Concrete Beam Deflection Checker

Design checks

Calculation

1. 1Concrete modulus and modulus of ruptureACI318 Eq. (8-5)
   E_c = 57000\sqrt{f'_c}, \quad f_r = 7.5\sqrt{f'_c}
   E_c = 3605, \quad f_r = 474
   E_c=3605 ksi
2. 2Cracking momentACI318 Eq. (9-9)
   M_{cr} = \frac{f_r I_g}{y_t}
   M_{cr} = 45.5
   M_{cr}=45.5 kip·ft
3. 3Cracked transformed section
   \frac{b c^2}{2} = n A_s (d - c), \quad I_{cr} = \frac{b c^3}{3} + n A_s (d-c)^2
   c = 7.50, \quad I_{cr} = 6418 \text{ in}^4 \quad (I_g = 13824 \text{ in}^4)
   I_{cr}=6418

   Modular ratio n = 8.04. The cracked section retains 46% of the gross stiffness.

4. 4Effective stiffness at each load levelACI318 Eq. (9-8)
   I_e = \left(\frac{M_{cr}}{M_a}\right)^3 I_g + \left[1 - \left(\frac{M_{cr}}{M_a}\right)^3\right] I_{cr} \le I_g
   I_{e,D} = 7502, \quad I_{e,D+L} = 6652 \text{ in}^4
   I_{e,D+L}=6652

   The section cracks under the full service load, so the reduced stiffness applies.

5. 5Immediate deflection under live loadACI318 9.5.2
   \Delta_L = \Delta_{D+L} - \Delta_D
   \Delta_L = 0.623 - 0.331 = 0.291
   \Delta_L=0.291 in

   Each case is evaluated with its own effective stiffness, then subtracted — not evaluated once with a shared one.

6. 6Long-term multiplierACI318 Eq. (9-11)
   \lambda_\Delta = \frac{\xi}{1 + 50\rho'}
   \lambda_\Delta = \frac{2}{1 + 50 \times 0} = 2
   \lambda_\Delta=2.000

   With no compression steel the multiplier is the time factor itself.

7. 7Deflection checked for this member: deflection occurring after the elements were attachedACI318 Table 9.5(b)
   \Delta_{check} = \lambda_\Delta \Delta_{sus} + \left(\Delta_{D+L} - \Delta_{sus}\right)
   \Delta_{check} = 1.031 \quad \text{vs } \frac{L}{480} = 0.600
   \Delta_{check}=1.031 in

Assumptions

• Simply supported single span, rectangular section, uniform load over the whole span.
• Normalweight concrete. The modulus and the modulus of rupture are the normalweight expressions.
• The gross section ignores the reinforcement, which is the usual simplification for a deflection check.
• Loads are service loads, unfactored. Deflection is a serviceability question.