Verification: Web Shear Cracking - Sam Eurocode UK Pretressed Beam Sample Report

Design code: EN 1992-2:2005 with UK National Annex (modified) Analysis: Shears at time t=infinity

EN 1990 Equation 6.15 SLS Frequent Exposure Class: XD1, XD2, XS1, XS2, XS3

Load case: Traffic gr1a TS - for Shear design 1 Section Ref 2 at 21m from left end of beam

WARNING - This analysis assumes that all tension steel (A_s) is adequately anchored to resist the required tensile forces.

(Refer to clause 6.2.1(7), Figure 6.3, and clause 6.2.3(7) of EN1992-1-1).

Section details:

Ref 2 "Section 2"

at 1 x span = 21 m from left end of beam Analysis:

Traffic Actions: Shear for gr1a, loading I.D. 1 At time considered, t = ∞

Serviceability Limit State: Frequent - EN 1990 Equation 6.15

SUMMARY OF ACTIONS

Bestech Systems Limited 2 Slaters Court

Princess Street Knutsford WA16 6BW

Job: Sample Reports Job No.: 6.5d

Calc. By: dlg

Beam: Prestress Beam - Inner span 1 Checked:

Eurocode + UK NA

Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44

SAM v6.50d 06/02/2012 16:36:18 Page: 2

PERMANENT ACTIONS

ACTION TYPE SHEAR MOMENT AXIAL

kN kN.m kN Beam erection before composite = -140.97 0.0 0.0 Construction stage 1A = -74.95901 0.0 0.0 Construction stage 1B = -27.08487 0.0 0.0 Surfacing = -37.72461 -130.6559 0.0 Differential Shrinkage / creep = 9.715637 -203.8871 0.0

TOTAL PERMANENT ACTIONS, G_k -271.0229 -334.543 0.0 VARIABLE ACTIONS

ACTION TYPE SHEAR MOMENT AXIAL

kN kN.m kN ψ₀ ψ₁ ψ₂ Traffic gr1a TS - for Shear design = -355.16 -301.52 27.4134 0.75 0.75 0.0 Traffic gr1a UDL - for Shear desig = -142.99 -526.24 78.0224 0.75 0.75 0.0 Traffic gr1a Footway - for Shear d = -4.4529 -34.778 -11.642 0.4 0.4 0.0^[1]

TOTAL VARIABLE ACTIONS, ψ₁_,₁ x Q_k_,₁ "+" Σψ₂_,_i x Q_k_,_i

Traffic leading: ψ₁ x Traffic -373.6102 -620.8208 79.076887 ψ₂ x Other 0.0 0.0 0.0

Total -373.6102 -620.8208 79.076887 Other leading: ψ₂ x Traffic 0.0 0.0 0.0 ψ₁ x Other 0.0 0.0 0.0

Total 0.0 0.0 0.0 Critical case is with traffic leading

TOTAL COMBINATION

-644.633 -955.3638 79.076887

WEB SHEAR CRACKING EN 1992-2 Annex QQ

Characteristic strength of concrete in web, f_c_k = 40.0 MPa Characteristic tensile strength 5% fractile, f_c_t_k_;₀_,₀₅ = -2.4562 MPa Design shear on precast section before composite, V_E_d_,₁ = 243.01388 kN Total design shear on precast section, V_E_d = 654.34866 kN Design shear on precast section after composite, V_E_d_,₂ = 411.33479 kN Stress in precast from Prestress P and bending M_E_d:

at the top of the precast section, σ_a = -1.1362 MPa at the bottom of the precast section, σ_b = 3.32705 MPa Height of precast section, h = 1300.0 mm

Principal tensile stress is checked at the level of the centroid, and in addition at 100 points through the depth of the section to find the critical level.

