Technical Manual for Design and Construction of Road Tunnels - Civil Elements

Appendix C - Cut-and-Cover Tunnel Design Example

9.2 Shear Design (S5.8.3.3)

The nominal shear resistance, V_n shall be determined as the lesser of

LRFD eq. 5.8.3.3-1: V_n = V_c + V_s

or

LRFD eq. 5.8.3.3-2: V_n = 0.25 × f'_c × b_v × d_v

Note V_p is not considered

Where:

LRFD eq. (5.14.5.3-1)

Where

For slab concrete shear (V_c), refer to LRFD Section 5.14.5.

LRFD eq. (5.8.3.3-4)

Where for α = 90° and θ = 45°

Where:

A_s = area of reinforcing steel in the design width (in²)

b = design width (in)

d_e = effective depth from extreme compression fiber to centroid of tensile force in tensile reinforcement (in)
d_e= 27.75

V_u = shear from factored loads (kip)

M_u = moment from factored loads (kip-in)

A_v = area of shear reinforcement within a distance s (in²) = 0 in²

s = spacing of stirrups (in) = 12 in

b_v = effective web width taken as the minimum web width within the depth d_v (in)

d_v = effective shear depth taken as the perpendicular distance to the neutral axis (in)

d_v = 0.9 × d_e or 0.72 × h (LRFD section 5.8.2.9)

d_v = 24.98 in

Maximum shear and associated moment from analysis output:

V_u = 28 kip M_u = 63.0 kip-ft

V_c = 63.42 kip value controls

or V_c = 83.92 kip

V_s = 0.00 kip

V_n = 63.42 kip

V_n = 299.70 kip therefore V_n = 63.42 kip

Φ = 0.90

ΦV_n = 57.08 kip > V_u OK

10. Bottom Slab Design

10.1 Slenderness Check (S5.7.4.3)

K = 0.65

lu = 37.25 ft = 447 in

d = 1.75 ft = 21.0 in

I = 9261 in⁴

r = 6.06 in

β₁ = 0.85

d_s = 18.75 in

d'_s = 3.25 in

#8 bar dia. = 1.00 in

k × (l_u / r) = 47.93

From analysis output

where M₁ = 13 kip-ft P₁ = 23.6 kip

M₂ = 57.1 kip-ft P₂ = 23.6 kip

34 - 12 (M₁/M₂) = 31.27

Consider slenderness since k × (lu / r) is greater than 34 - 12 (M₁/M₂)

Calculate EI:

E_c = 3834.25 ksi

I_g = 9261 in⁴

c = 8 in EI = 3427836.25 kip-in²

I_s = 202.34 in⁴ EI = 6855672.51 kip-in²

M_no = 61.20 kip-ft

M₂ = 57.10 kip-ft

Note: M_no does not include effects of vertical live load surcharge

β_d = M_no/M₂ = 1.07

Approximate Method (LRFD 4.5.3.2.2)

P_e =

P_e = 801.51 kip

Moment Magnification

(The components for sidesway will be neglected. Bracing moment will not include lateral force influence. Live load surcharge is excluded also.)

C_m = 0.6 + 0.4 (M₁/M₂) = 0.69

P_u = 23.6 kip

δ_b = 1.00

M_c = δ_b × M_2b + Δ_s × M_2s

M_c = 28.32 kip-ft

Factored flexural resistance

Do not consider compression steel for calculating M_n.

c = 2.73 in

a = 2.32 i

M_n = 1667.36 kip-in = 138.95 kip-ft

M_r = ΦM_n = 125.05 kip-ft OK (≥ Mc) M_r>M_u

Create interaction diagram

Assume ρ_min = 1.0%

A_smin = 2.52 in²

A_sprov (total) = 3.16 in² choose #8 at 6"

E_s = 29000 ksi

β₁ = 0.85

Y_t = 10.5 in

0.85 × f'_c = 3.4 ksi

A_g' in² = 252 in²

A_s = A'_s = 1.6 in²

At zero moment point

P_o = 1036 kip

ΦP_o = 725 kip

At balance point calculate P_rb and M_rb

c_b = 11.25 in

a_b = 9.56 in

f'_s = 62 ksi
f's > fy; set f's = fy

A_comp = 114.75 in²

y' = 4.78125 in

ΦP_b = 271 kip

ΦM_b = 3303 kip-in = 275 kip-ft

At zero 'axial load' point (conservatively ignore compressive reinforcing)

a = 2.3 in

ΦM_o = 1500.6 kip-in = 125 kip-ft

At intermediate points

Note Φ may decrease from 0.90 to 0.75 as a increases from 0.0 to ab. Use 0.75 to be conservative.

