FEM-Design performs design calculations for reinforced concrete-, steel- and timber structures according to Eurocode. The following design considers EC1992-1-1 (standard) with the following National Annexes (NA) for:

• Denmark
• Estonia
• Finland
• Germany
• Hungary
• Latvia
• Norway
• Poland
• Romania
• Sweden
• United Kingdom.

Design forces

The design forces are the forces that the reinforcements should be designed for in the reinforcement directions. The term design forces have meaning only in surface structures like plate, wall or 3D plate. In beam structures the design forces are equivalent to the internal forces. The necessary reinforcement calculations are based on the design forces.

The way of calculating the design forces is common in all modules and in all standards.

In FEM-Design the design forces calculation is based on the mechanism of optimal reinforcement calculation for skew reinforcements made by M.P. Niel- sen, Wood-Armer and Dr. Ferenc Németh, see [2]. The following description will show the way of calculation for moments but the way of the calculation is the same for normal forces too. Just substitute the m signs with n and you will have the calculation for normal forces.

The results will be the design moments:

In the first step we are taking a ξ-ϑ coordinate system and transform the internal forces into this system:

Now the design forces will be chosen from four basic cases called a), b), ξ) and η). The possible design moment pairs of the cases:

a) case:

b) case:

ξ) case:

η) case:

From the four cases the one is invalid where:

• the signs are different: m_ξ*m_η < 0
• the crack tensor invariant is less than the internal forces invariant:

The valid positive pair will be the design moment for bottom reinforcement; the valid negative pair will be the design moments for the top reinforcement (positive means positive and zero values; negative means negative and zero values).

So the result will be four values in a certain point: two moment values for each reinforcement directions. It can sound strange that the reinforcements are used for both positive and negative moment in one direction at the same time, but if we are looking at a plate where the m_x is positive and the m_y is negative and the reinforcements have an angle of 45 degree to the x direction we could imagine that the bottom reinforcement bars make equilibrium to the m_x and the top reinforcement bars make equilibrium to the m_y. So a certain reinforcement direction takes positive and negative loads at the same time.

Shrinkage as load action

The effect of the shrinkage for the surface reinforcement bars in one direction (here X) (it is also valid in other bar directions):

The specific normal force causing the given shrinkage value (ε_cs [‰] at concrete materials) in the concrete zone of the section is (here in X direction):

N_X = E_c A_c ε_cs [k&Nu;/m]

The position change of centre of gravity considering reinforcement bars (here X-direction; see dashed line):

where:   

n = E_s / E_c and S_s is the statical moment of (here) X-directional bars around the Y axis of the calculation plane.

The moment around the Y axis of the calculation plane from N_X because of the position change of centre of gravity:

M_Y = N_X z_s

The specific rotation (curvature) from M_Y for 1 meter wide section:

Design calculations for surface structures

Ultimate limit state

The design of the slab is performed with respect to the design moments described in Design forces.

In order to minimize cracking in the slab a good way is to reinforce according to the elastic moments which normally also leads to good reinforcement economy. The required bending reinforcement is designed according to EC2 3.1.7, where a rectangular stress distribution as shown below has been assumed.

λ = 0,8                                              for  f_ck ≤ 50 MPa

λ = 0,8 - (f_ck - 50)/400                     for  50 < f_ck ≤ 90 MPa

and:

η = 1,0                                              for  f_ck ≤ 50 MPa

η = 1,0 - (f_ck - 50)/200                     for  50 < f_ck ≤ 90 MPa

Note, that the required bending reinforcement is at design level primary not affected of the presence of user defined reinforcement. However, when user defined applied reinforcement is selected the stiffness will be effected, which in most cases will influence the moment distribution and thus secondary the required bending reinforcement.

Shear capacity

The shear capacity is calculated according to EC2 6.2.2 and 6.2.3 considering applied bending reinforcement when the option Checking has been selected. Otherwise, the required bending reinforcement according to this chapter. The design criteria for the shear capacity is:

VSd < VRd1
 

where:   

V_Sd is the design shear force;

V_Sd = Q, which is calculated as  

V_Rd is the shear capacity.

If the section in which the shear force is acting has an angle with respect to the reinforcement directions the shear capacity is calculated as:

Punching

The punching capacity is calculated according to EC2 6.4.3 - 6.4.5.
 

Checking

Punching without shear reinforcement

A concrete compression check on u0 is made according to 6.4.5 (6.53). A concrete shear check on u₁  is made for a capacity calculated according to 6.4.4 (6.47).

