Live Loads

LRFD Art. 3.6.1.3.1specifies that for both negative moment between points of contraflexure and reaction at interior piers only, 90% of the effect of two design trucks spaced a minimum of 15000 mm between the lead axle of one truck and the rear axle of the other truck, combined with 90% of the effect of the design lane load should be considered. The distance between the 145 kN axles of each truck should be taken as 4300 mm. A uniform load on all spans determines the points of contraflexure.

Distribution Factor For A Typical Interior Beam

The live load bending moments and shear forces are determined by using the simplified distribution factor formulas [LRFD Art. 4.6.2.2.2&3]. To use the simplified live load distribution factor formulas, the conditions of [LRFD Art. 4.6.2.2.1] must be met. For precast concrete I or Bulb-Tee beams with cast-in-place concrete deck slab, the bridge type is (k). [LRFD Table 4.6.2.2.1-1]

1. The number of design lanes:

   Number of design lanes equals the integer part of the ratio of (w/3600), where (w) is the clear roadway width, in mm, between the curbs (Art. 3.6.1.1.1).

   w = 12600 mm

   Number of design lanes = integer part of$\frac{12600}{3600}$ = 3 lanes

Distribution Factor For Bending Moment

The live load bending moments and shear forces are determined by using the simplified distribution factor formulas [LRFD Art. 4.6.2.2.2&3]. To use the simplified live load distribution factor formulas, the conditions of [LRFD Art. 4.6.2.2.1] must be met. For precast concrete I or Bulb-Tee beams with cast-in-place concrete deck slab, the bridge type is (k). [LRFD Table 4.6.2.2.1-1]

1. For Positive Moment:

   For two or more lanes loaded: [LRFD Table 4.6.2.2.2b-1]

   $DFM=0.075+{\left(\frac{S}{2900}\right)}^{0.6}{\left(\frac{S}{L}\right)}^{0.2}{\left(\frac{K_g}{L{t}_s^3}\right)}^{0.1}$

   Provided that:

   • $1100\le S\le 4900$
   • $110\le t_s\le 300$_s
   • $6000\le S\le 73000$
   • $N_b\ge 4$

   where:

   DFM = distribution factor for moment for interior beam

   S = beam spacing, mm

   L = beam span, mm

   t_s = depth of concrete slab, mm

   K_g = longitudinal stiffness parameter, mm⁴$=n(I+A{e}_g^2)$

   n = modular ratio between beam and deck materials$=\frac{E_c}{E_{cs}}=\frac{33978.6}{26802.7}=1.2677>1$

   A = cross section area of the beam (non-composite section), mm²

   I = moment of inertia of the beam (non-composite section), mm⁴

   e_g = distance between the centers of gravity of the beam and slab, mm=$\frac{190}{2}+10+878=983mm$

   Therefore,

   $K_g=\left(1.2677\right)\left(1.908\right({10}^{11})+523107{\left(983\right)}^2)=8.827\times {10}^{11}{mm}^4$

   As per 2002 Interim (Table 4.6.2.2.2b-1) we also have an applicability range for Kg. It must be between 4 × 10⁹ mm⁴ and 3 × 1012 mm⁴. OK

   Therefore, the distribution factor for positive moment is:

   $DFM=0.075+{\left(\frac{3600}{2900}\right)}^{0.6}{\left(\frac{3600}{33000}\right)}^{0.2}{\left(\frac{8.827\times {10}^{11}}{33000\times {190}^3}\right)}^{0.1}=0.075+1.139\times 0.642\times 1.146=0.9125lanes/beam$

   For one design lane loaded:

   $DFM=0.06+{\left(\frac{S}{4300}\right)}^{0.4}{\left(\frac{S}{L}\right)}^{0.3}{\left(\frac{K_g}{12.0L{t}_s^3}\right)}^{0.1}$

   LRFD Table 4.6.2.2.2b-1

   $DFM=0.060+{\left(\frac{3600}{4300}\right)}^{0.4}{\left(\frac{3600}{33000}\right)}^{0.3}{\left(\frac{8.827\times {10}^{11}}{33000\times {190}^3}\right)}^{0.1}=0.060+0.931\times 0.514\times 1.146=0.609llanes/beam$

   Thus, the case of the two design lanes loaded controls, DFM = 0.9125 lanes/beam

2. for Negative Moment

   According to the AASTHO LRFD Specification, table C4.6.2.2.1-1, when computing the DF for negative moment near interior supports of continuous spans between points of contraflexure, L is taken as the average length of the two adjacent spans.

   In this case, for beam 2, span 2: L= (30000 + 33000)/2 = 31500mm

   Therefore, the distribution factor for negative moment is:

   For two or more lanes loaded: [LRFD Table 4.6.2.2.2b-1]

   $DFM(-)=0.075+{\left(\frac{3600}{2900}\right)}^{0.6}{\left(\frac{3600}{31500}\right)}^{0.2}{\left(\frac{8.827\times {10}^{11}}{31500\times {190}^3}\right)}^{0.1}=0.075+\left(1.1385\right)\left(0.648\right)\left(1.151\right)=0.9243lane/bea,$

For one design lane loaded:

Thus, the case of the two design lanes loaded controls, DFM(-) = 0.9243 lanes/beam

Distribution Factor For Shear Force

For two or more lanes loaded:

• $1100\le S\le 4900$
• $110\le t_s\le 300$_s
• $6000\le S\le 73000$
• $4\times {10}^9\le K_g\le 3\times {10}^{12}$¹¹⁴
• $N_b\ge 4$

where: DFV = Distribution factor for shear for interior beam.

Therefore, the distribution factor for shear force is:

For one design lane loaded:

Thus, the case of two design lanes loaded controls, DFV = 1.087 lanes/beam.

Dynamic Allowance

IM = 33% [LRFD Table 3.6.2.1-1]

where: IM = dynamic load allowance, applied only to truck or tandem.

Unfactored Shear Forces and Bending Moments Per Beam.

Unfactored HL-93 shear forces and bending moments per beam due to design truck are:

VLT = (shear force per lane) (DFV) (1 + IM)

= (shear force per lane) (1.087) (1 + 0.33)

= (shear force per lane) (1.446)

MLT = (bending moment per lane) (DFM)(1 + IM)

= (bending moment per lane) (0.9125) (1 + 0.33)

= (bending moment per lane) (1.214)

for contraflexure region (near interior supports)

MLT(-) = (bending moment per lane) (DFM(-))(1 + IM)

= (bending moment per lane) (0.9243) (1 + 0.33)

= (bending moment per lane) (1.229)

Unfactored HL-93 shear forces and bending moments per beam due to design lane are:

VLL = (shear force per lane) (1.087)

MLL = (bending moment per lane) (0.9125)

MLL(-) = (bending moment per lane) (0.9243) – contraflexure region

The three span structure was analyzed using the CONSYS 2000 program, which has the ability to generate live load shear force and bending moment envelopes in accordance with LRFD on a per lane basis. The span lengths used are the continuous span lengths.