# PACIFIC ENGR 121 - Shear and Bending Moment Diagrams (32 pages)

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## Shear and Bending Moment Diagrams

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## Shear and Bending Moment Diagrams

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Pages:
32
School:
University of the Pacific
Course:
Engr 121 - Mechanics Of Materials
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Bending Shear and Bending Moment Diagrams covered in Statics and self reviewed in Mechanics of Materials Know terminology simply supported versus fixed cantilevered overhanging Know sign conventions for positive versus negative shear and moments If you can t remember memorize the shear and bending moment diagram for a simply supported beam with a point load A graphical method can be used to quickly visualize shear bending moment diagrams Bending Deformation of a Straight Member Assumptions straight prismatic beam made of homogeneous material cross sectional area of beam is symmetrical with respect to an axis and bending moment is applied perpendicular to axis of symmetry Illustration of Bending for a Bar Distortion of a Rubber Bar due to Bending Assumptions Longitudinal axis does not charge length Cross sections remain plane Any deformation of cross section within its own plane can be neglected Longitudinal normal strain depends on y position and radius of curvature Undeformed Element Deformed Element Normal Strain Distribution Poisson effects that cause y z can be neglected Recall assumption any deformation of cross section within its own plane can be neglected The Flexure Formula From Hooke s law a linear variation in strain corresponds with a linear variation in stress Bending Stress Variation 6 32 Known The ski supports the 180 lb weight of the man The snow loading on its bottom surface is trapezoidal Find Determine the intensity w and draw the shear and bending moment diagrams for the ski V lb x M ft lb x 6 49 Known A steel beam has the cross section shown Find Determine the largest internal moment the beam can resist if the moment is applied as follows a About the z axis b About the y axis Data The allowable stress for the steel is allow 24 ksi 6 54 Known An aluminum strut has a crosssectional area in the form of a cross Find Determine the bending stress acting at a Point A and b Point B Show the stresses acting on volume elements at these points Data The moment M 8 kN m 6 59 Determine the largest bending stress developed in the member if it is subjected to an internal bending moment of M 40 kN m Unsymmetric Bending Moment Applied Along Principal Axis The flexure formula applies for sections that are unsymmetric if the moment is applied about one the principal axes Note that one of the principal axes is along an axis of symmetry and the other principal axis is perpendicular to it My Iz Unsymmetric Bending continued Moment Arbitrarily Applied A moment M applied in the yz plane at an angle has components Mz M cos and My M sin as shown below The flexure formula can be applied to each moment component to obtain the resultant normal stress at any point on the cross section The stresses combine to form the resultant stress distribution below Orientation of Neutral Axis N A 6 102 The box beam is subjected to a bending moment of M 15 kip ft directed as shown Determine the maximum bending stress in the beam and the orientation of the neutral axis y B y C z B y C z A D B C z A D A D 6 108 The 30 mm diameter shaft is subjected to the vertical and horizontal loadings of two pulleys as shown It is supported on two journal bearings at A and B which offer no resistance to axial loading Furthermore the coupling to the motor at C can be assumed not to offer any support to the shaft Determine the maximum bending stress developed in the shaft Composite and Reinforced Concrete Beams Composite beams consisting of layers with fibers or rods strategically placed to increase stiffness and strength can be designed to resist bending The transformed section method can be used to analyze composite beams First step determine transformation factor n by calculating the ratio of the modulus of the stiff material relative to that of the less stiff material Second step use transformation factor n to develop an equivalent beam consisting of the same material Use the flexure formula to determine the stress at each point on the transformed beam Then use the transformation factor n as a multiplier to determine the stress on the actual beam Alternative Approach to transformed section method First step determine alternative transformation factor n by calculating the ratio of the modulus of the less stiff material relative to that of the stiffer material Second step use alternative transformation factor n to develop an equivalent beam consisting of the same material Use the flexure formula to determine the stress at each point on the transformed beam Then use alternative transformation factor n as a multiplier to determine stress on the actual beam The transformed section method can also be used for reinforced concrete beams 6 127 Known A reinforced concrete beam is made using 2 steel reinforcing rods Find Determine the maximum moment M that can be applied to the section Data Allowable tensile strengths st allow 40 ksi and conc allow 3 ksi Elastic moduli Est 29 103 ksi and Econc 3 8 103 ksi Assumption the concrete cannot support a tensile stress Stress Concentrations Notches holes or abrupt changes in the outer dimensions of a member s cross section can lead to stress concentrations at the discontinuity when a bending moment is applied to the member Inelastic Bending The flexure equation is valid up to the yield point and before this point is reached the material behaves in a linear elastic manner After yield elasto plastic behavior can be assumed to determine normal stresses acting in a member For both elastic and plastic analyses three conditions must be met Linear Normal Strain Distribution Resultant Force Equals Zero Resultant Moment must be equivalent to moment caused by stress distribution about the neutral axis 6 160 Determine the plastic section modulus shape factor of the beam s cross section 6 167 Determine the plastic moment MP that can be supported by a beam having the cross section shown Y 30 ksi 6 106 Known The resultant moment acting on the aluminum strut has a magnitude of M 520 N m and is directed as shown Find Determine the bending stress at points A and B The location y of the centroid C of the strut s cross sectional area must be determined Also specify the orientation of the neutral axis 6 126 Known A composite beam is made of A36 steel A bonded to C83400 red brass B and has the cross section shown Find Determine the maximum moment M that can be applied to the beam Data allow st 180 MPa allow br 60 MPa

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