Parasismic verification of a building according to Eurocode 8 Make - - PowerPoint PPT Presentation

Parasismic verification of a building according to Eurocode 8

Make the seismic verification of the new building of the sport center located in Liege according to the Eurocode EN 1998‐1 The building characteristics are:

• Rectangular building of dimensions 28.2 X 12 m
• 1 ground level + 1 level and an underground level  3 levels
• Columns and central nucleus in concrete. All the other walls are in masonry except the underground walls

which are in concrete

• The slabs are in concrete
• The building is located in Liege with a soil class B according to EN 1998‐1
• The building is an office building with meeting rooms
• Concrete is C 30/37 with an instantaneous non cracked Young’s modulus of 35 000 N/mm²
• The dead masses of the floor are 700 kg/m² and 600 kg/m² for the roof
• The live loads are 500 kg/m²
• The vertical component of the earthquake can be neglected

Plane view ‐ underground

Plane view – level 0

Plane view – level 1

Plane view – roof

Elevation

Elevation

Facades

Facades

• 1. Simplify the static scheme

Ask yourself the following questions:

• In which direction do the earthquake loads act?
• What are the resisting elements under earthquake ?
• Do the columns sustain earthquakes loads ?
• What are the flexible parts of the building ?
• Which elements must be modelled ?
• The masonry has a very small resistance to horizontal loads  no resistance
• Make the simplest model to represent the structure behaviour under earthquake, minimise the dofs

Model the building with beam elements with only a few nodes, minimise the number of nodes

Static scheme – which element sustains the horizontal loads ?

Only the concrete central core !

Static scheme – improve the behaviour under horizontal loads

Big torsion in the central core. What can be done to reduce the torsion ?

Static scheme – improve the behaviour under horizontal loads

To add a concrete wall at the opposite side to increase the lever arm

Static scheme – underground level not taken into account

masonry concrte Very stiff In the soil  not taken into account

Static scheme – plane view

510 x 200 Th = 20 510 x 20 25 m 3.3 m 12 m

Econcrete,cracked = Econcrete/2

Static scheme – elevation view

1200 x 40 25 m 3.3 m 4 m 1200 x 40 5 m

Econcrete,cracked = Econcrete/2

Static scheme – elevation view

1200 x 40 25 m 3.3 m 4 m 1200 x 40 5 m No effect on the horizontal stiffness  not modelled But masses to take into account  Mnode = m.l

Static scheme – Masses

25 m 3.3 m 4 m 5 m

Masses = dead loads + 0.24 live loads 2 = 0.3 offices  = 0.8 Storeys with correlated occupancies

M = 600 kg/m² + 0.24 * 500 kg/m² = 720 kg/m² M = 700 kg/m² + 0.24 * 500 kg/m² = 820 kg/m²

• 2. Frequencies and eigen modes
• Establish and Compute the stiffness matrix of the system
• Establish and Compute the mass matrix of the system
• Compute the eigen modes and the frequencies
• Draw the deformed shape of the first 2 modes (by hand)
• How many modes do you have ?

Beam Stiffness Matrix

Matrix in the local axe of the element L : length A : section Iz : in‐plane inertia Iy : out‐of‐plane inertia J: torsional stiffness

Beam Static Characteristics

b h A = b.h I = b.h³ /12 J = h.b³ /3 b h A = b.h – (b‐2.t).(h‐2.t) I = b.h³ /12‐(b‐2.t).(h‐2.t)³ /12 J =

.

• 𝑒𝑡 =

. .

.

• S = surface inside the red line
• = length of the red line divided by the thickness

t

Local axes  Global axes

Apply a rotation matrix

1 2 3 4 5 6 1 C S 2 ‐S C 3 1 4 C S 5 ‐S C 6 1

r = R = r r

Kglo = RT * Kloc * R

6x6 6x6 6x6 6x6 6x6 6x6 6x6 6x6 6x6 6x6 6x6 6x6

Matrix assembly

1 1 3 2 2 4 3 K1= K2= K3=

1 2 3 4 1 6x6 6x6 2 6x6 6x6 6x6 3 6x6 6x6 4 6x6 6x6 6x6

NODES NODES

Supports

1 2 4 3 NODES NODES

1 2 3 4 1 6x6 6x6 2 6x6 6x6 6x6 3 6x6 6x6 4 6x6 6x6 6x6

Suppress the dofs related to supported nodes

Mass Matrix

1 2 3 4 5 6 7 8 9 10 11 12 1 m.L/2 2 m.L/2 3 m.L/2 4 5 6 7 m.L/2 8 m.L/2 9 m.L/2 10 11 12

m = mass by unit length L = length m,L x y Mloc  Mglo ; the same as the stiffness matrix Assembly of matrices; the same as the stiffness matrix

• 3. Modal properties
• Compute the modal stiffness and mass of each mode
• Compute the effective modal mass in both horizontal direction
• Compute the modal share ratio for both horizontal direction

Depend of the mode Depend of the mode and the seism direction Depend of the mode and the seism direction

Where:

• M is the mass matrix
• {ud}i is the ith eigen mode
• {e}k is a vector with 1 for each dof in the considered direction and 0 for the others

• 4. Parasismic calculation
• Establish the design acceleration spectrum according to EN 1998‐1

The building is located in Liege with a soil class B according EN 1998‐1 The building is an office building with meeting rooms The q factor is taken equal to 1.5

• Compute the response (displacements) of each mode in each direction
• Compute the maximum response in each direction (SSRS) of the building top
• Which modes govern the total seismic response for each direction ?
• Compute the support forces
• Combine the support forces in the X and Y direction
• Find a linear combination of the modes to obtain concomitant value of support forces at the foot of the

central core 𝛽

• 𝑥𝑗𝑢ℎ 𝑆

∑ 𝑆

• 𝑆 ∑ 𝛽 . 𝑆

Design Spectrum definition

According to EN 1998‐1: chapter 3.2.2.5

Design Spectrum definition

Design Spectrum definition

ag = i*agr agr=0.1.g

Design Spectrum definition

0,000 0,500 1,000 1,500 2,000 2,500 3,000 0,000 0,200 0,400 0,600 0,800 1,000 1,200 1,400 1,600 1,800 2,000

Sd (m/s²) T (s)

design Spectrum Response

Behaviour factor : q = 1.5

Seismic response of one mode

𝜑 2𝜊𝜕𝜑 𝜕𝜑 𝜑

• 1 dof system

n dof system

0,000 0,500 1,000 1,500 2,000 2,500 3,000 0,000 0,500 1,000 1,500 2,000

Sd (m/s²) T (s)

design Spectrum Response T a

𝜃 2𝜊𝜕𝜃 𝜕𝜃 𝑆𝑁 ∗ 𝜑

• accmax = Sd(,)

dmax = Sd(,)/² accmax,i = RMik * Sd(,) dmax,i = RMik * Sd(,)/²

Seismic response of all modes

For each mode: the maximum displacement = di All the mode are not maximum at the same time  total displacement ≠ ∑ 𝑒

•  total displacement = ∑

𝑒

• ; Square Root of the Sum of the Squares = SRSS

Support Forces

For each modes: [Kglo] * {ud}i= {Fi} where Fi = nodal forces If we have a support, the nodal force Fi: support forces Ri The maximum support force under earthquake: RSRSS = ∑ 𝑆

• To combine, the 2 horizontal directions :