10.2 Shear Design (S5.8.3.3)

V_n = V_c + V_s or V_n = 0.25 × f'_c × b_v × d_v
d_v = 16.88 in

For slab concrete shear (V_c), see LRFD Section 5.14.5

Maximum shear and associated moment from analysis output:

V_u = 19.4 kip M_u = 30.3 kip-ft

V_c = 44.96 kip value controls
or V_c = 56.70 kip

Where A_v = 0 in² and s = 12 in

V_s = 0.00 kip

V_n = 44.96 kip

V_n = 202.50 kip therefore V_n = 44.96 kip

ΦV_n = 40.46 kip > V_u OK

11. Exterior Wall Design

11.1 Slenderness Check (LRFD 5.7.4.3)

K = 0.65

lu = 22.13 ft = 265.5 in

d = 2.00 ft = 24.0 in

I = 13824 in⁴

r = 6.93 in

β₁ = 0.85

d_s = 21.75 in

d'_s = 3.25 in

#8 bar dia. = 1.00 in

k × (l_u / r) = 24.91

From analysis output

where M₁ = 171.4 kip-ft P₁ = 34.4 kip

M₂ = 137.2 kip-ft P₂ = 34.4 kip

34-12 (M₁/M₂) = 19.01

Consider slenderness since k × (l_u / r) is greater than 34-12 (M₁/M₂)

Calculate EI:

E_c = 3834.25 ksi

I_g = 13824 in⁴

c = 9.5 in       EI = 7330894.82 kip-in²

I_s = 285.29 in⁴       EI = 14661789.6 kip-in²

M_no = 61.20 kip-ft

M₂ = 137.20 kip-ft

Note: M_no does not include effects of vertical live load surcharge

β_d = M_no/M₂ = 0.45

Approximate Method (LRFD 4.5.3.2.2)

P_e =

P_e = 4858.82 kip

Moment Magnification

(The components for sidesway will be neglected. Bracing moment will not include lateral force influence. Live load surcharge is excluded also.)

C_m = 0.6 + 0.4 (M₁/M₂) = 1.10

P_u = 34.4 kip

δ_b = 1.11

M_c = δ_b × M_2b + δ_s × M_2s

M_c = 38.46 kip-ft

Factored flexural resistance

Do not consider compression steel for calculating M_n.

c = 2.73 in

a = 2.32 in

M_n = 1951.76 kip-in = 162.65 kip-ft

M_r = ΦM_n = 146.38 kip-ft OK (≥ M_c) M_r > M_u

Create interaction diagram

Assume ρ_min = 1.0%

A_smin = 2.88 in²

A_sprov (total) = 3.16 in² choose #8 at 6"

E_s = 29000 ksi

β₁ = 0.85

Y_t = 12 in

0.85 × f'_c = 3.4 ksi

A_g' in² = 288 in²

A_s = A'_s = 1.6 in²

At zero moment point

P_o = 1158 kip

ΦP_o = 811 kip

At balance point calculate P_rb and M_rb

c_b = 13.05 in

a_b = 11.09 in

f'_s = 65 ksi

f's > fy; set f's = fy

A_comp = 133.11 in²

y' = 5.54625 in

ΦP_b = 313 kip

ΦM_b = 4176 kip-in = 348 kip-ft

At zero 'axial load' point (conservatively ignore compressive reinforcing)

a = 2.3 in

ΦM_o = 1756.6 kip-in = 146 kip-ft

At intermediate points

Note Φ may decrease from 0.90 to 0.75 as a increases from 0.0 to ab. Use 0.75 to be conservative.

11.2 Shear Design (S5.8.3.3)

Maximum shear from analysis output:

V_u = 20.76 kip

Where β = 2

b_v = 12 in

d_v = 19.58 in

V_c = 0.0316 × β × f'_c^0.5 × b_v × d_v     LRFD eq. (5.8.3.3-3)

V_c = 29.69 kip

Where A_v = 0 in² and s = 12 in

V_s = 0.00 kip

V_n = 29.69 kip

V_n = 234.90 kip therefore V_n = 29.69 kip

ΦV_n = 26.72 kip > Vu OK