Structural Design Methods

Various structural design methods, of differing degrees of complexity and accuracy, are used to design floor diaphragms in structures. These include the; equivalent static analysis (ESA), displacement based design, parts and components, modal analysis, pushover analysis and elastic and inelastic time history analysis methods. Brief descriptions of these methods and

how they are used in the design of floor diaphragm design are provided in the following sections.

5.2.1.1 Equivalent Static Analysis

The Equivalent Static Analysis (ESA) method is based on the assumption that the building deforms in its fundamental translational free vibration mode and that essentially all the mass contributes to this mode. On the basis of this assumption, the design base shear force is found by multiplying the mass by a lateral force coefficient, which is determined from factors such as: the seismic hazard factor, the soil class factor, the structural importance factor, the near fault factor, the calculated or assumed fundamental period and the structural ductility factor (Standards New Zealand, 2004a).

The individual equivalent static forces at each level are found by distributing the design base shear force over the height of the structure. The New Zealand Structural Design Actions Standard NZS1170.5 (Standards New Zealand, 2004a) 92% of the base shear is distributed according to the maximum acceleration that the mass of each level would be expected to sustain if the building remains in the elastic range. The accelerations at each level are proportioned to the lateral deflections. For simplicity lateral deflections are assumed to be proportioned to height above the base. The remaining 8% of the base shear is added to the mass at the highest level to allow for higher elastic modes. For simplicity in the NZS1170.5, the lateral deflection of the structure relative to the ground is assumed to increase linearly with height.

Modal analyses of regular multi storey buildings indicate that for taller buildings less of the mass contributes to the first mode compared to shorter buildings. As a very high proportion of the lateral deflection is accounted with the first mode, it follows that the ESA method, which assumes 100% participation, overestimates lateral deflections when compared with modal analyses. In NZS1170.5 a displacement modification factor is used to reduce the discrepancy of displacements obtained from the ESA method and the modal analysis method.

Empirically derived coefficients are used to amplify lateral deflections to allow for the anticipated level of inelastic deformation corresponding to the chosen structural ductility factor used in assuming the design forces.

5.2.1.2 Modal Analysis

The modal response spectrum analysis is generally used when the higher mode affects are considered to be significant. The property of orthogonallity of mode shapes is used to

transform the coupled equations of the Multi Degrees of Freedom (MDOF) system, from the equation of motion, to uncoupled Single Degree of Freedom (SDOF) systems. The SDOF systems represent various modes of the structure. For each mode the: period, effective mass and participation factors are found. These values are then used with the response spectra, similar to the response spectra used for the ESA method, to determine the spectral accelerations and displacements for each mode.

To find the design actions from a modal response spectrum analysis, it is necessary to combine the actions from the different modes. As the peak mode values occur at different times, the resultant actions cannot be found by a simple addition. Design values are determined by methods such as the Sum Root Sum of the Squares (SRSS) or the Complete Quadratic Combination (CQC) method. The number of mode shapes to be incorporated, to determine the response of the structure, is determined by ensuring that in excess of 90% of the total mass of the structure participates in the response of the structure, in the direction under consideration. The deformations obtained from this method are increased to allow for inelastic actions in a similar manner to that in the ESA method.

This method is based on linear elastic engineering theory, but has some assumptions which simplify the method. However, the assumptions make it less accurate than more complicated approaches such as time history analysis. The assumptions of this method include:

The structure remains linearly elastic;

Higher modes, which provide insignificant effective mass contribution, are assumed to cause insignificant affects to the response of the structure;

The use of the response spectra is appropriate for the design;

The combination method used to determine the responses is practical.

The assumption that the system remains linearly elastic is questionable. The majority of buildings today are designed to sustain some level of inelastic deformation to reduce the costs.

Ignoring insignificant higher modes is a reasonable assumption. If a large proportion of the effective mass is included in the response of the structure, then errors should be small due to omission of higher modes with a small proportion of effective mass.

