Strength II Limit State

With the selected permit load the live load factor from rating factor equation and based on Table 6-6 from the LRFR manual: γ_L=1.5

For a special permit use, a single-lane loaded distribution factor: DFM = 0.512569 and DFV = 0.700895 which will be divided out with 1.2 (multiple presence factor). For a routine permit, we use a multi-lane loaded distribution factor.

As was already specified in the Legal Load Rating section above, it is presumed that in our example we have minor surface deviations or depressions, therefore IM = 0.2.

1. Flexural Strength (Strength II Limit State)
   $RF=\frac{{\phi }_c\times {\phi }_S\times \phi \times M_n-{\gamma }_{DC}\times DC-{\gamma }_{DW}\times DW}{{\gamma }_L\times M_{LL+IM}}=\frac{1\times 1\times 1\times 6245.2-1.25\times 1716.7-1.5\times 162.0}{1.8\times 1506.95}=1.71$
2. Shear Strength (Strength II Limit State)
   Shear evaluation is required for Permit Load Rating:

   Dead Loads (kips) at critical section		Permit
   DC	Total	70.26
   DW	Future Wearing Surface (Comp DW)	6.81

   Also, Vn differs in critical Location and these are the values:

   	Permit
   Vn	425.63

   $RF=\frac{{\phi }_c\times {\phi }_S\times \phi \times M_n-{\gamma }_{DC}\times DC-{\gamma }_{DW}\times DW}{{\gamma }_L\times M_{LL+IM}}=\frac{1\times 1\times 1425.63-1.25\times 70.26-1.5\times 6.81}{1.8\times 113.18}=2.95$

Service Limit States (Inventory and Operating Level)

These calculations are for Midspan location. Because we have total Moment greater than 0, we have compression at top and tension at bottom. For both cases we will compute the Rating Factors.

1. Compression at top (Compression Stress RFs are for Service I Limit State)
   ${RF}_{INV}=\frac{f_R^{top}-{\gamma }_D{f}_D^{top}}{{\gamma }_L\times f_{LL+I}^{top}}$
   In the following table we list the stresses due to dead load values, midspan location at the top.

   Dead Loads (kips) at midspan top
   DC	Self-Weight	0.886
   	Deck & Haunch	0.996
   	Diaphragms	0.162
   	Comp DC	0.039
   DW	Comp DW	0.032
   	Total	2.115

   And for Live Load:

   Stresses due to live loads (ksi) at midspan top
   LL+IM	fLL+I	0.556

   f_R - Flexural resistance at top

   f_R = f_pb + allowable compression stress at top

   f_pb - stress due to effective prestress: -0.856 ksi

   Allowable compression stress - 0.6 x 5 = 3.00 ksi

   Therefore, f^R = 3.00 ksi - (-0.856 ksi) = 3.856 ksi

   γ_L=1.00; γ_D=1.00;

   ${RF}_{COMP}^{TOP}=\frac{f_R^{top}-{\gamma }_D{f}_D^{top}}{{\gamma }_L\times f_{LL+L}^{top}}=\frac{3.856-1.00\times \left(2.115\right)}{1.00\times 0.556}=3.13$

2. Tension at bottom (Tension Stress RFs are for Service III Limit State)
   ${RF}_{TENS}^{BOT}=\frac{f_R^{bot}-{\gamma }_D{f}_D^{bot}}{{\gamma }_L\times f_{LL+L}^{bot}}$

   We can have these values from File > Print Positive Envelope Stresses. In the following table we have the stresses from dead load values - Midspan location at the bottom.

   Stresses due to dead loads (ksi) at midspan bottom
   DC	Self-Weight	-0.748
   	Deck & Haunch	-0.841
   	Diaphragms	-0.137
   	Comp DC	-0.137
   DW	Comp DW	-0.111
   	Total	-1.974

   And for Live Load:

   Stresses from live loads (ksi) at midspan bottom
   LL+IM	fLL+I	-1.94

   f_R - flexural resistance

   f_R = allowable tensile stress - f_pb

   f_pb = compressive stress due to effective prestress: 2.562 ksi

   Allowable compression stress $-0.19\times \sqrt{f_c}=-0.19\times \sqrt{5.5}=-0.425ksi$

   Therefore, f_R = -0.425 ksi - 2.562 ksi = -2.987 ksi

   γ_L=1.00; γ_D=1.00;

   ${RF}_{INV}=\frac{f_R-{\gamma }_D\times f_D}{{\gamma }_L\times (f_{LL+IM)}}=\frac{-2.987-1.00\times (-1.974)}{1.00\times 1.94}=0.52<1NG$