Load balancing is the concept of counteracting a portion or all of the external loads on a concrete member by placing the prestressed tendon along a determined curvature so that the stress in the member is kept within an allowable limit. It also helps to lower deflection and eliminate or reduce cracks. Load balancing is particularly widely used for the design of indeterminate post-tensioned members; it can also be used for the design of simple determinate beams and slabs. This technique of post-tension concrete design was popularized by T. Y. Lin in the early 1960s.
Understanding the Concept of Load Balancing
Consider the beam in the figure below, a compressive force P is exerted on the beam due to prestressing of the tendons, then another upward transverse force W is applied on the beam due to the curvature of the tendon. This transverse load W helps to counteract some of the gravity load on the beam so that the stress in the beam is only due to the unbalanced transverse load and the prestressing force. By this the stress in the member is better kept within limit. The upward transverse force exerted on the concrete by the curvature of the tendon is called “equivalent load” or “balanced load”
Equivalent load of typical Cable Profiles
Equivalent load concept can be better conceptualized when a prestressed concrete member is seen as a self-equilibrating system of equal and opposite forces counteracting one another. In this system, the forces acting on a concrete is the same but opposite to forces acting on the tendons; hence a balance system is created.
The design of post-tensioned structures for serviceability can be better simplified and understood when the tendon is fictitiously taken out of the system and replaced with equivalent load. The equivalent loads include the loads imposed on the concrete at the tendon anchorage (which may include the axial prestress, the shear force resulting from a sloping tendon and the moment due to an eccentrically placed anchorage) and the transverse forces exerted on the member wherever the tendon changes direction. The equivalent load acting on concrete when the tendons are fictitiously removed for some common tendon profile is shown below:
Straight eccentric tendon
If the cable is straight but eccentric, the free body diagram of forces acting on the tendon and the equivalent load on the concrete member is shown below:
Deflected tendon
If the cable is deflected, the free body diagram of forces acting on the tendon and the equivalent load on the concrete member is shown below:
Parabolic Tendon Profile
If the prestressing cable has a curved profile, the cable exerts transverse forces on the concrete throughout its length. The free body diagram of forces acting on the tendon and the equivalent load on the concrete member is shown below:
Equations for equivalent load of different Cable Profiles
The equivalent load is a key parameter in the concept of load balancing as it is the amount of external load on the beam to be balanced off. This informed designers of how much prestressing force is needed to achieve desired structural behavior. Below is the mathematical expression that relate the desired load to be balanced (equivalent load) to the quantity of prestress required.
Deflected tendon
Moment due to prestress = Moment due to equivalent load
P x e = W x L/4
W = P x 4e/L
Where e is the drape which is discussed below
Parabolic Tendon Profile
Moment due to prestress = Moment due to equivalent load
P x e = w l²/8
w = P x 8e/l²
NB: The above expression for equivalent load is for one-segment parabolic cable profiles that is mainly used for simply supported beams. Although there are expressions for other types of cable profile such as 3-parabolic segment, 4-parabolic segment etc. which are more suitable for estimating the equivalent load and amount of prestressed required in continuous structures, the one segment parabolic profile however can be used as an approximation for continuous structures also.
Drape
The drape of a tendon is the total sag or net eccentricity measured from a line joining the ends of the tendon. It is always limited by physical restraint such as the section depth and minimum cover to tendon. The drape of the tendon is shown below considering the position of the tendon relative to the centroid of the member.
When the eccentricities are positive all through (i.e: eccentricities below the centroid of the member) and symmetric, the drape is (e2 – e1). However, when the tendons are asymmetric, the drape is [e2 – (e1+e3)/2].
Verification of Allowable Stress in simply supported Beam by Load balancing Method
A 12m span simply supported beam of 600 mm depth is required to span an opening. Determine the prestress required to balance off 70% of its self-weight and determine the stress in the beam due to the unbalanced weight. Assume parabolic tendons have zero eccentricity at each support.
