Collapse of the Brazos River Bridge, Brazos, Texas, during erection of the 973-ft, continuous steel plate girders that support the roadway. The failure was initiated by overstress of the connections between the web and flange during erection. Structures are particularly vulnerable to failure during erection because stiffening elements—for example, floor slabs and bracing—may not be in place. In addition, the structure’s strength may be reduced when certain connections are partially bolted or not fully welded to permit pre- cise alignment of members. CHAPTER 9 Deflections of Beams and Frames 9.1 Introduction When a structure is loaded, its stressed elements deform. In a truss, bars in tension elongate and bars in compression shorten. Beams bend and cables stretch. As these deformations occur, the structure changes shape and points on the structure displace. Although these deflections are nor- mally small, as part of the total design, the engineer must verify that these deflections are within the limits specified by the governing design code to ensure that the structure is serviceable. For example, large deflections of beams can lead to cracking of nonstructural elements such as plaster ceil- ings, tile walls, or brittle pipes. The lateral displacement of buildings pro- duced by wind forces must be limited to prevent cracking of walls and windows. Since the magnitude of deflections is also a measure of a mem- ber’s stiffness, limiting deflections also ensures that excessive vibrations of building floors and bridge decks are not created by moving loads. Deflection computations are also an integral part of a number of ana- lytical procedures for analyzing indeterminate structures, computing buck- ling loads, and determining the natural periods of vibrating members. In this chapter we consider several methods of computing deflections and slopes at points along the axis of beams and frames. These methods are based on the differential equation of the elastic curve of a beam. This equation relates curvature at a point along the beam’s longitudinal axis to the bending moment at that point and the properties of the cross sec- tion and the material. 9.2 Double Integration Method The double integration method is a procedure to establish the equations for slope and deflection at points along the longitudinal axis (elastic curve) of a loaded beam. The equations are derived by integrating the 294 Chapter 9 Deflections of Beams and Frames differential equation of the elastic curve twice, hence the name double integration. The method assumes that all deformations are produced by moment. Shear deformations, which are typically less than 1 percent of the flexural deformations in beams of normal proportions, are not usu- ally included. But if beams are deep, have thin webs, or are constructed of a material with a low modulus of rigidity (plywood, for example), the magnitude of the shear deformations can be significant and should be investigated. To understand the principles on which the double integration method is based, we first review the geometry of curves. Next, we derive the dif- ferential equation of the elastic curve—the equation that relates the cur- vature at a point on the elastic curve to the moment and the flexural stiff- ness of the cross section. In the final step we integrate the differential equation of the elastic curve twice and then evaluate the constants of integration by considering the boundary conditions imposed by the sup- ports. The first integration produces the equation for slope; the second integration establishes the equation for deflection. Although the method is not used extensively in practice since evaluating the constants of inte- gration is time-consuming for many types of beams, we begin our study of deflections with this method because several other important proce- P dures for computing deflections in beams and frames are based on the a ( ) differential equation of the elastic curve. y Geometry of Shallow Curves ds B A To establish the geometric relationships required to derive the differen- x tial equation of the elastic curve, we will consider the deformations of dx x the cantilever beam in Figure 9.1a. The deflected shape is represented in Figure 9.1b by the displaced position of the longitudinal axis (also called (b) the elastic curve). As reference axes, we establish an x-y coordinate sys- tem whose origin is located at the fixed end. For clarity, vertical dis- o tances in this figure are greatly exaggerated. Slopes, for example, are d␪ typically very small—on the order of a few tenths