Strand7 Webnotes - ST7-1.57 Theory

ST7-1.57 Theory
ST7-1.57.10 Theory / Materials
0.7 MB	ST7-1.57.10.1 Mohr-Coulomb Yield Criterion The Mohr-Coulomb yield criterion is often used in the analysis of frictional materials such as soil and concrete. The behaviour of these materials is governed by material cohesion and the internal friction angle. In this Webnote we develop the equations used by Strand7 to calculate failure and render results using this yield criterion, based on the fundamental failure criterion.
0.5 MB	ST7-1.57.10.2 Brick Element Material and Geometric Nonlinearity The stress values reported at the Gauss points in material nonlinear brick elements can exceed the corresponding values on the stress-strain curve when geometric nonlinearity is included in the analysis, even for a fully converged increment. This Webnote uses the simple example of a cylindrical bar subjected to a uniaxial load to describe the reasons for this. ...
0.9 MB	ST7-1.57.10.3 Nonlinear Elastic Material Both elastoplastic and nonlinear elastic material models use stress-strain tables to define the nonlinear relationship between stress and strain in the material. However, the two material models interpret the stress-strain tables differently. The most significant difference, at least for uniaxial situations, is in the loading-unloading behaviour of the two models.
0.8 MB	ST7-1.57.10.5 Elastic and Plastic Strain This Webnote gives an overview of elastic and plastic strain as reported in the Strand7 plate and brick elements for material nonlinear analysis. An elasto-plastic material with the von Mises yield criterion and isotropic hardening is used to illustrate these quantities.
ST7-1.57.20 Theory / Elements
5.1 MB	ST7-1.57.20.2 User Defined Beam Matrix Strand7 allows for the definition of a user defined beam element (UDB) to be used to represent a component in place of modelling the component itself. In some cases the stiffness of the component will be available from manufacturers data or in others from a detailed finite element model. This document will briefly discuss the theory behind the UDB and also give examples on how to determine what stiffness values should be used. This Webnote should be accompanied by the following models: ...
0.3 MB	ST7-1.57.20.5 Virtual Overlap Virtual overlap refers to the non-physical overlap of two or more plate/shell or beam elements at a joining node. The following figure demonstrates the concept as pertaining to plate elements. The primary effect of virtual overlap is to increase the mass of the structure. This can introduce error into self-weight calculations and dynamics (natural frequencies, inertia relief analysis and transient dynamic analysis). Virtual overlap in plate elements is typically ignored, because plates...
1.2 MB	ST7-1.57.20.6 Plate Drilling Degrees of Freedom The classical theory for the structural analysis of plates describes the strains in the plate in terms of the two in-plane displacements (u,v), the transverse displacement (w), and rotations about two axes in the plane of the plate. The rotation about the normal to the plate does not appear in the theory. This degree of freedom is often referred to as the drilling degree of freedom because of the analogy with a drill used to drill a hole through a plate.
0.7 MB	ST7-1.57.20.7 Plate Bending Theories This Webnote introduces the different plate theories found in the literature, most of which are available in Strand7. These are: thin plate theory (Kirchhoff), thick plate theory (Mindlin), classical laminate theory (CLT) and high order shear deformation theory (HOSDT). This Webnote covers their advantages, disadvantages, limitations and how they are applied in Strand7. In Strand7, Tri3 and Quad4 elements are formulated based on classical thin plate theory. For a thin plate in the xy-plane,...
0.9 MB	ST7-1.57.20.8 Contact Element Fundamental Behaviour Contact between two bodies can be defined in Strand7 using connecting beam elements which have been assigned a Point Contact beam property. This Webnote covers the fundamental behaviour of all contact types and options. An overview of contact types and parameters is given first. Contact element behaviour is independent of the elements to which they are attached. Contact elements can be placed between any combination of element types, but must be attached to nodes or by attachment links to...
0.6 MB	ST7-1.57.20.9 Beam Displacement Behaviour due to Shear Centre Offsets The Strand7 beam element captures advanced beam deflection and stiffness behaviour including influences from shear centre offsets, beam offsets and beam end releases. A non-zero shear centre offset means that a lateral force will produce an axial (torsional) rotation in the beam. Shear centre offsets are the theoretical centre of rotation of rotations induced by a lateral force. Shear centre offsets are automatically calculated by Strand7 when a beam cross section is referenced. If the shear...
1.7 MB	ST7-1.57.20.11 User Defined Plate-Shell Matrix Strand7 supports the definition of User Defined Plate (UDP) elements, which allow the definition of plates with arbitrary stiffness characteristics - as defined by a plate stiffness matrix. The stiffness properties of such plates may be derived in an analytic manner, supplied from a manufacturer, or calculated by a numerical sub-model. The last option is explored in this Webnote. User defined plates can be used to model corrugated cladding and other tesselated structure.
1.2 MB	ST7-1.57.20.12 Beam and Shell Offsets Offset attributes are a convenient way of incorporating eccentricities into a model without modifying the connectivity of the mesh. This Webnote explains how Strand7 implements offsets for beam and plate elements; comparisons are made between models with offsets and equivalent models using rigid links. The Webnote also shows how displacements in offset elements are calculated from the nodal results. The information presented applies to Strand7 Release 2.4.5 and later. Consider a node on the...
0.7 MB	ST7-1.57.20.14 Connection Type Beam Element Behaviour The information in this Webnote relates to Strand7 Release 2.4.5 and later. Connection elements in Strand7 can be thought of as flexible links. This Webnote focuses only on mechanical analysis. For mechanical analysis, connection elements are defined by up to six independent stiffness values, three translational and three rotational. Unlike the beam element, there is no coupling between different degrees of freedom. The three directions 1, 2 and 3 shown on the property dialog for the connection...
