PRESTRESSED CONCRETE STRUCTURES (CE 407) 1
Partiall Par tiallyy Prestr P restresse essedd Concrete Conc rete Beams Bea ms By
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Contents 2
Objectives of the present lecture Full Prestressing Partial Prestressing Flexural Flexu ral stresses stresses in full prestressed prestressed beams Flexural Flexu ral stresses in partially partially prestressed prestressed beams Problems Further reading
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Objective of the Present lecture 3
To define define partial partial prestressing prestressing and learn learn the steps steps for the calculation of flexural stresses in partially prestr pre stress essed ed bea beams. ms.
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Full Prestressing 4
The kind of design, where the limiting tensile stress in the concrete at full service load is zero, is generally known as full prestressing. Full prestressing is required in those cases in which it is necessary to avoid all risk of cracking. Such cases include tanks or reservoirs r eservoirs where leaks must be avoided, submerged structures or those subject to a highly corrosive environment where maximum protection of reinforcement must be insured.
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Partial Prestressing 5
The kind of design in which flexural tension and thus some cracking are permitted in the concrete at normal service load is called partial prestressing.
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A Classification 6
A prestressed concrete concrete structure may be grouped in one of the three three classes depending on the extent of cracking. Class 1 Structures: Tensile stresses are not permitted in these structures, hence, no cracking under service loads. Such structures may be referred to as fully as fully prestressed prestressed structures. structures. Class 2 Structures: Limited tensile stresses are permitted but there should be no visible cracking under service loads. Tensile stresses should be less than 3 MPa. Class 3 Structures: Tensile stresses, and therefore, cracking under service loads is permitted in these structures. Such members are prestressed members. members. referred to as partially as partially prestressed
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Partia Par tially lly Prestre Prestressed ssed Mem Members bers 7
Prestressed concrete members behave well in the postcracking load range, provided they contain sufficient bonded reinforcement to control the cracks. A cracked prestressed concrete section section under service loads is significantly stiffer than a cracked reinforcement concrete section of similar size and containing similar quantities of bonded reinforcement. Members that are designed to crack at the full service loads are called partially prestressed. The width of surface cracks should not exceed 0.1 mm for members in aggressive environments and 0.2 mm for all other members.
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Elastic Flexural Stresses in full prestressing (Uncra (Un cracke cked d Bea Beams) ms) 8
As long as the beam remains uncracked, and both steel and concrete are stressed only within their elastic ranges, then concrete stresses can be found using equations of mechanics.
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Stress distribution due to initial prestress alone 9 If the memebr is subjected only only to the initial prestressi p restressing force P i , the concrete stress f 1 at the top face of the member an and d f 2 at the t he bottom bott om face can be found by superimposing axial an and d bendin bending g effects : f t
P i M Ac
f b
Z t
P i M Ac
Z b
P i Ac
P i Ac
P i e I c / cb
P i e I c / cb
P i P i ect Ac
I c
P i P i ect Ac
I c
P ec i 1 2t Ac r
P i ecb 1 2 Ac r
Stresses caused by initial prestress:
Concrete centroid
e
y
P i Ac
e tendon eccentricity measured downward from the concrete centroid Ac area of the concrete cross section I c moment of inertia of the concrete cross section r radius of gyration I c /Ac
ect 1 2 Ac r P i
P i ect I c
ct
c b Steel centroid
Section CE 407-Prestressed 407-Prestressed Co
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P i
P i ecb
Ac
I c ALGHRAFY
P i ecb 1 2 Ac r 2016
Stress distribution due to initial prestress +self-weight 10 If the memebr is subjected to the initial prestressi p restressing force P i and the dead load of the itself, the concrete stress f 1 at the top face of the member and f 2 at the bottom bott om face can be found by superimposing axial and bendin bending g effectsimm ediately :
moment caused by the self weight of the beam S 1 Section modulus with respect to the top surface of the beam S 2 Section modulus with respect to the bottom surface of the beam Ac area of the concrete cross section I c moment of inertia of the concrete cross section
M 0
f t
P i ect M 0 1 Ac r 2 Z t
f b
P i ecb M 0 1 2 Ac r Z b
r radius of gyration
P i Ac
Concrete centroid
e y
Steel centroid
Z t
c2
Ac Structures
P i ect M 0 1 2 Ac r Z t
M 0
c1
P i
Section CE 407-Prestressed 407-Prestressed Co
ect 1 2 r
I c /Ac
ec1 1 2 r
M 0 Z b
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P i ecb M 0 1 2 Ac r Z b 2016
Stress distribution due to final prestress +full service load 11 When the memebr is subjected to the effective prestressi p restressing force P i and the self weight ( M 0 ) superimposed dead load( M d ) superimposed live load ( M l ) the concrete stress f 1 at the top face of the member and f 2 at the bottom bott om face can be found by superimposing axial an and d bendin bending g effects : f t
