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:	*N_Ed*	is the design value of the vertical load
	*N_Rd*	is the design value of the vertical resistance of the wall

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

where:	Φ	is the capacity reduction factor
	A	is the total area of cross-section
	*f_d*	is the design compressive strength of the masonry

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

kde je:

is the total area of cross-section

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

where:	*Φ_i*	is the reduction factor for slenderness and eccentricity
	*e_i*	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 e_i is calculated using equation

where:	*M_id*	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
	*N_id*	is the design value of the vertical force at the top or bottom of the wall
	*e_init*	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:

h_ef

is the effective height of the wall

The reduction factor within the middle height of the wall Φ_m 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 e_mk is calculated using equation

where:	*e_m*	is the eccentricity due to loads
	*e_k*	is the eccentricity due to creep
	t	is the wall thickness

The eccentricity due to loads e_m is calculated using equation

where:	*M_md*	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
	*N_md*	is the design value of the vertical force at the middle height of the wall
	*e_init*	is the initial eccentricity

The eccentricity due to creep e_k is calculated using equation

where:	*h_ef*	is the effective height of the wall
	*t_ef*	is the effective thickness of the wall
	*Φ_∞*	is the final creep coefficient
	t	is the wall thickness
	*e_m*	is the eccentricity due to loads

For walls fulfilling condition

is the eccentricity due to creep e_k 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:	*M_Ed*	is the design value of the moment
	*M_Rd*	is the design value of the bending resistance

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

where:	*f_xd*	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:	*f_xd*	is the design flexural strength of masonry with the plane of failure parallel to the bed joints
	*σ_d*	is the design compressive stress on the wall, not taken to be greater than 0.2f_d

Unreinforced masonry walls subjected to shear loading

Shear is analysed according to 6.2, basic equation is

where:	*V_Ed*	is the design value of the shear force
	*V_Rd*	is the design value of the shear resistance

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

where:	*V_Rd*	is the design value of the shear resistance
	*f_vd*	is the design value of the shear strength of masonry
	*A_c*	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 W_y/W_z 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:	*N_Edc*	design value of concentrated vertical load
	*N_Rdc*	design value of the vertical concentrated load resistance

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

where:	β	is an enhancement factor for concentrated loads
	*A_b*	is the loaded area
	*f_d*	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)
	h_c	is the height of the wall to the level of the load
	A_ef	is the effective area of bearing, i.e. l_efmt
	l_efm	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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