Rebar Parametric Stress-Strain Curves

Transcription

1 COMPUTERS AND STRUCTURES, INC., JUNE 2008 TECHNICAL NOTE MATERIAL STRESS-STRAIN CURVES General All material types have stress-strain urves that are deined by a series o user-speiied stress-strain points. In addition, onrete, rebar and strutural steel and tendon materials have several speial types o parametri stressstrain urve deinitions. For onrete, Simple and Mander parametri deinitions are available. For rebar, Simple and Park parametri deinitions are available. For strutural steel, a Simple parametri deinition is available. For tendons, a 250Ksi strand and a 250Ksi strand deinition are available. User Stress-Strain Curves User stress-strain urves apply to all material types. They are deined by a series o stress-strain points (, ). One o the stress-strain points must be at (0,0). User stress-strain urves may be input and viewed as standard stressstrain urves or as normalized urves. Normalized urves plot / y versus / y, where y = y /E. The program stores user stress-strain urves as normalized urves. Thus, i the E or y value or a material is hanged, the stress-strain urve or that material automatially hanges. Rebar Parametri Stress-Strain Curves Two types o parametri stress-strain urves are available or rebar. They are Simple and Park. The two are idential, exept in the strain hardening region where the Simple urves use a paraboli shape and the Park urves use an empirial shape. The ollowing parameters deine the rebar parametri stressstrain urves: 1

2 Rebar Parametri Stress-Strain Curves 2 Stress, Stress, / y Strain, Standard Curve Strain, / y = /( y /E) Normalized Curve Figure 1 Stress-Strain Curves E y u = Rebar strain = Rebar stress = Modulus o elastiity = Rebar yield stress = Rebar ultimate stress apaity sh = Strain in rebar at the onset o strain hardening u = Rebar ultimate strain apaity The rebar yield strain, y, is determined rom y = y /E. The stress-strain urve has three regions. They are an elasti region, a peretly plasti region, and a strain hardening region. Dierent equations are used to deine the stress-strain urves in eah region. The rebar parametri stress-strain urves are deined by the ollowing equations:

3 Rebar Parametri Stress-Strain Curves 3 u Rebar Stress, y Strain hardening is paraboli or Simple and empirially based or Park Peretly plasti Elasti y sh Rebar Strain, u Figure 2 Rebar Parametri Stress-Strain Curve For y (elasti region) = E For y < sh (peretly plasti region) = y For sh < u (strain hardening region) For Simple parametri urves, = y + ( ) For Park parametri urves, u m = y 60 where, r = u sh y ( sh ) ( ) sh u sh sh + 2 ( )( ) ( ) sh 60 m r + 1

4 Simple Strutural Steel Parametri Stress-Strain Curve 4 m = ( )( 30r + 1) u y r 60r 1 Both the Simple and the Park parametri stress-strain urves have the option to use Caltrans deault strain values or the urves. Those deault values are dependent on rebar size. With A s denoting the area o a rebar, the Caltrans deault strains used by the program are as ollows: u = or A s 1.40 in 2 u = or A s > 1.40 in 2 sh = or A s 0.85 in 2 sh = or 0.85 < A s 1.15 in 2 sh = or 1.15 < A s 1.80 in 2 sh = or 1.80 < A s 3.00 in 2 sh = or A s > 3.00 in 2 In terms o typial bar sizes, the deault values are as ollows: u u sh sh sh sh sh = or #10 (#32m) bars and smaller = or #11 (#36m) bars and larger = or #8 (#25m) bars = or #9 (#29m) bars = or #10 and #11 (#32m and #36m) bars = or #14 (#43m) bars = or #18 (#57m) bars Simple Strutural Steel Parametri Stress-Strain Curve The Simple strutural steel parametri stress-strain urve has our distint regions. They are an elasti region, a peretly plasti region, a strain hardening region, and a sotening region.

5 Simple Strutural Steel Parametri Stress-Strain Curve 5 u Sotening Steel Stress, y Strain hardening Peretly plasti Elasti y sh Steel Strain, u r Figure 3 Simple Strutural Steel Parametri Stress-Strain Curve The ollowing parameters deine the strutural steel Simple stress-strain urve. E y u = Steel strain = Steel stress = Modulus o elastiity = Steel yield stress = Steel maximum stress sh = Strain at onset o strain hardening u = Strain orresponding to steel maximum stress r = Strain at steel rupture The steel yield strain, y, is determined rom y = y /E. The strutural steel Simple parametri stress-strain urve is deined by the ollowing equations: For y (elasti region), = E