of a degree. If we were to show the deflected shape to scale, it would appear as a straight line. line tangent at B To establish the geometry of a curved element, we will consider an infinitesimal element of length ds located a distance x from the fixed end. ␳ As shown in Figure 9.1c, we denote the radius of the curved segment by r. At points A and B we draw tangent lines to the curve. The infinitesimal B angle between these tangents is denoted by du. Since the tangents to the ds ␪ d␪ curve are perpendicular to the radii at points A and B, it follows that the A angle between the radii is also du. The slope of the curve at point A line tangent at A equals (c) dy ϭ tan u Figure 9.1 dx Section 9.2 Double Integration Method 295 If the angles are small (tan u Ϸ u radians), the slope can be written dy ϭ u (9.1) dx From the geometry of the triangular segment ABo in Figure 9.1c,we can write r du ϭ ds (9.2) Dividing each side of the equation above by ds and rearranging terms give du 1 c ϭ ϭ (9.3) ds r where du/ds, representing the change in slope per unit length of distance along the curve, is called the curvature and denoted by the symbol c. Since slopes are small in actual beams, ds Ϸ dx, and we can express the curvature in Equation 9.3 as du 1 c ϭ ϭ (9.4) dx r Differentiating both sides of Equation 9.1 with respect to x, we can dx express the curvature du/dx in Equation 9.4 in terms of rectangular coor- dinates as du d 2y x ϭ (9.5) dx dx 2 (a) Differential Equation of the Elastic Curve ␳ d␪ To express the curvature of a beam at a particular point in terms of the moment acting at that point and the properties of the cross section, we N.A. will consider the flexural deformations of the small beam segment of length dx, shown with darker shading in Figure 9.2a. The two vertical lines rep- resenting the sides of the element are perpendicular to the longitudinal axis M A d␪ (b) ␳ dl F E ⑀ ␴ M ␪ c c B d ␺ Figure 9.2: Flexural deformations of seg- N.A. D ment dx:(a) unloaded beam; (b) loaded beam and moment curve; (c) cross section of beam; (d) flexural deformations of the small beam dx segment; (e) longitudinal strain; ( f ) flexural (c) stresses. (d) (e) ( f ) 296 Chapter 9 Deflections of Beams and Frames of the unloaded beam. As load is applied, moment is created, and the beam bends (see Fig. 9.2b); the element deforms into a trapezoid as the sides of the segment, which remain straight, rotate about a horizontal axis (the neutral axis) passing through the centroid of the section (Fig. 9.2c). In Figure 9.2d the deformed element is superimposed on the original unstressed element of length dx. The left sides are aligned so that the deformations are shown on the right. As shown in this figure, the longi- tudinal fibers of the segment located above the neutral axis shorten because they are stressed in compression. Below the neutral axis the lon- gitudinal fibers, stressed in tension, lengthen. Since the change in length of the longitudinal fibers (flexural deformations) is zero at the neutral axis (N.A.), the strains and stresses at that level equal zero. The variation of longitudinal strain with depth is shown in Figure 9.2e. Since the strain is equal to the longitudinal deformations divided by the original length dx, it also varies linearly with distance from the neutral axis. Considering triangle DFE in Figure 9.2d, we can express the change in length of the top fiber dl in terms of du and the distance c from the neutral axis to the top fiber as dl ϭ du c (9.6) By definition, the strain P at the top surface can be expressed as dl P ϭ (9.7) dx Using Equation 9.6 to eliminate dl in Equation 9.7 gives du P ϭ c (9.8) dx Using Equation 9.5 to express the curvature du/dx in rectangular coordi- nates, we can write Equation 9.8 as d 2y P ϭ (9.9) dx 2 c If behavior is elastic, the flexural stress, s, can be related to the strain P at the top fiber by Hooke’s law, which states that s ϭ EP where E ϭ the modulus of elasticity Solving for P gives s P ϭ (9.10) E Using Equation 9.10 to eliminate P in Equation 9.9 produces d 2y s ϭ (9.11) dx 2 Ec Section 9.2 Double Integration Method 297 For elastic behavior the relationship between the flexural stress at the top fiber and the moment acting on the cross section is given by Mc s ϭ (5.1) I Substituting the value of s given by Equation 5.1 into Equation 9.11 pro- duces the basic differential equation of the elastic curve d 2y M ϭ (9.12) dx 2 EI In Examples 9.1 and 9.2 we use Equation 9.12 to establish the equa- tions for both the slope and the deflection of the elastic curve of a beam.