Learn : Three types of problems in singly reinforced beam working stress method :To determine the moment of resistance of the given section,To determine the stresses developed in concrete and steel under given loading,To design the section for given loading
Problems in singly reinforced beam working stress method
There are three types of problems in singly reinforced beam :
- To determine the moment of resistance of the given section.
- To determine the actual stresses developed in steel and concrete under given loading.
- To design the section for given loading.
Type 1:To determine the moment of resistance of the given section
Data Given:
(i) Dimensions, b and d of the section.
(ii) Area of steel reinforcement in tension (A_{st})
(iii) Material i.e., grade of concrete and steel.
Procedure:
(i) For the given grade of concrete and steel, determine the permissible stresses i.e., σcbc and σstfrom the Tables 2.1 and 2.2.
(ii) Calculate modular ratio m.
\[ m=\frac{280}{3\sigma _{cbc}}\]
(iii) Determine critical neutral axis (n)
\[\frac{m.\sigma _{cbc}}{\sigma _{st}}=\frac{n _{c}}{d-n_{c}}\]
(iv) Determine actual neutral axis (n)
\[b.\frac{n^{2}}{2}=m.A_{st}(d-n)\]
(v) Compare n and nc
(i) If n= nc, the section is balanced and the moment of resistance can be calculated by any of the following equation
\[M_{r}=\frac{1}{2}\sigma_{cbc}b.n_{c}\left ( d-\frac{n}{3} \right )\]
or \[ m_{r}=\sigma_{st}.A_{st}\left ( d-\frac{n}{3} \right )\]
(ii) If n< nc, the section is under reinforced and the moment of resistance is calculated as
\[m_{r}=\sigma_{st}.A_{st}\left ( d-\frac{n}{3} \right )\]
(iii) If n> nc, the section is over reinforced and
\[ m_{r}=\frac{1}{2}\sigma_{cbc}.b.n\left ( d-\frac{n}{3} \right )\]
Note : Sometimes it is required to find out the safe load (w) which the beam can carry. For this, the maximum bending moment due to the loads is calculated and equated to the moment of resistance of the section.
The maximum bending moment values for some beams are written below:
- Simply supported beam, for (u.d.l.) = \[\frac{wl^{2}}{8} (Sagging)\]
- Cantilever beam, for (u.d.l.) =\[ \frac{wl^{2}}{2} (Hogging)\]
where l is the effective span of the beam.
Type-II : To determine the stresses developed in concrete and steel under given loading
Given Data :
(i) Dimensions of beam (b and d)
(ii) Area of steel, Ast
(iii) Material i.e., grade of concrete and steel.
(iv) External loads or bending moment.
Procedure :
(i) Determine the permissible stresses from Tables 2.1 and 2.2.
(ii) Calculate actual neutral axis
\[b.\frac{n^{2}}{2}=m.A_{st)}(d-n)\]
(iii) Calculate maximum bending moment (M) due to loads (external loads as well as self-weight) if not given.
(iv) Calculate stresses by equating the maximum bending moment to the moment of resistance
\[ M=M_{r}\]
\[ =\sigma_{st}.A_{st}\left ( d-\frac{n}{3} \right )\]
(v) Knowing σst from above stepσc or actual stress in concrete is calculated as
\[\frac{m.\sigma_{c}}{\sigma_{st}}=\frac{n}{d-n}\]
Type-III : To design the section for given loading
Given data :
(i) External loads or bending moment.
(ii) Material-grade of concrete and steel.
(iii) Span of the beam.
Procedure :
- Determine the permissible stresses for materials from Table 2.1 and 2.2.
- Determine design constant k, j and R.
- Assume suitable value of b/d ratio and calculate the moment of resistance using
\[ M_{r}=Rbd^{2}\left [ \frac{b}{d}varies from \frac{1}{2}to\frac{2}{3} \right ]\]
- For the given loads and approximate self weight, compute the maximum bending (M).
- Determine d by equating M and M_{r}
\[ M_{r}=M\]
\[ Rbd^{2}=M\]
\[d=\sqrt{\frac{M}{R.b}}\]
- Calculate b from assumed \[\frac{b}{d}\] ratio.
- Calculate (Ast) area of steel as follows:
\[M=\sigma_{st}.A_{st}jd\]
\[ A_{st}=\frac{M}{\sigma_{st}jd}\]
- Provide suitable number of bars for the required area of steel, Ast
TABLE 2.1. Permissible Stresses in Concrete (Refer to Table 21, IS 456)
Grade of Concrete | Permissible Stress in Compression | Permissible Stress in Bond (Average) | ||
Bending σcbc (N/mm^{2}) | Direct σcc (N/mm^{2}) | For Plain Bars in Tensionτbd (N/mm^{2}) | For HYSD Bars (N/mm^{2}) | |
– | – | – | – | – |
M15 | 5.0 | 4.0 | 0.6 | 0.96 |
M20 | 7.0 | 5.0 | 0.8 | 1.28 |
M25 | 8.5 | 6.0 | 0.9 | 1.44 |
M30 | 10.0 | 8.0 | 1.0 | 1.6 |
M35 | 11.5 | 9.0 | 1.1 | 1.76 |
M40 | 13.0 | 10.0 | 1.2 | 1.92 |
M45 | 14.5 | 11.0 | 1.3 | 2.08 |
M50 | 16.0 | 12.0 | 1.4 | 2.24 |
Notes:
(i) The bond stress given above for tension is increased by 25% for bars in compression.
(ii) The bond stress for plain bars is increased by 60% for deformed bars.
TABLE 2.2. Permissible Stress in Steel Reinforcement (Refer Table 22, IS 456)
S. No. | Type of Stress in Steel Reinforcement | Permissible Stresses in N/mm^{2} | High yield strength deformed bar (HYSD) conforming to IS 1786 (Grade Fe 415) | |
Mild steel bars conforming to Grade I of IS 432 (Part I) | Medium tensile steel conforming to IS 432 (Part I) | |||
1. | Tension ( σ_{st} or σ_{sv} )
(i) Upto and including 20mm (ii) Over 20mm |
140
130 |
Half the guaranted yield stress subject to maximum of 190
190 |
230
230 |
2. | Compression in column bars σ_{sc} | 130 | 130 | 190 |
3. | Compression in bars in beam or slab when compressive resistance of concrete is taken into account | The calculated compressive stresses in the surrounding concrete multiplied by 1.5 times the modular ratio or s_{sc} whichever is lower=1.5m_{c} or s_{sc} |