Beam Security Check - An Analytical Model

45b9c870-c38f-4fe3-b470-7e700c001c52 21 Beam Security Check - An Analytical Model João Pedro Cortês Pereira 4 5 false false 0 0 decimal

&[DATE] &[TIME] - &[FILENAME]

Author: João Pedro Cortês Pereira

2.1 Beam Data

2.1. Geometry

Beam length:

l 7500 mm * :

Height of the cross-section:

h 600 mm * :

Cross-section width:

b 300 mm * :

Concrete cover top:

c.top 30 mm * :

Concrete cover bottom:

c.bot 30 mm * :

Half of the beam support width:

a.w 250 mm * :

Steel Reinforcements:

Stirrups:

Bottom (tension):

Top (compression):

Diameter:

ϕ.s 6 mm * :

Diameter:

ϕ.s1 20 mm * :

Diameter:

ϕ.s2 0 mm * :

Spacing:

s.w 200 mm * :

Number of bars:

n.s1 4 :

Number of bars:

n.s2 0 :

Angle:

α.s 90 ° * :

Bond factors of internal steel reinforcement:

bond.s good :

(good or medium)

shape.s ribbed :

(ribbed or smooth)

Internal arm of the reinforcements:

d.s1 h c.bot - ϕ.s - ϕ.s1 2 / - : mm 554

d.s2 c.top ϕ.s + ϕ.s2 2 / + : mm 36

A.s1 π 4 / ϕ.s1 2^* n.s1 * : mm 2^1256.6371

A.s2 π 4 / ϕ.s2 2^* n.s2 * : mm 2^0

Total area:

ρ.s1 A.s1 b d.s1 * / : 0.0076

ρ.s2 A.s2 b d.s2 * / : 0

Steel reinforcement ratio:

A.sw 2 π 4 / * ϕ.s (2^* : mm 2^56.5487

Area of shear reinforcement (stirups):

Internal lever arm for the steel reinforcement:

z.s 0.9 d.s1 * : mm 498.6

Area of the beam section :

A.c b h * : mm 2^1.8 105^*

2.2. Load Conditions

2.2.1 ULS Combinations:

2.2.2 SLS Combinations:

M.Ed.ULS 210 kN * m * :

M.Ed.k 175.78 kN * m * :

V.Ed.ULS 50 kN * :

M.Ed.qp 120 kN * m * :

M.Ed.a 180 kN * m * :

2.3. Properties of Existing Materials:

2.3.1 Concrete:

f.ck 20 MPa * :

Characteristic value of the compressive strength:

Y.c 1.5 :

Material partial safety factor for persistent and transient load conditions:

ε.cu f.ck 50 MPa * ≥ 2.6 3590 MPa * f.ck - (100 MPa * / (4^* + 3.5 if : 3.5

Ultimate compressive strain (in ‰):

Coefficient to take into account the long term unfavorable effects on the tensile strength:

α.ct 1 :

Design compressive strength:

f.cd f.ck Y.c / : MPa 13.333

Mean value of compressive strength:

f.cm f.ck 8 MPa * + : MPa 28

Mean value of elastic modulus:

E.cm 22 f.cm 10 MPa * / (0.3^* GPa * : MPa 29961.951

Mean tensile strength:

f.ctm 0.3 MPa * f.ck MPa / (23 /^* : MPa 2.21

Minimum characteristic tensile strength:

f.ctk_min 0.7 f.ctm * : MPa 1.547

Design tensile strength:

f.ctd α.ct f.ctk_min * Y.c / : MPa 1.032

Stress - Strain relationship:

ε.c2 f.ck 50 MPa * ≥ 2.0 0.085 f.ck MPa / 50 * (0.53^* + 2.0 if : 2

n f.ck 50 MPa * ≥ 1.4 23.4 90 f.ck - 100 / (* + 2.0 if : 2

ε.c σ.c ε.c 2 < f.cd 11 ε.c ε.c2 / - (2^- (* f.cd if :

ε.c σ.c MPa /

2.3.2 Steel Reinforcements

Characteristic strength of the longitudinal reinforcement:

f.yk 400 MPa * :

Characteristic strength of the shear reinforcement:

f.ywk 400 MPa * :

Modulus of elasticity for both longitudinal and transversal reinforcement:

E.s 200 GPa * :

Material partial safety factor for persistent and transient load conditions:

Y.s 1.15 :

f.yd f.yk Y.s / : MPa 347.826

f.ywd f.ywk Y.s / : MPa 347.826

Design strength:

Characteristic yielding strain:

ε.yk f.yk E.s / : 0.002

Modular ratio between steel and concrete:

n.s E.s E.cm / : 6.675

Design yielding strain:

ε.yd f.yd E.s / : 0.0017

ε.s σ.s ε.s 1000 f.yd * E.s / ≥ f.yd ε.s 1000 / E.s * if :

