The tensile-reinforced rectangular beam shown below is made using steel with fy-

Question

The tensile-reinforced rectangular beam shown below is made using steel with fy-60 ksi and modulus of elasticity, E-29000 ksi. A perfectly plastic response after yielding can be assumed for steel reinforcement. The concrete used has the stress-strain curve shown below with limit of elastic response at a strain of 0.0005, maximum stress at 0.002, and ultimate strain of 0.003. The concrete elastic modulus is E 4230 ksi, and modulus of rupture is f 556 psi Based on the above given information, plot a curve relating applied moment to unit curvature at a section subjected to flexural cracking. Label points corresponding to: 2. 3· 4· 5. 6· First cracking (for uncracked and cracked transformed sections) Limit of concrete elastic response Ec=0.001 Ec-0.00 15 First yielding of steel Ec-0.002 Ec 0.003 Show all your calculations f,,-5.5 ksi- f4.50 ksi 12" 24" 21 4#9 e 4230 ksiI 0.0005 0.002 0.003 f,-556 psi

Explanation / Answer

Ans.

x = Depth of Neutral axis
b = breadth of section
d = effective depth of section
The depth of neutral axis can be obtained by considering the equilibrium of the normal forces , that is,
Resultant force of compression = average stress X area
= 0.36 fck bx
Resultant force of tension = 0.87 fy At
Force of compression should be equal to force of tension,
0.36 fck bx = 0.87 fy At
The distance between the lines of action of two forces C & T is called the lever arm and is denoted by z.
Lever arm z = d – 0.42 x
z = d – 0.42
z = d –(fy At/fck b)
Moment of resistance with respect to concrete = compressive force x lever arm
= 0.36 fck b x z
Moment of resistance with respect to steel = tensile force x lever arm
= 0.87 fy At z
Maximum depth of neutral axis

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