Concrete Design Part 1

REINFORCED CONCRETE MATERIALS: CONCRETE: f’ c  =

compressive strength at 28 days E c  = modulus of elasticity of concrete  Ec  w

1.5 c

c

0.043 f  'c

 wc  2500 kg/m3  for   3 w  1500 kg/m  c 

3  4700  f  'c for normal weight concrete wc  2300 kg/m 

REINFORCING STEEL: Structural Grade (ASTM Gr.33/PS Gr.230) Intermediate Grade (ASTM Gr.40/PS Gr.275) High Carbon Grade (ASTM Gr.60/PS Gr.415) E s  = modulus of elasticity of steel E s  = 200,000 MPa

f y  =

230 MPa f y  = 275 MPa f y  = 415 MPa

REINFORCED CONCRETE DESIGN WORKING STRESS DESIGN (Alternate Design Method) Design Principle: actual stress, f < allowable stress, f allowable Code Spec’s: allowable f c = 0.45

f’ c 

Investigation of Beam Section 

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1. Transform the beam section into an equivalent homogenous section. 2. Locate the Neutral Axis (NA) of the section. 3. Analyze using: c a) flexure formula,  f    , or b) internal couple method (statics of internal forces)  x

(c)

#1. Problem  A rectangular reinforced concrete beam section 300mm wide with an effective depth of 600mm is subjected to a bending moment of 200 kN-m. The beam is reinforced with 4-32mm . The modular ratio , n = 8. 

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1. Find the distance of the NA from the top of the section. 2. Find the total compressive force in concrete. 3. Calculate the maximum stress in concrete. 4. Determine the maximum stress in steel.

300 mm

600 mm

4-32mm

25728/246.3/386175/10.45/120.1

#2. Problem  A reinforced concrete beam is 400 mm wide with an effective depth of 530 mm. It has a tensile reinforcement of 4 – 25mm . f’ c  = 21 MPa and f s  = 140 MPa. 

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1. Calculate the modular ratio, n 400 mm (up to one decimal place). 2. Determine the the moment of inertia of the transformed section. 530 mm 3. Find the moment capacity of the 4-25mm section. 4. Calculate the total safe uniform load the beam could support over a 6m span in kN/m. 9.3/18265/179/3015e6/159.2/129.3/29.73

T-BEAM ANALYSIS: Case I: NA is in the flange

Case II: NA is in the web

b tf

NA NA

d As

nAs

nAs

bw

Analyze as a rectangular section

Analyze as a real T-beam section

#3. Problem  A concrete beam section has the following dimensions: b  = 600mm, b w  = 300mm, d  = 500mm and t f  = 80 mm. It is reinforced with 3-28mm tension steel. It is subjected to a bending moment of 100 kN-m. f’ c  = 21 MPa, f s  = 165 MPa and n = 9. 

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1. Calculate the maximum stress in concrete. 2. Find the concrete stress at the bottom of the flange. 3. Determine the total compressive force in the concrete.

600 mm 80 mm

500 mm

16632/147.7 2678e6/5.52

3-28mm

2.53/118.4 218803/218896

300 mm

RC Beams with Compression Steel To take into account the creep of concrete and non-linearity of the stress-strain relation, compression steel A s ’  is transformed as 2nA s ’ . Note: tension steel A s  is still transformed as nA s .  Area Tension Steel  A s  Compression Steel  A s ’

Transformed Area n A s 2n A s ’

#4. Problem  A 300mm x 600mm reinforced concrete beam section is reinforced with 4-28mm tension steel at d  = 536mm and 2-28mm compression steel at d’ = 64mm. The section is subjected to a bending moment of 150 kN-m. f’ c  = 21 MPa, f s  = 165 MPa and n = 9. 

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1. Find the maximum stress in concrete. 2. Determine the stress in the compression steel. 3. Calculate the stress in the tension steel.

300 mm 64 mm 2-28mm

22176 20944

536 600 mm mm

186.2 3672e6

4-28mm

7.61 89.85 128.6

For Practice:

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 A 400mm x 600mm box section has a uniform wall thickness of 60 mm except at the bottom which is 140mm thick. It is reinforced with 3 – 25mm at d = 540mm. n = 9, f’ c  = 20 MPa and f s  = 124 MPa. 1. Calculate the moment capacity of the section considering the concrete part of the beam. 400 mm (Mc = 123.7 kN-m)

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2. Find the moment capacity of the section considering the steel part of the beam. (Ms = 90.35 kN-m)

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3. Determine the maximum simply supported span of the beam if it is to carry a midspan load of 80 kN. (L = 4.153m)

540 mm

600 mm

3-25mm

END of LECTURE