I am grateful to have been the recipient of the Harold Hellwig Graduate Scholarship in Structural Engineering. Two buildings damaged by the Northridge earthquake were investigated to verify the program's ability to match the level of response and the extent and location of measured damage.
List of Figures
List of Tables
Chapter 1
Introduction
- Objectives
- Background
- Fractured Connections
- Ground Motions
- Analysis
- Organization of the Thesis
- Notation
- Chapter 2
Adequate levels of inelastic behavior were achieved, mainly due to plastic deformations of the panel zone. The programs discussed so far are limited to modeling plastic hinges without explicitly modeling the propagation of plasticity.
Overview of the Planar Frame Module
- Introduction
- Planar Frame Model
- Beam-Columns
- Fiber Model
The monotonic stress-strain relationship is shown in figure 2.5, and its hysteretic behavior is discussed in detail in (Challa and Hall 1994). Once the fiber has reached the yield plateau (1-3' in figure 2.6), future reversals will be from the virgin curve.
Modelled
Connection Fracture
Once the sets of fracture inhibitions and the groups of fibers are established, the fracture strains are randomly assigned to the groups within each type using the specified occurrence probabilities of the associated set. Even though each group uses the same fracture set, the fracture stresses are randomly assigned so that one beam can have four different fracture regions (upper left, lower left, upper right, and lower right).
Panel Zones
The panel zone has its own dimensions, so beams and columns are considered to be of the length that extends between the edges of the panel zones. The node can be modeled without a panel zone using distinct span dimensions of beams and columns or the panel zone can be used but considered rigid.
2. 7 Foundation Springs
Chapter 3
Mathematical Details of the Planar Frame Module
Introduction
Degrees of Freedom
The local and global degrees of freedom for beams, columns and braces can be seen in Figures 3.2 and 3.3. A panel area has two rotational degrees of freedom which are determined by the end rotations of the beams and columns that frame it (figure 3.5).
Stiffness Formulation
- Fiber Stiffness
- Segment Stiffness
- Beam-Column Stiffness
The geometric contribution to the local stiffness matrix can be determined by moving the fiber transversely (Figure 3.7). Segment stiffness is composed of the contributions to fiber stiffness plus a separate contribution to shear stiffness.
The forces of the elements act on the end of the element that is at the front of a panel zone. These forces must be converted into frame knot forces applied in the center of the panel zone.
Beam forces
The incremental displacements can be found by superimposing small displacements from each of the frame nodal degrees of freedom (figure 3.14).
Column forces Frame forces
Brace forces Frame forces
Panel Zone Stiffness
The stiffness of the panel zone relates the shear load /PZ of the steel volume considered to act in the plane of the panel zone and the moment MPZ resisting this load. The volume cut is dcdbt, where they are the depth of the column, db is the depth of the deeper beam impinging on the column, and t is the contributing thickness. The shear load is related to the beam and column rotations in stepwise form by.
Foundation Stiffness
Frame Stiffness
Three-Dimensional Modeling
Constraint Methods
The average of frame node horizontal displacements is constrained to be equal to the master node displacement at each floor. If the constraint equations are not exactly satisfied, there will be some non-zero entries on the right-hand side of the equation, so the actual term. In the above derivation, it was assumed that the stiffness matrix Kr is not a function of displacements x.
The method of Lagrange multipliers forms the scalar product of the homogeneous constraint equations with an equal number of unknown multipliers, >.i.
Constraint Equations
The concern is that the main diagonal stiffness values will vary by many orders of magnitude. Each of the above matrices will have a ratio of post-elimination (new) to pre-elimination (old) for the lower diagonal of. This would imply the same level of ill-conditioning, so the pivot dofs of the main node are ignored when determining the conditioning of the stiffness matrix based on the range of diagonal main values.
The program will stop when the average of the values of the main diagonals (excluding those of koo) is greater than 1010 times the smallest diagonal.
Shear Spring Elements
- Gravity Columns
- Frame Columns Out-of-Plane
The displacements at the shear spring location must first be written in terms of the master node displacements using the three-dimensional small rotation transformation. Note that a shear spring on story j contributes to the master node dofs on story j + 1 above and story j below. This method will automatically establish specific values for the torsional stiffness and strength in the story due to the shear springs.
The reduction of out-of-plane geometric stiffness is not taken into account when updating the global tangent stiffness matrix.
Geometric Nonlinearity
- Small Angle Assumption
- Small Rotation Assumption
The minor rotation assumption has been widely used to linearize the equations of motion (Weaver and Nelson 1966; Wilson and Allahabadi 1975; Wakabayashi 1986; Chopra 1995) by removing trigonometric functions of the rotations from the transformation equation. Some buildings are eccentric and show a torsional response due to the lateral excitations, but even in the inelastic range for heavy records, the cumulative rotations are not large enough to affect the accuracy, with the increments assumed to be small rotation. If the assumption of a small rotation is used, the transformation equation (4.19) is repeated here in equation form for comparison.
The small angle assumption refers to the total angle, so even if the small rotation assumption does not result in large cumulative errors, the constraint equation and P-D are out of plane.
