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THIS IS ALL ABOUT THE THESIS
Str. Mech. Lab
Job 05
09-CIV-172
TO STUDY THE BENDING BEHAVIOR OF Z SECTION Objective: To confirm the theory that loading a beam along principal axis always produces symmetrical bending. Apparatus: i. ii. iii. iv.
Cantilever beam (Z section) with rotating arrangement Dial gauges (two in number each having L.C. = 0.001”) Loading pans and weights Steel tape
Related theory: Axis of symmetry: If an axis divides a section into two parts in such a manner that the parts are mirror images of each other, such an axis is called axis of symmetry. Symmetrical sections: A section that has at least one axis of symmetry is called symmetrical section. Examples are: i. ii. iii.
W section; has two axes of symmetry T section; has one axis of symmetric so called as singly symmetric Channel section; is also singly symmetric
Unsymmetrical sections: A section which doesn’t have any axis of symmetry is called unsymmetrical section. Examples are: i. ii.
Z section L section (angle section)
Principal axes: Principal axes are set of rectangular axes about which moments of inertia (i.e. 2 nd moments of area) are either maximum or minimum but the product moment of inertia is zero about the same axes. It is a geometrical property. Axes of symmetry are always the principal axes but the converse is not necessarily true. Orientation of principal axes: In case of a symmetrical section the axes of symmetry are the principal axes but for an unsymmetrical section we’ve to determine their orientation by using the following relation.
1|Page 14 Mar 2012
(performance date)
Str. Mech. Lab
Job 05
09-CIV-172
Note: Here α is measured positive in C.C.W direction and vice versa. Here x is longitudinal axis while y and z axes are lying on the cross-sectional plane. Symmetrical bending: When loading and deflection are parallel or along the principal axis, the bending is said to be symmetrical bending or uniaxial bending. Unsymmetrical bending: When deflection occurs in more than one plane, the bending is termed as unsymmetrical bending. Unsymmetrical bending may occur in symmetrical as well as unsymmetrical sections.
Notes: a) If load is passing through shear center and acting on principal axis or parallel to it, it will not cause any torsion and twist but if it is not passing through the shear center although acting on principal axis or parallel to it, it will cause torsion and twist in the member. b) If member is set so that loading applied on it will cause only uniaxial bending then the section is economical, have aesthetics as well as encountering less stresses.
2|Page 14 Mar 2012
(performance date)
Str. Mech. Lab
Job 05
09-CIV-172
Bending theory: Assumptions: 1. 2. 3. 4. 5.
The plane sections of the beam remains plane. The material in the beam is homogenous and obeys Hook’s law. The moduli of elasticity for tension and compression are equal. The beam is initially straight and of constant cross section. The plane of loading must contain a principal axis of the beam cross section and the loads must be perpendicular to the longitudinal axis of the beam.
Bending theory is the relationship:
M I σ y
= = = =
E R
= =
bending moment (generally in N-mm) Second moment of area (mm4) bending stress (also called flexural stress) in N/mm4 distance of fiber form N.A. of cross section to the location at which bending stress is to be calculated w.r.t. principal axis, measured in mm modulus of elasticity (N/mm2) radius of curvature in mm
Experimental performance settings: i.
Web vertical:
Web of the Z section will be vertical. Load will be applied vertically. Horizontal and vertical deflections will be measured, if both the deflections are present then there will be bi-axial bending and if only one deflection is there the Z section will be bending uniaxial. ii.
Web rotated by α:
The web will be rotated by α degree so that one of the principal axis of Z section becomes horizontal and the other vertical.
Comments: Referring to the results tabulated in ‘Comparison Table’ on page # 06 of this job; i.
ii.
The magnitude of experimentally measured both of the horizontal and vertical deflections in setting # 01 and the vertical deflections in setting # 02 are higher than the theoretically calculated ones (the deviation is obvious from G-01, G-02 and G-04, on page 06 and 07). Also, experimentally, we’ve got (minor) horizontal deflection readings though these should be zero according to the theory (the deviation is obvious from G-03, on page # 07).
Both of these deviations from the ideal cases are referred to following reasons: i.
ii.
Bending theory is not fully applicable here since the plane of loading neither contains nor is parallel to any principal axes of the beam cross section. However the loads are resolved into rectangular components along the vertical and horizontal axes in setting # 01 and along the principal axes in setting # 02. That’s why the results are not accurate. In the bending theory it’s assumed that E compression= E tension. It’s valid for ideal case. Though it’s true for steel but not 100% true for the steel material we’ve conducted test on. That’s why our results are deviated from the theoretical ones since the theoretical value of E is used in computing the experimental deflections. However ignoring the minor deviation of results from the theoretical ones, we’ve concluded that loading a beam along principal axis always produces symmetrical bending.