First check at the composite section centroid:

At the composite section centroid, z_f_,_m_a_x = 912.41288 mm At this height:

width of precast section, b = 354.58722 mm direct stress, σ_c₁ = σ_b + z_f_,_m_a_x/h*(σ_a-σ_b)

= 3.32705 + 912.413/1300.0*(-1.1362-3.32705) = 0.19452 MPa

For precast section:

area beyond level z_f_,_m_a_x A₁ = 1.547E5 mm² first moment of area (A.z)₁ = 8.269E7 mm³ second moment of area I_y_y_,₁ = 9.2977E10 mm⁴

Shear stress at height z_f_,_m_a_x τ_y_z_,_E_d_,₁ = V_E_d_,₁ * (A.z)₁/(I_y_y_,₁.b) = 243.014 * 8.269E7 / (9.2977E10*354.587)

= 0.60953 MPa

For composite section:

area beyond level z_f_,_m_a_x (transformed) A₂ = 3.825E5 mm² first moment of area (A.z)₂ = 2.113E8 mm³ second moment of area I_y_y_,₂ = 2.3848E11 mm⁴

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From Expression QQ.101, and using Annex QQ sign convention, σ₁ = 1.54323 MPa

Shear stress at level z_f_,_m_a_x τ_y_z_,_E_d_,₂ = V_E_d_,₂ * (A.z)₂/(I_y_y_,₂.b) = 411.335 * 1.828E8 / (2.3848E11*218.2)

= 1.44478 MPa

total shear stress, τ_y_z_,_E_d = 1.18646 + 1.44478 = 2.63124

From Mohr's circle:

centre, σ_c₀ = 0.5*(1.80956+0.0) = 0.90478 MPa radius, σ_r = √[τ_y_z_,_E_d² + (σ_c₁-σ_c₀)²] = 2.78246 MPa σ₃ = 0.90478 + 2.78246 = 3.68724 MPa

σ₁ = 0.90478 - 2.78246 = -1.8777 MPa

From Expression QQ.101, and using Annex QQ sign convention, σ₁ = 1.87767 MPa

f_c_t_b = [1 - 0,8.(σ₃/f_c_k)].f_c_t_k_;₀_,₀₅ = 0.92625 * 2.45617

= 2.27505 MPa

which is greater than σ₁, so minimum reinforcement in accordance with 7.3.2 should be provided.

________

[1] EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 [2] Nearest multiple of depth/100

Pre-tensioned Pre-stressed Beam Bridge Design Example

Appendix - National Annex NDP Values

NATIONALLY DETERMINED PARAMETERS European Standard EN1990

Description Current United

Kingdom

NDPs

Annex A1 (Buildings)

Clause A1.2

• A1.2.2(1) Table A1.1 ψ₀ Combination factors Imposed loads in buildings, category Category A: domestic; residential area 0.7 0.7 Category B: office areas 0.7 0.7 Category C: congregation areas 0.7 0.7 Category D: shopping areas 0.7 0.7 Category E: storage areas 1 1 Category F: traffic area; vehicle weight <=30kN 0.7 0.7 Category G: traffic area; vehicle weight <=160kN 0.7 0.7 Category H: roofs 0.7 0.7 Snow Load Snow: for sites at altitude H>1000m 0.7 0.7 Snow: for sites at altitude H<=1000m 0.5 0.5 Wind forces 0.5 0.5 Thermal 0.6 0.6 • A1.2.2(1) Table A1.1 ψ₁ Frequent reduction factors Imposed loads in buildings, category Category A: domestic; residential area 0.5 0.5 Category B: office areas 0.5 0.5 Category C: congregation areas 0.7 0.7 Category D: shopping areas 0.7 0.7 Category E: storage areas 0.9 0.9 Category F: traffic area; vehicle weight <=30kN 0.7 0.7 Category G: traffic area; vehicle weight <=160kN 0.5 0.5 Category H: roofs 0 0 Snow Load Snow: for sites at altitude H>1000m 0.5 0.5 Snow: for sites at altitude H<=1000m 0.2 0.2 Wind forces 0.2 0.2 Thermal 0.5 0.5 • A1.2.2(1) Table A1.1 ψ₂ Quasi-Permanent factors

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Traffic and pedestrian actions LM1 gr1a Tandem System, gr1a TS 0.75 0.75 UDL, gr1a UDL 0.75 0.75 Pedestrian + cycle-track loads, gr1a Footway 0.4 0.4

