To determine The required dimension d of concrete beam and the maximum allowable internal moment M applied to beam. Explanation Given information: The diameter of rod is 20 mm. The allowable compressive stress for concrete ( σ allow ) con is 12.5 MPa The allowable compressive stress for steel ( σ allow ) st is 220 MPa The modulus of elasticity for concrete E c o n is 25 GPa . The modulus of elasticity for steel E st is 200 GPa . Calculation: Show the cross section of the concrete beam as shown in Figure 1. Find the transformation factor ( n ) as follows: n = E st E con (1) Here, E st is modulus of elasticity for steel and E c o n is modulus of elasticity for concrete. Substitute 200 GPa for E st and 25 GPa for E c o n in Equation (1). n = 200 25 = 8 Both concrete and steel achieve their allowable stress simultaneously and its required. Show the allowable bending stress for concrete as follows: ( σ allow ) con = M c con I (2) Here, M is the moment, I is the moment of inertia, and c is the centroid distance. Substitute 12.5 MPa for ( σ allow ) con in Equation (2). 12.5 × 10 6 = M c con I M = 12.5 × 10 6 ( I c con ) (3) Show the allowable bending stress for steel as follows: ( σ allow ) steel = n ( M c s t I ) (4) Substitute 220 MPa for ( σ allow ) st in Equation (4). 220 × 10 6 = 8 [ M ( d − c c o n ) I ] M = 27.5 × 10 6 ( 1 d − c c o n ) (5) Equate the Equation (3) and (5). 12.5 × 10 6 ( I c con ) = 27.5 × 10 6 ( 1 d − c c o n ) c c o n = 0.3125 d (6) Section properties: Find the area of steel bars ( A s t ) as follows: A s t = 3 [ π 4 r 2 ] (7) Substitute 20 mm for r in Equation (7). A s t = 3 [ π 4 20 2 ] = 300 π mm 2 ( 1 m 2 10 6 mm 2 ) = 0.3 × 10 − 3 π m 2 Find the area of concrete bars ( A c o n ) as follows: ( A c o n ) t = n A s (8) Here, A s is area of stress. Substitute 8 for n and 0.3 × 10 − 3 π m 2 for A s in Equation (8)

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Chapter 6.9, Problem 6.131P

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The required dimension d of concrete beam and the maximum allowable internal moment M applied to beam.

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Chapter 6.9, Problem 6.130P

Chapter 6.9, Problem 6.132P

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