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Civil Engineering Material

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1. Objectives

To determine the bending stress distribution along the flange of a cantilever beam.

To determine the shear stress distribution across the web of a cantilever beam.

To determine principal stresses and principal plane from the strain rosetteЎЇs reading.

2. Theory

A point load F at the free-end of a cantilever beam as shown below will cause the beam to bend, producing a linear bending moment and a constant shear force along the beam. In the grider with thin web, the bending moment will be mainly resisted by the bending stress in the flanges, while the shear force will be resisted by the shear stress in the web of the beam.

Cantilever Beam (Sideview)

(Cross-section view)

The distribution of bending stress and shear stress across the cross section is shown below respectively.

Bending stress distribution Shear stress distribution

In this experiment, strain gauges and strain rosettes will be used to obtain the longitudinal strains in the flanges and strains in the web of a girder, respectively. The principal stresses at the web of the girder will be determined based on the reading of the strain rosettes. Bending and shear stresses obtained from experiment will be compared with the calculations based on the theory. Detailed theorys are shown blow.

Stress Distribution

A tip loaded cantilever beam of any constant cross-section is as shown below.

An elemental section of the beam of length dx is shown below.

Consider the equilibrium of an element of cross-sectional area dA above an imaginary cutting plane ABCD.

From the basic flexure formula , ,where ¦Ð¢ = longitudinal stress due to bending at some distance y from the neutral axis, and I is the second moment of area of the cross section.

At section 1-1, the bending moment = M

At section 2-2, the bending moment = M+dM

At section 1-1, longitudinal force on the element

At section 2-2, longitudinal force on the element

Hence, there is an out of balance force of magnitude on

the elemental area dA. This must be balanced by a shear force acting along the face of the imaginary cutting plane.

Shear stress ¦Ð£ in this cutting plane = =

Since (the applied shear force), ¦Ð£ =

This shear formula gives the value of the shear stress at any distance a from the neutral axis of the beam.

The girder used in the shear web apparatus has an idealized cross-sectional as shown in the figure(cantilever beam).

The prime constituents of girder used in the shear web apparatus are:

a) Longitudinal flange members

b) Thin shear webs

The application of the flexure formula to a beam with a thin web will yield the bending stress as shown in the figure(bending stress distribution). Hence it is feasible to assume that the bending stress produces a uniform compressive stress in the top flange and a uniform tensile stress in the bottom flange and that the web resists no bending stress, i.e. offers no moment of resistance.

If shear formula is applied tot his beam, the shear stress distribution across its depth is shown in the figure(shear stress distribution). Hence, it is quite feasible to assume that the shear stress is reacted completely in the web, and that is magnitude is uniform along the depth of the web.

The bending moment associated with moving the point of application of the applied force produces a compressive stress in the top flange and a tensile stress in the bottom flange,


where P = longitudinal flange force at any section

AF = flange cross-sectional area

M = bending moment at the corresponding



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