Stresses, Strains and Deflections of Steel Beams in Pure Bending Experiment # 3 Stresses, Strains and Deflections of Steel Beams in Pure Bending

Objectives : Compare the theoretical strain predictions with the strain measurements obtained from electrical gage method. 2) To experimentally determine the location of the neutral axis in the beam cross section and compare it with the theoretical predicted value. 3) To examine the validity of the assumption made in flexural analysis of beams that cross sections remain plane during bending. 4) To compare the theoretically predicted deflections with measured experimental values.

Scope of the Experiment : ■ For Objectives 1 to 3 : Provide a Pure Bending Condition Measure the strains along a particular surface with strain gages Calculate the strain distribution along that surface from theory Compare

Pure Bending Condition: a ■ In Pure Bending there will be no transverse shear force along the beam section considered. ■ How can we provide this condition? P P + _ V P -P x + M P.a x

Pure Bending Deformation : ■ Assumptions: - Longitudinal straight lines become curved. - Vertical transverse lines remain straight (but undergo rotations). - Top part is compressed. - Bottom part is elongated. - Member is divided by a neutral surface. Before Deformation After Deformation

Neutral Axis : ■ A surface within a member under bending which is not deformed. ■ Longitudinal fibers do not undergo any changes in length. ■ For all points on the neutral surface normal strain is equal to zero. ■ Neutral Axis passes through the geometric centroid of the section. (If the material is homogenous and stress-strain relationship is linear.)

Bending Formulas: Neutral axis Linear Distribution of Stress Linear Distribution of Strain

Maximum Deflection: y ■ The differential equation of the deflection curve of a beam is: ■ After integrating and also applying the boundary conditions the equation for the deflection is: a is the distance at which P is away from the supports ■ And the maximum deflection (at x=L/2) is:

The Loads in the Experiment : ■ In this test three levels of loading are applied on the beam: P=2000 lbs P=4000 lbs P=6000 lbs How much does a Ford Focus weigh ? ■ How are such large loads going to be applied on the beam?

Experiment Equipment: O P1 P2

Experiment Procedure: ■ The Experiment is conducted in two phases :  First Phase : three stages of Loading P= 0  2000  4000  6000  Second Phase: three stages of Unloading P= 6000  4000  2000  0 ■ At every stage two types of measurement is performed : - Deflection : using deflection gage - Strain : using electrical strain gages Deflection gage Electrical gage Mechanical gage holes ■ After measuring the strains and processing the raw data calculate the corresponding stress for each location:

Stresses Calculated From Strains Measured by Electrical Case Data Analysis: y (in) # 8.08 6.69 5.93 5.12 4.50 3.77 3.00 2.18 1.40 0.40 1 2 3 4 5 6 7 8 9 10 Stress (σ) y (depth) Stresses Calculated From Strains Measured by Electrical Case ■ Here for each loading condition the validity of the assumption that the cross sections remain plane during bending can be examined.

Stresses Calculated From the Theory (Formula) Data Analysis: y (depth) y (depth) z z′ Stress (σ) y´ z´ Stresses Calculated From the Theory (Formula) d (in) bf (in) A (in2) Iz′ (in4) S18×18.4 8.00 4.00 5.41 57.6 E = 29,000,000 psi

Data Analysis: ■ Finally, compare the theoretical and experimental Stress (σ) y (depth) electrical theoretical ■ For each loading condition plot all the two lines in the same graph and then : 1) Compare the two graphs together. 2) Compare the maximum tensile stresses together. (Use Table 3-3 of the manual) 3) Compare the locations of N.A. obtained by all the graphs. ■ Finally, compare the theoretical and experimental mid-span deflections. For this purpose use Table 3-4 of the manual.

Thank you ! Question?