Basic Equations - DIRECT INTEGRATION METHOD

DIRECT INTEGRATION METHOD

3.1 Basic Equations

In this chapter, we focus attention on a beam structure that is made from a linearly elastic material whose constitutive behavior is completely characterized by a single material parameter termed the Young’s modulus E (this parameter can be obtained via conducting proper laboratory experiments).

In the development of differential equations governing behavior of such beam, we follow Euler-Bernoulli beam theory. More precisely, this theory is based on following key assumptions: (i) beam is made from a linearly elastic material whose properties are uniform across the section, (ii) a plane section remains plane after undergoing deformation, (iii) shear deformation is negligible, (iv) the rotation of the beam is relatively small, and (v) equilibrium equation is set up in the undeformed state. These assumptions play a central role in derivation presented below.

Figure 3.1: Schematic of a beam subjected to a set of transverse loads

Consider a beam of length L occupying a line defined by x = 0 and y  [a, a + L] as shown schematically in Figure 3.1. Note that a constant a, which defines a coordinate of the left end of the beam, can be chosen arbitrarily as a matter of preference. Without loss of generality, we may choose a = 0 and, as a result, the left end of the beam is located at x = 0 while its right end is located at x = L. The beam is subjected to a set of transverse loads as illustrated in Figure 3.1; this set of

x dx

q = q(x)

X Po

M_o

a A B C D

dx^*

applied loads may consist of a distributed transverse force q = q(x) and concentrated forces and moments acting at certain points; q is positive if it directs upward or in Y-direction, otherwise it is negative. Under these actions, the beam moves to a new deformed state as indicated by a red dash line. More specifically, a point occupying the coordinate (x, 0) in the undeformed configuration displaces to a point occupying the coordinate (x + u, v) in the deformed configuration where u = u(x) and v = v(x) denote the longitudinal displacement and transverse displacement at point x, respectively. Let V = V(x) and M = M(x) denote the shear force and bending moment at the cross section located at any point x.

3.1.1 Kinematics

Figure 3.2: Schematic of undeformed and deformed infinitesimal elements

Let ds be an infinitesimal element connecting a point (x, 0) to a neighboring point (x + dx, 0) in the undeformed configuration and dx^* be the same infinitesimal element in the deformed configuration as shown in Figure 3.2. In particular, dx^* is a curve element connecting a point (x + u, v) to a point (x + dx + u + du, v + dv) in the deformed configuration as shown in Figure 1(a). From geometric consideration of the element dx^* and the fact that there is no axial deformation for the entire beam (i.e. dx^* = dx), components of the displacement u and v can readily be related to the rotation  =

(x) at any point (x, 0) by sinθ dv

dx (3.1) cosθ 1 du

 dx (3.2) From the definition of the curvature (a quantity that is commonly used to represent the deformation of flexural members) and the inextensibility condition dx^* = dx, we then obtain a relation between the curvature  =  (x) and the rotation  = (x) by

d dx

   (3.3)

From the assumption (iv), following approximations are sound and commonly recognized





 

 







 

120 sin 6

5 3

(3.4) Y

(x, 0) X

(x + dx, 0) dx

(x + u, v) dx^* (x + dx u + du, v + dv)



24 1 1 2

cos ²  ⁴  (3.5) With the approximations (3.4) and (3.5), the relations (3.1) and (3.2) simply reduce to

θ dv

dx (3.6) du 0

dx  (3.7) Equation (3.7) simply implies that the longitudinal displacement must be constant throughout the beam and, when supplied by a proper constraint to prevent rigid translation in the longitudinal direction, u must vanish identically for the entire beam. As a result, based on a small rotation assumption, any point of the beam undergoes only a transverse displacement v and it is commonly termed a deflection. Further, by combining (3.3) and (3.6), it leads to a linear relation between the deflection v and the curvature :

2 2

(x) d v

  dx (3.8) Equation (3.8) is commonly termed a linearized kinematics of a beam that undergoes a small rotation. It is noted by passing that for beams undergoing large displacement and rotation, the approximations (3.4) and (3.5) cannot be employed to accurately capture responses of those structures. Various investigations of such problems using exact kinematics (i.e., exact relationship between the deflection and the curvature) can be found in the literature (e.g., Tangnovarad, 2008;

Tangnovarad and Rungamornrat, 2008; Tangnovarad and Rungamornrat, 2009; Rungamornrat and Tangnovarad, 2011; Douanevanh, 2011; Douanevanh et al, 2011).