Punching with shear reinforcement

A concrete compression check on u₀ is made according to 6.4.5 (6.53). Reinforcement is calculated with regard to critical perimeters u₁, u₂, ... u_nReinf  according to 6.4.5 (6.52 ).

(u_i are control perimeters above the reinforced region, distance between them is ”Perimeter distance”, defined in the calculation parameter).

A concrete shear check on u_out is made for a capacity calculated according to 6.4.4 (6.47)

(u_out is either the first perimeter that does not need reinforcement, or if it is not found, the perimeter that is k d_eff distance from the outer perimeter of the reinforcement).

Warnings

A warning message is shown, if reinforcement does not comply with the detailing rules in 9.4.3.

Design

1. Check, if reinforcement is needed at all,
2. If reinforcement is needed, it is designed to satisfy the detailing rules in 9.4.3, if possible,
3. Design fails and a warning message is displayed if, u_out is not found within 6 d_eff distance from the column perimeter.

Comments, limitations

- openings are not considered when control perimeters are generated.
- the position of the column relative to the plate is considered only in the generation of control perimeter. It means, (user or program defined) reinforcement may be partly out of the plate, but it won't affect the calculation.
- If ”Calculate β automatically” is set in the calculation parameter, β is calculated according to equation 6.4.3 (6.39).

Serviceability limit state

Method of solution

The program performs crack- and deflection control for all load combinations according to EC2 7.3 and 7.4. Two limiting conditions are assumed to exist for the calculations: Stadium I (the uncracked condition) and Stadium II (the fully cracked condition).

Stadium I Uncracked condition

If the user does not activate the option Cracked section analysis, the calculation will be performed with respect to the total stiffness of the slab.

Stadium II Fully cracked condition

If the option Cracked section analysis, is activated the program will consider the decrease in slab stiffness on behalf of cracking. This means an iterative calculation where the slab in the beginning is assumed to be uncracked when the section forces are calculated. Sections which are not loaded above the level which would cause the tensile strength of the concrete to be exceeded will be considered to be uncracked (Stadium I). Sections which are expected to crack will behave in a manner intermediate between the uncracked and fully cracked conditions and an adequate prediction of behavior used in the program is shown below.

The stiffness calculation is performed considering the required or the applied reinforcement depending on what option has been selected. If applied reinforcement has been selected this is used in all load combinations. If applied reinforcement is not present or not selected the required reinforcement is used instead. In the latter case the required reinforcement in every element is calculated as the maximum value from all load combinations, which means that all calculations of serviceability limit values are performed with the same reinforcement.

In the next step a new calculation based on the new stiffness distribution is performed and so on. When the deflection values resulting from two calculations does not differ more than a defined percentage of the first one or the maximal number of allowed calculations has been reached the calculation is stopped.

Crack width

Crack width is according to EC2 7.3.4 calculated as:

w_k = S_r,max (ε_sm - ε_cm) 

where:

S_r,max is the maximum crack spacing,

ε_sm is the mean strain in the reinforcement under the relevant com- bination of loads, including the effect of imposed deformations and taking into account the effects of tension stiffening. Only the additional tensile strain beyond the state of zero strain of the concrete at the same level is considered,

ε_cm is the mean strain in the concrete between cracks.

ε_sm - ε_cm may be calculated from the expression:

where:

σ_s is the stress in the tension reinforcement assuming a cracked section. For pretensioned members, σ_s may be replaced by &Delta;σ_p the stress variation in prestressing tendons from the state of zero strain of the concrete at the same level,

α_e= is the ratio Es / Ecm

A_p and A_c,eff are as defined in 7.3.2 (3),

ξ₁ according to Expression (7.5),

k_t is a factor dependent on the duration of the load. 

k_t = 0,6 for short term loading

k_t = 0,4 for long term loading. For long term loads (k_t = 0,4):

Ap´ = 0,0 (pre or post-tensioned tendons)

Ac,eff:

h_cef = min (2,5 * (h - d), (h - x) / 3, h / 2 )

s_r,max = k₃c + k₁ k₂ k₄φ /ρ_p,eff

where:

φ is the bar diameter. Where a mixture of bar diameters is used in a
section, an equivalent diameter, φ_eq, should be used. For a section with n₁ bars of diameter φ₁ and n₂ bars of diameter φ₂, the following expression should be used,

c is the cover to the longitudinal reinforcement,

k₁ is a coefficient which takes account of the bond properties of the bonded reinforcement:

k₁ = 0,8 for high bond bars,

k₁ = 1,6 for bars with an effectively plain surface (e.g. prestressing tendons),

k2 is a coefficient which takes account of the distribution of strain:

k2 = 0,5 for bending,

k2 = 1,0 for pure tension,

k₂ = (ε1 + ε₂) / 2ε₁

where ε₁ is the greater and ε₂ is the lesser tensile strain at the boundaries of the section considered, assessed on the basis of a cracked section.
Recommended values of k₃ = 3,4 and k₄ = 0,425 are used.

