MINI PROJECT REPORT

Verification of Bending Stress and Shear Stress Distribution across the Depth for Slender Beam

NAME: Siddharth R Karde MIS: 111201023

BRANCH: Civil Engineering

1) PHYSICS OF THE PROBLEM : A beam of length l m and cross-section b*d is subjected a uniformly distributed load. When a beam having an arbitrary cross section is subjected to a transverse loads the beam will bend. In addition to bending the other effects such as twisting and buckling may occur, and to investigate a problem that includes all the combined effects of bending, twisting and buckling could become a complicated one. Thus we are interested to investigate the bending effects alone, in order to do so, we have to put certain constraints on the geometry of the beam and the manner of loading.

INTRODUCTIONA beam deforms and stresses develop inside it when a transverse load is applied on it. In a horizontal beam supported at the ends and loaded uniformly, the material at the over-side of the beam is compressed while the material at the underside is stretched. There are two forms of internal stresses caused by lateral loads: Shear stressparallel to the lateral loading plus complementary shear stress on planes perpendicular to the load direction; Directcompressive stressin the upper region of the beam, and directtensile stressin the lower region of the beam.These last two forces form acoupleormomentas they are equal in magnitude and opposite in direction. Thisbending momentresists the sagging deformation characteristic of a beam experiencing bending. The stress distribution in a beam can be predicted quite accurately even when some simplifying assumptions are used.

Assumptions: theory of bendingThe constraints put on the geometry would form theassumptions:1. Beam is initiallystraight, and has aconstant cross-section.2. Beam is made ofhomogeneous materialand the beam has alongitudinal plane of symmetry.3. Resultant of the applied loads lies in the plane of symmetry.4. The geometry of the overallmemberis such that bending not buckling is the primary cause of failure.5. Elastic limit is nowhere exceeded andE'is same in tension and compression.6. Plane cross - sections remains plane before and after bending.

GOVERNING DIFFERENTIAL EQATIONFOR BENDING STRESSIn theEulerBernoulli theoryof slender beams, a major assumption is that 'plane sections remain plane'. In otherwords, any deformation due to shear across the section is not accounted for (no shear deformation). Also, this linear distribution is only applicable if the maximum stress is less than theyield stressof the material. For stresses that exceed yield, refer to articleplastic bending. At yield, the maximum stress experienced in the section (at the furthest points from theneutral axisof the beam) is defined as theflexural strength.The Euler-Bernoulli equation for the quasistatic bending of slender, isotropic, homogeneous beams of constant cross-section under an applied transverse loadW(X)is Where, : Youngs modulus, : Area moment of inertiaof the cross-section, : Deflection of the neutral axis of the beam.After a solution for the displacement of the beam has been obtained, the bending moment () and shear force () in the beam can be calculated using the relations

FOR SHEAR STRESSShear Stress Distribution is given by:

Where:V(x) : Shear force carried by the section, found from the shear force diagram.I : Second moment of area.t(y) : Sectional width at the distance y from the N.A. A is the top (or bottom) portion of the members cross-sectional area, defined from the section where t(y) is measured, and y is the distance to the centroid of A, measured from the Neutral Axis.

BOUNDRY CONDITIONS The beam equation contains a fourth-order derivative in. To find a unique solutionwe need four boundary conditions. The boundary conditions usually modelsupports, but they can also model point loads, distributed loads and moments. Thesupportor displacement boundary conditions are used to fix values of displacement () and rotations () on the boundary. X : DIFLECTION: SLOPE

X=00

X=L0

ELEMENTS USED2 dimensional beam element