Load balancing was first introduced by T.Y. Lin as an alternative method for analyzing prestressed members. This powerful method provides a simple process for checking presstressed members using hand calculations. This method of analysis seeks to remove the presstressing stands from the structural member and replace them with a set of equivalent forces that act on the member.
In traditional design, the equivalent forces act in a direction that is opposite the applied loading on the member – it is possible to “balance” a portion of the applied loads with equivalent prestress forces, sometimes referred to as the “balanced load” as referenced in Figure 1.
Fig. 1: Balanced load example:
- (A) PT beam
- (B) Free body diagram of PT tendon
- (C) Net load on PT beam (Note: 1 in. = 25 mm; 1 ft = 0.3 m; 1 kip or K = 4.4 kN)
Experienced engineers recommend checking the balanced dead load percentage during design of a member as this can provide insight into the efficiency and reliability of the design. Balanced loads exceeding 100% dead load are often acceptable and even desirable, as long as the design is serviceable and code-compliant. There is no specific code requirement on the percentage of load that should be balanced with post-tensioning, and we do not recommend using a prescribed percentage as a design criterion.
We have often seen instances where designers have attempted to balance more than 300% of the dead load in some spans – this is a sure bet for failure during stressing. For transfer members such as transfer girders, transfer plates, and podium slabs, it is not unusual to have balanced loads that exceed 150% of the dead load. These cases can be complex and it is vital that the engineering team pay close attention to initial stresses, service load stresses, initial and long-term deflections, and detailing of reinforcing steel. In many transfer members; it is necessary to stage stress by stressing the member at successive intervals as load is being added to the member.
Just as overbalancing may be an indicator that the slab or beam does not have adequate thickness or depth, under-balancing may be an indicator that the member depth is overly conservative. Again, checking the percentage of dead load that is balanced is an important tool that should be used to refine and verify the design.
For typical slabs in buildings, we recommend using the values in Table 1.
Table 1: Pragmatic guidelines for balanced load range as a function of building type
|Building type||Balanced load, %|
|Apartments, condominiums, and hotels (LL = 40 psf)||40 to 70|
|Office buildings (LL > 75 psf)||50 to 80|
Note: 1 psf = 0.00005 MPa
In our experience, the load balancing method of analysis is perhaps the most powerful tool at the disposal of an engineer who designs prestressed structures. If you have any questions as you consider utilizing this method yourself, consult an expert!
Sidebar: Practical Hand Calculations for Post-Tensioning
by Donald Kline
While reliance on computer software for design of prestressed concrete members is nearly universal, it is possible to design these members using hand calculations. After all, this is how buildings were designed before the proliferation of computers. It is important for the design engineer to be able to perform hand calculations to check a design or even to make a last minute tendon adjustment in the field before a pour. So how is this possible? The answer lies in the load balancing method together with Equation 1.
- Wpre is the balanced load due to prestress (kips/ft)
- P is prestress force (kips)
- l is span length (ft)
- a is tendon drape defined in the figure (in)
Equation 1 defines the relationship between prestress force (P), tendon drape (a), and balanced load (wpre), and is based on the assumption of an idealized parabolic tendon profile. If any 2 of these variables is known, then the third can be calculated using this equation. For example, it is possible to determine the tendon force for a design strip in a two-way slab with tributary width of 20 ft, span length of 30 ft, slab thickness of 8 inches, and the assumption that the post-tensioning should balance 80 percent of the self-weight of the slab.
wpre= 0.8 x 8 in x 0.15 k/ft3 x 20 ft/12 in/ft = 1.6 k/ft
l= 30 ft
a= 8 in – 1 in – 1 in = 6 in
P= 12 x wpre x l2/8a = 360 kips
Equation 1 provides a simple and elegant way to perform preliminary design or to check a design to ensure that it is reasonable. Clearly, once the tendon force and profile are established, the design must be checked against all of the serviceability and strength requirements in the Code. But it can be shown that for members that are sized using customary span-to-depth ratios and that are subjected to typical superimposed dead and live loads, equation 1 will provide a reasonable solution. Be advised, however, that this equation does not apply for cantilevers or for spans with harped tendons. Also, it is not advisable to use Equation 1 for designing transfer members, including podium slabs. As can be seen in the example above, Equation 1 is one of the most useful tools at the disposal of engineer designing post-tensioned concrete.