The use of the combination methods, the SRSS or the CQC, results in a loss of information, such as the sign of the action. Consequences of this are:

− Design moments are not consistent with design shear, torsion or displacements.

The output values obtained from this method are maxima for each level of the structure and they are not equivalent to a set of actions that occur simultaneously. Consequently, these values should not be applied to the structure simultaneously, in an envelope fashion like the ESA method, as the forces are not in equilibrium.

5.2.1.3 Parts and Components

The “Parts and Components” method is a force based design approach used for the design of “parts” or “components” of a structure. This method was previously used to design the forces which develop within floor diaphragms, as traditionally these elements were considered a part (secondary in nature to the “primary” structure resisting lateral forces) and their influence on the overall behaviour of the structure was neglected. The current New Zealand Structural Design Actions Standard NZS1170.5:2004 (Standards New Zealand, 2004b) states that floor diaphragms must be designed by considering the interaction of the floor with the structure as a whole; therefore the “parts” method is no longer appropriate for the design of floor diaphragms.

The Parts and Components method is carried out in a similar way to the ESA method. Firstly, the elastic design spectrum for the part (or component) is determined with consideration of the vertical location of the part within the structure and the fundamental period of the part. The ductility, risk and weight of the part or component are taken account in determining the design actions on the part. This method does not to consider force equilibrium, or the interaction of the part or component with the remainder of the structural system.

5.2.1.4 Pushover Analyses

Pushover analyses are generally carried out by applying a monotonically increasing force pattern to an elastic or inelastic responding structure. The lateral forces are increased until either the total capacity of the structure is exceeded or a predetermined displacement of the structure is reached. Damping of the structural response is not considered during pushover analyses.

There are two difficulties with the general pushover method, they include: choosing an appropriate load pattern to apply to the structure and determining an appropriate displacement termination point. The results of general pushover analyses can be sensitive to the load pattern used in the analysis and the termination point.

A variation of the general pushover is the adaptive pushover method. The phrase “adaptive” is used because the parameters at each step of the analysis are updated. This includes parameters such as lateral force distribution, period and displacement of the structure.

The adaptive pushover method was developed to avoid the uncertainties surrounding the pattern of forces to be applied to the structure. With this approach, the loading pattern adjusts itself to the level of deformation of the structure. The loading pattern is dependent on the mass, the equivalent frequency and displaced shape of the structure.

Comparisons of predictions from pushover and time history analyses show that the pushover values tend to under estimate inter storey drifts in the lower levels of buildings. This aspect is particularly significant in multi storey moment resisting frame buildings. The discrepancies arise as pushover analyses fail to allow for higher mode affects associated with the formation of plastic hinges in the lower reaches of buildings (Fenwick and Davidson (1991),(1997)).

5.2.1.5 Time History Analysis

Numerical integration non linear time history analysis (THA) is a complex and time consuming analytical method for determining the seismic response of structures. This method appears to be the most comprehensive technique as it makes allowance for the designed strength of plastic hinge zones and the hysteretic behaviour of these zones. However, due to the sensitivity of response to individual seismic ground motions, a number of different analyses are required, using different ground motions, to enable a safe design to be made.

Time history analyses use numerical integration to solve the response of a multi degree of freedom structure. It can directly modify the stiffness of the elements and joints based on the deformation states. The equation of motion incorporates the mass, damping and stiffness of the structure and the dynamic loading from the earthquake ground motion.

The design process requires a vast range of information for the elements of a structure such as: geometries, strength, yield and hysteretic properties. The time history records, which are applied when using THA, need to be carefully chosen to ensure they are representative of the likely ground motion for the location of structure of interest. For example, the records chosen should match the typical type of ground rupture, duration of shaking, frequency content and other effects associated with the site. To ensure the time history records are of reasonable magnitude, they are generally required to be scaled to match a response spectrum for the site.