1.7 MB	ST7-1.57.20.15 BXS Beam Elements This Webnote examines the user-defined beam section, or BXS section, in Strand7. This functionality gives the user the ability to create detailed beam cross section shapes to include features that are not available when using the standard section shapes, such as ribs, fillets, voids, and multi-part sections of different shape and orientation. In addition to standard section shapes, Strand7 allows the assignment of a customised section (or BXS) defined by the user using a 2D plate mesh representing...
0.6 MB	ST7-1.57.20.16 Comparison of the Moment Curvature and Fibre Stress Beam Nonlinear material behaviour in beams can be modelled in Strand7 using a table defining the beam's moment-curvature behaviour or a table defining the material stress-strain behaviour. These methods are illustrated and compared in this Webnote by way of examples.
0.6 MB	ST7-1.57.20.17 Curved Pipe Element Stress When a beam is curved, the neutral axis shifts in towards the centre of curvature. This results in higher stress on the inner edge of the curved beam. Strand7 has a curved beam element that considers this effect, which is the pipe element with a Pipe Radius attribute applied (Attributes/Beam/Pipe/Radius). The magnitude of the shift away from the neutral axis is generally calculated from elasticity theory.
ST7-1.57.30 Theory / Solvers
0.7 MB	ST7-1.57.30.1 Pivot Ratios and Ill Conditioning Digital computers work with a finite number of significant figures, which makes them prone to rounding errors when numbers of significantly different magnitudes are combined; this can lead to a loss of precision. Pivots are named for their participation in the reduction of the global stiffness matrix. The pivot ratio compares the maximum and minimum pivot values in the matrix during the factorisation stage, and is an indicator of a solutions numerical stability. This is also known as stiffness matrix conditioning.
0.8 MB	ST7-1.57.30.3 Damping in Strand7 Damping is a term used for the measure of the energy loss in a dynamic system. There are many mechanisms responsible for damping, e.g. material damping, friction at contact surfaces, etc. In Strand7, damping can be represented by different damping models. Three models are available: Rayleigh damping, modal damping and the viscous damping matrix models. The viscous damping model uses the following expression to calculate the element damping matrix: where The global damping matrix is obtained...
0.6 MB	ST7-1.57.30.4 Solution Restarts and Initial Conditions This Webnote summarises the initial conditions options available in the Strand7 solvers. Initial conditions describe the displacement and stress state of the structure prior to the commencement of the current analysis; they can be set by applying certain types of attributes, such as pre-stress and pre-strain, or for more flexibility, initial conditions for the current analysis can be based upon a previous analysis.
0.5 MB	ST7-1.57.30.11 Solver Scheme and Node Ordering The solver options Scheme and Node ordering can have a significant impact on the efficiency of your solution. This is especially true if your Strand7 licence does not include the direct sparse solver option. A good understanding of what these parameters mean and how to use them will allow your models to solve in less time using less memory. Solver scheme refers to the way in which the solver stores and accesses the global stiffness matrix. When only the non-zero entries are stored (this...
ST7-1.57.40 Theory / Results
0.7 MB	ST7-1.57.40.1 Calculation of von Mises Stress in Circular Beam Elements This Webnote demonstrates the calculation of von Mises stress in beam elements with circular cross-section. Currently Strand7 results for general beam elements do not include von Mises stress due to approximation to torsional shear stress. For a non-circular cross-section, torsional shear stress is approximated using parametric equations. For circular cross-sections, torsional shear stress is an exact solution and hence von Mises stress can be calculated accurately.
0.4 MB	ST7-1.57.40.2 Principal Stress Calculation for Beams with Circular Section Strand7 calculates the principal stresses for beams of circular cross-sections only. The stress components included in this calculation are: Fibre stress, which includes axial stress, and bending stresses in planes 1 and 2 (zz); Shear stress due to torque (xy). Shear stress in plane 1 (xz); Shear stress in plane 2 (yz). For the Pipe type element, Strand7 includes two additional stress components: Pipe radial stress (RR); Pipe hoop stress (). The principal stress...
0.7 MB	ST7-1.57.40.4 Review of Wood-Armer Moments in the RC Module Wood-Armer moments are a convenient way of expressing an arbitrary field of moments (Mxx, Myy and Mxy) in terms of the direction of the reinforcing bars in a concrete slab. They were derived from the theory of elasticity in the 1960s when a number of authors contributed to their development. A brief timeline is shown below (full reference details are provided at the end of this Webnote): 1953 Swedish engineer A. Hillerborg publishes Reinforcement of slabs and shells designed according to...
2.2 MB	ST7-1.57.40.5 Transverse Shear Stress in Plate Elements Transverse shear refers to the internal shearing force that develops on a plane perpendicular to the plate surface, due to out-of-plane loading on the plate (e.g. a plate with normal pressure). It is analogous to the shear force found in beam elements loaded in bending. A scissor cutting through paper is an extreme example of transverse shearing action on the paper. Transverse shear force is reported in units of force per width of plate element (e.g. N/mm) and produces a shear stress through...
0.9 MB	ST7-1.57.40.6 Deviatoric and Hydrostatic Stresses and Their Application This Webnote discusses deviatoric and hydrostatic stresses in the Strand7 environment. Starting from a basic definition of stress at a point, it describes the separation of the stress into deviatoric and hydrostatic components, and shows the physical application and implication of the deviatoric and hydrostatic stress components. Volumetric strain and bulk modulus are also introduced in the context along the way. These concepts are explained using Strand7 model examples.