ε.s σ.s MPa /

3. Verifications of Ultimate Limite States (ULS)

3.1. Common formulations for ULS:

Strain and Stresses in the steel reinforcements:

x ε.c ε.s1 ε.c d.s1 x - x / (* :

x ε.c ε.s2 ε.c x d.s2 - x / (* :

x ε.c σ.s1 x ε.c ε.s1 E.s * f.yd 12 mat min :

x ε.c σ.s2 x ε.c ε.s2 E.s * f.yd 12 mat min :

3.2. Flexural capacity of the existing member:

ε.c ε.cu 1000 / : 0.0035

ε.c2 ε.c2 1000 / : 0.002

3.2.1 Neutral axis depth is found through internal arm equilibrium (while ignoring top steel reinforcement (compression)):

x f f.cd x mm * ε.c2 ε.c / x * mm * - (* b * 23 / f.cd * ε.c2 ε.c / * x * mm * b * + A.s1 x mm * ε.c σ.s1 * - :

root x f x 0 h mm / solve : 134.984

x root 1 * mm * : mm 134.984

The stresses and strains in the steel reinforcement are:

x ε.c ε.s1 0.0109

(Tension)

x ε.c σ.s1 MPa 347.8261

x ε.c ε.s2 0.0026

(Compression)

x ε.c σ.s2 MPa 347.8261

3.2.2 The flexural resistance of the cross section can be found according to:

x c g f.cd x ε.c2 ε.c / x * - (* b * x ε.c2 ε.c / x * - 2 / (* 23 / f.cd * ε.c2 ε.c / * x * b * x ε.c2 ε.c / x * - 38 / ε.c2 ε.c / * x * + (* + f.cd x ε.c2 ε.c / x * - (* b * 23 / f.cd * ε.c2 ε.c / * x * b * + (c * mm * - :

root x c g c 0 h mm / solve : 56.1488

c root 1 * mm * : mm 56.1488

z.Mrd d.s1 c - : mm 497.8512

x M.Rd f.cd x ε.c2 ε.c / x * - (* b * 23 / f.cd * ε.c2 ε.c / * x * b * + (z.Mrd * : kN m * 217.6063

Checking if the resulting resisting moment is sufficient for the moment applied to the beam:

M.Ed.ULS x M.Rd > Flexural capacity of the beam is NOT sufficient Flexural capacity of the beam IS sufficient if Flexural capacity of the beam IS sufficient

3.3 Verification for robustness requirement

Moment of Inertia of the transformed cracked section:

I.BS 13 / b * x 3^* n.s 1 - (A.s2 * x d.s2 - (2^* + n.s A.s1 * d.s1 x - (2^* + : 0.0017 m 4^*

Flexural stiffness of the cracked section:

EI.BS E.cm kPa / I.BS m 4^/ * : 51495.803

Safety verification for an accidental load combination (utilizing characteristic values for the material properties):

x ε.c σ.s1.k x ε.c ε.s1 E.s * f.yk 12 mat min :

x ε.c σ.s2.k x ε.c ε.s2 E.s * f.yk 12 mat min :

x f f.ck x mm * ε.c2 ε.c / x * mm * - (* b * 23 / f.ck * ε.c2 ε.c / * x * mm * b * + A.s1 x mm * ε.c σ.s1.k * - :

root x f x 0 h mm / solve : 103.4878

x root 1 * mm * : mm 103.4878

The stresses and strains in the steel reinforcement are:

x ε.c ε.s1 0.0152

(Tension)

x ε.c σ.s1.k MPa 400

x ε.c ε.s2 0.0023

(Compression)

x ε.c σ.s2.k MPa 400

3.3.1 The flexural resistance of the cross section can be found according to:

x c g f.ck x ε.c2 ε.c / x * - (* b * x ε.c2 ε.c / x * - 2 / (* 23 / f.ck * ε.c2 ε.c / * x * b * x ε.c2 ε.c / x * - 38 / ε.c2 ε.c / * x * + (* + f.ck x ε.c2 ε.c / x * - (* b * 23 / f.ck * ε.c2 ε.c / * x * b * + (c * mm * - :

root x c g c 0 h mm / solve : 43.0474

c root 1 * mm * : mm 43.0474

z.Mrd d.s1 c - : mm 510.9526

x M.Rd.a f.ck x ε.c2 ε.c / x * - (* b * 23 / f.ck * ε.c2 ε.c / * x * b * + (z.Mrd * : kN m * 256.8328

Checking if the resulting resisting moment is sufficient for the moment applied to the beam:

M.Ed.a x M.Rd.a > Beam COMPLIES with the robustness requirement. Beam DOES NOT comply with the robustness requirement. if Beam DOES NOT comply with the robustness requirement.

3.4 Verification for Shear capacity of the member

Shear capacity of the beam will be checked according to Eurocode EN 1992, Section 6.2.

