3.4 Analysis of Members under Flexure (Part III)

Transcription

1 3.4 Analysis of Members nder Flexre (Part III) This section covers the following topics Analysis for Ultimate Strength Analysis of a Rectanglar Section Analysis for Ultimate Strength Introdction A prestressed member sally remains ncracked nder service loads. The analysis nder service loads assmes the material to be linear elastic. After cracking, the behavior of a prestressed member is similar to a non-prestressed reinforced concrete member. With increasing load, the stress verss strain behavior of concrete becomes non-linear. Close to the yielding of the prestressing steel, the stress verss strain behavior of steel also becomes non-linear. The analysis of a prestressed member for ltimate strength is similar to that of a reinforced concrete member. The analysis aims to calclate the ltimate moment capacity (ltimate moment of resistance). The capacity is compared with the demand at ltimate loads. There is an inconsistency in the traditional analysis at the ltimate state. The force demand is calclated based on elastic analysis, with sperposition for the different load cases sing the load factors. Bt the capacity is calclated based on the non-linear limit state analysis. The inconsistency is jstified by the following argments. 1) The moment verss crvatre relationship is almost linear till the yielding of the steel. The moment verss crvatre relationship is also referred to as the behavior and is explained in Section 3.6, Analysis of Member nder Flexre (Part V). 2) The moment at yield is only slightly lower than the ltimate moment capacity. Hence the behavior is practically linear for most of the range of the moment. 3) The calclated moment demand for a load case based on elastic analysis is well within the moment at yield. Hence, sperposition for the load cases is applied to find ot the moment demand nder combined loads.

2 Of corse, sperposition cannot be sed to calclate the deflection nder combined loads. Variation of Stress in Prestressing Steel In non-prestressed reinforced concrete members, the tension and conseqently the stress in steel increase almost proportionately with increasing moment till yielding. The lever arm between the resltant compression and tension remains almost constant. In prestressed concrete members, the tension and conseqently the stress in prestressing steel increase slightly with increasing moment till cracking of concrete. The increase in moment changes the lever arm significantly. This is explained in Section 3.2, Analysis of Member nder Flexre (Part I). After cracking, the stress in prestressing steel increases rapidly with moment. The following sketch explains the variations of the stress in prestressing steel (f p ) with increasing load. The variations are shown for bonded and nbonded tendons. After the prestress is transferred while the member is spported at the ends, the stress will tend to increase from the vale after losses (f p0 ) de to the moment nder self weight. Sbseqently the stress will tend to drop de to the time dependent losses sch as from creep, shrinkage and relaxation. The losses of prestress are covered in Section 2.3, Losses in Prestress (Part III). The effective prestress after time dependent losses is denoted as f pe. De to the moment nder service loads, the stress in the prestressing steel will slightly increase from f pe. The increase is more at the section of maximm moment in a bonded tendon as compared to the increase in average stress for an nbonded tendon. The stress in a bonded tendon is not niform along the length. Usally the increase in stress is neglected in the calclations nder service loads. If the loads are frther increased, the stress increases slightly till cracking. After cracking, there is a jmp of the stress in the prestressing steel. Beyond that, the stress increases rapidly with moment till the ltimate load. At ltimate, the stress is represented as f p. Similar to the observation for pre-cracking, the average stress in an nbonded tendon is less than the stress at the section of maximm moment for a bonded tendon.

3 f p f p Bonded p0 f f pe Losses Unbonded Self Cracking Load weight load Service load Ultimate load Figre Variation of stress in prestressing steel The above sketch assmes that the section is failing in flexre. Other types of failre are not considered. Conditions at Ultimate Limit State In the limit states method of analysis, the limit state of collapse (ltimate state) of a member nder flexre is defined as the state when the extreme concrete compressive strain reaches a vale of At ltimate, let the extreme concrete compressive strain be denoted as ε c. Ths, ε c = Depending on the amont of prestressing steel, a section can be nder-reinforced or over-reinforced. For an nder- reinforced section, the amont of prestressing steel is less and the steel yields before the extreme concrete strain reaches For an over-reinforced section, the amont of steel is high and the steel does not yield at ltimate. The transition sitation is called a balanced condition. The strain profiles across the depths of prestressed flexral members (p to the depth of CGS) for the three sitations are shown below.