6 Tendon 250Ksi Strand Stress-Strain Curve 6 For y < sh (peretly plasti region), = y For sh < r (strain hardening and sotening regions), u = y 1 + r y where, r sh = u sh ( 1 r 1 e ) The strain hardening and sotening expression is rom Holzer et al. (1975). Tendon 250Ksi Strand Stress-Strain Curve The ollowing parameters deine the 250Ksi Strand stress-strain urve. E = Tendon stress = Tendon strain = Modulus o elastiity y = Tendon yield stress u = Tendon ultimate strain The tendon ultimate strain, u, is taken as The tendon yield strain, y, is determined by solving the ollowing quadrati equation, where E is in ksi. The larger obtained value o y is used. 2 E y 250y = 0 The stress-strain urve is deined by the ollowing equations: For y, = E

7 Tendon 270Ksi Strand Stress-Strain Curve 7 For y < u 0.25 = 250 Tendon Stress, y Tendon Strain, u Figure 4 Tendon 250Ksi Strand Stress-Strain Curve Tendon 270Ksi Strand Stress-Strain Curve The ollowing parameters deine the 270Kksi Strand stress-strain urve. E = Tendon stress = Tendon strain = Modulus o elastiity y = Tendon yield stress u = Tendon ultimate strain The tendon ultimate strain, u, is taken as The tendon yield strain, y, is determined by solving the ollowing quadrati equation, where E is in ksi. The larger obtained value o y is used. 2 E y ( E ) = 0 y The stress-strain urve is deined by the ollowing equations:

8 Simple Conrete Parametri Stress-Strain Curve 8 Tendon Stress, y Tendon Strain, u Figure 5 Tendon 270Ksi Strand Stress-Strain Curve For y, = E For y < u 0.04 = Simple Conrete Parametri Stress-Strain Curve The ompression portion o the Simple onrete parametri stress-strain urve onsists o a paraboli portion and a linear portion. The ollowing parameters deine the Simple onrete parametri stress-strain urve. = Conrete strain = Conrete stress = Conrete ompressive strength = Conrete strain at u = Ultimate onrete strain apaity The onrete Simple parametri stress-strain urve is deined by the ollowing equations: For (paraboli portion),

9 Mander Conrete Parametri Stress-Strain Curve 9 2 = 2 For < u (linear portion), = u The tensile yield stress or the Simple onrete urve is taken at 7.5 psi. in Conrete Stress, u Conrete Strain, Linear Paraboli Figure 6 Simple Conrete Parametri Stress-Strain Curve Mander Conrete Parametri Stress-Strain Curve The Mander onrete stress-strain urve is doumented in the ollowing reerene: Mander, J.B., M.J.N. Priestley, and R. Park Theoretial Stress- Strain Model or Conined Conrete. Journal o Strutural Engineering. ASCE. 114(3) The Mander onrete stress-strain urve alulates the ompressive strength and ultimate strain values as a untion o the oninement (transverse reinoring) steel. The ollowing types o Mander stress-strain urves are possible. Mander Unonined Conrete

10 Mander Unonined Conrete Stress-Strain Curve 10 Mander Conined Conrete Retangular Setion Mander Conined Conrete Cirular Setion The Mander unonined onrete stress-strain urve an be generated rom material property data alone. The Mander onined onrete stress-strain urves requir both material property data and setion property data. The ollowing setion rame setion types have appropriate setion property data or Mander onined onrete: Retangular Setion Cirular Setion The ollowing setion objets in Setion Designer setions have appropriate property data or Mander onined onrete: Solid Retangle Solid Cirle Poly Caltrans Hexagon Caltrans Otagon Caltrans Round Caltrans Square When a material with Mander stress-strain urves is assigned to a setion that has appropriate setion property data or Mander onined onrete, the type o Mander stress-strain urve used or that setion is determined rom the setion property data. When the setion does not have appropriate data or Mander onined onrete, the Mander unonined onrete stress-strain urve is always used. Mander Unonined Conrete Stress-Strain Curve The ompression portion o the Mander unonined stress-strain urve onsists o a urved portion and a linear portion. The ollowing parameters deine the Mander unonined onrete stress-strain urve. = Conrete strain = Conrete stress

11 Mander Unonined Conrete Stress-Strain Curve 11 E = Modulus o elastiity = Conrete ompressive strength = Conrete strain at u = Ultimate onrete strain apaity The Mander unonined onrete stress-strain urve is deined by the ollowing equations: For For 2 (urved portion), xr = r r 1 + x where x = E r = E ( ) 2 < u (linear portion), 2 = r u r r u 2 where r is as deined previously or the urved portion o the urve. The tensile yield stress or the Mander unonined urve is taken at 7.5 in psi.