Single axle, gr1b 0 0

Horizontal forces, gr2 0 0

Pedestrian loads, gr3 0 0

LM4 - Crowd loading, gr4 0 0

Vertical forces - LM3 Special vehicles, gr5 0 0

Horizontal forces - LM3 Special vehicles, gr6 0 0

Wind forces Wind: - persistent, F_W_k 0.5 0.5 Wind: - execution, F_W_k 0.8 0.8 Wind: F_w_k^*, F_W_k^* 1 1

Thermal actions, T_k 0.6 0.6 A2.2.6(1) Table A2.1 ψ₁ Frequent reduction factors Traffic and pedestrian actions LM1 gr1a Tandem System, gr1a TS 0.75 0.75 UDL, gr1a UDL 0.75 0.75 Pedestrian + cycle-track loads, gr1a Footway 0.4 0.4 Single axle, gr1b 0.75 0.75 Horizontal forces, gr2 0 0

Pedestrian loads, gr3 0.4 0.4 LM4 - Crowd loading, gr4 0 0

Vertical forces - LM3 Special vehicles, gr5 0 0

Horizontal forces - LM3 Special vehicles, gr6 0 0

Wind forces Wind: - persistent, F_W_k 0.2 0.2 Wind: - execution, F_W_k 0 0

Wind: F_w_k^*, F_W_k^* 0 0

Thermal actions, T_k 0.6 0.6 A2.2.6(1) Table A2.1 ψ₂ Quasi-Permanent factors Traffic and pedestrian actions LM1 gr1a Tandem System, gr1a TS 0 0

UDL, gr1a UDL 0 0

Pedestrian + cycle-track loads, gr1a Footway 0 0

Single axle, gr1b 0 0

Horizontal forces, gr2 0 0

Pedestrian loads, gr3 0 0

LM4 - Crowd loading, gr4 0 0

Vertical forces - LM3 Special vehicles, gr5 0 0

Horizontal forces - LM3 Special vehicles, gr6 0 0

Wind forces Wind: - persistent, F_W_k 0 0

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Other operating actions Operating: Aerodynamic effects 0.8 0.8 Non-public footpath maintenance 0.8 0.8 Wind Forces Wind: persistent, F_w_k 0.75 0.75 Thermal actions Thermal, T_k 0.6 0.6 A2.2.6(1) Table A2.3 ψ₁ Frequent reduction factors Individual components of traffic actions LM71 0.8 0.8 SW/0 0.8 0.8

SW/2 1 1

Unloaded train 0 0

HSLM 1 1

Nosing forces 0.8 0.8 Non-public footpath loads 0.5 0.5 Real trains 1 1

Horizontal earth pressure 0.8 0.8 Aerodynamic effects 0.5 0.5 Main traffic actions Gr11-Gr17 0.8 0.8 Gr21-Gr24; Gr26; Gr27 0.7 0.7 Gr31 0.6 0.6 Other operating actions Operating: Aerodynamic effects 0.5 0.5 Non-public footpath maintenance 0.5 0.5 Wind Forces Wind: persistent, F_w_k 0.5 0.5 Thermal actions Thermal, T_k 0.6 0.6 A2.2.6(1) Table A2.3 ψ₂ Quasi-Permanent factors Individual components of traffic actions LM71 0 0