Since the beam is represented by a one dimensional line model or, equivalently, a cross section is represented by a single point, the deformation at any point within a cross section cannot completely be characterized only by the curvature  but requiring additional assumption on kinematics of the cross section. To investigate this issue, let us consider an infinitesimal element of length dx of the beam in the undeformed state and the corresponding element in the deformed state as shown in Figure 3.3. From the assumptions (ii) and (iii), the geometry of the deformed element must be a sector of hollow circular cross section. Next, let us define a neutral axis (NA) which is a locus of points that undergo no deformation in the deformed state or, equivalently, there is no change in length in the deformed state, i.e. dx^* = dx, and let  denote the radius of curvature of the neutral axis in the deformed state. Note by passing that the elastic curve of a beam (as shown by a red dash line in Figure 3.1) is in fact the neutral axis of the deformed beam. To obtain a formula for a normal strain at any point within the cross section, let us consider a fiber of length dx located at the distance y from the neutral axis. This fiber deforms to a curve fiber of length ds and, again from the assumption (ii) and (iii), this deformed fiber is in fact an arc segment of a circle of radius  – y.

From the definition of the engineering strain, the normal strain at a point with a distance y from the neutral axis is given by where x denotes the coordinate of the cross section. Upon using (3.3), we finally obtain the normal strain at any point within the cross section in terms of the curvature of that cross section as:

(x, y) y (x)

    (3.10)

Equation (3.10) implies that the normal strain at any point within the cross section varies linearly with respect to the distance from the neutral axis. The negative sign simply indicates that the positive curvature produces a compressive strain at all points above the neutral axis.

Figure 3.3: Schematic of undeformed infinitesimal element dx and corresponding deformed element

3.1.2 Constitutive law

From the assumption (i), the normal stress at any point within the cross section is related to the normal strain at the same point via a linear stress-strain relation:

(x, y) E(x) (x, y)

   (3.11) where E = E(x) is Young’s modulus at any cross section. From (3.11) along with (3.10), it can be deduced that the normal strain at any point within the cross section also varies linearly with respect to the distance from the neutral axis.

3.1.3 Equilibrium equations

From assumption (v), we obtain following two equilibrium equations in differential form (see derivation in subsection 2.5.5.1)

dV q(x)

dx  (3.12) dM V(x)

dx  (3.13) where V = V(x) and M = M(x) are the shear force and bending moment at any point x. Validity of equations (3.12) and (3.13) depends primarily on the smoothness of loading data at point x as elaborated in details below. For a point A that is free of concentrated force and moment and q is continuous, V, M, dV/dx and dM/dx are well-defined at this point and, as a result, both (3.12) and (3.13) are valid at this point. Also, it can readily be verified that there is no jump of the shear force and bending moment at point A:

where (x_A, 0) is a coordinate of a point A. For a point that is free of concentrated force and concentrated moment but q is discontinuous (such as a point B in Figure 3.1), V, M and dM/dx are well-defined while dV/dx is not defined at this point; as a result, only the relation (3.13) is valid at this point. In addition, by considering equilibrium at this point (see subsection 2.5.5.3), it can be shown that there is no jump of the shear force and no jump of the bending moment at point B, i.e.

[V]_B (3.16) 0

[M]_B (3.17) For a point that is subjected to a concentrated force Po (such as a point C in Figure 3.1), M is well-defined while V, dV/dx and dM/dx are not well-defined at this point and, similarly, it can be deduced from equilibrium at this point that the jump of the shear force and the bending moment satisfy

C P

[V]  (3.18) 0

[M]_C (3.19) For a point that is subjected to a concentrated moment Mo (such as a point D in Figure 3.1), V, M, dV/dx and dM/dx are not defined at this point. Again, by considering equilibrium at this point, we can conclude following jump conditions:

[V]_D  (3.20)

D M

[M]  (3.21) In addition, force and moment resultants of the normal stress  over the entire cross section yield the axial force F(x) and the bending moment M(x) as follows:

F(x) 



(x,y)dA (3.22)

M(x)  



y (x,y)dA (3.23) By substituting (3.10) and (3.11) into (3.22) and (3.23), it leads to

F(x) E(x)κ(x) ydA E(x)κ(x)y



 (3.24)

2 A

M(x) E(x)κ(x) y dA E(x)I(x)κ(x)



 (3.25) where y is the distance from the centroid of the cross section to the neutral axis and I is the moment of inertia of the cross section. From (3.24) and the fact that the axial force vanishes for the entire beam (i.e. F(x) = 0), it implies that the neutral axis is located at the centroid of the cross section.

Equation (3.25) that can be viewed as a constitutive relation in the cross section level is also known as a moment-curvature relationship. It is evident that the moment-curvature relationship (3.25) is linear; this results directly from the linear stress-strain relation (3.11). For nonlinear elastic and inelastic materials, the relation between the bending moment and the curvature is, in general, nonlinear. Analysis of beams by taking material nonlinearity into account can be found, for examples, in the work of Danmongkoltip (2009), Danmomgkoltip and Rungamornrat (2009) and Pinyochotiwong et al (2009).

In document Fundamental Structural Analysis.8765.1408344506 (Page 159-164)