ULS final stage - stiffened openings

 

The calculation is based on the publication SCI P355 - Design of composite beams with large web openings, R. M. Lawson, S. J. Hicks published in 2011.

The topic shows below only info specific to stiffened openings and modifications in calculations compared to openings without stiffeners.

Horizontal stiffeners are ment to be formed from rectangular plates, welded to the top and /or the bottom edges of the openings, on one or both sides. The input of the openings stiffeners may be done through "Member1D opening" dialog. The stiffeners are currently only used by the composite check.

Opening reinforcement group in the dialog is available if all conditions below are fulfilled:

• shape is not set to Cross-section.
• steel and concrete material are activated in the project
• composite beam module from CADS is not activated
• valid license is present
• opening is defined on a beam with steel material and Structure type of the member is plate rib (this condition is neglected during the input of the opening, when the target beam is not known)

Geometric recommendations

Chapters 5.2.1-2 specifies general geometry recommendations for openings with horizontal stiffeners. The list of tested recommendations may be seen below:

• The b_r/t_r ratio of the stiffeners should not exceed the limits for a Class 3 section (b_r/t_r ≤ 10*ε). Since the beam is designed as a Class 1 or 2 section, then any part of the outstand in excess of the Class 2 limit is considered as ineffective. (b_r ≤ 10*t_r*ε)

• Area of the stiffener (independent if single or double sided) A_r ≤ 0,5*h_o* t_w

• Distance to the inner surface of the flange ≥ 50mm

• Minimum offset from opening ≥ 8mm

• 8mm ≤ Thickness of the stiffener t_r ≤ 20mm

• 80 mm or l_o/10 ≤ Width of the stiffener b_r ≤ MIN (10*t_r*ε, 200mm)

• Weld throat thickness a ≥ MAX (0,5*t_r, 5mm)

• Distance from the closest support ≥ MAX (l_o, h)

• Distance between studs ≤ 300mm

• Length of the opening l_o ≤ 2,5*h_o

• Anchorage length l_v of the stiffener beyond each end of the opening should generally be taken as not less than 0.25*l_o or 2*b_r or a minimum of 150 mm and should satisfy the following criteria given by expressions (55):
  
  For shear resistance of the fillet welds:
  
  
  For shear resistance of the stiffeners:
  
  
  For shear resistance of the web
  
  

with:

h - height of steel cross-section

n - number of stiffeners (single-sided = 1, double-sided = 2)

b_r - width of stiffener

t_r - thickness of stiffener

t_w - thickness of the web

h_o - height of opening

l_o - length of opening

A_r - cross-sectional area of stiffener A_r = n*t_r*MIN(b_r;10*t_r*ε)

F_r - design force in stiffener F_r = A_r * f_yr / γ_M0

f_yr - yield strength of stiffener material

f_vw,d - design shear strength of a fillet weld. Material of fillet weld is assumed to be the same as the material of the stiffener.

f_ur - ultimate strength of stiffener material

f_ub - ultimate strength of beam material

β_w - correlation factor for fillet welds (EN 1993-1-8, Table 4.1)

γ_M0 - partial safety factor

γ_M2 - partial safety factor for welds

 

Chapter 5.2.3 specifies additional geometry recommendations for openings with single-sided horizontal stiffeners. The list of tested recommendations may be seen below:

• Web depth thickness t_w ≤ 70*ε

• Anchorage length l_v should be taken as MIN (2b_r, s_o - the distance to the edge of the adjacent opening) and should satisfy the limits in expression (55).
• Stiffener width b_r ≤ l_o/6 for fabricated beams and l_o /4 for rolled beams

• Thickness of the stiffener t_r should satisfy the following limit:

  

Not fulfilling any of the recommendations will cause a general warning to be displayed.

Global bending

Resistance

To determine the plastic bending resistance at the centreline of the opening, the forces in the Tees and the slab must be in equilibrium. Two situations are possible: with the plastic neutral axis in the slab and with the plastic neutral axis in the top Tee.

The possible cases of plastic stress blocks in a perforated composite beam:

Resistance of concrete slab including the resistance of shear studs is given as:

with:

f_cd - design resistance of concrete in compression ( = f_ck/γ_c)

b_eff,o - effective width of the composite beam at opening

n_sc - number of shear connectors placed over the distance from the nearer support to the centreline of the opening

P_Rd - design resistance of the shear connectors used with profiled sheeting

Resistance of bottom Tee in tension is given as:

with:

A_bT - cross-sectional area of bottom Tee

A_r - cross-sectional area of stiffener

 

Case 1: N_c,Rd ≥ N_bT,Rd

The compression resistance of the full thickness of the slab at the position of the opening is the lesser of the compression resistance of the effective width of slab and the resistance provided by the shear connectors between the end of the beam and the centreline of the opening. In this case plastic neutral axis is in the concrete slab.

