Design of  Prestressed Concrete Structures

This section covers the following topics

• Calculation of Demand
• Design of Sections for Axial Tension 

Introduction 

The design of prestressed concrete members can be done by the limit states method as given in Section 4 of IS: 1343 - 1980. First, the force demand in a member under the design loads is determined from a structural analysis. A preliminary size of the member is assumed for analysis. Next, the member is designed to meet the demand. If necessary, another cycle of analysis and design is performed. The following material explains the calculation of the demand in a member under the design loads.  

1.1 Calculation of Demand

In the limit states method, the design loads are calculated from the characteristics loads by multiplying them with load factors (f). Several types of loads are considered to act together under the selected load combinations.  The load factors are included in the load combinations as weightage factors. The demand in a member for a particular type of load is obtained from the analysis of the structure subjected to the characteristic value of the load. The demands for the several load types are then combined under the load combinations, based on the principle of superposition.

 Characteristics Loads 
For dead loads, a characteristic load is defined as the value which has a 95% probability of not being exceeded during the life of the structure. This concept assumes a normal distribution of the values of a particular dead load.  In the following figure, the shaded area above the characteristic value represents 5% probability of exceedance of the load in the design life of the structure.

 

For live load, wind load and earthquake load, a characteristic load is defined based on an extreme value distribution.  For example, the characteristic wind load is defined as the value which has a 98% probability of not being exceeded during a year.  

 

    

The characteristics loads can be obtained from IS:875 - 1987 (Code of Practice for Design Loads for Buildings and Structures) and IS:1893 - 2002 (Criteria for Earthquake Resistant Design of Structures) as follows.

For special loads, there are some guidelines in IS: 875 - 1987, Part 5. In addition, specialised literature may be referred to for these loads. The special loads are listed below. 

§ Temperature
§ Hydrostatic
§ Soil pressure
§ Fatigue
§ Accidental load
§ Impact and collision
§ Explosions

§ Fire

For special situations, the loads are determined from testing of  prototype specimens. Dynamic load tests, wind tunnel tests, shake table tests are some types of tests to determine the loads on a structure.  Finite element analysis is used to determine the stresses due to concentrated forces and dynamic loads.

Load Factors and Load Combinations

The load factors and the combinations of the various types of loads are given in Table 5 of IS:1343 - 1980. The following are the combinations for the ultimate condition. 

a)      1.5 (DL + LL) 
b)      1.2 (DL + LL +  WL) 
c)       1.2 (DL + LL +  EL)  
d)      1.5 (DL +  EL) 
e)      1.5 (DL +  WL)  
f)       0.9 DL +  1.5 EL 

 The load combinations for service conditions are as follows. 
a)      DL + LL 
b)      DL + 0.8 (LL   EL) 
c)       DL +  EL 
d)      DL +  WL 
 

Analysis of Structures 

Regarding analysis of structures, IS:1343 - 1980  recommends the same procedure as stated in  IS:456 - 2000. A structure can be analysed by the linear elastic theory to calculate the internal forces in a member subjected to a particular type of load.

Design of Members

There can be more than one way to design a member. In design, the number of unknown quantities is larger than the number of available equations. Hence, some quantities need to be assumed at the beginning. These quantities are subsequently checked.

The member can be designed either for the service loads or, for the ultimate loads. The procedure given here is one of the possible procedures. The design is based on satisfying the allowable stresses under service loads and at transfer.  Initially, a lumpsum estimate of the losses is considered under service loads.  After the first round of design, detailed computations are done to check the conditions of allowable stresses.  Precise values of the losses are computed at this stage.   The section is then analysed for the ultimate capacity. The capacity should be greater  than the demand under ultimate loads to satisfy the limit state of collapse. 

1.2 Design of Sections for Axial Tension

Introduction

Prestressed members under axial loads only, are uncommon. Members such as hangers and ties are subjected to axial tension.  Members such as piles may have bending moment along with axial compression or tension. 

Design of Prestressing Force

First, a preliminary dimension  of the member is selected  based on the architectural requirement. The prestressing force at transfer (P₀) should be such that the compressive stress in concrete is limited to the allowable value.  At service, the designed prestressing force (P_e) should be such that the tensile stress in concrete should be within the allowable value.  The amount  of prestressing steel (A_p) is determined from the designed prestressing force based on the allowable stress in steel. 

At transfer, in absence of non-prestressed reinforcement, the stress in concrete (fc) is given as follows. 

Here,

A_c  = net area of concrete 

P₀  = prestress at transfer after short-term losses. 

In presence of non-prestressed reinforcement, the stress in the concrete (f_c) can be calculated as follows. 

Here,

A_s = area of non-prestressed reinforcement 

E_s = modulus of elasticity of steel 

E_c = modulus of elasticity of concrete. 

At service, the stress in concrete (f_c) can be calculated as follows.

Here,

A_t = transformed area of section

P = external axial force 

P_e = effective prestress.

The external axial force is considered positive if it is tension and negative if it is compression.  In the above expression, non-prestressed reinforcement is not considered.  If there is non-prestressed reinforcement, A_c is to be substituted by (A_c + (E_s/E_c) A_s) and A_t is to be calculated including A_s.

Analysis of Ultimate Strength

The ultimate tensile strength of a section (P_uR) is calculated as per Clause 22.3, IS: 1343 - 1980. The ultimate strength should be greater than the demand due to factored loads. 

In absence of non-prestressed reinforcement, the ultimate tensile strength of a section (P_uR) is given as follows.

In presence of non-prestressed reinforcement,

In the previous equations, 

f_y  = characteristic yield stress for non-prestressed reinforcement with mild steel bars

f_y = characteristic 0.2% proof stress for non-prestressed reinforcement with high yield strength                             deformed bars. 

f_pk = characteristic tensile strength of prestressing tendons.