Element analysis 287 9.1.1 Equations of static equilibrium From Newton’s law of motion, the conditions under which a body remains in static equilibrium can be expressed as follows: • The sum of the components of all forces acting on the body, resolved along any arbitrary direction, is equal to zero. This condition is completely satisfied if the components of all forces resolved along the x, y, z directions individually add up to zero. (This can be represented by SP x = 0, SP y = 0, SP z = 0, where P x , P y and P z represent forces resolved in the x, y, z directions.) These three equations represent the condition of zero translation. • The sum of the moments of all forces resolved in any arbitrarily chosen plane about any point in that plane is zero. This condition is completely satisfied when all the moments resolved into xy, yz and zx planes all individually add up to zero. (SM xy = 0, SM yz = 0 and SM zx = 0.) These three equations provide for zero rotation about the three axes. If a structure is planar and is subjected to a system of coplanar forces, the conditions of equilibrium can be simplified to three equations as detailed below: • The components of all forces resolved along the x and y directions will individ- ually add up to zero (SP x = 0 and SP y = 0). • The sum of the moments of all the forces about any arbitrarily chosen point in the plane is zero (i.e. SM = 0). 9.1.2 The principle of superposition This principle is only applicable when the displacements are linear functions of applied loads. For structures subjected to multiple loading, the total effect of several loads can be computed as the sum of the individual effects calculated by applying the loads separately. This principle is a very useful tool in computing the combined effects of many load effects (e.g. moment, deflection, etc.). These can be calculated separately for each load and then summed. 9.2 Element analysis Any complex structure can be looked upon as being built up of simpler units or components termed ‘members’ or ‘elements’. Broadly speaking, these can be classified into three categories: • Skeletal structures consisting of members whose one dimension (say, length) is much larger than the other two (viz. breadth and height). Such a line element is Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ 288 Introduction to manual and computer analysis variously termed as a bar, beam, column or tie. A variety of structures are obtained by connecting such members together using rigid or hinged joints. Should all the axes of the members be situated in one plane, the structures so produced are termed plane structures. Where all members are not in one plane, the structures are termed space structures. • Structures consisting of members whose two dimensions (viz. length and breadth) are of the same order but much greater than the thickness fall into the second category. Such structural elements are called plated structures. Such structural elements are further classified as plates and shells depending upon whether they are plane or curved. In practice these units are used in combination with beams or bars. Slabs supported on beams, cellular structures, cylindrical or spherical shells are all examples of plated structures. • The third category consists of structures composed of members having all the three dimensions (viz. length, breadth and depth) of the same order. The analysis of such structures is extremely complex, even when several simplifying assumptions are made. Dams, massive raft foundations, thick hollow spheres, caissons are all examples of three-dimensional structures. For the most part the structural engineer is concerned with skeletal structures. Increasing sophistication in available techniques of analysis has enabled the eco- nomic design of plated structures in recent years. Three-dimensional analysis of structures is only rarely carried out. Under incremental loading, the initial defor- mation or displacement response of a steel member is elastic. Once the stresses caused by the application of load exceed the yield point, the cross section gradually yields. The gradual spread of plasticity results initially in an elasto-plastic response and then in plastic response, before ultimate collapse occurs. 9.3 Line elements The deformation response of a line element is dependent on a number of cross- sectional properties such as area, A, second moment of area (I xx =Úy 2 dA; I yy =Úx 2 dA) and the product moment of area (I xy =ÚxydA). The two axes xx and yy are orthogonal. For doubly symmetric sections, the axes of symmetry are those for which Úxy dA = 0. These are known as principal axes. For a plane area, the principal axes may be defined as a pair of rectangular axes in its plane and passing through its centroid, such that the product moment of area ÚxydA = 0, the co-ordinates referring to the principal axes. If the plane area has an axis of symmetry, it is obviously a principal axis (by symmetry Úxy dA = 0). The other axis is at right angles to it, through the centroid of the area. Tables of properties of the section (including the centroid and shear centre of the section) are available as published data (e.g. SCI Steelwork Design Guide, Vol. 1). 