Strength Analysis of Rotating Cylindrical Shells
Abstract Generated abstract
This paper presents a strength calculation procedure for thin-walled rotating cylindrical shells used in chemical equipment, such as rotating autoclaves and kilns, under combined internal pressure, bending, and torsion. Treating the shell as a hollow beam on elastic supports, it separates the assessment into static load-carrying capacity under short-term maximum loads and endurance under long-duration cyclic loading caused by loads rotating relative to the shell. The method gives formulas for equivalent stress, plastic-deformation safety margin, and fatigue safety factor, including allowances for asymmetric stress cycles, weld stress concentration, surface condition, and scale effects. It concludes with a recommended calculation sequence and suggested minimum safety margins for static and fatigue strength.
Full Text
STRENGTH ANALYSIS OF ROTATING CYLINDRICAL SHELLS
D. K. Nechaev
In the chemical industry, units are often used whose principal elements are thin-walled axisymmetric shells subjected to excess pressure and, during operation, undergoing rotational motion at a small angular velocity. Such shells include the shells of rotating autoclaves, the shells of clinker-burning kilns in which pressure is absent, etc.
The rotating shells of such units are subjected to the combined action of bending and torsion (Fig. 1) and must be calculated with allowance for the complex stress state.
The loads taken up by parts of equipment in chemical production may be regarded as practically constant, since their magnitude deviates from some mean value within the limits of ±10–15%.
Fig. 1. Loading diagram of the shell
Loads that are constant in magnitude (from internal pressure, the shell’s own weight, and the weight of the product) cause both constant and variable stresses in the shell section. Loads from internal pressure, rotating together with the shell (i.e., stationary relative to the shell), cause constant stresses in its section. Loads from the shell’s own weight and from the weight of the product are stationary in space (i.e., rotate relative to the shell) and, consequently, cause variable stresses in the shell section.
Thus, the calculation of the load-carrying capacity of a rotating shell under the indicated type of loading must consist of a calculation of static load-carrying capacity and a calculation for endurance.
For the strength analysis of shells that are in a complex stressed state, the method proposed in [1] for the strength analysis of shafts may be used.
The shell may be regarded as a hollow beam, loaded by bending and twisting moments, resting in bearings as supports. The sliding bearings used for autoclaves are deformed by the load from the shell; therefore it may be considered as a beam resting on elastic foundations.
Calculation of the Shell for Static Strength
The calculation of the shell for static strength should be carried out for the greatest short-term loads, taking into account dynamic forces and vibrations.
For driving machines in chemical production, as a rule, ordinary three-phase electric motors are used; for these, the maximum torques developed by the motor (starting and overturning torques) are approximately twice the values corresponding to the continuous power. This makes it possible, in a first approximation, to take as the greatest short-term load twice the value of the continuous load. Accordingly, in the general case, in order to determine the internal forces it is necessary to calculate the bending and twisting moments separately for the greatest short-term loads and separately for the greatest continuously acting loads.
The static load-carrying capacity of shells made of ductile steels is determined by the limiting deformation loads, i.e., by the loads at which the greatest nominal reduced stresses in the shell section reach the yield point. Thus, the criterion for the static load-carrying capacity of shells should be taken as the prevention of plastic deformation.
The calculation of the shell for static strength is reduced to determining the stress from rotating and nonrotating loads and calculating the strength margin according to the selected load-carrying-capacity criterion.
For the case of shell loading under consideration (Fig. 1), the equivalent stress in its section, according to the energy theory of strength, may be found from the expression
\[ \sigma_{\mathrm{e}}=\sqrt{\sigma_x^2+\sigma_y^2-\sigma_x\sigma_y+3\tau_{xy}^2}. \]
In our case, \(\sigma_x=\sigma_1+\sigma_u;\ \sigma_y=\sigma_2;\ \tau_{xy}=\tau_k\). The values of these stresses are determined by the formulas:
a) meridional (axial) stress in the shell due to internal pressure
\[ \sigma_1=\frac{pD_b}{4S\varphi}; \]
b) circumferential (hoop) stress in the shell due to internal pressure
\[ \sigma_2=\frac{pD_b}{2S\varphi}; \]
In these formulas: \(p\) is the excess pressure of the medium, \(D_b\) is the internal diameter of the shell, \(S\) is the shell-wall thickness, \(\varphi\) is the coefficient of weakening of the wall by the welded seam.
c) bending stress
\[ \sigma_u=\frac{M_{u\max}}{W_u}, \]
where \(M_{u\max}\) is the maximum briefly acting bending moment,
\(W_u\) is the section modulus of the shell section in bending,
d) torsional stress
\[ \tau_k=\frac{M_{k\max}}{W_k}, \]
where \(W_{k\max}\) is the maximum briefly acting torque,
\(W_k\) is the section modulus of the shell section in torsion.