SW/0 0 0

SW/2 0 0

Unloaded train 0 0

HSLM 0 0

Nosing forces 0 0

Non-public footpath loads 0 0

Real trains 0 0

Horizontal earth pressure 0 0

Aerodynamic effects 0 0

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Footbridges Traffic and pedestrian actions, γ_Q 1.35 1.35 Wind forces Wind: persistent, γ_Q 1.7 1.7 Wind: NOTE5 - persistent, γ_Q 1.55 1.55 Thermal actions Thermal, γ_Q 1.55 1.55 Thermal: NOTE6, γ_Q 1.45 1.45 Rail bridges Individual components of traffic actions LM71, γ_Q 1.45 1.45 SW/0, γ_Q 1.45 1.45 SW/2, γ_Q 1.4 1.4 Unloaded train, γ_Q 1.45 1.45 HSLM, γ_Q 1.45 1.45 Nosing forces, γ_Q 1.45 1.45 Non-public footpath loads, γ_Q 1.35 1.35 Real trains, γ_Q 1.7 1.7 Horizontal earth pressure, γ_Q 1.45 1.45 Aerodynamic effects, γ_Q 1.45 1.45 Main traffic actions Groups with LM71; SW/0; HSLM, γ_Q 1.45 1.45 Groups associated with SW/2, γ_Q 1.4 1.4 Other operating actions, γ_Q 1.5 1.5 Wind forces Wind: persistent, γ_Q 1.7 1.7 Wind: NOTE5 - persistent, γ_Q 1.55 1.55 Thermal actions Thermal, γ_Q 1.55 1.55 Thermal: NOTE6, γ_Q 1.45 1.45 • A2.3.1 Table A2.4(B) STR/GEO (Set B) Reduction factor - bridges, ξ 0.85 0.85 Permanent actions Bridges Weight of soil, γ_G_,_s_u_p 1.35 1.35 γ_G_,_i_n_f 0.95 0.95 Dead load, γ_G_,_s_u_p 1.2 1.2 γ_G_,_i_n_f 0.95 0.95 Settlement: linear analysis, γ_G_,_s_u_p 1.2 1.2 γ_G_,_i_n_f 0 0 Settlement: non-linear analysis, γ_G_,_s_u_p 1.35 1.35 γ_G_,_i_n_f 0 0 Shrinkage, γ_G_,_s_u_p 1 1

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Non-public footpath loads, γ_Q 1.35 1.35 Real trains, γ_Q 1.7 1.7 Horizontal earth pressure, γ_Q 1.45 1.45 Aerodynamic effects, γ_Q 1.45 1.45 Main traffic actions Groups with LM71; SW/0; HSLM, γ_Q 1.45 1.45 Groups associated with SW/2, γ_Q 1.4 1.4 Other operating actions, γ_Q 1.45 1.45 Wind forces Wind: persistent, γ_Q 1.7 1.7 Wind: NOTE5 - persistent, γ_Q 1.55 1.55 Thermal actions Thermal, γ_Q 1.55 1.55 Thermal: NOTE6, γ_Q 1.45 1.45 • A2.3.1 Table A2.4(C) STR/GEO (Set C) Permanent actions Bridges

Weight of soil, γ_G_,_s_u_p 1 1

γ_G_,_i_n_f 1 1

Dead load, γ_G_,_s_u_p 1 1

γ_G_,_i_n_f 1 1

Settlement: linear analysis, γ_G_,_s_u_p 1 1

γ_G_,_i_n_f 0 0

Settlement: non-linear analysis, γ_G_,_s_u_p 1 1

γ_G_,_i_n_f 0 0

Concrete, γ_G_,_s_u_p 1 1

γ_G_,_i_n_f 1 1

Steel, γ_G_,_s_u_p 1 1

γ_G_,_i_n_f 1 1

Surface, γ_G_,_s_u_p 1 1

γ_G_,_i_n_f 1 1

Ballast, γ_G_,_s_u_p 1 1

γ_G_,_i_n_f 1 1

Other, γ_G_,_s_u_p 1 1

γ_G_,_i_n_f 1 1 Variable actions Road Bridges Traffic and pedestrian actions, γ_Q 1.15 1.15 Wind forces Wind: - persistent, γ_Q 1.45 1.45 Wind: - execution, γ_Q 1.45 1.45 Wind: F_w_k^*, γ_Q 1.45 1.45

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European Standard EN1991-2

Description Current United

Kingdom

NDPs

Clause 4

• 4.3.2(3) LM1 Tandem System adjustment factors α_Q₁ 1 1 α_Q₂ 1 1 α_Q₃ 1 1

• 4.3.2(3) LM1 UDL adjustment factors α_q₁ 0.61 0.61 α_q₂ 2.2 2.2 α_q₃ 2.2 2.2 α_q_n 2.2 2.2 α_q_r 2.2 2.2

