ULS final stage - web-post

The calculation is based on publication SCI P355 - Design of composite beams with large web openings, R. M. Lawson, S. J. Hicks published in 2011. The publication limits the usage of the theory to simply supported beams and only to positive bending. If an opening next to the calculated web-post is skipped by the check from any reason, the web-post is skipped as well.

As given by Chapter 3.5 of the publication, the web-post between adjacent openings is subject to high stresses due to:

Longitudinal (horizontal) shear on its narrowest width.
Bending developed due to Vierendeel action.
Compression due to transfer of vertical shear (buckling).
In addition Chapter 4 specifies an additional check of vertical shear resistance for beams with closely spaced openings. The check is executed in the midway between the centrelines of adjacent openings.

Web-post between openings will be qualified for the check if the web-post width s_o ≤ 5*l_o. Max value of lo should be used in case openings are different. Larger web-post will be discarded from the design and a Note will be printed on the output.

If the geometry of adjacent openings differs or one of the openings is stiffened and the other unstiffened, then the mean of the values for the two openings will be used in the respective calculations. The shape of the adjacent openings for web-post calculations is in case of different opening shapes assumed as:

Rectangular opening + Circular opening -> Rectangular openings

Rectangular opening + Elongated opening -> Rectangular openings

Elongated opening + Circular opening -> Elongated openings

In case of multiple web-posts are checked, user may choose to see results of all web-posts or only the extreme one, by using "Openings and web posts" in Output setting properties.

Longitudinal shear

Design

Design longitudinal shear force acting on the web-post is given by chapter 3.5.1 as:

with:

V_Ed - the design absolute value of the average of the shear forces at the centrelines of adjacent openings

s - the centre-to-centre spacing of the openings

h_eff - the effective depth of the beam between the centroids of the Tees

z_t - the depth of the centroid of the top Tee from the outer edge of the flange

h_c - height of full concrete

h_s - height of composite slab

The incremental compression force that can be mobilized in the slab due to the resistance of the shear connectors between openings is given by:

with:

n_sc,s - the number of shear connectors between the centrelines of adjacent openings

P_Rd - the design resistance of a shear connector used with profiled sheeting

If there is insufficient shear connection between the openings (There is sufficient connection if V_wp.Ed ≤ ∆N_cs,Rd), the longitudinal shear force developed in the web-post is increased, and becomes:

The larger of the values given by expressions above should be used.

Resistance

Shear resistance of web-post is given by chapter 3.5.4 as:

with:

s_o - the edge-to-edge spacing of the openings

t_w - thickness of the web

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

Bending

Design

As given by chapter 3.5.3, the web-post is subject to bending because of Vierendeel bending action. When the shear forces resisted by the Tee sections above and below the opening are equal (which is the case for a symmetric steel beam with central openings, i.e. e_o = 0), there is zero moment at the mid-height of the web-post. This will lead to unity check of zero.

When the shear forces are not equal, an additional moment M_wp,Ed acts at mid-height of the web-post, given by:

with:

V_t,Ed - the design value of the shear force in the top Tee

V_b,Ed - the design value of the shear force in the bottom Tee

V_c,Ed - the design value of the shear force in the concrete slab

V_wp,Ed - the design value of the longitudinal shear force acting on the web-post

e_o - the eccentricity of the centreline of the opening measured from mid-height of the beam

N_cs - the incremental compression force developed by the shear connectors over a length, s between adjacent openings. The value may be taken as ∆N_cs,Rd

Resistance

As given by chapter 3.5.6, the design bending resistance of the web-post should be taken as its elastic value. This is given as:

For rectangular openings, the bending resistance of the web-post should be such that:

with:

V_wp,Ed - the design value of the longitudinal shear force acting on the web-post

M_wp,Ed - the value of the web-post moment at the mid-height of the opening

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

Buckling

Closely / Widely spaced openings

If s_o ≤ h_o for circular openings or s_o ≤ l_o for rectangular and elongated openings, the openings are recognized as closely spaced openings. In other cases the openings are recognized as widely spaced openings.

