Elastic Shortening

Loss of prestress due to elastic shortening is a result of elastic shortening of a girder after release. When transformed section properties are used, the loss of prestress due to elastic shortening does not have to be evaluated explicitly since the equations for evaluation of stress already includes the effect of elastic shortening. This is because the area and moment of inertia of the cross-section includes the transformed steel, as specified in Reference 6, Design of Prestressed Concrete Structures, Chapter 5 p. 126-132.

When using transformed steel, Precast/Prestressed Girder shows ES equal to zero on the printout. This does not mean that there is no elastic shortening; it simply means that the elastic shortening is included as part of the stress equations and is not calculated separately.

When not using the transformed section properties option, the gross section properties method follow what has been industry practice for many years. This is because elastic gains are not included and the result may be a reduction of compression in the beam bottom at mid-span.

Stress in Concrete Due to Prestress

Notation:

Terms	Definition
A_c	Net area of concrete
A_s	Area of tension reinforcement
A_g	Gross area of cross section (without steel)
fΔf_c	Change of stress in concrete occurring during transfer
Δf_s	Change of stress in steel occurring during transfer
E_c	Modulus of elasticity of concrete
E_sc	Modulus of elasticity of steel
ES	Loss of prestress due to elastic shortening (difference between stress in prestressing steel immediately before and after release)
T₀	Prestressing force applied at the centroid of the pretensioned member
T_f	Final tensile force in the tendons just after elastic shortening has occurred
Δε_s	Change of strain in concrete during transfer (difference between strain immediately before and after transfer)
Δε_c	Change of strain in steel during transfer (difference between strain immediately before and after transfer)
	Ratio of elastic modulus

Standard Procedure

Stress in concrete due to prestress is computed by elastic theory, which assumes that there is a linear relationship between the stress and the strain. The change of the stress in concrete can be expressed as

Equation 1:

Δf_s=E_cΔε_c

and the change of the stress in steel as

Equation 2:

Δƒ_s=E_sΔε_s

For simplification of the problem, some other assumptions are also made, e.g., the area of steel of prestressing strands, A_s, remains the same immediately before and after the transfer.

For pretensioned members, when the prestress in the steel is transferred from the bulkheads to the concrete, the force, which was resisted by the bulkheads, is transferred to both the steel and concrete. Let T₀ be the prestressing force that is applied at the centroid of the concrete section in a pretensioned member. After the transfer, this force can be divided into two components as follows:

Equation 3:

T₀=T_ƒ+Δƒ_sA_s

where T¦=final tensile force in the tendons just after elastic shortening has occurred and Δƒ_s is the loss of prestress times area of steel. Please not that the total force, T₀, did not change the value during transfer and only the component due to elastic shortening was introduced. The change in strain (unit shortening) in the tendons as a result of losses can be expressed as

Equation 4:

$image\ebx_1795007003.gif$

The increase of strain in concrete can be expressed as

Equation 5:

$image\ebx_840948865.gif$

Since the decrease in strain in tendons caused shortening of concrete, Eq. 5. and Eq. 6 can be equated:

Equation 6:

$image\ebx_2009660962.gif$

which gives

Equation 7:

$image\ebx_419117564.gif$

Equation 8:

$image\ebx_203575798.gif$

where

Equation 9:

$image\ebx_1554318203.gif$

The above assumption implies that the concrete acts with the steel as a homogenous material and it already suggests that the concept of transformed section properties can be used. Practically, however, gross section area is used instead. Please note also that the area of concrete is equal to the gross area minus the area of steel, A_c=A_g-A_s.

The loss of prestress can be computed utilizing Eq. 3 as follows

Equation 10:

$image\ebx_-773078757.gif$

and substituting Eq. 7 yields

Equation 11:

$image\ebx_-1784522988.gif$

and utilizing Eq. 8 gives

Equation 12:

$image\ebx_192262419.gif$

Once the loss in prestress is calculated, the next step is to determine T_f by virtue of Eq. 3. Therefore the stress in concrete, Δƒc, can be determined by substituting Eq. 5 into Eq. 2 leading to

Equation 13:

$image\ebx_268603787.gif$

To recapitulate the above procedure the following steps are introduced:

Consider the effect of loss of prestress due to elastic shortening as an element of the prestressing force, Eq. 3.

Evaluate elastic shortening, Eq. 12.

Calculate tensile force in steel immediately after transfer, Eq. 8.

Determine stress in concrete, Eq. 13.