PreStressed Concrete (PSC) Analysis
The behaviors of prestressed concrete structures depend on the effective prestress. When a prestressed concrete structure is analyzed, the change of tensions in prestressing tendons must be accurately calculated for a load history through every construction stage. Tension losses in PreStressed (PS) tendons occur due to many different factors including the tensioning method.
In the case of pretensioning, tension losses are attributed to shrinkage and tendon relaxation before tensioning and elastic shortening, creep, shrinkage, tendon relaxation, loading, and temperature after tensioning.
In the case of posttensioning, tension losses are attributed to frictions between tendons and sheaths, anchorage slip, creep, shrinkage, tendon relaxation, loading, and temperature.
MIDAS/Gen reflects the following tension losses for analyzing prestressed concrete structures:

Instantaneous losses upon the release

Timedependent losses after the release
MIDAS/Gen uses net crosssections for calculating the section properties such as crosssectional areas and bending stiffness, which account for duct areas deducted from the gross cross sections prior to jacking PS tendons. After tensioning, converted sections are used reflecting the tendon crosssections.
The stiffness of the tendons is relatively larger than the concrete, and it results in the shift of the centroid. The eccentricities of the tendons are then calculated relative to the new centroid and their tension forces are calculated.
Rather than modeling PS tendons as truss elements or the like, MIDAS/Gen treats the tendons as equivalent prestressing loads, while the stiffness of the tendons is reflected in the section properties as noted above. The tensions in the tendons, which are used to calculate the equivalent loads must be based on the prestress losses at every construction stage caused by various factors.
MIDAS/Gen adopts the following procedure for analyzing a prestressed concrete structure:
 Model the structure.
 Activate (create) construction stages by defining timedependent material properties and construction stages followed by defining elements, boundary conditions and loadings for each construction stage.
 Define the tendon properties; crosssectional area, material properties, ultimate strength, duct diameter, frictional coefficients, etc.
 Assign the desired tendons to the section and define the tendon placement profile.
 Define the tensions applied to the tendons and enter the tensions in the appropriate construction stages.
 Perform the analysis
PreStress Losses

Instantaneous losses after the release

Anchorage slip

Friction between PS tendons and sheaths

Elastic shortening of concrete

Longterm timedependent losses after the release

Creep in concrete

Shrinkage in concrete

Relaxation of PS tendons
The frictional loss is considered in the posttensioning but not in the pretensioning. The total losses for both instantaneous and longterm losses normally range in the 15~20% of the jacking force.
The most important factor for calculating the stresses of PSC (prestressed & posttensioned concrete) members is the final effective prestress force in the tendons reflecting all the instantaneous losses and longterm losses. A relationship between and can be expressed as follows:
R is referred to as the effective ratio of prestress, which is generally for pretensioning and for posttensioning.
The following outlines the prestress losses considered in MIDAS/Gen:
Instantaneous Losses
When a PS tendon is tensioned and released, the prestress is transferred to the anchorage. The friction wedges in the anchorage fixtures to hold the wires will slip a little distance, thus allowing the tendon to slacken slightly. This movement, also known as anchorage takeup, causes a tension loss in the tendon in the vicinity of the anchorage. This phenomenon occurs in both post and pretensioning, and overstressing the tendon can compensate it.
The loss of prestress due to the anchorage slip is typically limited to the vicinity of the anchorage due to the frictional resistance between the PS tendon and sheath. The effect does not extend beyond a certain distance away from the anchorage.
The tendon length l_{set} in the anchorage zone in Figure 1 represents the zone in which tension loss is experienced. The length is a function of the friction; if the frictional resistance is big, the length becomes shorter and vice versa. If we define the anchorage slip as Δl, tendon crosssection area as A_{P,} and modulus of elasticity as E_{P}, the following equation is established. The equation represents the shaded area in Figure 1.
If we further define the frictional resistance per unit length as p, the prestress loss ΔP in Figure 1 can be expressed as
From the equations (1) and (2) above, we can derive the equation for l_{set}, which represents the length of the tendon being subjected to prestress loss due to anchorage slip.
Figure 1 shows a linear distribution of tension along the length of the tendon for an illustrative purpose. MIDAS/Gen, however, considers a true nonlinear distribution of tension for calculating the prestress loss due to the anchorage slip.
Figure 1. Effect on PreStressing Force Due to Anchorage Slip
2. Loss due to friction between PS tendons and sheaths
In posttensioning, frictions exist between the PS tendon and its sheathing. The prestressing force in the tendon decreases as it gets farther away from the jacking ends. The length effect and the curvature effect can be classified. The length effect, also known as the wobbling effect of the duct, depends on the length and stress of the tendon and refers to the friction stemming from imperfect linear alignment of the duct.
The loss of prestress due to the curvature effect results from the intended curvature of the tendon in addition to the unintended wobble of the duct. Frictional coefficients, μ (/radian) per unit angle and k (/m) per unit length are expressed.
If a prestressing force P_{0} is applied at the jacking end, the tendon force P_{x} at a location, l away from the end with the angular change can be expressed as follows:
Prestressing Material 
Wobble coefficient, 
Curvature coefficient, 