12 Mander Conined Conrete Stress-Strain Curve 12 Conrete Stress, Conrete Strain, 2 u Curved Linear Figure 7 Mander Unonined Conrete Stress-Strain Curve Mander Conined Conrete Stress-Strain Curve For the ompression portion o the Mander onined onrete stress-strain urves, the ompressive strength and the ultimate strain o the onined onrete are based on the oninement (transverse reinoring) steel. The ollowing parameters deine the Mander onined onrete stress-strain urve: E = Conrete strain = Conrete stress = Modulus o elastiity (tangent modulus) E se = Seant modulus o elastiity = Compressive strength o unonined onrete = Compressive strength o onined onrete; this item is dependent on the oninement steel provided in the setion and is explained later = Conrete strain at u = Ultimate onrete strain apaity or unonined onrete and onrete spalling strain or onined onrete = Conrete strain at

13 Mander Conined Conrete Stress-Strain Curve 13 Conrete Stress, E E se 2 u Conrete Strain, u Figure 8 Mander Conined Conrete Stress-Strain Curve u = Ultimate onrete strain apaity or onined onrete; this item is dependent on the onined steel provided in the setion and is explained later The Mander onined onrete stress-strain urve is deined by the ollowing equations: = where, xr r 1 + x = x = r

14 Mander Conined Conrete Compressive Strength, 14 E se = r = E ( E ) E se Mander Conined Conrete Compressive Strength, The ollowing parameters are used in the explanation o : A = Area o onrete ore measured rom enterline to enterline o oninement steel A = Conrete ore area exluding longitudinal bars; A = A (1-ρ ) A e = Conrete area that is eetively onined A s = Area o a irular hoop or spiral oninement bar A sl = Total area o all longitudinal bars A sx = Area o retangular hoop legs extending in the x-diretion A sy = Area o retangular hoop legs extending in the y-diretion b = Centerline to enterline distane between retangular perimeter hoop legs that extend in the y-diretion d = Centerline to enterline distane between retangular perimeter hoop legs that extend in the x-diretion d s = Diameter o irular hoops or spirals o oninement steel measured rom enterline to enterline o steel = Unonined onrete ompressive strength L = Lateral pressure on onined onrete provided by the oninement steel L = Eetive lateral pressure on onined onrete provided by the oninement steel yh = Yield stress o oninement steel K e = Coeiient measuring the eetiveness o the oninement steel s = Centerline to enterline longitudinal distane between hoops or spirals s = Clear longitudinal distane between hoops or spirals

15 Mander Conined Conrete Compressive Strength, 15 w = Clear transverse distane between adjaent longitudinal bars with ross ties ρ = Longitudinal steel ratio; ρ = A sl /A ρ s = Volumetri ratio o transverse oninement steel to the onrete ore ρ x = Steel ratio or retangular hoop legs extending in the x-diretion; ρ x = A sx /sd ρ y = Steel ratio or retangular hoop legs extending in the y-diretion; ρ y = A sy /sb For irular ores: ρ s = L = 4As dss ρ s yh π A = d ( 1 ρ ) s A e = 2 π s 4 2 d s or tied hoops π s A e = ds ds or spirals 4 2 K e = Ae A L = K e L = For retangular ores ρ x = L 2 L Asx sd

16 Mander Conined Conrete Ultimate Strain Capaity, (u 16 ρ y = Asy sb Lx = ρ x yh Ly = ρ y yh n A e ( w ) = b d i A = b d i 1 2 s s 1 1 2b 2d 6 K e = Ae A Ater Lx = K e Lx Ly = K e Ly Lx and Ly are known, is determined using a hart or the multiaxial ailure riterion in terms o two lateral onining stresses that is published in the previously reerened artile, Mander et al. (1984). Mander Conined Conrete Ultimate Strain Capaity, u The Mander onined onrete ultimate strain apaity, u, is a untion o the oninement steel. The ollowing igure shows the Mander stress-strain urves or onined and unonined onrete. The dierene between the onined and unonined urves is shown shaded. The shaded area shown in Figure 9 represents the additional apaity provided by the oninement steel or storing strain energy.

17 Mander Conined Conrete Ultimate Strain Capaity, (u 17 Conined Unonined Conrete Stress, 2 u Conrete Strain, u Figure 9 Mander Conined and Unonined Stress-Strain Curves This area is limited to the energy apaity available in the area under the oninement steel stress-strain urve up to the ultimate steel strain, u. Suppose A 1 is the shaded area between the Mander onined and unonined urves and A 2 is the area under the oninement steel stress-strain urve. Further suppose ρ s is the volumetri ratio o oninement steel to the onrete ore. Then, equating energies under the onrete and oninement steel stress-strain urves gives: A 1 = ρ S a 2 The program determines the appropriate value o the onined onrete ultimate straining, u, by trial and error, equating energies as desribed previously. When the A 1 = ρ s A 2 relationship is satisied, the orret value o u has been ound.

18 Reerenes 18 The tensile yield stress or the Mander onined urves is taken as 7.5 psi. in Reerenes Holzer et al SINDER. A Computer Code or General Analysis o Two-Dimensional Reinored Conrete Strutures. Report. AFWL-TR Vol. 1. Air Fore Weapons Laboratory, Kirtland, AFB, New Mexio. Mander, J.B., M.J.N. Priestley, and R. Park Theoretial Stress-Strain Model or Conined Conrete. Journal o Strutural Engineering. ASCE. 114(3)