Initial Stress And Strain Initial Stress And Strain Initial Stress And Strain Initial Stress And Strain LoadingLoadingLoadingLoading

When Are They Used? Initial stresses/strains in a structure are due to the initial load conditions, e.g. the application of initial stresses or strains without equilibrating forces results in deformation. Their effect is the same and the use of one or the other is dependent on the known loading that is to be simulated. An example in which initial stresses would be utilised is the generation of initial equilibrating loads and stresses in a soil excavation analysis. The use of initial stresses is illustrated in the LUSAS 11 verification manual example 1.4.1. Initial Stresses And Strains In MYSTRO In Mystro Initial Stresses/Strains may be specified for elements (but not for nodes or Gauss points). This loading type can make use of the variations facility in order to vary the loading over an element i.e. to vary the loading at the nodes. For information on the values to be input into the loading definition form in Mystro refer to the section on the relevant element in the Lusas Element Library, and simply enter the values. How Does LUSAS Deal With Them? The finite element solution algorithm solves the general equation

F = K δδδδ Where F is a force, K the element stiffness and δ the displacement. The specification of initial stress or strain loading will, therefore, require an initial transformation from these specified loads into their equivalent forces at node/Gauss points. This force vector is defined as the sum of the following volume integral performed over all loaded elements

Ro = - ∫∫∫∫v BT (σ0 - D ε0) dV Where

σo: Initial stress value (if specified) εo: Initial strain value (if specified) B: Strain-displacement matrix (see the LUSAS theory manual) D: Element modulus matrix (see the LUSAS theory manual)

Thus, in general, a tensile initial stress loading will generate a compressive equivalent force according to

Ro = - ∫∫∫∫v BT σ0 dV and a tensile initial strain loading will generate a tensile equivalent force according to

Ro = ∫∫∫∫v BT D ε0 dV The following sections are a summary of some simple examples to explain the expected behaviour of the initial stress/strain loading facility. The results are the same for both linear and nonlinear regimes. The examples were carried out using a beam element with Young's Modulus (E) =1.0, area (A) = 1.0 and length (L) = 0.1. One end of the bar is fixed and the other free... Loading:

Apply a positive initial axial stress of 0.1 Results:

Displacements = -0.1 Stress = 0.0 Strain = -0.1 Reactions = 0.0

Explanation:

The tensile initial stress loading is transformed into a compressive equivalent load as above. These compressive forces cause a compressive displacement (-0.1) and, hence, strain (-0.1) in the beam. The stress computed for the beam, evaluated from this strain value, is therefore -0.1 (equal and of opposite sense to the initial stress loading). The final stress output is then computed according to

Final Stress = Computed Stress + Initial Stress

and is, for this case, zero. This is physically equivalent to assuming that the bar is acted upon by the applied tensile stress, but without any external influence to maintain that state. As a result the bar will want to contract - producing negative displacements and strains. An example of this would be a bar initially stretched and then released.

Loading:

Apply a positive initial strain of 0.1 Results:

Displacements = 0.1

Stress = 0.0 Strain = 0.1 Reactions = 0.0

Explanation:

The tensile initial strain loading is transformed into a tensile equivalent load as above. These tensile forces cause a tensile displacement (0.1) and, hence, strain (0.1) in the beam. The stress computed for the beam, evaluated from this strain value, is therefore, 0.1 (equal and of same sign as the initial strain loading). The final strain output is then computed according to

Final Strain = Computed Strain - Initial Strain which is zero.

Both ends are fixed... Loading:

Apply a positive initial stress of 0.1 Results:

Displacements = 0.0 Stress = 0.1 Strain = 0.0 Reactions = ±0.1

Explanation:

The tensile initial stress loading is transformed into a compressive equivalent load as above. These compressive forces cannot produce compressive displacement or strain in the beam because of the support conditions. The stress computed for the beam, evaluated from this strain value, is therefore 0.0. The final stress output is then computed according to

Final Stress = Computed Stress + Initial Stress

and is, for this case, 0.1. This is physically equivalent to assuming that the bar is in an initially stressed state and is restrained to remain in that state.