1 Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ V y U x Line elements 289 If the section has no axis of symmetry (e.g. an angle section) the principal axes will have to be determined. Referring to Fig. 9.1, if uOu and vOv are the principal axes, the angle a between the uu and xx axes is given by (9.1) (9.2) The values of a, I uu and I vv are available in published steel design guides (e.g. Reference 1). 9.3.1 Elastic analysis of line elements under axial loading When a cross section is subjected to a compressive or tensile axial load,P, the result- ing stress is given by the load/area of the section, i.e. P/A. Axial load is defined as one acting at the centroid of the section. When loads are introduced into a section in a uniform manner (e.g. through a heavy end-plate), this represents the state of stress throughout the section. On the other hand, when a tensile load is introduced via a bolted connection, there will be regions of the member where stress concen- trations occur and plastic behaviour may be evident locally, even though the mean stress across the section is well below yield. If the force P is not applied at the centroid, the longitudinal direct stress distri- bution will no longer be uniform. If the force is offset by eccentricities of e x and e y measured from the centroidal axes in the y and x directions, the equivalent set of actions are (1) an axial force P, (2) a bending moment M x = Pe x in the yz plane and (3) a bending moment M y = Pe y in the zx plane (see Fig. 9.2). The method of eval- uating the stress distribution due to an applied moment is given in a later section. I II II I vv xx yy xx yy xy = + - - + 22 22cos sinaa I II II I uu xx yy xx yy xy = + + - - 22 22cos sinaa tan 2 2 a = - - I II xy xx yy Fig. 9.1 Angle section (no axis of symmetry) Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ y ey y 290 Introduction to manual and computer analysis The total stress at any section can be obtained as the algebraic sum of the stresses due to P, M x and M y . 9.3.2 Elastic analysis of line elements in pure bending For a section having at least one axis of symmetry and acted upon by a bending moment in the plane of symmetry, the Bernoulli equation of bending may be used as the basis to determine both stresses and deflections within the elastic range. The assumptions which form the basis of the theory are: • The beam is subjected to a pure moment (i.e. shear is absent). (Generally the deflections due to shear are small compared with those due to flexure; this is not true of deep beams.) • Plane sections before bending remain plane after bending. • The material has a constant value of modulus of elasticity (E) and is linearly elastic. The following equation results (see Fig. 9.3). (9.3) where M is the applied moment; I is the second moment of area about the neutral axis; f is the longitudinal direct stress at any point within the cross section; y is the distance of the point from the neutral axis; E is the modulus of elasticity; R is the radius of curvature of the beam at the neutral axis. M I f y E R == Fig. 9.2 Compressive force applied eccentrically with reference to the centroidal axis Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ -* cross section stress diagram neutral axis Line elements 291 From the above, the stress at any section can be obtained as For a given section (having a known value of I) the stress varies linearly from zero at the neutral axis to a maximum at extreme fibres on either side of the neutral axis: (9.4) The term Z is known as the elastic section modulus and is tabulated in section tables. 1 The elastic moment capacity of a given section may be found directly as the product of the elastic section modulus, Z, and the maximum allowable stress. If the section is doubly symmetric, then the neutral axis is mid-way between the two extreme fibres. Hence, the maximum tensile and compressive stresses will be equal. For an unsymmetric section this will not be the case, as the value of y for the two extreme fibres will be different. For a monosymmetric section, such as the T-section shown in Fig. 9.4, subjected to a moment acting in the plane of symmetry, the elastic neutral axis will be the centroidal axis. The above equations are still valid. The values of y max for the two extreme fibres (one in compression and the other in tension) are different. For an applied sagging (positive) moment shown in Fig. 9.4, the extreme fibre stress in the flange will be compressive and that in the stalk will be tensile.The numerical values of the maximum tensile and compressive stresses will differ. In the case sketched in Fig.9.4,the magnitude of the tensile stress will be greater, as y max in tension is greater than