BETON PRATEGANG TKS - 4023
Session 9:
Members Analysis under Flexure (Part I)
Dr.Eng. Achfas Zacoeb, ST., MT. Jurusan Teknik Sipil Fakultas Teknik Universitas Brawijaya
Geometric Properties A prestressed member may also have non-prestressed reinforcement to carry the forces. This type of members is called partially prestressed members. The commonly used geometric properties of a prestressed member with non-prestressed reinforcement are defined as follows: = gross cross-sectional area A A = area of concrete c = area of non-prestressed reinforcement A s A = area of prestressin prestressing g tendons p = transformed area of the section A t = Ac + (E s / E c ) As + (E p / E c ) Ap
Geometric Properties (cont’d)
Fig. 1 shows the commonly used areas of a prestressed member with non-prestressed reinforcement.
Transformation ation Areas Fig. 1 Transform
Introduction The analysis of members refers to the evaluation of the following conditions: 1. Permissible
prestress based on allowable stresses at
transfer. 2. Stresses under service loads. These are compared with allowable stresses under service conditions. 3. Ultimate strength. This is compared with the demand under factored loads. 4. The entire loads versus deformation behaviour.
Introduction (cont’d) The analysis of members under flexure considers the following assumptions: 1.
Plane sections remain plane untill failure (known as Bernoulli’s hypothesis).
2.
Perfect bond between concrete and prestressing steel for bonded tendons.
Introduction (cont’d) The analysis of behavior involves three principles of mechanics: 1.
Equilibrium of internal forces with the external loads. The compression in concrete ( C ) is equal to the tension in the tendon ( T ). The couple of C and T are equal to the moment due to external loads.
Introduction (cont’d)
Compatibility of the strains in concrete and in steel for bonded tendons. The formulation also involves the assumption of plane section remaining plane after bending and a perfect bond between the two materials. For unbonded tendons, the compatibility is in terms of total deformation. 3. Constitutive relationships relating the stresses and the strains in the materials. The relationships are developed based on the material properties. (Collins & Mitchell, Prestressed Concrete Structures ) 2.
Variation of Internal Forces In reinforced concrete members under flexure, the values of compression in concrete ( C ) and tension in the steel (T ) increase with increasing external load. The change in the lever arm ( z ) is not large. In prestressed concrete members under flexure, at transfer of prestress C is located close to T . The couple of C and T balance only the self weight. At service loads, C shifts up and the lever arm ( z ) gets large. The variation of C or T is not appreciable.
Variation of Internal Forces (cont’d)
Fig. 2 explains this difference schematically for a simply supported beam under uniform load.
Fig. 2 Variations of internal forces and lever arms
Variation of Internal Forces (cont’d) For the reinforced concrete member C 2 is substantially large than C 1, but z 2 is close to z 1. For the prestressed concrete member C 2 is close to C 1, but z 2 is substantially large than z 1, where:
C 1, T 1 C 2, T 2 w 1 w 2 z 1 z 2
= compression and tension at transfer due to self weight = compression and tension under service loads = self weight = service loads = lever arm at transfer = lever arm under service loads
Analysis at Transfer and at Service The analyses at transfer and under service loads are similar. Hence, they are presented together. A prestressed member usually remains uncracked under service loads. The concrete and steel are treated as elastic materials. The principle of superposition is applied. The increase in stress in the prestressing steel due to bending is neglected.
Analysis at Transfer and at Service (cont’d) There are three approaches to analyse a prestressed member at transfer and under service loads. These approaches are based on the following concepts: a.
Based on stress concept.
b.
Based on force concept.
c.
Based on load balancing concept.
Analysis at Transfer and at Service (cont’d)
Based on Stress Concept In the approach based on stress concept, the stresses at the edges of the section under the internal forces in concrete are calculated. The stress concept is used to compare the calculated stresses with the allowable stresses.
Analysis at Transfer and at Service (cont’d)
Fig. 3 shows a simply supported beam under a uniformly distributed load (UDL) and prestressed with constant eccentricity ( e ) along its length.
Fig. 3 A simply supported beam under UDL
Analysis at Transfer and at Service (cont’d)
Fig. 4 Stress profiles at a section due to internal forces
Fig. 4 shows the internal forces in concrete at a section and the corresponding stress profiles. The first stress is due to the compression P , the second is due to the eccentricity of the compression, and the third is due to the moment. The moment is due to self weight at transfer, and due to service loads at service.