0.7 MB

ST7-1.57.10.1 Mohr-Coulomb Yield Criterion
The Mohr-Coulomb yield criterion is often used in the analysis of frictional materials such as soil and concrete. The behaviour of these materials is governed by material cohesion and the internal friction angle. In this Webnote we develop the equations used by Strand7 to calculate failure and render results using this yield criterion, based on the fundamental failure criterion.

0.5 MB

ST7-1.57.10.2 Brick Element Material and Geometric Nonlinearity
The stress values reported at the Gauss points in material nonlinear brick elements can exceed the corresponding values on the stress-strain curve when geometric nonlinearity is included in the analysis, even for a fully converged increment. This Webnote uses the simple example of a cylindrical bar subjected to a uniaxial load to describe the reasons for this. ...

0.9 MB

ST7-1.57.10.3 Nonlinear Elastic Material
Both elastoplastic and nonlinear elastic material models use stress-strain tables to define the nonlinear relationship between stress and strain in the material. However, the two material models interpret the stress-strain tables differently. The most significant difference, at least for uniaxial situations, is in the loading-unloading behaviour of the two models.

0.8 MB

ST7-1.57.10.5 Elastic and Plastic Strain
This Webnote gives an overview of elastic and plastic strain as reported in the Strand7 plate and brick elements for material nonlinear analysis. An elasto-plastic material with the von Mises yield criterion and isotropic hardening is used to illustrate these quantities.

5.1 MB

ST7-1.57.20.2 User Defined Beam Matrix
Strand7 allows for the definition of a user defined beam element (UDB) to be used to represent a component in place of modelling the component itself. In some cases the stiffness of the component will be available from manufacturers data or in others from a detailed finite element model. This document will briefly discuss the theory behind the UDB and also give examples on how to determine what stiffness values should be used. This Webnote should be accompanied by the following models: ...

0.3 MB

ST7-1.57.20.5 Virtual Overlap
Virtual overlap refers to the non-physical overlap of two or more plate/shell or beam elements at a joining node. The following figure demonstrates the concept as pertaining to plate elements. The primary effect of virtual overlap is to increase the mass of the structure. This can introduce error into self-weight calculations and dynamics (natural frequencies, inertia relief analysis and transient dynamic analysis). Virtual overlap in plate elements is typically ignored, because plates...

