Homework Earthquake

3 HIGH INTENSITY EXCITATION ANALYSIS

The goal of this chapter is to perform an elastic dynamic analysis of the structure under a high

intensity unidirectional seismic excitation (parallel to the frame main direction X). Two differ-

ent approaches were considered:

• Time history (TH) analysis;

• Response spectrum (RS) method.

3.1 Initial considerations

Time history analysis requires a design accelerogram and obtains the response of the structure

by means of a numerical integration of the equation of motion of the system:

)=−ü(

ẋ x

ẍ( )+ ( )+ ( )

Where x=x(t) is the displacement vector with respect to the reference frame that moves with the

̈ ()

base of the structure, describing the relative displacements of the structure, while is the

ground acceleration as given by the design accelerogram.

The design accelerogram for our project is shown in the figure below.

Figure 3.1: Design accelerogram

For the RS analysis, the response spectrum used is taken from EC8 for 5% damping, Soil Class

B and PGA=0.4g, with a T of 0.6s instead of 0.5s. In this way response spectrum is consistent

C

with the one obtained directly from the accelerogram we used for the time history analysis. 19

Figure 3.2 EC8 Response Specturm

Moreover, in order to compare the numerical results with the experimental ones, base shear and

top displacement were evaluated. However, since the model in Midas is purely elastic and the

real structure behaves in an elastoplastic manner, behaviour and ductility factors were used to

correct the numerical solution.

The assigned value of the behaviour factor, considering the structure’s ductility class and regu-

larity, is q=5. =̈ /̈

Starting from the behaviour factor, it is possible to evaluate the ductility factor

using the expressions below: if T>T (equal displacement rule)

= C

if T<T (equal energy rule)

= (−1)+1 C

*

/

To be consistent with the response spectrum used, T was assumed to be 0.6s.

C

Using the obtained values of q and μ, shear and displacement were corrected using the following

expressions:

=

̈

̈ =

–

20 EARTHQUAKE RESISTANT DESIGN A.Y. 2023-24 Prof. Chesi & Martinelli

3.2 Model H1

This model is just the model chosen in the first chapter (i.e. Model 1.2), where the masses have

been modified in order to take into account of the different masses present in the experimental

tests. In this case, the floor diaphragm assumption has been made again and the new masses are

shown before: Table 3.1: Floor masses

Floor Mp Mr 2

ton ton*m

Roof 54 655.88

3 59 716.61

2 59 716.61

1 60 728.76

The results of Model H1 are shown below:

Table 3.2: Model H1 results μ

q Tc T

mode3

- s s -

5 0.6 0.3204 8.49

RESPONSE SPECTRUM

Vel Vpl TEST ERR uel upl TEST ERR

kN kN kN % mm mm mm %

1845.2 369.0 1161.9 -68.2 33.863 57.503 100.689 -42.8

TIME HISTORY

Vel Vpl TEST ERR uel upl TEST ERR

kN kN kN % mm mm mm %

1828.7 365.7 1161.9 -68.5 32.312 54.869 100.689 -45.5

Figure 3.3: Model H1 results

As expected, this model is far off if compared to the real behaviour. 21

3.3 Model H2

For the PsD test simulation, FEMA 356 suggests stiffness for reinforced concrete members: they

are presented in the table below.

Figure 3.4: Effective stiffness values for concrete

In this model walls were considered as uncracked and to implement in Midas this reduction the

command Stiffness Scale Factor was used; this command affects shear area A and flexural inertia

w

I for each section with no effects on the elastic parameters E and G.

Results are shown below: Table 3.3: Model H2 results μ

q Tc T

mode3

- s s -

5 0.6 0.417 6.75

RESPONSE SPECTRUM

Vel Vpl TEST ERR uel upl TEST ERR

kN kN kN % mm mm mm %

1845.5 366.5 1161.9 -68.45 57.63 77.87 100.689 -22.67

TIME HISTORY

Vel Vpl TEST ERR uel upl TEST ERR

kN kN kN % mm mm mm %

2072.4 414.5 1161.9 -64.33 57.03 77.05 100.68 -23.47

–

22 EARTHQUAKE RESISTANT DESIGN A.Y. 2023-24 Prof. Chesi & Martinelli

Figure 3.5: Model H2 results

It’s worth noting that the reduction of stiffness induced an increase of displacement values for

both time history and response spectrum analysis. As a matter of fact the error in terms of dis-

placements decreased, while the one related to shear force remains quite constant with respect to

del previous model.

3.4 Model H3

This model, instead, consider cracked walls, hence the reduction factor for walls flexural rigidity

turned from 0.8 into 0.5. Table 3.4: Model H3 results μ

q Tc T

mode3

- s s -

5 0.6 0.445 6.391

RESPONSE SPECTRUM

Vel Vpl TEST ERR uel upl TEST ERR

kN kN kN % mm mm mm %

1866.2 373.2 1161.9 -67.88 65.05 83.14 100.68 -17.42

TIME HISTORY

Vel Vpl TEST ERR uel upl TEST ERR

kN kN kN % mm mm mm %

1978.9 395.7 1161.9 -65.94 64.72 82.72 100.68 -17.84 23

Figure 3.6: Model H3 results

3.5 Model H4

Experimentally it is proven that walls are able to reduce their stiffness much more that as indicated

by FEMA.

