Question

Explain the mathematical and physical relationship (if any) between the following pairs of functions and terms: Curvature K(x) AND Slope (x) Deflection v(x) AND Slope (x) Bending Moment M(x) AND Shear distribution V(x) Second moment of area section f(x) and deflection v(x) Normal stress o(x) AND normal strain $(x) Mass per unit length (e.g. in kg/m) AND self-weight(e.g. in kN/m) of a beam where the variable x indicates the position along a the centreline of a structural member (either a beam or a column).

Answer 1

Curvature XCx) and Slope o (2) We know that iP a beam V() = deflection of at a Section then curvature is given en as mathemathecally Kn) d²v dna And slope of beam at a section is dv da 2 By comparing both ear ① 42 we consay that . K(n)= do da That means more is the rate of change of beam more is the angwar displacement of curvature OR bend. 4 A cato v Less curve More curve Esscanned with CamScanner

e) Deflection V) and Blopedcm). Rate of change of displacement along length of a beam at as slope at that section. du da is Anow A = 3) Bending Moment Mo) and sheas distribution Ven). Shear distribution V(n) am वत्र Rate of change of momen: Hence Bending momen mainly Shear is develop due to change in bending moment. for a beam of bending moment Bron) diogram is of parabolic Shape BM on 2 Shear force variation will be d M dn 2 an da - 3 " Shear force variation will be lincas Esscanned with Camscanner

4) Second moment of area section I (2) and deflection vin). from_bending equation n = e E R OR M 1 12 EI. day dne 2 EI. - da EI. IS m Deflection is invessely proportional to Second moment of of-area section. I() ... More is moment of inertia of section dess deflection. will be its is. of Even the inertia a_section to resist More is inertia defined as tendency the change in position. less. رازدار be deflection 5 any material Normal stress And Normal strain. Upto propottionality limit of We can say that Normal stress T Normal strain. Normal_stress_is_directlypropoetional to Normal_strain and constant ob proportionality known as Young's modulues i 0(2) E. Ecm) Esscanned with CamScanner is

OR in other words. If a a beam 6 (2) = E. Ecn) Strain. of Ecom) is generated in then there will_stress of magnitude Mass per unit length (kg/m) And self And self-weight of (kN/m) beam. Self weight of the beam Mašs of beam mg X acceleration due to gravite w W M teflim) (kg/m) * (m/s2) ESScanned with CamScanner