4 ε c ε c ε c ε p ε p ε p ε p > ε p,bal ε p = ε p,bal ε p < ε p,bal Under-reinforced Balanced Over-reinforced Figre Strain profiles along the depths of three prestressed members In the above sketch, ε p = strain in the prestressing steel at the level of CGS at ltimate condition ε p,bal = strain in the steel for a balanced section. The strain difference ( ε p ) is the strain in the prestressed tendons when the adjacent concrete has zero strain (ε c = 0). The strain difference gets locked dring the transfer of prestress. The vale can be determined as follows. For pre-tensioned members, the strain difference gets locked when the tendons are ct. The strain difference at that instant is given as follows. ε p = ε pi 0 (3-4.1a) Here, ε pi = strain in tendons jst before transfer ε c = strain in concrete is zero. For post-tensioned members, the strain difference gets locked when the tendons are anchored. The strain difference at that instant is given as follows. ε p = ε p0 ε c0 (3-4.1b) Here, ε p0 = strain in tendons de to P 0, the prestress after transfer ε c0 = strain in concrete de to P 0. In general at any load stage, ε p = ε pe ε ce (3-4.1c)

5 Here, ε pe = strain in tendons de to P e, the prestress at service ε ce = strain in concrete de to P e. As mentioned nder material properties, the prestressing steel does not have a definite yield point. The 0.2% proof stress is defined when the steel reaches an inelastic strain of 0.2%. Hence, nlike reinforced concrete, the transition from nder-reinforced to overreinforced section is gradal and there is no definite balanced condition. IS: does not explicitly enforce an nder-reinforced section. Bt the IRS Concrete Bridge Code reqires that the strain in the otermost tendon shold not be less than the following. 0.87f E p pk The above vale can be considered to be the strain in the steel at balanced condition. Assmptions for Analysis The analysis of members nder flexre for ltimate strength considers the following. 1) Plane sections perpendiclar to the axis of the member remain plane till the ltimate state. 2) Perfect bond is retained between concrete and prestressing steel for bonded tendons. 3) Tension in concrete is neglected. 4) The design stress verss strain crves of concrete and steel are considered. The methods of analysis will be presented for three types of sections. 1) Rectanglar section: A rectanglar section is easy to cast, bt it is not an efficient section. 2) Flanged section: A precast flanged section, with flanges either at top or bottom needs costlier formwork. Bt the section is efficient in flexre. A flanged section can also be made of precast web and cast-in-place slab. 3) Partially prestressed section: A section in a member containing both prestressed and non-prestressed reinforcement.

6 3.4.4 Analysis of a Rectanglar Section The following sketch shows the beam cross section, strain profile, stress diagram and force cople at the ltimate state. b ε c = f ck 0.42x d x C A p Cross-section Figre ε p ε p T f p Strain Stress Force Sketches for analysis of a rectanglar section The variables in the above figre are explained. b = breadth of the section d = depth of the centroid of prestressing steel (CGS) A p ε p x ε p f p = area of the prestressing steel = strain difference = depth of the netral axis at ltimate = strain in prestressing steel at the level of CGS at ltimate = stress in prestressing steel at ltimate The stress block in concrete is derived from the constittive relationship for concrete. The relationship is explained in Section 1.6, Concrete (Part II). The compressive force in concrete can be calclated by integrating the stress block along the depth. The stress in the tendon is calclated from the constittive relationship for prestressing steel. The relationship is explained in Section 1.7, Prestressing Steel. In the force diagram, C = T = A f 0.36 ck p p f x b The strengths of the materials are denoted by the following symbols. f ck = characteristic compressive strength of concrete f pk = characteristic tensile strength of prestressing steel (3-4.2) (3-4.3)

