Shear Force & Bending Moment This section covers one of the most practical concepts in Strength of Materials — Shear Force and Bending Moment. You’ll understand how beams carry loads, how internal forces develop, and how to analyze bending behavior — essential for designing bridges, shafts, machine frames, and structural supports. 1 / 20 In bending of beams, the top fibers are under Tension Compression Shear Torsion In a simply supported beam loaded downward, top fibers shorten (compression) while bottom fibers elongate (tension).This is fundamental for stress distribution in bending. 2 / 20 The bending stress at a section is directly proportional to Shear force Moment of inertia Bending moment Modulus of elasticity From flexure formula: σ=My​/I.So, bending stress increases directly with bending moment and distance from neutral axis. 3 / 20 In a beam under pure bending, the neutral axis Passes through the centroid Lies above centroid Lies below centroid Shifts to tension side Under pure bending (no shear), neutral axis passes through the centroid of the cross-section, where bending stress is zero. 4 / 20 The slope of the bending moment diagram at any point equals the Load intensity Reaction Shear force None From differential relation, dM/dx=VThus, the slope of bending moment curve = shear force at that section. 5 / 20 For a simply supported beam with UDL, the SFD is Rectangular Parabolic Linear Triangular In a UDL case, shear decreases linearly from maximum positive at one support to maximum negative at the other, forming a triangle shape. 6 / 20 The SFD (Shear Force Diagram) for a cantilever with a UDL is Rectangular Triangular Parabolic Trapezoidal For a UDL, shear varies linearly (V = –w × x).Hence, the shear force diagram forms a triangle, starting maximum at fixed end and zero at free end. 7 / 20 The bending moment at the point of zero shear force is Minimum Zero Constant Maximum Where shear force is zero, bending moment reaches a maximum or minimum value.This point is critical in beam design since bending stress peaks there. 8 / 20 In a simply supported beam with a central load, shear force just to the left and right of the load are Equal and opposite Same and positive Zero Unequal The load acts exactly at mid-span, causing shear force to shift from +ve to –ve side with equal magnitude.This indicates zero shear at that point. 9 / 20 The maximum bending moment for a simply supported beam with a central point load W is WL/4 WL/8 WL/2 W/L For simply supported beam,Maximum moment at midspan = WL/4.This formula is used frequently in basic bending stress calculations. 10 / 20 A beam supported at both ends and loaded at the center behaves as Cantilever Overhanging beam Simply supported beam Fixed beam A beam with two supports and load at the center is simply supported.Maximum bending moment occurs at mid-span (M = WL/4). 11 / 20 The bending moment diagram for a cantilever beam with UDL is Triangular Rectangular Parabolic Trapezoidal For a cantilever beam with uniformly distributed load (w):Bending moment at distance x, M = –(w x²)/2 → a parabolic curve.Maximum moment at the fixed end = –(wL²)/2. 12 / 20 In a shear force diagram, a sudden vertical jump indicates A change in bending moment A point load A uniform load No load A point load creates a sudden change (jump) in shear force value at that section, as load acts at a single point. 13 / 20 The relationship between load (w), shear force (V), and bending moment (M) is V = dM/dx and w = dV/dx M = dV/dx and V = dw/dx M = w × x and V = M/x None Fundamental beam relations: dMdx=V\frac{dM}{dx} = VdxdM​=V dVdx=−w\frac{dV}{dx} = -wdxdV​=−wThese differential equations form the base for shear and bending diagrams. 14 / 20 For a cantilever beam carrying a point load at the free end, the maximum bending moment occurs at the Free end Fixed end Middle Both ends For a cantilever of length L with load W:Maximum bending moment = WL at the fixed end, since it resists both load and moment. 15 / 20 Bending moment at supports of a simply supported beam is Zero Negative Equal to UDL Maximum At supports of a simply supported beam, no moment can be developed since the beam can freely rotate — hence M = 0. 16 / 20 Shear force at the midpoint of a simply supported beam with a uniform load is Zero Maximum positive Maximum negative Equal to reaction For a symmetrically loaded beam, shear force changes sign at the midpoint — meaning zero shear force and maximum bending moment. 17 / 20 In a simply supported beam with a uniformly distributed load, the bending moment is maximum at Support Free end Mid-span Quarter span For a UDL over the entire span (L):Mmax = wL²/8 at the mid-span.This point experiences maximum bending stress — crucial for beam section design. 18 / 20 The point where bending moment changes its sign is called Neutral point Stress point Point of inflection Elastic point At the point of contraflexure (inflection), the bending moment changes sign — from positive (sagging) to negative (hogging).This helps in designing continuous beams. 19 / 20 The bending moment at the free end of a cantilever beam is Maximum Zero Minimum Equal to shear force At the free end, no load can resist bending, hence the bending moment is zero.Maximum moment occurs at the fixed end due to the reaction moment. 20 / 20 Shear force at a section of a beam is the Rate of change of bending moment Sum of all forces acting Moment of all loads Reaction at support Shear force (V) = dM/dx.It represents the internal transverse force resisting shear. A sudden change in load causes a sudden change in shear force but not in bending moment. Your score isThe average score is 0% 0% Restart quiz