Bonded tendons 
Wire tendons Highstrength bars 7wire strand 
0.0033~0.0050 0.0003~0.0020 0.0015~0.0066 
0.15~0.25 0.08~0.30 0.15~0.25 

Unbonded tendons 
Mastic coated 
Wire tendons 7wire strand 
0.0033~0.0066 0.0033~0.0066 
0.05~0.15 0.05~0.15 
Pregreased 
Wire tendons 7wire strand 
0.0010~0.0066 0.0010~0.0066 
0.05~0.15 0.05~0.15 
Table 1. Friction Coefficients k & µ (ACI318)
3. Loss due to elastic shortening of concrete
As a prestress force is transferred to a concrete member, the concrete is compressed. The length of the concrete member is reduced, and the tendon shortens by the same amount thus reducing the tension stress. The characteristics of the elastic shortening differ slightly from pretensioning to posttensioning although the two methods share the same principle.
In the case of pretensioning, an instantaneous elastic shortening takes place as soon as the tendon is released from the anchorage abutments, that is, when the jacking force is applied. This results in a shorter tendon and a loss of pretension. As shown in Figure 2, the pretension (Pj) in the tendon differs from the prestressing force (P_{i}) applied to the concrete member.
In the case of posttensioning, prestressing is directly imposed against the concrete member. The process of elastic shortening is identical to the pretensioning method, but the tension force in the tendon is measured after the shortening has already taken place. Therefore, no tension loss occurs as a result of elastic shortening in posttensioning. MIDAS/Gen does not consider prestress loss due to elastic shortening. As such, when a prestress force is specified in a concrete member where a pretensioning method is used, the prestress load (P_{i}) must be entered in lieu of the jacking load (Pj).
In a typical posttensioned member, multiple tendons are placed, stressed and anchored in a predefined sequence. A series of concrete elastic shortenings takes place in the same member, and the prestress loss in each tendon changes as the prestressing sequence progresses. There is no tension loss in the first tendon being stressed in Figure 3 (b).
When the second tendon is tensioned as shown in Figure 3(c), a tension loss is observed in the first tendon due to the subsequent shortening from the second tensioning. Not only does MIDAS/Gen account for prestress loss due to elastic shortening at every construction stage, but it also reflects all prestress losses due to elastic shortenings caused by external forces.
Figure 3. PreStress Losses in MultiTendons Due to Sequential Tensioning (PostTensioned Member)
TimeDependent Losses
Prestress losses also occur with time due to concrete creep, shrinkage, and PS tendon relaxation. MIDAS/Gen reflects the time dependent material properties of concrete members and calculates the corresponding creep and shrinkage for all construction stages. It also accounts for prestress losses in PS tendons due to the changing member deformation. The prestress loss history can be examined for each construction stage by graphs.
Stress relaxation in steel, also termed as creep, is the loss of its stress when it is prestressed and maintained at a constant strain for a period of time. Prestress loss due to relaxation varies with the magnitude of initial stress, elapsed time in which the stress is applied, and product properties. MIDAS/Gen adopts the Magura equation for tendon relaxation.
ƒ_{si }is initial stress; ƒ_{s} is the stress after loading for a period of time t ; ƒ_{y} is ultimate stress (0.1% offset yield stress); and C is productspecific constant. C=10 for general steel and C=45 for low relaxation steel are typically used. The above equation assumes that the stress in the tendon remains constant. In real structures, stresses in PS tendons continuously change with time due to creep, shrinkage, external loads, etc., and as such equation (5) cannot be directly applied.
Accordingly, MIDAS/Gen calculates the change in prestress loads in tendons due to all causes except for the relaxation itself for every construction stage and calculates the relaxation loss based on fictitious initial prestress for each construction stage.
PreStress Loads
MIDAS/Gen converts the prestress tendon loads applied to a structure into equivalent loads as described in Figure 4 below.
Figure 4. Conversion of PreStress into Equivalent Loads
Figure 4 illustrates a tendon profile in a beam element. 2Dimension is selected for the sake of simplicity, but the process of converting into equivalent loads in the xy plane is identical to that in the xz plane. MIDAS/Gen divides a beam element into 4 segments and calculates equivalent loads for each segment as shown in Figure 4.
The tendon profile in each segment is assumed linear. The tension forces P^{i} and P^{j} in the tendon are unequal due to frictional loss. 3 Concentrated loads (P_{x}, m_{y}m P_{z)} at each end, i and j, alone cannot establish an equilibrium, and hence distributed loads are introduced for an equilibrium. Equations (1) and (2) are used to calculate the concentrated loads at each end, and Equation (3) and (4) are used to calculate the internal distributed loads.
(1)
(2)
(3)
(4)
MIDAS/Gen calculates timedependent prestress losses due to creep, shrinkage, relaxation, etc. for every construction stage as well as other prestress losses due to external loads, temperature, etc. First, the change in tension force in the tendon is calculated at each construction stage, and the incremental tension load is converted into equivalent loads, which are then applied to the element as explained above.