P e ect M t 1 2 Ac r Z t
f b
P e ecb M t 1 2 Ac r Z b
Concrete centroid
e
y
M t M 0 M d M l
P i ect M 0 1 Ac r 2 Z t
M d
M l
Z t
P e ect M t 1 Ac r 2 Z t
c1
c2
Steel centroid
P i ecb M 0 1 2 Ac r Z b
Section CE 407-Prestressed 407-Prestressed Co
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M d M l S 2
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P e ecb M t 1 2 Ac r Z b 2016
Flexural Stresses in Partially Prestressed Beams 12
Load stage (1) corresponds t o application of effective pre p rest stress ress P e alone. At this this stage, t he stress in the tendon is f p1 f pe
P e A p
;
and an d t he compressive strain in the bar bar reinforcement is f s1 E s s 2
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Decompression Stage 13
It is useful to consider a fictitious load stage (2) corresponding to complete decompression of t he concrete, at which there is zero concrete strain thr ough t he entire depth. At t his his hypothetical load stage, the stress in the bar bar reinforcem ent, neglecting the effects of shrinkage an and d creep, is f s E s ( s 2 s 2 ) 0 The change in strain in the tendon is t he same as that in the concrete at that level, and an d can be calculated on the basis basis of t he uncracked concrete section prop p ropertie ertiess: ε
p 2
P e e2 1 2 f p 2 E p ε p 2 Ac E c r
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The bar reinforcem ent is unstressed at stage (2), but t o produce p roduce t he zero stress state in the concrete, the tendon must be pull p ulled ed with a fictitious external force : F A p ( f p1 f p 2 ) The effect of this fictitious decompresssing force is now cancelled by applying an equal and oppositeforce F . This force, together with with the t he external moment M t due t o elf elf - weight and superimposed loads, can be represented by a resultant force R applied with eccentrici ty e above the uncracked concrete centroid. R F F Re Fd M t Fe e CE 407-Prestressed 407-Prestressed Co
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M t Fe R ALGHRAFY
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Equivalent concrete 15
T he incrementa l steel stresses, as well as t he stress in the concrete, can be found using the transformed section concept. T he tendon is replaced by an equivalent area of tensile concrete n p A p and the bar reinforcem ent is replaced by the area n s A s , where n p
E p E c
and n s
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E s E c
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Depth of Neutral Axis
T he depth of neutral axis y from the t op surface, can be found from t he equilibriu m condition that that the moment of all all internal forces about the line of action of R must be zero.This results in a cubic equation for y that can be solved by successive trials. -
d p y d y y n s A s (d e c1 ) 0 ( ) A e c1 f c 3 n p A p d p e c1 f c 3 2 3 y y
f c 3
d y 1 y d p y n p A p (d p e c1 ) n s A s (d e c1 ) 0 - A e c1 2 3 y y y known CE 407-Prestressed 407-Prestressed Co
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16
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Centro Cen troid id of Crack Cracked ed concr concrete ete 17
Once y is known, t he effective transformed area Act an and d moment of inertia I ct of t he cracked section, about its own centro id c1* from t he t op surface, ca can n be found.
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Contd..
The incrementa l stresses sought, as loading passes p asses from stage (2) to stage (3), are f c 3
R Act
R.e*c1* I ct
Note Not e : Bending moment R.e* & Bending stress R.e* /S
R R.e* (d p c1* ) f p 3 n p I ct Act R R.e* (d s c1* ) f s 3 n s A I ct ct CE 407-Prestressed 407-Prestressed Co
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18
Final Stresses 19
Tendon : f p f p1 f p 2 f p 3 Bar reinforcem ent f s f s 3
Note Not e : f s1 f s 2 0
Concrete stress at the t op surface of t he bea beam : f c f c 3 CE 407-Prestressed 407-Prestressed Co
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Problem (in US Customary Unit) 20
The partially prestressed T beam shown in cross section in Fig. below is subjected to superimposed dead and service live load moments of 38 and 191 ft-kips, in addition to a moment of 83 ft-kips resulting from its own weight. An effective prestress force of 123 kips is applied using six Grade 250 ½ -inch diameter strands. Two non-prestressed Grade 60 no. 8 bars are located close to the tension face of the beam, The elastic elastic moduli for the concrete, concrete, tendon tendon steel, and bar steel are respectively 3.61 ×10 6 psi, 27 ×106 psi and 29 ×106 psi. The modulus of rupture of the concrete is 500 psi. Find the stresses in the concrete, prestressed steel, and bar reinforcement at the full service load. CE 407-Prestressed 407-Prestressed Co
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Note: 1ft-kips = 1.36 kN.m kips=4.45 kN 1 ksi ksi = 6.9 6.9 MPa 1 inch =25.4 mm 1 psi = 0.0069 MPa ALGHRAFY
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Solution 21
First, t he tensile stress in the concrete at t he bottom bott om of t he beam beam will be checked, assuming t he memebr is uncracked. Properties of t he uncracked cross section are Ac 212 in 2 S 1 1664 in 3 S 2 1290 in 3 c1 13.1 in c2 16.9 in r 2 103 in 2
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Contd.
