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Ultimate load states

The ultimate limit state design is based on rules given in EN 1996-1-1, chapter 6.

Unreinforced masonry walls subjected to mainly vertical loading

Basic equation for design of unreinforced masonry walls subjected to mainly vertical loading is (in accordance with paragraph 6.1.2):

where:

NEd

  • is the design value of the vertical load

NRd

  • is the design value of the vertical resistance of the wall

The design value of the vertical resistance NRd is calculated using formula

where:

Φ

  • is the capacity reduction factor

A

  • is the total area of cross-section

fd

  • is the design compressive strength of the masonry

The design compressive strength fd of the cross-sections with area smaller than 0.1m2 is multiplied in accordance with 6.1.2.1.(3) by factor

kde je:

A

  • is the total area of cross-section

The calculation of reduction factor for slenderness and eccentricity Φi at the top and bottom of the wall is based on on a rectangular stress block:

where:

Φi

  • is the reduction factor for slenderness and eccentricity

ei

  • is the eccentricity at the top or bottom of the wall

t

  • is the wall thickness

The eccentricity at the top or bottom of the wall ei is calculated using equation

where:

Mid

  • is the design value of the bending moment at the top or bottom of the wall caused by the eccentricity of the floor load at the support

Nid

  • is the design value of the vertical force at the top or bottom of the wall

einit

  • is the initial eccentricity

t

  • is the wall thickness

The initial eccentricity is calculated in accordance with 5.5.1.1(4) using equation

where:

hef

  • is the effective height of the wall

The reduction factor within the middle height of the wall Φm is calculated according to the annex G. Following equation is used for the walls with rectagular cross-sections:

where is

and

where is

The eccentricity at the middle height of the wall emk is calculated using equation

where:

em

  • is the eccentricity due to loads

ek

  • is the eccentricity due to creep

t

  • is the wall thickness

The eccentricity due to loads em is calculated using equation

where:

Mmd

  • is the design value of the bending moment at the middle of the height of the wall resulting from the moments at the top and bottom of the wall

Nmd

  • is the design value of the vertical force at the middle height of the wall

einit

  • is the initial eccentricity

The eccentricity due to creep ek is calculated using equation

where:

hef

  • is the effective height of the wall

tef

  • is the effective thickness of the wall

Φ

  • is the final creep coefficient

t

  • is the wall thickness

em

  • is the eccentricity due to loads

For walls fulfilling condition

is the eccentricity due to creep ek equal to zero.

The design value of the vertical force is calculated using iteration of the deformation along the cross-section area under conditions written in  6.1.1(2) for more complicated shapes of the cross-sections. the stress-strain relationship diagram is taken to be rectangular. Normal force can't be equal to zero and can't be located outside the cross-section.

Unreinforced masonry walls subjected to lateral loading

Basic equation for design of unreinforced masonry walls subjected to lateral loading is (in accordance with paragraph 6.3.1):

where:

MEd

  • is the design value of the moment

MRd

  • is the design value of the bending resistance

The design value of the bending resistance MRd is calculated using equation

where:

fxd

  • is the design flexural strength appropriate to the plane of bending

Z

  • is the elastic section modulus

When a vertical load is present, the favourable effect of the vertical stress is considered using equation in accordance with 6.3.1(4)(i):

where:

fxd

  • is the design flexural strength of masonry with the plane of failure parallel to the bed joints

σd

  • is the design compressive stress on the wall, not taken to be greater than 0.2fd

Unreinforced masonry walls subjected to shear loading

Shear is analysed according to 6.2, basic equation is

where:

VEd

  • is the design value of the shear force

VRd

  • is the design value of the shear resistance

The design value of the shear resistance MRd is calculated using equation

where:

VRd

  • is the design value of the shear resistance

fvd

  • is the design value of the shear strength of masonry

Ac

  • is the area of compressed part of the cross-section

Buckling of columns with more complicated cross-sections

Buckling verification of columns, that have more complicated shapes of cross-sections, is performed with the help of an effective cross-section. The effective cross-section is a rectangle, that is selected according to the following rules:

  • The area of the effective cross-section is identical to the area of the real cross-section
  • The ratio Wy/Wz is identical for the real and effective cross-sections

Walls subjected to concentrated loads

Basic equation for design of walls subjected to concentrated vertical load (in accordance with paragraph 6.1.3):

where:

NEdc

  • design value of concentrated vertical load

NRdc

  • design value of the vertical concentrated load resistance

Design value of the vertical concentrated load resistance NRdc is calculated using equation:

where:

β

  • is an enhancement factor for concentrated loads

Ab

  • is the loaded area

fd

  • is the design compressive strength of the masonry

Enhancement factor for concentrated load β is calculated using equation:

where:

a1

  • is the distance from the end of the wall to the nearer edge of the loaded area (see figure 6.2)

hc

  • is the height of the wall to the level of the load

Aef

  • is the effective area of bearing, i.e. lefm t

lefm

  • is the effective length of the bearing as determined at the mid height of the wall or pier (see figure 6.2)

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