Axial force in the cross-section due to loading or prestressing:

N.Ed 0 kN * :

Coefficient - National Annex:

k.1 0.15 :

Checking for members without shear reinforcement:

b.w b : mm 300

s s.w : mm 200

z z.s : mm 498.6

k 1200 mm * d.s1 / sqrt + 212 mat min : 1.6008

ρ.1 ρ.s1 0.02 12 mat min : 0.0076

C.Rdc 0.18 Y.c / : 0.12

Concrete compressive stress at the centroidal axis due to axial loading:

σ.cp N.Ed A.c / 0.2 f.cd * 12 mat min : MPa 0

The design shear resistance of the member without shear reinforcement can be found:

V.Rdc_1 C.Rdc k * 100 f.ck MPa / * ρ.1 * (13 /^* k.1 σ.cp * + (b.w * d.s1 * MPa * : kN 78.9521

EN 6.2a

V.min 0.035 k 32 /^* f.ck MPa / (12 /^* MPa * : MPa 0.317

EN 6.3N

V.Rdc_2 V.min k.1 σ.cp * + (b.w * d.s1 * : kN 52.691

EN 6.2b

V.Rd_unreinforced V.Rdc_1 V.Rdc_2 12 mat max : kN 78.9521

V.Rd_unreinforced V.Ed.ULS ≤ Shear reinforcement is required Shear reinforcement is NOT required if Shear reinforcement is NOT required

The design shear resistance of the member with shear reinforcement can be found:

Inclination of the shear failure crack:

θ 39.8 deg * :

Coefficient taking into account the state of stress in the compresison chord:

α.cw 1 :

v.1 0.6 :

Strength reduction factor for concrete cracked in shear:

V.Rds A.sw s / z * f.ywd * θ cot α.s cot + (* α.s sin * : kN 58.8537

EN 6.13

V.Rd.max α.cw b.w * z * v.1 * f.cd * θ cot α.s cot + (* 1 θ cot 2^+ (/ : kN 588.4905

EN 6.14

V.Rd_reinforced V.Rds V.Rd.max 12 mat min : kN 58.8537

V.Rd_final A.sw 0 ≡ V.Rd_unreinforced V.Rd_reinforced if : kN 58.8537

V.Rd_final V.Ed.ULS ≤ Safety in shear DOES NOT comply. Safety in shear DOES comply. if Safety in shear DOES comply.

4. Verifications of Serviceability Limite States (SLS)

4.1. Stress Limitations:

1ª Equação de equilíbrio de forças:

x ε.c SF 0.8 f.cd * b * x * A.s1 x ε.c σ.s1 * - :

2ª\[REGION[6ed3f9d670554d44856dd11dbca67a85]]\

Equação de equilíbrio de momentos:

x ε.c SM 0.8 f.cd * b * x * (d.s1 0.8 x 2 / * - (* A.s1 x ε.c σ.s1 * (d.s1 0.8 x 2 / * - (* - M - :

Strains in the steel reinforcements:

x ε.c ε.s1 ε.c d.s1 x - x / (* :

x ε.c ε.s2 ε.c x d.s2 - x / (* :

Derivatives of Stress functions:

x ε.c dσ1_dx E.s d.s1 ε.c * x 2^/ - (* :

x ε.c dσ2_dx E.s d.s2 ε.c * x 2^/ (* :

x ε.c dσ1_dε E.s d.s1 x / 1 - (* :

x ε.c dσ2_dε E.s 1 d.s2 x / - (* :

Jacobian Matrix:

x ε.c J.11 0.8 f.cd * b * x ε.c dσ1_dx A.s1 * (- (:

x ε.c J.12 A.s1 x ε.c dσ1_dε * (- :

x ε.c J.21 0.8 f.cd * b * d.s1 0.8 x 2 / * - (* 0.8 f.cd * b * x * (0.4 - (* + A.s1 x ε.c dσ1_dx * (d.s1 0.8 x 2 / * - (* - A.s1 x ε.c σ.s1 * (0.4 - (* - :

x ε.c J.22 A.s1 x ε.c dσ1_dε * (d.s1 0.8 * x 2 / * (* - :

x Jacobiano x 1 el x 2 el J.11 x 1 el x 2 el J.12 x 1 el x 2 el J.21 x 1 el x 2 el J.22 22 mat :

x f x 1 el x 2 el SF x 1 el x 2 el SM 21 mat :

Force tolerance tol.F 1 N * :

Moment tolerance tol.M 1 N * m * :

Maximum number of iterations nmáx 104^:

M NewtonRaphson.JP a.ini 0.165 h * 11000 / 21 mat : r a.ini f : step 0 : r 1 el det tol.F > (r 2 el det tol.M > (| (step nmáx < (& J a.ini Jacobiano : δ.f J 11 el m N / * J 12 el N / J 21 el N / J 22 el N m * / 22 mat (1 -^r 1 el N / r 2 el N m * / 21 mat * : a.2 a.ini δ.f 1 el m * δ.f 2 el 21 mat - : a.ini a.2 : r a.ini f : step step 1 + : 61 line while step a.ini 1 el a.ini 2 el 13 mat 51 line :

M.Ed.k NewtonRaphson.JP #