Element Library
The analysis engine supports a variety of 1 and 2-dimensional finite elements typically used in the modeling of bridge structures. The 1-dimensional element is always linear elastic while the 2-dimensional element may be either linear elastic or be assigned material properties that vary with time. All elements may be placed into and removed from the system at any solution interval. Once an element is removed from the system it cannot be returned. All elements can either be plane or space type elements. Plane frame elements lie in the global X-Y plane and are formed identically as their space frame counter-parts.
| Setting | Description |
|---|---|
| 1-Dimensional Element | The 1-dimensional element has either 1 or 2 defining nodes and is used to model supports or the connection between adjacent structural elements. |
| Space Frame Element | The space frame element is a two-node linear bending element: a third node is associated with it, referred to as the K-node, in order to give direction to the local 2-2 and 3-3 axes. The 1-axis is oriented along the axis of the element. The element has twelve degrees-of-freedom (DOF): three translational and three rotational at each node point. Shear deformations are considered if a shear area is specified in either the 2 or 3 direction. The shear modulus is computed using the following equation: (0.5 * E) / (1+v), where v is the Poisson's Ratio. The plane frame element is identical to the space frame element accept that it has six DOF and it is constrained to the global X-Y plane, with positive "Y" being vertically upward. |
| Element Stiffness Matrix | The element stiffness matrix is developed by computing the initial-end flexibility coefficient matrix, inverting to obtain the initial end stiffness matrix and then using equilibrium equations to obtain the remaining stiffness coefficients. This method of computing the element stiffness matrix permits section properties to change along the length of the element since the end flexibility coefficients are computed using a summation procedure over the length of the element. |
| Non-Prismatic | Section properties will almost always change within a span element when those components are non-prismatic. Non-prismatic spans are modeled as a series of prismatic segments. The number of prismatic segments and their lengths are determined by the user and are independent of the number of intermediate node points on each span. Therefore, node points will not usually coincide with changes in section properties and thus the need for the flexibility approach for the element stiffness matrix. |
| Element Load Vectors | The element load vectors are computed using a similar approach to the element stiffness matrix. Initial end displacements are obtained due to loads over each prismatic segment, and then transformed into initial end using the element stiffness matrix. Terminal end forces are obtained using equilibrium equations. |
| Time-Dependent Material Behavior | Time effects are incorporated into the analysis by dividing the solution into discrete solution intervals, each with a finite amount of time, using time dependent material properties for each interval and computing restraining forces on the elements due to a variety of effects. The implementation is based on a paper by Tadros, Ghali, and Dilger (Ref. 11) in which the creep, shrinkage and relaxation is transformed into a pseudo linear elastic incremental solution. |
| Effective Modulus | The quantity E(i)/(1 + *(i+1/2,i)) is referred to as the effective modulus and represents the effects of creep over a solution interval. The quantities in braces represents the effects of creep from the first time a cross section part has applied stress up until the current solution interval and is similar to an initial strain. Shrinkage and relaxation (and temperature and cross section parts leaving the system) effects also form part of the initial strain. |
| Elastic Modulus | The elastic modulus at the mid-point of an interval is computed using some code specification equation, such as AASHTO. These are usually based on some fixed quantity such as the modulus at 28-days and an associated time function. In the presence of creep, the effective modulus is used in the formation of the element stiffness matrix and element load vectors, as can be seen by equation (1). The initial strain due to creep uses the time variation modulus with modification for creep. |
| Creep | The effects due to creep are calculated by using the concept of a creep factor which is based on an ultimate creep coefficient multiplied by a time function. The creep strain is the amount of additional strain within the concrete element that takes place after the initial elastic strains. The creep coefficient is the ratio of the creep strain to the elastic strain. The ultimate creep is a function of the section and material properties, age at loading and the time elapsed after loading. The creep within any time interval is the difference in the creep up to the start and the end of the interval. For calculation purposes, the time of loading for any load applied in an interval is the time at the middle of the interval. |
| Shrinkage | The effects due to shrinkage are calculated by including a shrinkage strain as initial strain within the element. The shrinkage strain is based on an ultimate shrinkage value multiplied by a time function and is a function of material properties and the time after shrinkage starts. The shrinkage strain during any time interval is the difference in the shrinkage strain at the start and the end of the interval. |
| Relaxation | Relaxation of strand steel is the reduction in stress with time due to sustained stress. The relaxation loss of stress during any time interval is the difference of the relaxation loss at the beginning and the end of the interval. If the strain is zero, the loss of stress is called the intrinsic relaxation, however, if the strain is reduced (due to other loads), the actual relaxation is different. The intrinsic relaxation must be modified to calculate the actual value. This is performed by using the stress in the strand during any time interval to calculate the ultimate relaxation, and subsequently adjusting it for the given time interval. |