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Unformatted text preview: Chapter Two
_____________________________________________________________________ Analysis of Sections of Reinforced Concrete Members 2.1 Design Stress-Strain Curves Short-term stress-strain curves in BS 8110: Part 1: 1997 are presented in
Figs 2.1 and 2.2. These curves are in an idealised form and are used in the analysis of R.C. member sections.
0.67 f cu γm Strain Fig. 2.1-1 BS8110 design stress/strain curves for concrete in compression • Maximum stress: 0.67 f cu γm (when γm = 1.5, 0.67⋅fcu / 1.5 = 0.45fcu ) The factor 0.67 allows for the difference between the bending
strength and the cube crashing strength of concrete.
• Concrete is assumed to fail at an ultimate strain, εcu , of 0.0035. 33 2.1.2 Reinforcing steel
fy / γ m
Es = 200 kN/mm2 fy / γ m Fig. 2.1-2 BS8110 design stress/strain curves for steel reinforcement • Maximum stress: fy /γm (when γm = 1.05, fy /1.05 = 0.95fy )
(Hong Kong code: γm = 1.15; fy /1.15 = 0.87fy )
Design yield strain: ε y = f y /γ m
Es (For fy = 460 N/mm2, εy = 0.00219)
2.2 General Behaviour of R.C. Beams in Bending 2.2.1 Types of cross sections
The three common types of R.C. beam sections:
(1) Rectangular section with tension steel only;
(2) Rectangular section with both tension and compression steel;
(3) Flanged sections of T or L shapes. Fig. 2.2-1 (a) Rectangular beam and slab, tension steel only; (b) rectangular
beam, tension and compression steel; (c) flanged beams.
34 2.2.2 Structural behaviour in bending (a) Yield load Crack load (b) (c)
Fig. 2.2-2 (a) Cracked beam; (b) load-deflection curve; (c) cracked
section and compressive stress distributions of concrete • Concrete − resist compression
Reinforcing steel − resists tension due to bending
• Path of failure:
Concrete cracking Steel yielding Concrete crashing
35 2.2.3 A general theory for ultimate flexural strength
Current design methods for R.C. beams in British and American codes
are based on the general theory described below.
• Basic assumptions
(a) Plane sections remain plane.
(b) The ultimate limit state of collapse is reached when concrete
strain reaches εcu = 0.0035. (ACI code: εcu = 0.003)
(c) Tensile strength of concrete is ignored.
• Strain and stress distributions of cross-section at failure Fig. 2.2-3 Strain and stress distributions at failure Notation: ′
As − tension reinforcement; As − compression reinforcement;
εcu − ultimate strain of concrete (0.0035); εs − strain in steel;
d − effective depth (h − overall depth); x − neutral axis depth. 36 • Notes
o Form Assumption (a), the strain distribution in a beam cross-action is linear.
o Form Assumption (b), the maximum concrete compressive strain has a specified value εcu at the instant of collapse. The corresponding values of εs and ε’s are εs = d−x
x (2.1) ′
εs = x − d′
x (2.2) o From Assumption (c), the forces on the beam section are Concrete tension: Fct = 0
Concrete compression: Fcc = k1fcubx
Reinforcement tension: Fst = Asfs
Reinforcement compression: Fsc = As′ f s′
For the condition of equilibrium, ′
k1 f cu bx + As f s′ = As f s (2.3) Taking moments about the level of tension steel, ′
M u = (k1 f cu bx)(d − k 2 x) + As f s′(d − d ′) (2.4) Or taking moments about the centroid of the concrete stress block, ′
M u = As f s (d − k 2 x) + As f ′(k 2 x − d ′) (2.5) where Mu is the ultimate flexural strength or maximum moment of
37 2.2.4 Stress blocks
(1) Hognestad et al.’s block Fig. 2.2-4 Characteristics of Hognestad et al.’s stress block (2) Whitney’s equivalent rectangular block Fcc = Accfcu Fst = Asfs Fig. 2.2-5 Whitney’s equivalent rectangular stress block
38 o Both BS 8110 and the ACI Building Code make use of the concept of an equivalent rectangular stress block, which was
pioneered by Whitney.
o The actual stress block may be replaced by a fictitious rectangular block of intensity of 0.85 f c′ with a depth xw:
Area of 0.85 f c′x w = that of the actual block
Centroids of two blocks are very nearly at the same level
o For equilibrium: Fcc = Fst
or 0.85 f c′bxw = As f s (2.6) When fs = fy
xw = As f y
0.85 f c′b = fy
0.85 f c′ ρd (2.7) where ρ is reinforcement ratio, defined by ρ= As
bd (2.8) The ultimate moment of resistance is M u = As f y (d − xw
2 (2.9) This is a theoretical value of the maximum moment of
resistance – No any safety factor has been included. 39 (3) BS 8110 stress block
o The rectangular−parabolic stress block
(a) d (b) Fig. 2.2-6 (a) Section in bending with a rectangular-parabolic stress block;
(b) BS 8110 design stress block for ultimate limit state
o Determine the mean concrete stress, k1 From the strain diagram
x ε cu = w With εcu = 0.0035 and εo = 2.4 × 10 w= xε 0 ε cu = (2.10) ε0
−4 f cu
1.5 x f cu
17.86 (2.11) 40 0.45fcu For the stress block (Fig. 2.2-6a),
k1 = Area ( pqrs ) − Area (rst )
f cu x k1 = 0.45 f cu x − 0.45 f cu w / 3
f cu x or
(2.12) Substituting for w from Eq. (2.11) into Eq. (2.12) gives k1 = 0.45 [1 − f cu / 53.5] (2.13) o Determine the depth of the centroid k2x Taking moments about the neutral axis:
(area of stress block)⋅( x − k 2 x )
2 = Area (pqrs)⋅ − Area (rst)⋅
∴ x − k2 x = w
4 0.45( x 2 / 2 − w 2 / 12) 0.45 x 2
k1 x f cu ⎤
⎢0.5 − 3828 ⎥
k2 = 1 − f cu ⎤
⎢0.5 − 3828 ⎥
⎦ (2.14) 41 Fig. 2.2-7 Characteristic values of k1 and k2 for different concrete grades
o BS 8110 simplified rectangular block (Clause 126.96.36.199) (a) Strain (b) Stress Fig. 2.2-8 Simplified stress block for concrete at ultimate limit state The simplified rectangular block is a special case of BS 8110
stress block when k1 = 0.405 and k2 = 0.45.
Fig. 2.2-6(b) Fig. 2.2-8(b) k1fcubx = (0.45fcu)b·0.9x = 0.405fcubx k2 x = 0 .9 x
= 0.45 x
2 i.e. k1 = 0.405 and k2 = 0.45.
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This note was uploaded on 12/24/2011 for the course CIVL 232 taught by Professor Jskuang during the Spring '06 term at HKUST.
- Spring '06