Stress Evaluation
Since the basic result quantities for beams are the integrated stresses in terms of section forces and moment, special considerations are needed for the evaluation of actual stresses. When combining various stress contributions, a conservative approach is taken in every step.
Axial Stress
The axial stress in the pipe has the following contributions:
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Axial force
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Bending moment
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Local bending stress caused by internal pressure
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Local bending stress caused by temperature gradient through the wall.
Normal Force
The normal stress from axial force is constant over the section, and computed as
Bending Moments
The normal stress from bending is computed in four user-selected points (ylk, zlk) in the cross section as
In 2D, only two points, specified by their local y-coordinates are used.
Internal Pressure
The effects of internal pressure are described in detail in Effects of Internal Pressure above. The axial bending stress component σn,b is always positive due to the assumption that the gauge pressure is positive. In reality, it represents a bending stress, so the true sign differs between the inside and the outside of the pipe.
Thermal Expansion
The effects of thermal expansion are described in detail in Effects of Thermal Expansion above. The axial bending stress component σt,b is always positive due its definition. In reality, it represents a bending stress, so the true sign differs between the inside and the outside of the pipe.
Total Axial Stress
The total axial stress in evaluation point k is defined as
If the true distributions of σn,b and σt,b have opposite signs, this will be conservative.
The peak normal stress in the section is defined as
A special method is used for the Pipe cross section. Since there are no extreme positions around a circle, a maximum bending stress is computed as
where do is the outer diameter. This value replaces the stress from the stress evaluation points (σbk) in maximum stress expressions, thus ensuring that the correct peak stress is evaluated irrespective of where it appears along the circumference.
The following variables are used for the axial stress:
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Stress caused by axial force, σn: pipem.sn
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Stress caused by bending moments, σbk(evaluation points 1-4): pipem.sb1, …, pipem.sb4
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Axial stress from gauge pressure, σn,b: pipem.shb
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Axial stress from temperature gradient, σt,b: pipem.shb
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Total axial stress in evaluation points, σk: pipem.s1, …, pipem.sb4
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Maximum axial stress, σmax: pipem.smax
Hoop Stress
The hoop stress acts in a direction which is orthogonal to the axial stress. Hoop stresses can be caused by the internal pressure and by a temperature gradient through the thickness.
Internal Pressure
The hoop stress caused by the pressure is
where γm and γb are defined in Effects of Internal Pressure. The bending part is defined as positive. In reality, it represents a bending stress, so the true sign differs between the inside and the outside of the pipe. Under the assumption of a positive gauge pressure, the positive sum will however exist on either the inner or the outer boundary.
Thermal Expansion
The temperature gradient stress is caused by difference in thermal strain between the inside and the outside. The details are discussed in Effects of Thermal Expansion.
The total hoop stress is computed as
Shear Stress
The shear stress from twist in general has a complex distribution over the cross section. The maximum shear stress due to torsion is defined as
where Wt is the torsional section modulus. This result is available only in 3D.
The section shear forces are computed in two different ways depending on the beam formulation. For Euler-Bernoulli theory, the section forces proportional to the third derivative of displacement, or equivalently, the second derivative of the rotation.
where Tlz is available only in 3D. In the case of Timoshenko theory shear force is computed directly from the shear strain.
The average shear stresses are computed from the shear forces as
(12-1)
Since the shear stresses are not constant over the cross section, the maximum shear stresses are also available, using section dependent correction factors:
(12-2)
As the directions and positions of maximum shear stresses from shear and twist are not known in a general case, upper bounds to the shear stress components are defined as
Equivalent Stress
The maximum von Mises equivalent stress for the cross section is then defined as
Since the maximum values for the different stress components in general occur at different positions in the cross section, the equivalent stress thus computed is a conservative approximation.
The Tresca equivalent stress is computed from the principal stresses. Determining the principal stresses is however a nontrivial operation. The first assumption is that the through-thickness direction is a principal orientation with zero principal stress. This is consistent with ignoring the radial stress in the von Mises stress expression.
For the remaining axial-hoop plane a Mohr’s circle argument is used. The two non-zero principal stresses in that plane are computed for evaluation point k as
The principal stresses are ordered in descending order, and the Tresca stress is obtained as the highest value in any of the evaluation points.
 
The following variables are used for the equivalent and principal stresses:
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von Mises equivalent stress, σmises: pipem.mises
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Tresca equivalent stress, σtresca: pipem.tresca
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First principal stress, σ1k (evaluation points 1-4): pipem.sp1_1, …, pipem.sp1_4
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First principal stress, σ2k (evaluation points 1-4): pipem.sp2_1, …, pipem.sp2_4
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First principal stress, σ3k (evaluation points 1-4): pipem.sp3_1, …, pipem.sp3_4