Maximum crack spacing:

s_r,max = 1,3 (h - x)

Equivalent quantities perpendicular to crack direction:

• Reinforcement area:

• Number of bars:

• Diameter:

Deflections

The calculations is performed according to EC2 7.4.3.

Stadium I Uncracked condition

Load depended curvature is calculated as:

1 / r_f = M / E_c,ef I₁

where:

M is current moment,

I₁ is Moment of Inertia in Stadium I,

E_c,ef is the modulus of elasticity with respect to creep.

The modulus of elasticity is calculated as:

E_c,eff = E_cm / (1 + φ)

where φ is the creep coefficient.

Curvature with respect to shrinkage is considered according to 2.2.2 above.

Stadium II Fully cracked condition

Load depended curvature is calculated as:

1 / r_f = M / E_c,ef I₂

where:

E_c,ef is the modulus of elasticity as shown above,

I₂ is the moment of Inertia in Stadium II,

M is current moment.

Curvature with respect to shrinkage is considered according to 2.2.2 above.

Sections which are expected to crack will behave in a manner intermediate between the uncracked and fully cracked conditions and an adequate prediction of this behavior is given by:

α = ζ α_II + (1 - ζ) α_I

where:

α is in this case the curvature calculated for the uncracked and fully cracked conditions,

ζ is a distribution coefficient given by ζ = 1 - β (σ_sr / σ_s)²

ζ is zero for uncracked sections,

β is a coefficient taking account of the influence of the duration of the loading or of repeated loading on the average strain,

σ_s is the stress in the tension steel calculated on the basis of a crack- ed section,

σ_sr is the stress in the tension steel calculated on the basis of a cracked section under the loading which will just cause cracking at the section being considered.

Design calculations for bar structures

Material properties

Concrete

• Ultimate limit states:
  Continuous line is used.
   
• Servicibility limit states:
  Stage II is used (dashed line, without horizontal section).

Steel

• Ultimate limit states:
  B graph with horizontal line is used.
   
• Servicibility limit states:
  The same as ultimate but without safety factor.

Longitudinal reinforcement

Analysis of second order effects with axial load

According to EC2 5.8.

• For calculation of 2nd order effect Nominal curvature method (5.8.8) is used.
• If there is no compression force in the section the eccentricity is equal to 0,0.
• Buckling lengths l_0x and l_0y are specified by the user.
• Curvature:

1 / r = kr kϕ 1 / r0

where:

k_r is a correction factor depending on axial load,

κ_ϕ is a factor for taking account of creep,

1 / r0 = ε_yd / (0,45 d),

d is the effective depth,

d = (h / 2) + is

where i_s is the radius of gyration of the total reinforcement area.

k_r = (n_u - n) / (n_u - n_bal) ≤ 1

where:

n = N_Ed / (A_c f_cd), relative axial force,

N_Ed is the design value of axial force, n_u = 1 + ω,

n_bal is the value of n at maximum moment resistance; the value 0,4 is used,

ω = A_s f_yd / (A_c f_cd),

A_s is the total area of reinforcement,

A_c is the area of concrete cross section,

k_ϕ = 1 + β ϕ_ef ≥ 1

where:

ϕ_ef is effective creep ratio, defined by the user,

β = 0,35 + fck /200 - λ / 150,

λ is the slenderness ratio.

• 2nd order effect is ignored, if:

λ ≤ λ_lim

λ_lim = 20 A B C / √n

where: A = 1 / (1 + 0,2 ϕ_ef),

B = √1 + 2 ω,

C = 0,7

ϕ_ef is effective creep ratio,

ω = A_s f_yd / (A_c f_cd), mechanical reinforcement ratio,

A_s is the total area of longitudinal reinforcement,

n = N_Ed / A_c f_cd), relative normal force,

r_m = M₀₁ / M₀₂, moment ratio,

M₀₁, M₀₂ are the first order en moments |M₀₂| ≥ |M₀₁|.