The plastic bending resistance is then given by:

with:

h_eff - effective depth of the beam between the centroids of the Tees (stiffener area included in the calculation)

z_t - depth of the centroid of the top Tee from the outer edge of the top flange (stiffener area included in the calculation)

z_c - depth of concrete in compression

The axial force in bottom Tee is calculated as:

with:

M_Ed - design bending moment at opening

 

Case 2: N_c,Rd < N_bT,Rd

In this case, the compression resistance of the full depth of the effective width of slab is less than the tension resistance of the bottom Tee and equilibrium is achieved by developing compression in the top Tee. For this situation, it is conservative to assume that the top Tee is uniformly stressed and subject to a force equal to the difference between the tension resistance of the bottom Tee and the compression resistance of the slab, i.e. it provides a resistance equal to N_bT,Rd − N_c,Rd. In this case plastic neutral axis is in steel Top tee.

The compression resistance of unstiffened top Tee is limited by cross-section class 2 limit, thus:

with:

A_tT - cross-sectional area of compressed top Tee for global bending (Tee is assumed to be class 2)

h_wt - height of the web out stand of a top Tee

h_w - clear distance between flanges of unperforated section

h_o - height of opening

e_o - eccentricity of opening

 

The plastic bending resistance is then given by:

with:

z_c - depth of concrete in compression ( = hc)

 

To determine the axial force in the bottom Tee for this case, expression from Case 1 should be used unless this gives N_bT,Ed > N_c,Rd. In such case the force may be taken as:

As given by SCI P355 Article 3.2.4, for the composite design of beams with large openings close to the supports, it is likely that a shear connection resistance equal to 40% of the tensile force developed in the bottom Tee (i.e. N_c,Rd ≥ 0.4*N_bT,Ed) will be required to ensure adequate composite bending resistance. If this is not achieved by the application of current minimum degree of shear connection rules, then additional shear connectors should be provided for critical parts of the span or composite action should be neglected when calculating the global bending resistance at opening locations with an inadequate number of shear connectors.

In such a case the contribution of concrete is neglected in the calculation of bending resistance:

To determine the axial force in the bottom Tee for this case only steel section is used:

 

If the design bending moment M_Ed is greater than the given resistance M_o,Rd, the status of this check is set to "Not OK" and the warning that " The bending moment resistance of the section is not adequate." is displayed.

Vierendeel bending

Bending resistance of Tee

Classification of a Tee

The web is assumed to be Class 2 when stiffened, therefore no reduction of Tee height is not performed.

Resistance

Vierendeel bending is the means by which shear force is transferred across a large opening. The sum of the Vierendeel bending resistances at the four corners of the opening, plus the contribution due to composite action between the top Tee and the slab, must therefore not be less than the design value of the difference in bending moment from one side of the opening to the other due to that shear force; this may be expressed as:

with:

M_bT,NV,Rd - bending resistance of the bottom Tee, reduced for coexisting axial tension and shear

M_tT,NV,Rd - bending resistance of the top Tee, reduced for coexisting axial tension and shear

M_vc,Rd - local composite Vierendeel bending resistance

V_Ed - design value of the vertical shear force (taken as the value at the lower moment side of the opening)

l_e - effective length of the opening for Vierendeel bending

 

The plastic bending resistance of a top or bottom Tee section in the absence of axial force (including possible reduction due to vertical shear) is given by the following expression, assuming that the plastic neutral axis is in the flange of the Tee:

with:

A_w,T - cross sectional area of web of the respective Tee ( = h_w,T*t_w)

h_w,T - depth of web of the respective Tee

A_f - cross sectional area of the flange

t_f - thickness of the steel flange

b_f - width of the steel flange

A_r - cross-sectional area of stiffener

e_r - is the offset distance of centre of stiffener from tip of web

z_pl - distance between the plastic neutral axis in the flange and top of the flange

Similar formula assuming that the plastic neutral axis is in the web of the Tee:

with:

z_pl - distance between the plastic neutral axis in the web and bottom of the flange

Influence of normal force

The reduction of plastic bending resistance for co-existent axial force may be determined according to the following approximate formulas:

For top Tee:

For bottom Tee:

with:

M_pl,V,Rd - bending moment resistance of a Tee including possible reduction from vertical shear

M_pl,NV,Rd - bending moment resistance of a Tee including possible reduction from vertical shear and axial force

N_pl,Rd - axial resistance of the Tee

N_Ed - design value of the axial force in the Tee due to the global bending action, either the compression force in the top Tee or the tension force in the bottom Tee.

Top Tee:

For Case 1 N_Ed= 0 and for Case 2: N_Ed= max(N_bT,Rd - N_c,Rd).

Bottom Tee:

N_Ed= N_bT,Rd

 

Local composite resistance

A contribution to the Vierendeel bending resistance occurs due to local composite action of the top Tee with the slab.The bending resistance due to composite action of the top Tee with the slab is given

conservatively by:

with:

ΔN_c,Rd -compression force developed by the shear connectors placed over the opening (= n_sc,o*P_Rd)

n_sc,o - number of shear connectors placed over the opening

P_Rd - design resistance of the shear connectors used with profiled sheeting

z_t - depth of the centroid of the top Tee from the outer edge of the flange (as an approximate = t_f)

k_o - reduction factor due to the flexibility of the opening, which takes account of second order effects together with the combination of shear and tension forces at the edge of the opening, generally given as below, but take as 1.0 if l_o ≤5*h_wt

l_o - length of opening

h_wt - depth of the top Tee

 

If the design effect of Vierendeel bending is greater than the given resistance, the status of this check is set to "Not OK" and the warning that " The bending moment resistance of the section is not adequate." is displayed.