The axial reaction at the support, which is computed directly from the solution of the equation (linear) or from the integration of the internal stress at the point (nonlinear) will equal ± 0.1.

Loading:

Apply a positive initial strain of 0.1 Results:

Displacements = 0.0 Strain = 0.0 Stress = -0.1 Reactions = ±0.1

Explanation:

This one is a bit of an anomaly. Here we are saying that we want to prescribe an initial strain but are also restraining all movement. Hence, the computed strains are going to be 0.0. The final strains then become

Final Strain = Computed Strain - Initial Strain

and is, for this case, -0.1. These strains are then used to compute stresses so that stress = -0.1. LUSAS Command Format The input data required for each element is obtained from the LUSAS element library manual in the main section for the proposed element and then in the subsection headed, INITIAL STRESSES AND STRAINS. The command format is located in the LOADING section of the LUSAS User manual. There are a number of alternative methods of data input for this loading type, as follows

SSI Initial Stresses and Strains at Nodes SSIE Initial Stresses and Strains for Elements SSIG Initial Stresses and Strains at Gauss points

The choice depending upon the loading distribution of the loading and the relative ease of input. For instance a constant magnitude of loading would be most easily applied using the element-based load command, whilst a varying loading would require either the nodal or the Gauss point data command. The command syntax for initial stress and strain at nodes (SSI) is, for example, as follows

SSI n l N Nlast Ndiff < Vi >i=1,n

where

n The required number of initial stresses or strains at a node. For example, if

only an initial stress (σx) is required, set n=1. Application of both σx and σy would require n=2.

l The starting location of the first input value in the stress/strain data list (default, l=1 for stress input). The value l=ndse+1 gives the starting location for strain input, where ndse is the number of stress components for the loaded element type. For example, the QPM4 element has the following entry in the element library; σx, σy, σxy, εx, εy, τxy. In this case, ndse=3.

N Nlast Ndiff The first node, last node and difference between nodes of the series of

nodes with identical initial stresses/strains Vi The initial stress/strain values at a node.

For further information and examples refer to the LUSAS User manual. Where Is The Loading Applied? Initial stress and strain loading is actually applied at a Gauss point level which means that, for nodal and element loading types, extrapolation from the node points to the Gauss points will be performed (using the element shape functions). For steep gradients of loading, slight inaccuracies in results may be obtained when using the nodal loading type. Consider the following diagram

x

x x

x

0

0 100

100

Gauss Points

Nodal InitialStresses/Strains

Initial Stress/StrainVariation Across Element

σ/ε = 100σ/ε = 0

20

20

0 100

The loading is applied at the nodes as shown. As a result of the quadratic displacement assumption used in higher order elements, the interpolation to the Gauss points yields the variation across the length of the element as shown in the second diagram. This may not yield the values of loading at the Gauss points that are actually required. In practice, however, the inaccuracy is insignificant compared to the loading specified. Mesh refinement in the area of the greatest variations is the most appropriate method for overcoming this problem. Gauss point loading values will be used directly at the Gauss point positions.

Final Notes • Initial stresses/strains must be specified in the first load case and cannot be altered throughout

the analysis. The only exception to this is when activating elements during the analysis, when an initial stress/strain will then be applied to these elements.

• Initial stresses/strains loading cannot be applied using automatic incrementation or load curves. The entire loading magnitude is applied in the first increment only.

• Initial stress and strain loading may be specified at either nodes, elements or Gauss points in the Lusas data file. Whether the values are specified at elements, nodes or Gauss points purely depends on what is most convenient to the user, i.e. what information they have available. The actual variables applicable for use with a specific elements is given under the appropriate element section in the element library.

• To start an analysis with a known initial stress or strain but with zero deformation, it is sometimes possible to apply the required initial stress/strain loading together with an equal but opposite external load. This external load may be available from inspection but, typically, would be obtained by performing the initial stress load case with all external nodes loaded - the reactions from this analysis would be the required external equilibrating loading (in the form of Concentrated Loads)