1.2 MB

ST7-1.57.20.6 Plate Drilling Degrees of Freedom
The classical theory for the structural analysis of plates describes the strains in the plate in terms of the two in-plane displacements (u,v), the transverse displacement (w), and rotations about two axes in the plane of the plate. The rotation about the normal to the plate does not appear in the theory. This degree of freedom is often referred to as the drilling degree of freedom because of the analogy with a drill used to drill a hole through a plate.

0.7 MB

ST7-1.57.20.7 Plate Bending Theories
This Webnote introduces the different plate theories found in the literature, most of which are available in Strand7. These are: thin plate theory (Kirchhoff), thick plate theory (Mindlin), classical laminate theory (CLT) and high order shear deformation theory (HOSDT). This Webnote covers their advantages, disadvantages, limitations and how they are applied in Strand7. In Strand7, Tri3 and Quad4 elements are formulated based on classical thin plate theory. For a thin plate in the xy-plane,...

0.9 MB

ST7-1.57.20.8 Contact Element Fundamental Behaviour
Contact between two bodies can be defined in Strand7 using connecting beam elements which have been assigned a Point Contact beam property. This Webnote covers the fundamental behaviour of all contact types and options. An overview of contact types and parameters is given first. Contact element behaviour is independent of the elements to which they are attached. Contact elements can be placed between any combination of element types, but must be attached to nodes or by attachment links to...

0.6 MB

ST7-1.57.20.9 Beam Displacement Behaviour due to Shear Centre Offsets
The Strand7 beam element captures advanced beam deflection and stiffness behaviour including influences from shear centre offsets, beam offsets and beam end releases. A non-zero shear centre offset means that a lateral force will produce an axial (torsional) rotation in the beam. Shear centre offsets are the theoretical centre of rotation of rotations induced by a lateral force. Shear centre offsets are automatically calculated by Strand7 when a beam cross section is referenced. If the shear...

1.7 MB

ST7-1.57.20.11 User Defined Plate-Shell Matrix
Strand7 supports the definition of User Defined Plate (UDP) elements, which allow the definition of plates with arbitrary stiffness characteristics - as defined by a plate stiffness matrix. The stiffness properties of such plates may be derived in an analytic manner, supplied from a manufacturer, or calculated by a numerical sub-model. The last option is explored in this Webnote. User defined plates can be used to model corrugated cladding and other tesselated structure.

1.2 MB

ST7-1.57.20.12 Beam and Shell Offsets
Offset attributes are a convenient way of incorporating eccentricities into a model without modifying the connectivity of the mesh. This Webnote explains how Strand7 implements offsets for beam and plate elements; comparisons are made between models with offsets and equivalent models using rigid links. The Webnote also shows how displacements in offset elements are calculated from the nodal results. The information presented applies to Strand7 Release 2.4.5 and later. Consider a node on the...

0.7 MB

ST7-1.57.20.14 Connection Type Beam Element Behaviour
The information in this Webnote relates to Strand7 Release 2.4.5 and later. Connection elements in Strand7 can be thought of as flexible links. This Webnote focuses only on mechanical analysis. For mechanical analysis, connection elements are defined by up to six independent stiffness values, three translational and three rotational. Unlike the beam element, there is no coupling between different degrees of freedom. The three directions 1, 2 and 3 shown on the property dialog for the connection...

1.7 MB

ST7-1.57.20.15 BXS Beam Elements
This Webnote examines the user-defined beam section, or BXS section, in Strand7. This functionality gives the user the ability to create detailed beam cross section shapes to include features that are not available when using the standard section shapes, such as ribs, fillets, voids, and multi-part sections of different shape and orientation. In addition to standard section shapes, Strand7 allows the assignment of a customised section (or BXS) defined by the user using a 2D plate mesh representing...

0.6 MB

ST7-1.57.20.16 Comparison of the Moment Curvature and Fibre Stress Beam
Nonlinear material behaviour in beams can be modelled in Strand7 using a table defining the beam's moment-curvature behaviour or a table defining the material stress-strain behaviour. These methods are illustrated and compared in this Webnote by way of examples.

