figure 3.
1. Maximum strut length for specified allowable load of 20,000 lbs:
From Figure 3: K = 7.1; therefore,
This allowable is 74 percent of the specified allowable for the strut.
2. Minimum strut length for specified allowable load of 20,000 lbs:
From Figure 3: K=9.3; therefore,
This allowable is only 43 percent of the specified allowable for the strut.
King-Yuen Chu, Dr-Eng, PE, is with On-Board Corp., where he is a project engineer responsible for structural, mechanical and pipe stress analysis.
1 Catalog No. 77NF, Bergen-Paterson Pipe Support Corp., Boston, Mass.
2 ASME Boiler and Pressure Vessel Code, Section III, Subsection NF, American Society of Mechanical Engineers, 1986 ed.
3 Manual of Steel Construction, Ninth Ed., American Institute of Steel Construction Inc.
Timoshenko, S.P., and J.M. Gere, Theory of
Elastic Stability, Second ed.
Buergermeister, G., and H. Steup, Stabilitaet- stheorie, Akademie-Verlag, Berlin, 1959.
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