Thus, the formula for determining the equivalent stress takes the form
\[ \sigma_e=\sqrt{(\sigma_1+\sigma_u)^2+\sigma_2^2-(\sigma_1+\sigma_u)\sigma_2+3\tau_k^2}. \]
The margin of strength with respect to plastic deformations is determined by the formula
\[ n_T=\frac{\sigma_T^t}{\sigma_e}\geq [n_T], \]
where \(\sigma_T^t\) is the yield strength of the shell material at operating temperature,
\(\sigma_e\) is the equivalent (maximum) stress in the most heavily loaded section of the shell.
\(n_T\) is the allowable margin of strength.
Taking into account the approximate design scheme and the absence of proper experimental verification, the value of the margin of strength should be taken as \([n_T]\geq 2\).
Calculation of the Shell for Fatigue Strength
The shell should be calculated for endurance under the greatest long-duration load, taking into account the loading regime.
The greatest long-duration load may be taken as the maximum load whose recurrence during the service life of the part is not less than \(10^3\) cycles [1].
As a result of the combined action on the shell of rotating and non-rotating loads, stresses arise in its sections that vary according to an asymmetric cycle (Fig. 2). Loads that do not rotate with respect to the shell (internal pressure) create in it the constant component of the stress cycle, i.e., the mean stress of the cycle, \(\sigma_c\). Loads rotating with respect to the shell (its own weight, the weight of the product) create in it the variable component of the bending-stress cycle—the amplitude of the cycle, \(\sigma_a\). Thus, the cycle amplitude is caused by loads rotating relative to the shell, while the mean stress of the cycle is caused by non-rotating loads.
Under the action of external loads, the following arise in the cross section of the shell: tensile (constant) stress $\sigma_1$—from internal pressure; bending (amplitude) stress $\sigma_i$—from the shell’s own weight and the weight of the product; and also shear stress $\tau_k$—from the torque, varying from zero to a maximum. The formulas for determining the nominal value of these stresses were given above.
Since the greatest short-term and the greatest long-term loads are proportional to a single parameter (bending moment, torque), the values of the nominal stresses in torsion and bending for long-term loads can be obtained by multiplying the stresses for short-term loads by the ratio of the long-term moment to the greatest short-term moment.
Fig. 2. Diagrams of stress-variation cycles.
The endurance calculation reduces to determining the safety factor with respect to the fatigue limit in the critical sections. Critical sections are those sections of the shell in which high nominal stresses act, as well as sections in the stress-concentration zone.
Under the combined action of bending and torsion, the safety factor is determined by the formula [1]
$$ n=\frac{n_\sigma \cdot n_\tau}{\sqrt{n_\sigma^2+n_\tau^2}}, $$
where $n_\sigma$ is the safety factor with respect to normal stresses;
$n_\tau$ is the safety factor with respect to shear stresses.
The values of these coefficients are found from the expressions:
$$ n_\sigma=\frac{\sigma_{-1}}{\dfrac{k_\sigma}{\varepsilon_\sigma \beta}\sigma_a+\psi_\sigma \sigma_c}, $$
$$ n_\tau=\frac{\tau_{-1k}}{\dfrac{k_\tau}{\varepsilon_\tau \beta}\tau_a+\psi_\tau \tau_c}. $$
In these formulas
$\sigma_{-1}$, $\tau_{-1k}$ are the fatigue limits of the material in bending and, respectively, in torsion;
$\sigma_a$, $\tau_a$ are the normal and, respectively, shear amplitude stresses;
\(\sigma_c,\ \tau_c\) — the normal and, respectively, shear mean stress;
\(k_\sigma,\ k_\tau\) — the effective stress-concentration factor in bending and, respectively, in torsion;
\(\varepsilon_\sigma,\ \varepsilon_\tau\) — the coefficient of influence of the absolute dimensions of the cross section (scale factor) in bending and, respectively, in torsion;
\(\beta\) — a coefficient taking account of the state of the surface;
\(\psi_\sigma,\ \psi_\tau\) — the coefficient for reducing an asymmetric cycle to an equally dangerous symmetric one in bending and, respectively, in torsion.