Stiffness Assembly
If there were no master nodes, the semi-bandwidth would be 36, so only a minimal increase is observed. The formulation calculates the semi-bandwidth by searching for the maximum difference in DOF number for each element type. By denoting the number of degrees of freedom per frame i per story j by n1 and the number of frames by n1, an upper bound to the semi-bandwidth NsB can generally be expressed as.
Similar to the 2D case, if there are braces in the structure, the semi-bandwidth will increase by four.
4. 7 Mass Formulation
Damping Formulation
The rate required to achieve a desired level of damping in the first state can be determined experimentally and will be discussed in Chapter 7. Because the history damping is calculated from initial elastic properties, the total damping matrix can be expressed as C from Eq. 4.57, assuming that the damping is linear. This is why the damping force equation can be written above for total motion, not incremental motion.
The forces can be summed for nj control point locations on each floor j and composed into a total damping force vector of the floor as
Chapter 5
Three-Dimensional Columns
Modeling Considerations
The displacements in the local dofs can be determined from displacements in the five dofs in the global coordinates. In Figure 5-3, the global displacements u2 and u3 are projected on the local X~ axis and u2 and u1 are projected on the local Xi axis. Global displacements u1, u2 and u3 can be projected onto the local X~ axis using the global direction cosine 'Yi.
Note that in Figure 5-2 the rotation dofs u4 and us are unchanged from global to local axes, since the local axes are in the planes perpendicular to the rotation axes.
Three-Dimensional Issues
The error in the size of the resulting will be small even with large deformations. Recall that there is no axial deformation within the panel zone, so all axial deformation in a column line occurs in the free spans of the column. The clear span must be uniquely defined for corner posts, so the panel zone is the depth of the deepest beam framing within it.
The area of the column flanges times the depth of the panel area is used for the volume of the panel area.
Stiffness Formulation .1 Fiber Stiffness
- Segment Stiffness
The shear range A1 is dtw for the major axis of a wide flange section or 5/6(2btt) for the minor axis. For a box column, A1 is 2dtw for the strong axis or 2btt for the weak axis. See figures 5.21 and 5.22 for the positive meaning of the segment end forces and to figures 5.1 and 5.2 for local and global parameters.
The incremental displacements of bars are solved with individual iterations of bars, using the outer dof displacement increments retrieved from the frame as input, as discussed for the 2D bars.
Incremental displacements
Column end forces (Equation 5.56) act at the end of the column located at the edge of the plate zone. The depth of the slab area d~ and widths dJ.i and d3i and the beam and column twists ot, (}~i' OJ.i and 03i at each end (i or j) of the columns are used to determine the nodal frame loading from the columns.
Effect of beam rotation
Panel Zone Stiffness
For 3D columns, there are two panel zones per column acting in the planes of the beams that frame into the column. Both directions (i = 1, 3) have a panel zone stiffness which relates to the shear load 'YfZ of the steel volume acting in the plane of the panel zone and the moment Mf z resisting this load. The shear load is related to the beam and column rotations in stepwise form by (5.72).
Assembly to Frame Stiffness 5.3.6 Frame Stiffness
Influence on Three-Dimensional Solution
This is an additional reason to require only orthogonal framing for 3D columns since the current constraint method would not be suitable for tilted framing. The foundation elements are modified to be able to track the nonlinear response in two orthogonal directions corresponding to the column. Recall that the mass associated with a frame node has an out-of-plane contribution to the main connection resulting in a non-diagonal mass matrix (section 4.7).
In view of the 3D column, out-of-plane motion for one plane will be in-plane motion for the orthogonal plane.
Equations of Motion
- The Newmark Method of Time Integration
- Stiffness Linearization
- Nonlinear Time-Stepping Formulation
- Static Solution
- Dynamic Solution
- Other Newton-Raphson Iterative Schemes
If the updated version of displacement growth, i.e., cumulative growth, is used, the load will remain at the yield plateau, just further to the left (figure 6.1(b)). The time step equation of the previous section can be shortened for static analysis. Assuming that the displacement from the previous step has converged, the estimate of the total displacement for iteration k + 1 is.
The first iteration of each load step is to apply a load and subsequent iterations are solved for the residual forces due to the non-linear nature of the problem.
6. 7 Displacement Control Solution
Chapter 7
Analysis Preliminaries
- Modeling Assumptions
- Mass and Gravity Loading
- Fiber Cracking Models
- Panel Zone Properties
- Foundation Properties
- Shear Building Stiffness
- Damping
- Comparison of Buildings
- Earthquake Records
- Chapter 8
Plastic rotation was assumed to occur in the first two segments of the member. The criterion used in this work to determine the panel zone thickness requires that the panel zone strength (i.e., yield moment) be at least 0.8 times the plastic capacity of the connecting beams. The damping force, F/J, and the shear rate, v§, are chosen to give reasonable elastic damping in the lowest mode of the building.
The following is a quantitative example of the nonlinear damping used in this work for a 13-story building.
17-Story Building
- Building Description
- Building Design
- Pushover Analysis
- Simulations
- Figure Descriptions Time Histories
Concrete basement walls are assumed to exist in each direction, and steel columns are assumed to extend to the base of the walls. The next model includes composite beam action (CM) and the final model includes poor fracture strain (FR). Note that the composite effect has little effect on the overall stiffness or strength of the building.
All analyzes include foundation elements, composite action, P-~ effects, and shear structure.
