Chapter 2 Beams (Part 2)
B. Analysis (a) Singly reinforced rectangular section (i) Actual depth of neutral axis (Xa)
Here, B = Width of beam D = Overall depth d = Effective depth
s = C = permissible stress in concrete s = t = st permissible stress in steel In working stress method, actual depth neutral axis is calculated by equating moment of area on both sides of neutral sides.
On compression side, Moment of area = 2
B.X . a
a On tension side = ( m.A st ) (d – X a ) For actual depth of NA mA (d X ) 2
(ii) Critical depth of Neutral Axis (Xc)
Critical depth of neutral axis is that depth at which stresses in concrete and steel are attained to its maximum permissible values at the same time.
From similar triangle
k is called critical neutral axis depth factor
(iii) Moment of resistance: Maximum capacity of taking moment of a given RCC sections is called moment of resistance. 1. For Balanced section (Xa = Xc):
Moment of resistance = compressive force x lever arm or total tensile force × lever arm
lever arm = distance between C and T
Moment of resistance for a balanced section,
or balanced section Design of Beam For a given BM = M Equating BM = Mr M = Q.B.d2
Where, Q = moment of resistance coefficient Area of steel for a balanced section BM = Mr
(For balance section)
2. For under reinforced section (Xa < Xc)
Xa < Xc Ca < s cbc ta = sst From similar triangle
Properties Steel gets its maximum permissible value first, concrete is under stressed. Failure of section will be due to steel. Failure of section is called ductile failure It provides sufficient time before failure this type of section is preferred. 3. For over-reinforcement section (Xa > Xc)
Properties: Concrete gets its maximum permissible value first, steel is under stressed. Failure takes place due to failure of concrete Failure of concrete is sudden (brittle failure) this type of structure should be avoided. (b) Doubly reinforced section If depth and width are restricted.
If this beam has to support a BM more than Mr of the balanced section.
Ther are 2 options.
Provide an over-reinforced section, or Provide a doubly Reinforcement-section Over re-inforced section has many disadvantage like brittle failure so always a doubly reinforced section is a better option.
Properties Steel is provided on both side of NA Permissible stress for compression steel = 1.5 mC where C' = stress in concrete around compression steel. Equivalent area of steel in terms of concrete for compression steel = 1.5 mAsc For tension steel equivalent area = mAst. (i) A ct ual Dep t h of N eut r al A xi s, (X a)
Equating moment of area on both sides of NA ( )( ) ( a ) st
c sc a
a 1.5m – 1 A X d mA d X 2
+ - =
Here, Xa = Actual depth of Neutral axis (ii) Critical Depth of Neutral Axis, (Xc) .d t mc m
(iii) Moment of Resistance (Mr)
C' can be calcualted using similar triangles For a balanced section Xa= Xc Ca = s cbc
• Steel gets its maximum permissible value first, concrete is under stressed.
• Failure of section will be due to steel.
• Failure of section is called ductile failure
• It provides sufficient time before failure this type of section is preferred.
3. For over-reinforcement section (Xa > Xc)
• Concrete gets its maximum permissible value first, steel is under stressed.
• Failure takes place due to failure of concrete
• Failure of concrete is sudden (brittle failure) this type of structure should be avoided.