• Geometric imperfection (5.2 (7) a):

e_i = l₀ / 400

• The minimum of all eccentricities (1st order + imperfection + 2nd order effect): max (20,0; h / 30,0).
• Imperfection and 2nd order effect considered in both directions.
• The eccentricity is calculated in four possible positions:
  • Stiff direction+, weak direction+
  • Stiff -, weak+
  • Stiff+, weak-
  • Stiff-, weak-

Torsion

• Necessary longitudinal reinforcement area (A_sl):
  T_Ed is the applied design torsion (see Figure 6.11):

The required cross-sectional area of the longitudinal reinforcement for torsion &Sigma;A_sl may be calculated from:

where:

u_k is the perimeter of the area A_k,

f_yd is the design yield stress of the longitudinal reinforcement A_sl,

θ is the angle of compression struts, θ = 45 deg.

Considering torsion in calculation of longitudinal bars:

Calculation of torsional capacity by edges, considering all bars placed in tef strip. The minimum of capacities gives the torsional capacity of the section. Utilization for torsion calculated for all bars placed in the strip one by one.

Area of these bars decreased in the calculation of axial effects (N, My, Mz) in proportion of utilization (see formula below):

where:

A is area of the bar,

A’ is decreased area used in calculation

ULS checking

SLS checking

Crack width calculated according to EC2 7.3.

• Crack width calculated as:

w_k = s_r,max (ε_sm - ε_cm)

where:

s_r,max is the maximum crack spacing,

ε_sm is the mean strain in the reinforcement under the relevant combination of loads, including the effect of imposed derforma- tions and taking into account the effects of tension stiffening. Only the additional tensile strain beyond the state of zero strain of the concrete at the same level is considered,

ε_cm is the mean strain in the concrete between cracks.

ε_sm - ε_cm may be calculated from the expression:

where:

σ_s is the tension reinforcement assuming a cracked section,

α_e is the ratio E_s / E_cm

ρ_p,eff = A_s / A_c,eff,

A_c,eff is calculated as below,

k_t is a factor dependent on the duration of the load,

k_t = 0,6 for short term loading,

k_t = 0,4 for long term loading (always supposed by the program),

A_c,eff:

h_c,ef = min (2,5 (h - d), (h - x) / 3, h / 2)

s_r,max = k₃ c + k₁ k₂ k₄ φ / ρ_p,eff

where:

φ is the bar diameter. Where a mixture of bar diameters is used in a section, an equivalent diameter, φ_eq, should be used. For a section with n₁ bars of diameter φ₁ and n₂ bars of diameter φ₂, the following expression should be used,

c is the cover to the longitudinal reinforcement,

k₁ is a coefficient which takes account of the bond properties of the bonded reinforcement:

k₁ = 0,8 for high bond bars,

k₁ = 1,6 for bars with an effectively plain surface (e.g. prestressing tendons),

k₂ is a coefficient which takes account of the distribution of strain:

k₂ = 0,5 for bending,

k₂ = 1,0 for pure tension

For cases of eccentric tension or for local areas, intermediate values of k2 should be used which may be calculated from the relation:

k₂ = (ε₁ + ε₂) / 2 ε₁,

where: ε1 is the greater and ε₂ is the lesser tensile strain at the boundaries of the section considered, assessed on the basis of a cracked section.

Recommended values of k₃ = 3,4 and k₄ = 0,425 are used.

• Maximum crack spacing:

s_r,max = 1,3 (h - x)

Space between bars

• Minimum distance:
  The clear distance (horizontal and vertical) between individual parallel bars or horizontal layers of parallel bars should be not less than the maximum of k₁ bar diameter, (d_g + k₂ mm) or 20 mm where d_g is the maximum size of aggregate.

• Maximum distance:
  The longitudinal bars should be so arranged that there is at least one bar at each corner, the others being distributed uniformly around the inner periphery of the links, with a spacing not greater than 350 mm.

Lengthening and anchorage

• Because of shear effect (shift rule):

a_i = 0,9 max (h, b)

The code prescribes d instead of h, but the difference can be ignored.

• Anchorage:

f_bd = 2,25 η₁ η₂ f_ctd 

where:

f_ctd is design value of concrete tensile strength. Due to the increasing brittleness of higher strength concrete, f_ctk,0,05 should be limited here to the value for C60/75, unless it can be verified that the average bond strength increases above this limit

η₁ is a coefficient related to the quality of the bond condition and the position of the bar during concreting:

η₁ = 0,7

η₂ is related to the bar diameter:

η₂ = 1,0 for φ ≤ 32 mm,

η₂ = (132 - φ) / 100 for φ > 32 mm

l_b,rqd = (φ / 4) (σ_sd / f_bd)

where:

σ_sd = f_yd (fully utilized bar supposed),

l_bd = α₁ α₂ α₃ α₄ α₅ l_b,rqd ≥ l_b,min,

α_i = 1,0

l_b,min is the minimum anchorage length if no other limitation is applied:

• for anchorages in tension:
  l_b,min > max (0,3 l_b,rqd; 10 φ; 100 mm),
   
• for anchorage in compression:
  l_b,min > max (0,6 l_b,rqd; 10 φ; 100 mm),
  Rule given for compression is used.

