111 CEE Design Exercise 6
Winter 2012
Due March 19, 2012
Work independently!!!
Consider the reinforced concrete beam shown below:
b
50 ft
P = 6K
d
100 ft
2.5 in
As
Beam Cross-Section
The beam is to be designed to resist a maximum moment of M D 300 K-ft at least cost.
For simplicity, total cost can be considered to be only the material cost of the concrete
and the steel reinforcing bars. The reinforcing bars are to be made of A615 Grade 60
steel at $500/ton ($0.85/ft2 per foot of beam) and 4,000 psi normal weight concrete at
$25/yd3 ($0.93/ft2 per foot of beam). Let b = the beam width in inches, d = depth of the
beam to the centroid of reinforcing steel, and AS = the total cross-sectional area of
reinforcing steel.
Based on these values, the total cost of the beam is given by:
d 2.5 b
C $85 AS $95
12 12
The resisting moment of the beam, M R ,
AS f y
M R AS f y d
1.7 fCb
where
a strength reduction factor that reduces the theoretical strength of the
beam to account for minor adverse field conditions, equal to 0.90 for bending.
f y yield strength of steel reinforcement (60,000 psi)
fC compressive strength of concrete (4,000 psi)
must be greater than or equal to the maximum moment of M D 300 K-ft , or, converting
to kip-ft and simplifying,
4.5 AS d 39.706
( ) AS 2
300
b
The design is constrained to meet the American Concrete Institute (ACI) code
requirements that:
1. Minimum amount of steel reinforcement to ensure that the beam is stronger when
acting as a reinforced beam than when acting as a plain concrete section (utilizing
the strength of concrete in tension), avoiding the possible sudden failure of a plain
concrete section. This limit is given by:
AS
0.00333
bd
2. Alternatively, the steel reinforcement must be less than that which would cause
sudden compressive failure of the concrete at the top of the beam, rather than a
slow excessive deflection caused by the failure of the steel. This limit is given by:
AS
0.0214
bd
3. In order to protect against buckling, there is a minimum ration of the depth to
width of the beam. This limit is given by:
b1
d3
Design Exercise:
1. UsingExcelSolverroutines,determinetheoptimaldesignofthereinforced
concretebeamundertheconditionsspecified.
2. FormulatetheLagrangianforthisproblemandsolvetheresultingsystemof
nonlinearequationsusingtheNewtonRaphsonalgorithm.Compareyour
resultswiththoseobtainedusingExcelSolver
Submission should be in Report format with complete problem formulation,
solution, and documentation (e.g., printouts of Excel spreadsheets containing
formulations and answers obtained.