Vibration Analysis of Pedestrian Bridge

VIBRATION ANALYSIS OF PEDESTRIAN BRIDGE - AISC DG11

1.0 INPUT PARAMETERS

Geometry

L = ft Span length

B_w = ft Walkway Width

N_b = Number of Beams

Slab & Deck Details

d_slab = in Total slab depth (top of slab to bottom of deck ribs)

h_deck = in Deck rib height

w_c = lb/ft³ Concrete Unit Weight

f'_c = lb/in² Concrete compressive strength

Figure 1: Footbridge Section

Composite Steel-Concrete Footbridge Section

Steel Beam Properties (Default: W18x35)

A_s = in² Cross sectional area of one beam

d_s = in Depth of steel beam

I_x = in⁴ Moment of inertia of one beam

E_s = ksi Modulus of Elasticity of Steel

Loading & Damping

w_sdl = lb/ft² Superimposed Dead Load (Railings, misc - NOT Live Load)

β = Damping Ratio (0.01 for Outdoor, 0.02 for Indoor)

2.0 DYNAMIC PROPERTIES CALCULATION

2.1 Dynamic Modulus of Concrete

Static Modulus of Elasticity of Concrete

E_c = w_c^1.5·√(f'_c)

E_c = 145^1.5 × √4000 = 3492 ksi

Dynamic Modulus (1.35 × E_c)

E_c,dyn = 1.35·E_c

E_c,dyn = 1.35 × 3492 = 4714 ksi

Dynamic Modular Ratio

n = E_s/E_c,dyn

n = 29000 / 4714 = 6.15

2.2 Transformed Section Properties

Effective Concrete Width

B_eff = B_w (Full width acts compositely for edge beams)

B_eff = 10 ft = 120 in

Transformed Concrete Width

b_tr = B_eff/n

b_tr = 120 / 6.15 = 19.51 in

Effective Depth of Concrete (above ribs)

t_c = d_slab − h_deck

t_c = 6 - 0 = 6 in

Transformed Concrete Area

A_c,tr = b_tr·t_c

A_c,tr = 19.51 × 6 = 117.07 in²

Neutral Axis Location (from top of slab)

Σ(A·y) = A_c,tr·(t_c/2) + N_b·A_s·(d_slab + d_s/2)

Σ(A·y) = 117.07×3 + 2×13×(6+10.35) = 776.31 in³

ΣA = A_c,tr + N_b·A_s

ΣA = 117.07 + 2×13 = 143.07 in²

Y_na = Σ(A·y)/ΣA

Y_na = 776.31 / 143.07 = 5.43 in

Transformed Moment of Inertia

I_c = (b_tr·t_c³)/12

I_c = (19.51×6³)/12 = 351.18 in⁴

I_s,total = N_b·I_x

I_s,total = 2 × 843 = 1686 in⁴

Parallel Axis Theorem Distances

d_c = Y_na − t_c/2

d_c = 5.43 - 3 = 2.43 in

d_s,dist = (d_slab + d_s/2) − Y_na

d_s,dist = (6 + 10.35) - 5.43 = 10.92 in

I_tr = I_c + A_c,tr·d_c² + I_s,total + N_b·A_s·d_s,dist²

I_tr = 351.18 + 117.07×2.43² + 1686 + 2×13×10.92² = 5839 in⁴

2.3 Weight Calculation (Dead Load Only)

Note: For vibration analysis, use the actual expected weight.

t_avg = d_slab − h_deck/2

t_avg = 6 - 0/2 = 6 in

Slab Weight

w_slab = t_avg·B_w·w_c

w_slab = (6/12)×10×145 = 725 lb/ft

Superimposed Dead Load

w_sdl,line = w_sdl·B_w

w_sdl,line = 4 × 10 = 40 lb/ft

Beam Self-Weight (490 lb/ft³ for steel)

w_beam = A_s·490/¹⁴⁴·N_b

w_beam = 13×490/144×2 = 88.47 lb/ft

Total Line Load

w_total = w_slab + w_sdl,line + w_beam

w_total = 725 + 40 + 88.47 = 853.47 lb/ft

2.4 Natural Frequency Calculation

Midspan Deflection due to Total Supported Weight

Δ_m = (5·w_total·L⁴)/(384·E_s·I_tr)

Δ_m = (5×853.47×(40×12)⁴)/(384×29000000×5839) = 0.293 in

Natural Frequency [DG11 Eq. 3.3]

f_n = 0.18·√(g/Δ_m) where g = 386 in/s²

f_n = 0.18×√(386/0.293) = 6.53 Hz

Effective Weight

W_eff = 1.0·w_total·L (For simple span)

W_eff = 1.0 × 853.47 × 40 = 34139 lb

3.0 VIBRATION EVALUATION

3.1 Walking Excitation Check

Parameters from DG11 Section 4.1 / Example 4.5

P_o,walk = lbf Constant force for walking

Limit_walk = %g Acceleration limit (5.0%g Outdoor, 1.5%g Indoor)

Resonant Build-up Factor Check

Harmonic ranges: H1: 1.6-2.4 Hz | H2: 3.2-4.8 Hz | H3: 4.8-7.2 Hz

Standard DG11 approach: Assume resonance if f_n < 9 Hz

Peak Acceleration [DG11 Eq. 4.1]

a_p/g = (P_o·e^−0.35·f_n)/(β·W_eff·g)

a_p/g = (92×e^(-0.35×6.53)) / (0.01×34139×32.2) = 0.0089

a_p,walk = 0.89 %g

Walking Check Result

Walking Check = PASS (a_p,walk = 0.89 ≤ Limit = 5.0 %g)

3.2 Running Excitation Check

Running creates significantly higher forces. DG11 2nd Ed suggests ~146 lbs for running if resonant.

P_o,run = 0.79 × 168 = lbf (From Table 4-1, depends on f_n)

Limit_run = %g Acceleration limit

Running Frequency Range: 2.0 to 3.5 Hz approx. If f_n > 2.0 Hz, running resonance is highly likely.

Peak Acceleration (Running)

a_p/g = (P_o,run·e^{−0.173·f_n})/(β·W_eff·g)

a_p/g = (132.72×e^(-0.173×6.53)) / (0.01×34139×32.2) = 0.0039

a_p,run = 0.39 %g

Running Check Result

Running Check = PASS (a_p,run = 0.39 ≤ Limit = 5.0 %g)

* OUTPUT SUMMARY *

Analysis Results

Natural Frequency, f_n = 6.53 Hz

Transformed Moment of Inertia, I_tr = 5839 in⁴

Midspan Deflection, Δ_m = 0.293 in

Effective Weight, W_eff = 34139 lb

Walking Acceleration = 0.89 %g

Running Acceleration = 0.39 %g

Walking Check: PASS

Running Check: PASS

RESULTS

ALL CHECKS PASSED - VIBRATION PERFORMANCE IS ACCEPTABLE (AISC DG11 COMPLIANT)

FOOTFALL ANALYSIS OF PEDESTRIAN BRIDGE