3.1 Analysis of Members under Axial Load

Transcription

1 3.1 Analysis of Members under Axial Load This setion overs the following topis. Introdution Analysis at Transfer Analysis at Servie Loads Analysis of Ultimate Strength Analysis of Behaviour Notations Geometri Properties A prestressed axial member may also have non-prestressed reinforement to arry the axial fore. This type of members is alled partially prestressed members. The ommonly used geometri properties of a prestressed member with non-prestressed reinforement are defined as follows. A = gross ross-setional area A A s A p A t = area of onrete = area of non-prestressed reinforement = area of prestressing tendons = transformed area of the setion = A + (E s / E ) A s + (E p / E ) A p The following figure shows the ommonly used areas of a prestressed member with non-prestressed reinforement.

2 = + + A A A s A p A A t Figure Areas for a prestressed member with non-prestressed reinforement Introdution The study of members under axial load gives an insight of the behaviour of a prestressed member as ompared to an equivalent non-prestressed reinfored onrete member. Prestressed members under axial loads only, are unommon. Members suh as hangers and ties are subjeted to axial tension. Members suh as piles may have bending moment along with axial ompression or tension. In this setion, no eentriity of the CGS with respet to CGC is onsidered. The definitions of CGS and CGC are provided in Setion 2.1, Losses in Prestress (Part I). The following figure shows members under axial loads.

3 Hangers Figure Piles Members under axial load The analysis of members refers to the evaluation of the following. 1) Permissible prestress based on allowable stresses at transfer. 2) Stresses under servie loads. These are ompared with allowable stresses under servie onditions. 3) Ultimate strength. This is ompared with the demand under fatored loads. 4) The entire axial load versus deformation behaviour. The stages for loading are explained in Setion 1.2, Advantages and Types of Prestressing Analysis at Transfer The stress in the onrete (f ) in a member without non-prestressed reinforement an be alulated as follows. P0 f =- A P 0 = prestress at transfer after short-term losses. (3-1.1) In presene of non-prestressed reinforement, the stress in the onrete an be alulated as follows.

4 P 0 f =- A + (E s /E )A s (3-1.2) The permissible prestress is determined based on f to be within the allowable stress at transfer Analysis at Servie Loads The stresses in onrete in a member without non-prestressed reinforement an be alulated as follows. Pe P (3-1.3) f =- ± A A t P = external axial fore (In the equation, + for tensile fore and vie P e = effetive prestress. versa.) If there is non-prestressed reinforement, A is to be substituted by (A + (E s /E ) A s ) and A t is to be alulated inluding A s. The value of f should be within the allowable stress under servie onditions Analysis of Ultimate Strength The ultimate tensile strength of a setion (P ur ) an be alulated as per Clause 22.3, IS: In absene of non-prestressed reinforement, P ur = 0.87fPk Ap (3-1.4a) In presene of non-prestressed reinforement, P ur = 0.87fy A s fPk Ap (3-1.4b)

5 In the previous equations, f y = harateristi yield stress for non-prestressed reinforement with mild steel bars = harateristi 0.2% proof stress for non-prestressed reinforement with high yield strength deformed bars. f pk = harateristi tensile strength of prestressing tendons. The ultimate tensile strength should be greater than the demand due to fatored loads. The ultimate ompressive strength of a setion (P ur ) an be alulated in presene of moments by the use of interation diagrams. For a member under ompression with minimum eentriity, the ultimate strength is given as follows. the ontribution of prestressing steel is negleted. P ur = 0.4 f k A f y A s (3-1.5) Analysis of Behaviour The analysis of behaviour refers to the determination of the omplete axial load versus deformation behaviour. The analyses at transfer, under servie loads and for ultimate strength orrespond to three instants in the above behaviour. The analysis involves three priniples of mehanis (Referene: Collins, M. P. and Mithell, D., Prestressed Conrete Strutures, Prentie-Hall, In., 1991). 1) Equilibrium of internal fores with the external loads at any point of the load versus deformation behaviour. The internal fores in onrete and steel are evaluated based on the respetive strains, ross-setional areas and the onstitutive relationships. 2) Compatibility of the strains in onrete and in steel for bonded tendons. This assumes a perfet bond between the two materials. For unbonded tendons, the ompatibility 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.

6 Equilibrium Equation At any instant, the equilibrium is given by the following equation. P = A f + A s f s + A p f p (3-1.6) f = stress in onrete f s = stress in non-prestressed reinforement f p = stress in prestressed tendons P = axial fore. Compatibility Equations For non-prestressed reinforement ε s = ε (3-1.7) For prestressed tendons ε p = ε + ε p (3-1.8) ε ε s ε p = strain in onrete at the level of the steel = strain in non-prestressed reinforement = strain in prestressed tendons ε p = strain differene in prestressed tendons with adjaent onrete The strain differene ( ε p ) is the strain in the prestressed tendons when the onrete has zero strain (ε = 0). This ours when the strain due to the external tensile axial load balanes the ompressive strain due to prestress. At any load stage, ε p = ε pe ε e (3-1.9) ε pe = strain in tendons due to P e, the prestress at servie ε e = strain in onrete due to P e. The strain differene is further explained in Setion 3.4, Analysis of Member under Flexure (Part III). Constitutive Relationships The onstitutive relationships an be expressed in the following forms based on the material stress-strain urves shown in Setion 1.6, Conrete (Part II), and Setion 1.7, Prestressing Steel.

7 For onrete under ompression f = F 1 (ε ) (3-1.10) For prestressing steel f p = F 2 (ε p ) (3-1.11) For reinforing steel f s = F 3 (ε s ) (3-1.12) The stress versus strain urve for onrete is shown below. The first and third quadrants represent the behaviour under tension and ompression, respetively. f ε Figure Stress versus strain for onrete The stress versus strain urve for prestressing steel is as shown below. f p ε p Figure Stress versus strain for prestressing steel The following stress versus strain urve is for reinforing steel.

8 f s ε s Figure Stress versus strain for reinforing steel The equilibrium and ompatibility equations and the onstitutive relationships an be solved to develop the axial fore versus deformation urve. The deformation an be alulated as ε L, where L is the length of the member. The following plot shows the axial fore versus deformation urves for prestressed and non-prestressed setions. The two setions are equivalent in their ultimate tensile strengths. Axial fore Craking Tensile strengths Deformation Compressive strengths Prestressed setion Non-prestressed setion Figure Axial fore versus deformation urves From the previous plot, the following an be inferred. 1) Prestressing inreases the raking load. 2) Prestressing shifts the urve from the origin. For the prestressed member, there is a ompressive deformation in absene of external axial fore. A ertain amount of external fore is required to deompress the member.

9 3) For a given tensile load, the deformation of the prestressed member is smaller. Prestressing redues deformation at servie loads. 4) For a given ompressive load, the deformation of the prestressed member is larger. Prestressing is detrimental for the response under ompression. 5) The ompressive strength of the prestressed member is lower. Prestressing is detrimental for the ompressive strength. 6) For a partially prestressed setion with the same ultimate strength, the axial load versus deformation urve will lie in between the urves for prestressed and nonprestressed setions. The above onlusions are generi for prestressed members.