7 For analysis of a prestressed section, three principles of mechanics are sed. First, the eqilibrim relates the external applied forces with the internal forces. Second, the compatibility condition relates the strain in the prestressing steel with the strain in concrete at the level of CGS. This also considers the first two assmptions given in the previos section. The third principle involves the constittive relationships of the materials. Based on the above principles of mechanics, the following eqations are derived. 1) Eqations of eqilibrim The first eqation states that the resltant axial force is zero. This means that the compression and the tension in the force cople balance each other. F= 0 T =C Af p p = 0.36 fck xb (3-4.4) The second eqation relates the ltimate moment capacity (M R ) with the internal cople in the force diagram. M =T (d-0.42x ) R p p ( 0.42 ) =Af d- x 2) Eqation of compatibility The depth of the netral axis is related to the depth of CGS by the similarity of the triangles in the strain diagram. (3-4.5) x = d ε - ε p p (3-4.6) 3) Constittive relationships a) Concrete The constittive relationship for concrete is considered in the expression C = 0.36f ck x b. This is based on the area nder the design stress-strain crve for concrete nder compression.

8 b) Prestressing steel f p ( p ) =F ε (3-4.7) The fnction F(ε p ) represents the design stress-strain crve for prestressing steel nder tension. The known variables in an analysis are: b, d, A p, ε p, f ck, f pk. The nknown qantities are: x, M R, ε p, f p. The objective of the analysis is to find ot M R, the ltimate moment capacity. The simltaneos eqations to can be solved iteratively. This procedre of analysis is called the strain compatibility method. The steps are as follows. 1) Assme x. 2) Calclate ε p by rearranging the terms of Eqn ) Calclate f p from Eqn ) Calclate T from Eqn ) Calclate C from Eqn If Eqn (T = C ) is not satisfied, change x. If T < C decrease x. If T > C increase x. 6) Calclate M R from Eqn The capacity M R can be compared with the demand nder ltimate loads. In the strain compatibility method, the difficlt step is to calclate x and f p. IS: allows to calclate these variables approximately from Table 11, Appendix B, based on the amont of prestressing steel. The later is expressed as a prestressed reinforcement index ω p. A f p pk ω p = bdf (3-4.8) ck Table 11 is reprodced as Table which is applicable for pre-tensioned and bonded post-tensioned beams. The vales of f p and x are given as f p /(0.87f pk ) and x /d, respectively.

9 Table Vales of x and f for pre-tensioned and bonded post-tensioned p rectanglar beams (Table 11, IS: ) f p /(0.87f pk ) x /d ω p Pre-tensioned Bonded posttensionetensioned Pre-tensioned Bonded post The vales of f p /(0.87f pk ) and x /d from Table are plotted in Figres and 3-4.5, respectively. It is observed that with increase in ω p, f p redces (beyond certain vales of ω p ) and x increases. This is expected becase with increase in the amont and strength of the steel, the stress in steel drops and the depth of the netral axis increases to maintain eqilibrim fp / 0.87 fpk Pre-tensioned Post-tensioned (bonded) Figre Variation of f p with respect to ω p (Table 3-4.1) ω p

10 x / d ω p Pre-tensioned Post-tensioned (bonded) Figre Variation of x with respect to ω p (Table 3-4.1) Ths given the vale of ω p for a section, the vales of f p and x can be approximately calclated from the above tables. Example A prestressed concrete beam prodced by pre-tensioning method has a rectanglar cross-section of 100 mm 160 mm (b h). It is prestressed with 10 nmbers of straight 2.5 mm diameter wires. Each wire is stressed p to a load of 6.8 kn. The design load verss strain crve for each wire is given in a tablar form. The grade of concrete is M 40. The vale of ε p is Estimate the ltimate flexral strength of the member by the strain compatibility method.

11 100 CGC Vales in mm Cross-section of member Design load (P) verss strain (ε p ) vales for the prestressing wire are given for the range nder consideration. ε p P (kn) Soltion Strain difference ε p = The effective depth of the CGS (d ) is 120 mm. The strain compatibility method is shown in a tablar form. Here, P = load in a single wire obtained from the table T = 10 P, for the ten wires.

12 x x /d ε p ε p ε p P T C Checking (mm) (kn) (kn) (kn) (3-4.6) (Table) (3-4.2) (3-4.4) T > C T < C T C The ltimate flexral strength is given as follows. M = T ( d-0.42 x ) R = 91.5 ( ) knmm =8.5kNm