Worked Example
Since the beam depth is 600mm, the maximum eccentricity of the tendons at mid span shall be constraint my factors such as cover to tendon, and allowance for shear reinforcement. Let’s take 50mm as allowance for these constraints
Eccentricity at mid span = 600 – 50 = 550mm
Drape of tendon = [e2 – e2] = [550 – 0] = 550mm
Self-weight of beam = 0.6 x 0.23 x 25 = 3.45KN/m
Balance 70% of self-weight = 70/100 x 3.45 = 2.415KN/m
Required prestressing force = Ps = (wunb x l²)/8 x e
Ps = (2.415 x 12²) / (8 x 0.55) = 9KN
For a practical design some percentage of prestress would be added to the required prestress to cater for prestress losses. In this example for illustration purpose, the required prestress shall be adopted.
Unbalanced load on beam = 3.45 – 2.415 = 1.035KN/m
Moment caused by unbalanced load = Munb = 1.035 x 12² / 8 = 18.63KNm
Check the stress at top of the beam
σt = $\left(\frac{-P}{A}+\frac{M_{\unb }}{Z_t}\right) $
Area = 0.6 x 0.3 = 0.18m² = 1.8 x 105 mm²
P = 79KN
Zt = = I/y (where y = h/2 = 0.6/2 = 0.3)
Zt = (b x h³) / (12 x y) = (0.23 x 0.6³)/ (12 x 0.3) = 0.014m³ = 1.4 x 105 mm³
σt = $\left(\frac{- 79 x 10³}{1.8 x 105} – \frac{18.63 x 106}{1.4 x 105}\right) $
σt = – 0.439 – 133
σt = – 0.439 – 133
σt = – 133.4MPa
Check the stress at bottom of the beam
σb = $\left(\frac{-P}{A}+\frac{M_{\unb }}{Z_b}\right) $
Zb = (b x h³) / (12 x y) = (0.23 x 0.6³)/ (12 x 0.3) = 0.014m³ = 1.4 x 105 mm³
σb = $\left(\frac{- 79 x 10³}{1.8 x 105} + \frac{18.63 x 106}{1.4 x 105}\right) $
σb= – 0.439 + 133
σb = – 0.439 + 133
σb = 133.4MPa
Indeterminate Post-tension Structures
For statically determinate structures, the moment at a section caused by prestress is the product of prestress and eccentricity (P x e) at that section. It is as simple as that! However, for indeterminate structures, the restraining actions from redundant supports to deformation caused by prestressing force would cause additional reactions – also called hyperstatic reactions – to develop at the supports, which in turn introduces additional shears and moments. These additional shears and moments are called secondary moments and shears.
Consequently, the total moment due to prestress occurring at a section of an indeterminate structure is the summation of the primary moment and secondary moment at the section. The primary moment is the product of prestressing force and the eccentricity (P x e), while the secondary moment is that which is caused by the constraints induced by the hyperstatic reactions.
Similarly, the total shear force caused by prestress on a cross-section in a statically indeterminate structure is the summation of the primary and secondary component. The primary shear force in the concrete is equal to the product of the prestressing force and the slope θ of the tendon at the cross-section under consideration. For a member containing only horizontal tendons (ie: θ = 0), the primary shear force on each cross-section is zero. The secondary shear force at a cross-section is caused by the hyperstatic reactions.
Analysis of Indeterminate PT Structures
The concept of determining hyperstatic moment is not readily absorb by many engineers. Nonetheless, analysis of highly indeterminate PT structures for effects of prestress can be as simple as analysing other non-prestressed structures when carried out using equivalent load method. This method is highly practical and less cumbersome. In this method, the equivalent loads (forces exerted on a concrete member by tendons) is considered as an external load and the structure is then analyzed using any method of choice (moment distribution method is recommended or any other stiffness method). The results of the analysis would give the total internal moment caused by prestress saving the designer the stress of determining the primary moment and then the secondary (hyperstatic) moment separately.
Additionally, when calculating the secondary moments is necessary, this can easily be obtained by subtracting the primary moment (which is the product of prestress and eccentricity at desired section) from the moments gotten from the equivalent load analysis (which is the total moment due to prestress)
Verification of Indeterminate Post-tension Structures
Serviceability verifications do not require any separation of the primary and secondary effects as only the total effects of prestress is used in calculations. However, for verification of ultimate limit state the primary and secondary effects must be separated because the later effects are treated as applied loads. To calculate the ultimate loading on an element, the secondary forces and moments multiplied by a load factor of 1.0 are combined with the ultimate forces and moments from dead and live loads.
References
Design of Prestressed Concrete to Eurocode 2 By Gilbert et al
restressed Concrete Design to Eurocodes by PRAB BHATT