that in compression. Caution has to be exercised in extending the pure bending theory to asymmetric sections. There are two special cases where no twisting occurs: f My I M Z Z I y max max max . == =where f My I = Fig. 9.3 Pure bending Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ y 292 Introduction to manual and computer analysis • Bending about a principal axis in which no displacement perpendicular to the plane of the applied moment results. • The plane of the applied moment passes through the shear centre of the cross section. When a cross section is subjected to an axial load and a moment such that no twisting occurs, the stresses may be determined by resolving the moment into com- ponents M uu and M vv about the principal axes uu and vv and combining the result- ing longitudinal stresses with those resulting from axial loading: (9.5) For a section having two axes of symmetry (see Fig. 9.2) this simplifies to Pure bending does not cause the section to twist. When the shear force is applied eccentrically in relation to the shear centre of the cross section, the section twists and initially plane sections no longer remain plane. The response is complex and consists of a twist and a deflection with components in and perpendicular to the plane of the applied moment. This is not discussed in this chapter. A simplified method of calculating the elastic response of cross sections subjected to twisting moments is given in an SCI publication. 2 9.3.3 Elastic analysis of line elements subject to shear Pure bending discussed in the preceding section implies that the shear force applied on the section is zero. Application of transverse loads on a line element will, in general, cause a bending moment which varies along its length, and hence a shear force which also varies along the length is generated. f P A My I Mx I xy xx xx yy yy , =± ± ± f P A Mv I Mu I uv uu uu vv vv , =± ± ± Fig. 9.4 Monosymmetric section subjected to bending Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ b shear stress distribution rectangular cross section (a) T qmax) I -I shear stress distribution I—section (b) Line elements 293 If the member remains elastic and is subjected to bending in a plane of symmetry (such as the vertical plane in a doubly symmetric or monosymmetric beam), then the shear stresses caused vary with the distance from the neutral axis. For a narrow rectangular cross section of breadth b and depth d, subjected to a shear force V and bent in its strong direction (see Fig. 9.5(a)), the shear stress varies parabolically from zero at the lower and upper surfaces to a maximum value, q max , at the neutral axis given by i.e. 50% higher than the average value. For an I-section (Fig. 9.5(b)), the shear distribution can be evaluated from (9.6) where B is the breadth of the section at which shear stress is evaluated. The integration is performed over that part of the section remote from the neutral axis, i.e. from y = h to y = h max with a general variable width of b. Clearly, for the I- (or T-) section, at the web/flange interface the value of the integral will remain constant.As the section just inside the web becomes the section just inside the flange, the value of the vertical shear abruptly changes as B changes from web thickness to flange width. q V IB by y yh yh = = = Ú d max q V bd max = 3 2 Fig. 9.5 Shear stress distribution: (a) in a rectangular cross section and (b) in an I-section Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ yield point / 0 strain hardening < strain hardening commences strain 294 Introduction to manual and computer analysis 9.3.4 Elements stressed beyond the elastic limit The most important characteristic of structural steels (possessed by no other material to the same degree), is their capacity to withstand considerable deforma- tion without fracture. A large part of this deformation occurs during the process of yielding, when the steel extends at a constant and uniform stress known as the yield stress. Figure 9.6 shows, in its idealized form, the stress–strain curve for structural steels subjected to direct tension. The line 0A represents the elastic straining of the material in accordance with Hooke’s law. From A to B, the material yields while the stress remains constant and is equal to the yield stress, f y . The strain occurring in the material during yielding remains after the load has been removed and is called plastic strain. It is important to note that this plastic strain AB is at least ten times as large as the elastic strain, e y , at yield point. When subjected to compression, various grades of structural steel behave in a similar manner and display the same property of yield. This characteristic is known as ductility of steel. 