As is known, fatigue resistance is affected by: the absolute dimensions of the cross section of the part, stress concentration, and the condition of the surface layer. In determining safety factors with respect to normal and shear stresses, these factors are taken into account by introducing the corresponding coefficients into the calculation formulas (given above).
In the literature sources, data on the stress-concentration coefficients \(k_\sigma,\ k_\tau\) and on the scale factor \(\varepsilon_\sigma,\ \varepsilon_\tau\) for shells are absent.
Welded (transverse) seams may be regarded as stress concentrators in a shell.
Obtaining a welded joint is inevitably associated with the formation of three zones: the base metal, the weld metal, and the transition—heat-affected—zone. The metal of these three zones may have different mechanical properties and, consequently, different fatigue-resistance characteristics, as well as different sensitivities to stress concentration.
According to [1], the values of the theoretical stress-concentration coefficient for butt welded joints may lie within the range \(1.3 \div 2.5\). This makes it possible, in calculating shells for fatigue strength, to take as approximate values of the effective stress-concentration coefficients for butt welds \(k_\sigma = 1.5 \div 2.0\).
The results of investigations [3] give grounds to assume that, for shells whose wall thickness does not exceed \(40 \div 50\) mm, the influence of the scale factor on fatigue resistance is insignificant, since in full-scale parts of the indicated thickness (diameter) the endurance limit proved to be lower by only \(10 \div 15\%\) in comparison with standard-size specimens (diameter 10 mm).
Thus, in calculating shells for endurance, the values of the scale factor may be taken as approximately equal to \(\varepsilon_\sigma \simeq \varepsilon_\tau \simeq 0\).
The influence of the state of the surface and of the properties of the surface layer on the endurance limit of the material is expressed by the coefficient \(\beta\), equal to the ratio of the endurance limit in the presence of any factors associated with the state of the layer to the endurance limit in their absence. For low-carbon steel with an untreated surface, one may take \(\beta = 0.75\).
When calculating shells for strength, it is recommended to adhere to the following sequence:
- Determine the initial data required for the calculation:
a) the design dimensions of the shell (internal diameter, wall thickness, distance between supports);
b) the greatest loads (internal excess pressure, bending and torsional moments);
c) the mechanical characteristics of the material \((\sigma_b,\ \sigma_t,\ \sigma_{-1},\ \tau_{-1k})\);
d) special operating conditions of the shell (requirement of increased reliability, operation under elevated-temperature conditions, corrosive environment, impact loads, etc.).
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Prepare the computational scheme of the shell.
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Construct diagrams of bending and torsional moments.
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Draw up a structural sketch of the shell and identify the dangerous sections.
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Determine the nominal stresses in the dangerous sections due to the maximum briefly acting load.
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In the dangerous sections, determine the safety factor with respect to static load-carrying capacity.
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Determine the nominal stresses in the dangerous sections due to internal pressure, bending, and torsion, and then the components of the stress cycle \(\sigma_a\), \(\tau_a\), \(\sigma_c\), \(\tau_c\) under the maximum long-acting external loads.
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Determine the stress-concentration factors \(k_\sigma\), \(k_\tau\), the coefficients of influence of absolute dimensions \(\varepsilon_\sigma\), \(\varepsilon_\tau\), and of the surface condition \(\beta\).
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In the dangerous sections, determine the strength margins with respect to the fatigue limit.
The magnitude of the strength margin with respect to the fatigue limit must be not less than \(n = 2.0 \div 2.5\).
REFERENCES
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S. V. Serensen et al. Load-Carrying Capacity and Strength Calculations of Machine Parts. Moscow, Mashgiz, 1963.
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S. D. Ponomarev et al. Strength Calculations in Mechanical Engineering. Vol. II, Mashgiz, 1956.
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G. V. Uzhik. Scale factor in connection with the assessment of the strength of metals and the calculation of parts. Izvestiya OTN, No. 11, 1955.