Design

Design compression force for widely spaced openings on the web-post is given by chapter 3.5.2 as magnitude of the compressive force taken as equal to the larger of the vertical shear forces in the top and bottom Tees:

with:

V_T,Ed - the larger of the shear forces in the two Tees (composite top Tee and steel bottom Tee)

Web post buckling effects are small and can be ignored for widely spaced openings for the following conditions:

h_o/t_w ≤ 25 for circular openings (also elongated openings)

h_o/t_w ≤ 20 for rectangular openings

In such case the buckling check of web post is skipped and unity check set to zero.

Design compression force acting on the web-post for closely spaced openings on the web-post is given by chapter 3.5.2 as:

Resistance

Buckling resistance for widely spaced openings is given by chapter 3.5.5. A simple compression field model has been calibrated over the range ho ≥ so ≥ 0.3*h_o and has been extended to consider openings placed asymmetrically in the depth of the beam. The analysis for rectangular and circular openings differs only in the buckling length l_w of the web-post in compression.

For widely spaced openings, a buckling length of l_w = h_o is appropriate for rectangular openings and l_w = 0.7*h_o for circular and elongated circular openings thus for the determination of the reduction factor on buckling resistance Χ, the non-dimensional slenderness of the web-post λ_rel may be expressed for these buckling lengths as:

For circular openings and elongated circular openings:

For rectangular openings:

with:

λ₁ - defined in EN 1993-1-1, 6.3.1.3 as:

The design buckling resistance of the web post, N_wp,Rd, is determined from buckling curve b in BS EN 1993-1-1, 6.3.1.2 for rolled sections and beams fabricated from rolled sections, or from buckling curve c for beams fabricated from plates. The buckling resistance of the web-post is given by:

with:

χ - buckling coefficient

For closely spaced openings, an appropriate buckling length l_w and he non-dimensional slenderness of the web-post λ_rel may then be expressed as:

For circular openings and elongated circular openings:

For rectangular openings:

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

Limiting vertical shear resistance for closely spaced openings

The expressions for the limiting value of vertical shear resistance of beams with closely spaced openings, dependent on web-post bending resistance and web-post buckling resistance.

The value of the shear resistance V_Rd should be taken as the smaller of the limiting values for web-post bending and web-post buckling given by chapters 4.1 and 4.2 but not greater than (V_pl,Rd + V_c,Rd ) for the perforated section.

Resistance governed by web-post bending for circular and elongated openings is given by chapter 4.1.1.

For partial shear connection, when V_wp,Ed > ∆N_cs,Rd, the limiting value of design vertical shear resistance of the perforated beam is given by:

with:

M_wp,Rd - the elastic bending resistance of the web-post at mid-height of the opening

M_bT,NV,Rd - the bending resistance of the bottom Tee, reduced due to the effect of axial tension

ΔN_cs,Rd - the incremental compression force that can be mobilised in the slab due to the resistance of the shear connectors between openings

For full shear connection, when V_wp,Ed ≤ ∆N_cs,Rd, the compression force in the web-post is taken as equal to V_wp,Ed, which leads to the following equation for limiting vertical shear resistance of the perforated beam:

Resistance governed by web-post bending for rectangular openings is given by chapter 4.1.2.

For partial shear connection, when V_wp,Ed > ∆_Ncs,Rd, the limiting value of design vertical shear resistance of the perforated beam is given by:

For full shear connection, when V_wp,Ed ≤ ∆N_c,s,Rd, the limiting value of design shear resistance is given by:

Resistance governed by web-post buckling is given by chapter 4.2

For partial shear connection as:

with:

N_wp,Rd - the web-post buckling resistance

N_bT,NV,Rd - the bending resistance of the bottom Tee, reduced due to the effect of axial tension and shear as appropriate

For full shear connection, the compression force ∆_Nc,Ed is taken as equal to V_wp,Ed, which leads to the following equation for limiting vertical shear resistance:

For verification of shear resistance, the value of V_Rd should be at least equal to the shear force V_z,Ed midway between the centrelines of adjacent openings.

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