0.6 MB

ST7-1.57.20.17 Curved Pipe Element Stress
When a beam is curved, the neutral axis shifts in towards the centre of curvature. This results in higher stress on the inner edge of the curved beam. Strand7 has a curved beam element that considers this effect, which is the pipe element with a Pipe Radius attribute applied (Attributes/Beam/Pipe/Radius). The magnitude of the shift away from the neutral axis is generally calculated from elasticity theory.

0.7 MB

ST7-1.57.30.1 Pivot Ratios and Ill Conditioning
Digital computers work with a finite number of significant figures, which makes them prone to rounding errors when numbers of significantly different magnitudes are combined; this can lead to a loss of precision. Pivots are named for their participation in the reduction of the global stiffness matrix. The pivot ratio compares the maximum and minimum pivot values in the matrix during the factorisation stage, and is an indicator of a solutions numerical stability. This is also known as stiffness matrix conditioning.

0.8 MB

ST7-1.57.30.3 Damping in Strand7
Damping is a term used for the measure of the energy loss in a dynamic system. There are many mechanisms responsible for damping, e.g. material damping, friction at contact surfaces, etc. In Strand7, damping can be represented by different damping models. Three models are available: Rayleigh damping, modal damping and the viscous damping matrix models. The viscous damping model uses the following expression to calculate the element damping matrix: where The global damping matrix is obtained...

0.6 MB

ST7-1.57.30.4 Solution Restarts and Initial Conditions
This Webnote summarises the initial conditions options available in the Strand7 solvers. Initial conditions describe the displacement and stress state of the structure prior to the commencement of the current analysis; they can be set by applying certain types of attributes, such as pre-stress and pre-strain, or for more flexibility, initial conditions for the current analysis can be based upon a previous analysis.

0.5 MB

ST7-1.57.30.11 Solver Scheme and Node Ordering
The solver options Scheme and Node ordering can have a significant impact on the efficiency of your solution. This is especially true if your Strand7 licence does not include the direct sparse solver option. A good understanding of what these parameters mean and how to use them will allow your models to solve in less time using less memory. Solver scheme refers to the way in which the solver stores and accesses the global stiffness matrix. When only the non-zero entries are stored (this...

0.7 MB

ST7-1.57.40.1 Calculation of von Mises Stress in Circular Beam Elements
This Webnote demonstrates the calculation of von Mises stress in beam elements with circular cross-section. Currently Strand7 results for general beam elements do not include von Mises stress due to approximation to torsional shear stress. For a non-circular cross-section, torsional shear stress is approximated using parametric equations. For circular cross-sections, torsional shear stress is an exact solution and hence von Mises stress can be calculated accurately.

0.4 MB

ST7-1.57.40.2 Principal Stress Calculation for Beams with Circular Section
Strand7 calculates the principal stresses for beams of circular cross-sections only. The stress components included in this calculation are: Fibre stress, which includes axial stress, and bending stresses in planes 1 and 2 (zz); Shear stress due to torque (xy). Shear stress in plane 1 (xz); Shear stress in plane 2 (yz). For the Pipe type element, Strand7 includes two additional stress components: Pipe radial stress (RR); Pipe hoop stress (). The principal stress...

0.7 MB

ST7-1.57.40.4 Review of Wood-Armer Moments in the RC Module
Wood-Armer moments are a convenient way of expressing an arbitrary field of moments (Mxx, Myy and Mxy) in terms of the direction of the reinforcing bars in a concrete slab. They were derived from the theory of elasticity in the 1960s when a number of authors contributed to their development. A brief timeline is shown below (full reference details are provided at the end of this Webnote): 1953 Swedish engineer A. Hillerborg publishes Reinforcement of slabs and shells designed according to...

2.2 MB

ST7-1.57.40.5 Transverse Shear Stress in Plate Elements
Transverse shear refers to the internal shearing force that develops on a plane perpendicular to the plate surface, due to out-of-plane loading on the plate (e.g. a plate with normal pressure). It is analogous to the shear force found in beam elements loaded in bending. A scissor cutting through paper is an extreme example of transverse shearing action on the paper. Transverse shear force is reported in units of force per width of plate element (e.g. N/mm) and produces a shear stress through...

0.9 MB

ST7-1.57.40.6 Deviatoric and Hydrostatic Stresses and Their Application
This Webnote discusses deviatoric and hydrostatic stresses in the Strand7 environment. Starting from a basic definition of stress at a point, it describes the separation of the stress into deviatoric and hydrostatic components, and shows the physical application and implication of the deviatoric and hydrostatic stress components. Volumetric strain and bulk modulus are also introduced in the context along the way. These concepts are explained using Strand7 model examples.