Analysis at Transfer and at Service (cont’d) The resultant stress at a distance y from the CGC is given by the principle of superposition as Eq. 1. For a curved tendon, P can be substituted by its horizontal component. But the effect of the refinement is negligible. (Eq. 1)
Analysis at Transfer and at Service (cont’d)
Based on Force Concept The approach based on force concept is analogous to the study of reinforced concrete. The tension in prestressing steel (T ) and the resultant compression in concrete (C ) are considered to balance the external loads. This approach is used to determine the dimensions of a section and to check the service load capacity. Of course, the stresses in concrete calculated by this approach are same as those calculated based on stress concept. The stresses at the extreme edges are compared with the allowable stresses.
Analysis at Transfer and at Service (cont’d)
Fig. 5 shows the internal forces in the section.
Fig. 5 Internal forces at a section
Analysis at Transfer and at Service (cont’d) The equilibrium equations are as follows: (Eq. 2)
(Eq. 3)
Analysis at Transfer and at Service (cont’d) The resultant stress in concrete at distance y from the CGC is given as follows: (Eq. 4)
Substituting C = P and Cec = M – Pe , the expression of stress becomes same as that given by the stress concept as Eq. 5. (Eq. 5)
Analysis at Transfer and at Service (cont’d)
Based on Load Balancing Concept The approach based on load balancing concept is used for a member with curved or harped tendons and in the analysis of indeterminate continuous beams. The moment, upward thrust and upward deflection (camber) due to the prestress in the tendons are calculated. The upward thrust balances part of the superimposed load.
Analysis at Transfer and at Service (cont’d) The expressions for three profiles of tendons in simply supported beams are give such as: For a Parabolic Tendon b. For Singly Harped Tendon c. For Doubly Harped Tendon a.
Analysis at Transfer and at Service (cont’d) a. Parabolic Tendon
Fig. 6 Simply supported beam with parabolic tendon
Analysis at Transfer and at Service (cont’d) The moment at the centre due to the uniform upward thrust (w up ) is given by Eq. 6.
(Eq. 6)
Analysis at Transfer and at Service (cont’d) The moment at the centre from the prestressing force is given as M = Pe . The expression of w up is calculated by equating the two expressions of M . The upward deflection ( Δ) can be calculated from w up based on elastic analysis.
(Eq. 7)
(Eq. 8)
Analysis at Transfer and at Service (cont’d) b. Singly Harped Tendon
Fig. 7 Simply supported beam with singly harped tendon
Analysis at Transfer and at Service (cont’d) The moment at the centre due to the upward thrust (W up ) is given by the following equation. It is equated to the moment due to the eccentricity of the tendon. As before, the upward thrust and the deflection can be calculated.
(Eq. 9) (Eq. 10)
Analysis at Transfer and at Service (cont’d) c. Doubly Harped Tendon
Fig. 8 Simply supported beam with doubly harped tendon
Analysis at Transfer and at Service (cont’d) The moment at the centre due to the upward thrusts (W up ) is given by the following equation. It is equated to the moment due to the eccentricity of the tendon. As before, the upward thrust and the deflection can be calculated.
(Eq. 11)
(Eq. 12)
Example 1
A concrete beam prestressed with a parabolic tendon is shown in the figure. The prestressing force applied is 1620 kN. The uniformly distributed load includes the self weight. Compute the extreme fibre stress at the midspan by applying the three concepts. Draw the stress distribution across the section at mid-span.
Solution a. Stress concept Area of concrete,
Moment of inertia,
Bending moment at mid-span,
Solution (cont’d) Top fibre stress,
Bottom fibre stress,
Solution (cont’d)
b. Force concept
Applied moment, Lever arm,
Solution (cont’d) Eccentricity of C ,
Top fibre stress,
Solution (cont’d) Bottom fibre stress,
Solution (cont’d) c. Load balancing method Effective upward load,
Residual load,
Residual bending moment,
Solution (cont’d) Residual bending stress,
Top fibre stress,
Solution (cont’d) Bottom fibre stress,
The resultant stress distribution at mid-span:
