Concept of Prestressing

Transcription

1 Concept of Prestressing Concept of Prestressing Prestressing the concrete is to transfer precompression (compressive stress) to the concrete How the prestressing force transmitted to concrete can be explained by concept of prestressing 1

2 Degree of Prestressing This classification introduced depend on the level of prestress introduced in the structural element to nullify the stress due to external load. Fully prestressed : all cracking should be avoided d restricted t by no tensile stress allowed under service load, the whole section in permanent state of compression Limited prestressing : tensile stresses do not exceed the cracking stresses of concrete Partial prestressing g: combination of tensioned and untensioned steel. Represent form of construction which is intermediate between reinforced and prestressed concrete Concept of Prestressing Stress concept Concept of Prestressing Force concept Load balancing concept 2

3 Stress Concept The concept that considering prestressing force transmitted to concrete as initial internal stress to counteract the internal stress developed due to external loads is known as stress concept The combination of the effect of external loads and prestressing are studied together as equivalent stresses and compared with permissible levels of stresses in the material The procedures of this concept can be divided into two stages: stress at transfer/stage 1 and stress at service/stage 2 For stage 1, the stresses across cross section due to self weight and prestressing are taken into account For stage 2, the stresses caused by prestressing, dead and live loads and other external loads are calculated together through the depth of cross section The stresses should be within the permissible limits Stresses at transfer = + - Axial stress eccentricity of prestressing force dead load 3

4 Stresses at transfer Stress at top fibre : Stress at bottom fibre : (Allowable tensile stress at transfer, Clause BS8110) (Allowable compressive stress at transfer) Stresses at service = Axial stress eccentricity of prestressing force dead load external load 4

5 Stresses at service Stress at top fibre : (Allowable compressive stress at service, Clause BS8110) Stress at bottom fibre : (Allowable tensile stress at service) Stress concept In stress concept, we used theory of bending throughout the analysis where: - it is assumed that plane sections remain plane before or after the moments are applied - the top and the bottom fibre of the structural elements are subjected to maximum stresses The permissible/allowable streses under compression and tension in the materials concrete and steel do have a major role to play in analysis and design of prestressed concrete structure based on stress concept 5

6 Example 1 A simply supported prestressed concrete beam of cross section 400mm x 600mm has a span of 10m. It is subjected to an uniformly distributed load of 30kN/m in addition to its selfweight and is prestressed with a force of 1740kN with a prestressing able of parabolic profile. The cable is anchored at the center of gravity of the cross section at support and has an eccentricity of 160mm below NA at the mid span cross section. Analyze the beam for the effects of prestressing and the loads at mid cross section using the philosophy p of stress concept. Solution Span of the beam = 10 m Cross section = 400mm x 600mm External load = 30 kn/m Unit weight of concrete = 24 kn/m3 Prestressing force = 1740 kn Cable profile = parabolic Eccentricity of mid cross-section = 160mm (below NA) Eccentricity at support section = 0 mm 6

7 Properties of section Area of cross section, A = 0.4 x 0.6 = 0.24 m2 Moment of inertia, I = Modulus of section, Zt = Zb = Selfweight of the beam, w/m = 24kN/m 3 x 0.24m 2 = 5.76kN/m Calculate stress due to axial load, moment from eccentricity of prestress force, bending moment from selfweight and external load Forces Axial force, P = 1740 kn Moment due to eccentricity of prestressing force = P x e Pe = 1740 kn x 0.16 m = knm Bending moment due to : selfweight = wl 2 /8 = 5.76 x 10 2 /8 = 72 knm external load = 30 x 102/8 = 375 knm All causes and effects are converted to stresses in stress concept p for further evaluation. 7

8 At transfer + + = Axial stress eccentricity of prestressing force dead load Stresses at top fibre : Stresses at bottom fibre : At service = Axial stress eccentricity of prestressing force Stress at top fibre : dead load external load Stress at bottom fibre : 8

9 Force Concept In this approach the structural element is considered as if it is a reinforced concrete element The total prestressing force is taken tensile force and the stresses generated in concrete will produced compression force of an equal value. The forces are collinear to keep the element in equilibrium if only prestressing force is considered Hence, the structural element at any cross-section will be subjected to tensile force in the prestressing element and a compressive force in the concrete which is the resultant force of all compressive stresses acting on that cross-section Force Concept If any additional load (say dead load) is considered, the tensile force in prestress element will be modified and the center of compression will also be shifted. For a case of positive sagging bending moment applied on the structure due to external loads the tensile force in the prestressed steel element is marginally increased and the compressive force which is the resultant stresses caused by the prestressing and by the loading will be shifted upwards from the line of action of tension. The tensile force or the compressive force multiplied by the shift between these two forces will be the external moment. This concept is used to design the structures and to get the moment resisting capacity of the cross section Since the capacity of the section is decided based on the total tension and compression it carries, this approach is called the force approach 9

10 Example Analyze the beam in Example 1 using force concept. Solution In force concept all causes and effects are considered as forces for evaluation. Bending moment at mid span due to : a) Selfweight = 72 knm (top comp, bottom tension) b) External load = 375 (top comp, bottom tension) Total bending moment = 447 knm Prestressing force = 1470 kn 10

11 When the prestressing force of 1740 kn (tensile) in the cables alone is acting, the stresses generated in concrete will lead to a resultant compression of equal value (1740 kn) and the compression also acts at the same level of prestressing force. The forces are collinear. Stage 1/at transfer When selfweight starts acting (which is immediately after prestress) there will be a small increase in the tensile forc in the cables. But this is neglected. The total tensile force in the cable = 1740 kn Total compressive force = 1740 kn (to keep the section in equilibrium) But the resultant compression will act at a different level, so that the compression and the tension will form a couple to resist dead load bending moment Dead load bending moment = 72 knm Distance between the tensile force (cable position) and the center of resultant compression, a a = M/P = 72/1740 = m Distance of compression from the NA of cross section = m = mm mm mm a = 41.38mm T NA 11

12 The resultant compression will act at 118.6mm from NA only for a given stress distribution. The stress distribution can be evaluated as detailed below : Stress at top = Stress at the bottom = = N/mm2 Stage II/ at service When the external load also starts acting the resultant (final) bending moment shall be resisted by the total compression and total tension with a lever arm. Total tension = 1740 kn Total moment to be resisted = dead load + bending moment due to other loads = = 447 knm Lever arm required, a = M/P = 447/1740 = m Resultant center of compression will be located at 256.9mm from the center of tension cable position 12

13 Hence position of compression will be located at 256.9mm from the center of tension cable position. Hence position of center of compression from center of NA = = mm (upward) This resultant compression (1740kN) will act at 96.9mm above NA only for one particular stress distribution across the section. The stress distribution is evaluated as follows Stress at top = Stress at bottom = The stresses are the same as we obtained in the stress concept. Load balancing concept Opposite type of loads in structural element (opposite in nature to the external loads) If the external loads cause a sagging curvature in the beam, any load which introduces the hogging curvature on to the beam, equal and opposite in nature to that caused by external loads is also called prestressing and this method of prestressing is recognized as load balancing concept. The external loads are treated only as loads and not converted as stress on the structure Prestressing also converted as equivalent load and this equivalent load must counteract or balance the external loads The load balancing concept is used for analysis of indeterminate prestressed concrete structures and complicated analysis where the effect of prestressing cannot easily depicted 13

14 For example the parabolic profile of prestressing cable with prestressing force, P can be considered equivalent to the upward force of Example 3 Analyze the prestressed concrete beam described in Example 1 using load balancing concept. Solution In this concept all the causes and the effects will be considered as loads and the member will be analyzed Total downward load = = kn/m The equivalent upward uniformly distributed load provided by prestress = Net downward on the beam = = kn/m 14

15 The bending moment caused by resultant downward force at center section Stresses at mid span caused by this moment (Compression at top, tension at bottom) Stresses at mid span caused by pretensioning force that acting at the centroid of the section (compression) Hence net stresses : At top fibre = stress due to prestress + stress due to downward force At bottom fibre The stresses are the same as obtained in stress concept and force concept at service 15