Stirrups

Shear

In Figure 6.5 below the following notations are shown:

α is the angle between shear reinforcement and beam axis perpendicular to the shear force (measured positive as shown in Figure 6.5),

θ is the angle between the concrete compression strut and the beam axis per- pendicular to the shear force,

F_td is the dessign value or the tensile force in the longitudinal reinforcement,

F_cd is the design value of the concrete compression force in the direction of the longitudinal member axis,

b_w is the minimum width between tension and compression chords,

z is the inner lever arm, for a member with constant depth, corresponding to the bending moment in the element under consideration. In the shear analy- sis of reinforced concrete without axial force, the approximate value z = 0,9 d may normally be used.

• Member do not require shear reinforcement, if:

The design value for the shear resistance V_Rd,c is given by:

V_Rd,c = [C_Rd,c k (100 ρ_l f_ck)1/3 + k₁ σ_cp] b_w d 

with a minimum of:

V_Rd,c = (v_min + k₁ σ_cp) b_w d 

where: f_ck is in MPa

A_sl is the area of the tensile reinforcement, which extends: ≥ (l_bd + d) beyond the section aonsidered (see Figure 6.3), b_w is the smallest width of the cross-section in the tensile area [mm],

σ_cp = N_Ed / A_c < 0,2 f_cd [MPa],

N_Ed is the axial force in the cross-section due to loading or prestressing [in N] (N_Ed > 0 for compression). The influence of imposed deformations on N_E may be ignored,

Ac is the area of concrete cross section [mm2],

V_Rd,c is [N]

The recommended value for C_Rd,c is 0,18 / γ_c, that for v_min is given by the expression below and that for k₁ is 0,15.

• Upper limit of shear:

V_Rd,max = α_cw b_w z ν₁ f_cd / (cotθ + tanθ) 

where:

A_sw is the cross-sectional area of the shear reinfocement,

s is the spacing of the stirrups,

f_ywd is the design yield strength of the shear reinforcement,

ν₁ is a strength reduction factor for concrete cracked in shear,

α_cw is a coefficient taking account of the state of the stress in the compression chord.

The recommended value of ν₁ is ν (see expression below). The recommended value of α_cw is as follows:1 for non-prestressed structures,

Capacity of stirrups:

where:

A_sw is the cross-sectional area of the shear reinforcement,

s is the spacing of the stirrups,

f_ywd is the design yield strength of the shear reinforcement.

Torsion

T_Ed is the applied design torsion (see Figure 6.11)

A_k is the area enclosed by the centre-lines of the connecting walls, including inner hollow areas,

τ_t,i is the torsional shear stress in wall i,

t_ef,i is the effective wall thickness. It may be taken as A/u, but should not be taken as less than twice the distance between edge and center of the longitudinal reinforcement. For hollow sections the real thickness is an upper limit,

A is the total area of the cross-section within the outer circumference, including inner hollow areas,

u is the outer circumference of the cross-section,

z_i is the side length of wall i defined by the distance between the intersection points with the adjacent walls,

θ = 45 deg, in all calculations.

• Member do not require torsional reinforcement, if:

T_Rd,c = f_cd t_ef 2 A_k ≤ T_Ed

• Upper limit of torsion:

T_Rd,max = 2 ν α_cw f_cd A_k t_ef,i sinθ cosθ 

where ν and α_cw are as above.

• Force in stirrups:

T_Rd,max=2να_cwf_cdA_kt_ef,isinθcosθ

The shear force V_Ed,i in a wall i due to torsion is given by:

V_Ed,i = τ_t,i t_ef,i z_i

z_i is section height used to be able to sum with shear.

• Capacity of stirrups:

See Shear.

Shear and torsion

• Forces in stirrups:

V_Ed = V_Ed(shear) / 2 + V_Ed(torsion)

• No stirrup required:

T_Ed / T_Rd,c + V_Ed / V_Rd,c ≤ 1,0 

• Upper limit of the effects:

T_Ed / T_Rd,max + V_Ed / V_Rd,max ≤ 1,0 

• Calculation is done in two directions y' and z' independently.