9.3.5 Bending of beams beyond the elastic limit For simplicity, the case of a beam symmetrical about both axes is considered first. The fibres of the beam subjected to bending are stressed in tension or compression according to their position relative to the neutral axis and are strained as shown in Fig. 9.7. While the beam remains entirely elastic, the stress in every fibre is proportional to its strain and to its distance from the neutral axis. The stress, f, in the extreme fibres cannot exceed the yield stress, f y . When the beam is subjected to a moment slightly greater than that which first produces yield in the extreme fibres, it does not fail. Instead, the outer fibres yield Fig. 9.6 Idealized stress–strain relationship for mild steel Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ (a) (b) (c) (d) plastic zones / (comp.) \ H plastic zones Y strain distribution (tension) (a) (b) (c) (d) Line elements 295 at constant stress, f y , while the fibres nearer to the neutral axis sustain increased elastic stresses. Figure 9.8 shows the stress distribution for beams subjected to such moments. Such beams are said to be partially plastic and those portions of their cross sections which have reached the yield stress are described as plastic zones. The depths of the plastic zones depend upon the magnitude of the applied moment. As the moment is increased, the plastic zones increase in depth, and it is assumed that plastic yielding will continue to occur at yield stress, f y , resulting in two stress blocks, one zone yielding in tension and one in compression. Figure 9.9 represents the stress distribution in beams stressed to this stage. The plastic zones occupy the whole area of the sections,which are then described as being fully plastic. When the cross section of a member is fully plastic under a bending moment, any attempt to increase this moment will cause the member to act as if hinged at that point. This point is then described as a plastic hinge. Fig. 9.7 Elastic distribution of stress and strain in a symmetric beam. (a) Rectangular section, (b) I-section, (c) stress distribution for (a) or (b), (d) strain distribution for (a) or (b) Fig. 9.8 Distribution of stress and strain beyond the elastic limit for a symmetric beam. (a) Rectangular section, (b) I-section, (c) stress distribution for (a) or (b), (d) strain distribution for (a) or (b) Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ [...]... the value is about 1. 15 increasing to 1 .5 for a rectangle Steel Designers' Manual - 6th Edition (2003) Line elements 297 (v1 00) 1.00 0.87 (i'1. 15) 0.67 (u=1 .50 ) M= jyMp This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/ curvature.. .Steel Designers' Manual - 6th Edition (2003) 296 Introduction to manual and computer analysis _ dEl nerfis PH _ fy 4 This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/ (a) (b) (c) (d) Fig 9.9... 9.12 Steel Designers' Manual - 6th Edition (2003) Introduction to manual and computer analysis 298 w plastic zone yield zone I I L I lii 1! I stiff length I I B M B moment—rotation curve This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/... resisted = , axial capacity of section A Steel Designers' Manual - 6th Edition (2003) Introduction to manual and computer analysis 300 B T total area = A This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/ Fig 9.13 a= f total stresses... 4flexibility of beam = 4- (0) Fig 9.18 Flexibility (b) Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/ 306 Fig 9.19 Introduction to manual and computer analysis Flexibility coefficients... B - q A + Ë L L¯ M AB = M FAB + M BA (9.18) Steel Designers' Manual - 6th Edition (2003) 312 Introduction to manual and computer analysis CA I I M MFAB This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/ Fig 9.23 Slope–deflection method... Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/ as propped cantilever r4:31 12 +!= 12 —16 +16 2 2 12 +!= 12 —16 +16 —1 6 (simply supported) step 5: release joint B +3.43 and distribute 0 +2.28 step 3: release joint C step 4: carry over to B 5 +4 .57 step 2: calculate the fixed end... U, hence x1 = ∂U ∂ P1 x2 = ∂U ∂ P2 (9.20) Steel Designers' Manual - 6th Edition (2003) Analysis of skeletal structures jPi B r2 jFs C 3 15 D iagram 11 B This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/ IA El qram Fig 9.27 Unit load... inter-element compatibility, since this condition is neces- Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/ 318 Fig 9.28 Introduction to manual and computer analysis Triangular and rectangular... Fig 9.31 and ex, ey, gxy are the corresponding strain Steel Designers' Manual - 6th Edition (2003) Introduction to manual and computer analysis This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/ 320 Fig 9.31 Stress components on a plane . (e.g. SCI Steelwork Design Guide, Vol. 1). 1 Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction. stress–strain relationship for mild steel Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute. 12/2/2007 To buy a hardcopy version of this document call 01344 8727 75 or go to http://shop.steelbiz.org/ (i'1. 15) (u=1 .50 ) (v1 .00) 1.00 0.87 0.67 M= jyMp curvature Line elements 297 Plastic