The model by Paulay and Priestley suggests further reducing the stiffness of shear walls (Chapter

“Seismic Concrete and Mansory Building”).

5.3, Design of Reinforced 100

= + ≅ 0.20

( )

′

Where I and I are respectively the reduced and the unreduced moment of inertia, is the yield-

e g

ing strength of steel, equal to 500 MPa, is the maximum axial load, averaged on the whole

̈

′ is the cylindrical tension strength of concrete and A

height of the building, is the area of

g

concrete.

Introducing this further reduction of bending stiffness for walls, we get:

Table 3.5: Model H4 results μ

q Tc T

mode3

- s s -

5 0.6 0.515 5.658

RESPONSE SPECTRUM

Vel Vpl TEST ERR uel upl TEST ERR

kN kN kN % mm mm mm %

1918.82 383.77 1161.9 -66.97 88.72 100.401 100.689 -0.29

TIME HISTORY

Vel Vpl TEST ERR uel upl TEST ERR

kN kN kN % mm mm mm %

1930.36 386.07 1161.9 -66.77 90.24 102.12 100.689 1.42

–

24 EARTHQUAKE RESISTANT DESIGN A.Y. 2023-24 Prof. Chesi & Martinelli

Figure 3.7: Model H4 results

This model is able to capture pretty well the displacement at the top, but, looking at the shear

forces at the base, error is still really big.

3.6 Model H5

In order to solve the above-mentioned problem, the q factor value was changed, according to the

formula:

=

Taking into account the shear values related to the response spectrum analysis of Model H4,

which is the one with the smallest error in terms of displacements.

Table 3.6: Model H4 results μ

q Tc T

mode3

- s s -

1.66 0.6 0.515 1.67

RESPONSE SPECTRUM

Vel Vpl TEST ERR uel upl TEST ERR

kN kN kN % mm mm mm %

1918.82 1154.96 1161.9 -0.60 88.72 94.53 100.689 -5

TIME HISTORY

Vel Vpl TEST ERR uel upl TEST ERR

kN kN kN % mm mm mm %

1930.36 1161.9 1161.9 0 90.24 96.15 100.689 -4.51

All the errors are smaller or equal to 5%, hence Model H5 is a good representation of the structure

under high intensity excitation. 25

4 PUSHOVER ANALYSIS

The purpose of this chapter is to set up a model for a pushover analysis on the building and com-

pare the resulting capacity curve with the experimental results. In fact, as mentioned before, the

structure is designed at ULS to resist the design earthquake by exploiting plastic phenomena,

that’s why it is important to check the structure response in nonlinear phase.

4.1 Introduction

Pushover analysis is a non-linear static analysis carried out under conditions of constant gravity

loads and monotonically increasing horizontal loads. These are distributed, at each level of the

building, proportionally to inertia forces, and have as resultant a shear base F .

b

Forces are scaled in such a way that the horizontal displacement d of a control point, placed at

n

the center of mass of the last level of the building, increases in a monotonic way, both in positive

–

and negative direction, up to the collapse conditions. The F d diagram represents the capacity

b n

curve of the structure. Figure 4.1 Pushover plastic hinge position

According to Eurocode 8 and NTC 2018, pushover analysis is usually performed for the following

reasons:

• To verify or revise the overstrength ratio values α / α

u 1

• To estimate the expected plastic mechanisms and the distribution of damage in build-

ings designed with the force reduction factor q

• To assess the structural performance of existing or retrofitted buildings

• As an alternative to the design based on linear-elastic analysis which uses the behav-

iour factor q. In this case, the target displacement should be used as the basis of the

design. Figure 4.2 Pushover total shear result –

26 EARTHQUAKE RESISTANT DESIGN A.Y. 2023-24 Prof. Chesi & Martinelli

The key steps in the non-linear static method are:

• Evaluation of vertical loads;

• Evaluation of horizontal forces distribution;

• Placement of reinforcement;

• Definition of plastic properties.

4.2 Vertical Loads

Starting from Model H5, vertical loads were modified in order to have more accuracy in this type

of analysis: to reproduce the effects of the axial force in the vertical elements, at each node has

been applied a vertical force computed using the tributary length of each vertical elements and a

uniform load of 22 kN/m, 24.4 kN/m, 24.4 kN/m and 24.9 kN/m acting respectively on the hori-

zontal elements for the levels from roof to foundation. These loads represent the structural self-

weight and the added weight, in the form of water filled containers, present at the rime of testing.

Figure 4.3 Additional nodal loads

This load distribution was implemented in Midas by using commands Self Weight and Element

Beam Loads.

Furthermore, an ultimate limit state load combination was generated.

4.3 Horizontal Loads

As suggested by the code, at least two distributions of horizontal forces are required:

• A “modal” pattern, proportional to lateral forces consistent with the lateral distribu-

tion in the direction under consideration determined in elastic analysis;

• A “uniform” pattern, based on lateral forces that are proportional to mass regardless

of elevation.

Both horizontal distributions must be considered in direction X and -X. 27

Figure 4.4 Pushover load distribution

4.4 Placement of reinforcement

To incorporate reinforcement in the model, the Midas Gen functi