For uncracked section we had derived tensile stresses in the bott om fiber f 2
P e ec2 M t 1 2 Ac r S 2
f 2
123000 11.9 16.9 312000 12 p si 1186 psi 1 212 103 1290
p si modulus of rupture (500 psi) p si) f 2 1186 psi been cracked. T he section has been p artiall lly y prestresse p restressed beam beam.. T he analysis will be carried out as discussed for partia CE 407-Prestressed 407-Prestressed Co
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Load stage 1 23
Load stage (1) corresponds t o application of effective prestress p restress P e alone. At this stage, t he stress in the tendon is f p1 f pe CE 407-Prestressed 407-Prestressed Co
P e A p
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123000 0.863
p si; 143000 psi ALGHRAFY
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Decompression stage 24
It is useful t o consider a fictitious load stage (2) corresponding ing to complete decompression of the concrete, at which there is zero concrete stra st rain in thr ough t he entire depth.
The change in strain in the tendon is the same as that in the concrete at that level, and an d can be calculated on the basis basis of t he uncracked concrete section propert p ropertie iess: ε
p 2
11.92 P e e2 123000 1 2 1 0.0004 6 Ac E c r 212 3.61 10 103
Therefore f p 2 E p ε p 2 27 106 0.0004 10800 psi p si
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To produ p roduce ce the zero stress state in the concrete, t he tendon must be pull p ulled ed with a fictitious external force : F A p ( f p1 f p 2 ) 0.863(143 10.8) 133 kips The effect of this fictitious decompresssing force is now cancelled by applying an equal an and d oppositeforce F . This force, together with the total moment of 312 ft - kips,can be represented by a resultant force R applied with eccentrici t y e above t he uncracked concrete centroid. e
M t Fe R
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312 12 133 11.9
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133
16.25 in
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Transformed area 26
The tendon is replaced by an equivalent area of tensile concrete n p A p and the bar bar reinforcem ent is replaced by the area n s A s , where n p
E p E c
27 106 3.61 10
6
7.48 and n s
E s E c
29 106 3.61 10
6
8.03
Transformed area of t he tendon n p A p 7.48 0.863 6.46 in 2 Transformed area of t he bars n s A s 8.03 1.57 12.61 in 2 CE 407-Prestressed 407-Prestressed Co
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Contd.. 27
The depth of neutral axis y from t he t op surface, can be found from t he equilibriu m condition that the moment of all all internal forces about the line of action of R must be zero. -
25 y y 6.46 (25 3.15) (4 y 60) 3.15 f c 3 2 3 y
f c 3
27 y 12.61 (27 3.15) 0 y 14.1 in f c 3 y CE 407-Prestressed 407-Prestressed Co
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Contd.. 28
Taking moments of t he par p artt ial ial areas about the top surface locates the centroid c1* 7.75 in from t he t op of the section. Section prop p roper ertt ies iesare Act Effective transformed area 12 5 4 y n p A p n s A s
Act 12 5 4 14.1 6.46 12.61 135 in 2 and an d I ct moment of inertia of t he cracked section, about its own centroid 9347 in 2 CE 407-Prestressed 407-Prestressed Co
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Contd.. 29
The eccentrici t y of t he force R with re resp speect to t he centroid of t he cracked transformed section is e* 16.25 13.1 7.75 10.90 in
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Contd.. 30
The incrementa l stresses sought, as loading passes p asses from stage (2) t o stage (3), are f c 3
R Act
R.e*c1* I ct
133000 135
133000 10.90 7.75 9347
p si -2190 psi
R R.e* (d p c1* ) 133000 133000 10.90 17.25 12600 psi f p 3 n p 7 . 48 p si I ct 9347 135 Act R R.e* (d s c1* ) 133000 133000 10.90 19.25 f s 3 n s 8 . 03 p si 16100 psi I ct 9347 135 Act
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Final Stresses 31
Tendon : f p f p1 f p 2 f p 3 143000 10800 12600 166400 psi p si Bar reinforcement f s f s 3 16100 psi p si Concrete stress at t he t op surface of t he bea beam : f c f c 3 2190 psi p si
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Further Reading 32
Read more about the partially prestressed prestressed concrete beams from: •
•
Design Desi gn of Pr Pres estr tres esse sed d Co Conc ncre rete te by A. H. Ni Nils lson on,, Jo John hn Wi Wiley ley an and d Sons, Second Edition, Singapore. Des Design of Prestressed Concrete by R. I. Gilbert and N. C. Mickleborough, First Edition, Edition, 2004, Routledge.
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Thank You 33
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