• 4.3.3(2) LM2 adjustment factor, β_Q 1 1

• 4.6.4(3) Fatigue LM3 second vehicle proportion 0.3 0.3

Clause 5

• 5.3.2.2(1) Footway concentrated load, Q_f_w_k 10 10

• 5.6.3(2) Footbridge accidental vehicle axle load, Q_s_v₁ 115 115

• 5.6.3(2) Footbridge accidental vehicle axle load, Q_s_v₂ 65 65

Clause 6

• 6.3.2(3), 6.3.3(5) Rail traffic adjustment factor for LM71 a... 1.1 1.1

• 6.4.5.2 Quality of track maintenance, Φ Standard Standard

Clause A.3

• A.3(6) Unloaded lane length for user-defined vehicle 5m 5m

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Clause 7

• 7.2(2) SLS concrete stress limit factor, k₁ 0.6 0.6 • 7.2(5) SLS reinforcement stress limit factor, k₃ 0.8 0.8 • 7.2(5) SLS prestress stress limit factor, k₅ 0.75 0.75 • 7.3.4(3) Crack calculation factor, k₃ 3.4 3.4 • 7.3.4(3) Crack calculation factor, k₄ 0.425 0.425

Clause 9

• 9.2.2(5) Numeric part of expression (9.5N), ρ_w_,_m_i_n 0.08 0.08

EN1992-2

Clause 3

• 3.1.2(102) Minimum concrete class, C_m_i_n 25MPa 25MPa • 3.1.2(102) Minimum concrete class, C_m_i_n_,_c_u_b_e 30MPa 30MPa • 3.1.2(102) Maximum concrete class, C_m_a_x 70MPa 70MPa • 3.1.2(102) Maximum concrete class, C_m_a_x_,_c_u_b_e 85MPa 85MPa • 3.1.6(101) ULS concrete strength coefficient, α_c_c 0.85 0.85

Clause 6

• 6.2.2(101) Value for C_R_d_,_c is hard-wired. Details are in the printout.

• 6.2.2(101) Shear stress / k^3/2.f_c_k^1/2, v_m_i_n 0.035 0.035 • 6.2.2(101) factor, k₁ 0.15 0.15 • 6.2.2(6) Value for v is hard-wired. Details are in the printout.

• 6.2.3(103) Values for v₁ and α_c_w are hard-wired. Details are in the printout.

Clause 7

• 7.2(102) SLS concrete stress limit factor, k₁ 0.6 0.6 • 7.2(102) Max increase in stress limit above k₁f_u_k 10% 10%

• 7.3.1(5) Prestressed decompression cover 65mm 65mm Decompression cover should be set to c_m_i_n_,_d_u_r from 4.4.1.2(5)

Bestech Systems Limited 2 Slaters Court

Princess Street Knutsford WA16 6BW

Job: Sample Reports Job No.: 6.5d

Calc. By: dlg

Beam: Prestress Beam - Inner span 1 Checked:

Eurocode + UK NA

Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44

SAM v6.50d 06/02/2012 09:33:32 Page: 14

European Standard EN1993

Description Current United

Kingdom

NDPs

EN1993-1-1

EN1993-1-5

Clause 5

• 5.1(2) factor for shear buckling, steel grade <= S460, η 1 1 • 5.1(2) factor for shear buckling, steel grade > S460, η 1 1

EN1993-2

Clause 6

• 6.1(1) Partial factor for excessive yielding, γ_M₀ 1 1 • 6.1(1) Partial factor for instability, γ_M₁ 1.1 1.1 • 6.2.2.5 Effects of local buckling for class 4, Method 1 1

Clause 7

• 7.3(1) Partial factor for Serviceability, γ_M_,_s_e_r 1 1

European Standard EN1994

Description Current United

Kingdom

NDPs

EN1994-2

Clause 6

• 6.6.3.1(1) Partial factor for shear connector resistance, γ_V 1.25 1.25 • 6.8.1(3) factor for headed stud shear connectors, k_s 0.75 0.75

Bestech Systems Limited 2 Slaters Court

Princess Street Knutsford WA16 6BW

Job: Sample Reports Job No.: 6.5d

Calc. By: dlg

Beam: Prestress Beam - Inner span 1 Checked:

Eurocode + UK NA

Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44

SAM v6.50d 06/02/2012 09:33:32 Page: 16

Bestech Systems Limited 2 Slaters Court, Princess Street Knutsford, WA16 6BW United Kingdom

General Enquiries: +44 (0) 1565 654 300 Help Desk: +44 (0) 1565 754 438 Fax: +44 (0) 1565 754 439 E-mail: [email protected]

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