Determine the bending resistance of the given composite crosssection. Perform
a) elastic analysis,
b) plastic analysis.
Compressive strength of concrete is fc =13.33 N/mm2, elastic modulus is Ec = 18.3 GPa. Section properties of the steel cross section are: A = 363.5 cm2, Iz= 481200 cm4, material properties are: fy =235 N/mm2, Es= 210 GPa. (In plastic analysis assume perfectly plastic material. The thicknesses of the web and the flanges are t w = 18.5 mm, t f = 35 mm.)
Solve Problem
Problem a) Bending resistance, MR,e [kNm]= Problem b) Bending resistance, MR,p [kNm]=Solve
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Steps
Problem a) Step 1. Calculate the section properties. The inhomogeneous cross section is replaced by an equivalent homogeneous one. The reference material is the concrete, the ratio of the elastic moduli is Tensile stiffness, position of the centroid and bending stiffness of the replacement homogeneous cross section are: Step 2. Determine elastic moment resistance. Failure occurs when the maximum stress of concrete reaches the its compressive strength, or the maximum stress of steel reaches its yield strength. The resistance can be expressed from the maximum stresses: The maximum allowed stress is reached in the steel first, the elastic moment resistance is Problem b) Step 1. Assume the location of the neutral axis. Draw the normal stress diagram. At plastic failure stresses reach the strength in each points of the cross section. Let us assume that stresses change sign in the web of the steel girder. Step 2. Calculate the location of the neutral axis. Check the assumption. The location of the neutral axis can be determined from the normal force equilibrium of the cross section. To check the assumption we compare the stress resultants of the cross sectional parts: which means that the web must be partially tensioned, our assumption is true, the neutral axis passes through the web. Step 3. Determine the plastic moment resistance Plastic moment resistance is determined from a moment equilibrium equation. Let us write the moment equilibrium about the neutral axis:Step by step
Check section properties
Check elastic resistance
Check diagram
Check neutral axis
Check plastic resistance
Results
Problem a) The inhomogeneous cross section is replaced by an equivalent homogeneous one. The reference material is the concrete, the ratio of the elastic moduli is Tensile stiffness, position of the centroid and bending stiffness of the replacement homogeneous cross section are: Failure occurs when the maximum stress of concrete reaches the its compressive strength, or the maximum stress of steel reaches its yield strength. The resistance can be expressed from the maximum stresses: The maximum allowed stress is reached in the steel first, the elastic moment resistance is Problem b) At plastic failure stresses reach the strength in each points of the cross section. Let us assume that stresses change sign in the web of the steel girder. The location of the neutral axis can be determined from the normal force equilibrium of the cross section. To check the assumption we compare the stress resultants of the cross sectional parts: which means that the web must be partially tensioned, our assumption is true, the neutral axis passes through the web.Worked out results
Plastic moment resistance is determined from a moment equilibrium equation. Let us write the moment equilibrium about the neutral axis: