68 Mechanics of Materials §4.3 4.3.Section modulus From eqn.(4.4)the maximum stress obtained in any cross-section is given by M cmax=了ymax (4.5) For any given allowable stress the maximum moment which can be accepted by a particular shape of cross-section is therefore M-10mat ymax For ready comparison of the strength of various beam cross-sections this is sometimes written in the form M=ZG max (4.6) where Z(=I/ym)is termed the section modulus.The higher the value of Z for a particular cross-section the higher the B.M.which it can withstand for a given maximum stress. In applications such as cast-iron or reinforced concrete where the properties of the material are vastly different in tension and compression two values of maximum allowable stress apply.This is particularly important in the case of unsymmetrical sections such as T-sections where the values of ymxwill also be different on each side of the N.A.(Fig.4.4)and here two values of section modulus are often quoted, Z1=I/y and Z2=I/y2 (4.7) each being then used with the appropriate value of allowable stress. Standard handbookst are available which list section modulus values for a range of girders,etc;to enable appropriate beams to be selected for known section modulus requirements. 4.4.Second moment of area Consider the rectangular beam cross-section shown in Fig.4.5 and an element of area dA, thickness dy,breadth B and distance y from the N.A.which by symmetry passes through the dA dy D NA Fig.4.5. t Handbook on Structural Steelwork.BCSA/CONSTRADO.London,1971,Supplement 1971,2nd Supplement 1976 (in accordance with BS449,The use of structural steel in building').Structural Steelwork Handbook for Standard Metric Sections.CONSTRADO.London,1976 (in accordance with BS4848,'Structural hollow sections').68 Mechanics of Materials $4.3 4.3. Section modulus From eqn. (4.4) the maximum stress obtained in any cross-section is given by M I omax= -Ymax (4.5) For any given allowable stress the maximum moment which can be accepted by a particular shape of cross-section is therefore M=- I Ymax For ready comparison of the strength of various beam cross-sections this is sometimes written in the form OmaX M = Za, (4.6) where Z( = I/ymax) is termed the section modulus. The higher the value of Z for a particular cross-section the higher the B.M. which it can withstand for a given maximum stress. In applications such as cast-iron or reinforced concrete where the properties of the material are vastly different in tension and compression two values of maximum allowable stress apply. This is particularly important in the case of unsymmetrical sections such as T-sections where the values of ymaxwi1l also be different on each side of the N.A. (Fig. 4.4) and here two values of section modulus are often quoted, Z, = I/y, and Z, =Ily, (4.7) each being then used with the appropriate value of allowable stress. Standard handbooks t are available which list section modulus values for a range of girders, etc; to enable appropriate beams to be selected for known section modulus requirements. 4.4. Second moment of area Consider the rectangular beam cross-section shown in Fig. 4.5 and an element of area dA, thickness dy, breadth B and distance y from the N.A. which by symmetry passes through the Fig. 4.5. t Handbook on Structural Steelwork. BCSA/CONSTRADO. London, 1971, Supplement 1971, 2nd Supplement 1976 (in accordance with BS449, ‘The use of structural steel in building’). Structural Steelwork Handbook for Standard Metric Sections. CONSTRADO. London, 1976 (in accordance with BS4848, ‘Structural hollow sections’)