Basic Energy Principles in Stiffness Analysis Stress-Strain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting our attention to linear elastic structural response. Further assuming that the material is homogenous and isotropic, we only need to know two of the following three material constants: E = Elastic (or Young s) modulus G = Shear modulus = Poisson ratio Normally, the shear modulus is expressed in terms of the elastic modulus and Poisson ratio as E G ( ) The most widely used civil engineering structural materials, steel and concrete, have uniaxial stress-strain diagrams of the types shown in Fig.. Mild steels yield Fig. : Typical Stress () Strain (e) Curves for (a) Steel and (b) Concrete with a pronounced permanent elongation at a stress ym (Fig.a). High strength steels yield gradually, which requires an arbitrary definition of its yield strength yh, offset criterion. Yield strengths for steel vary from less than 5 MPa to more than 7 MPa. For practical purposes, steel behaves as an ideal material in both tension and compression below the yield or buckling stress. The elastic modulus and Poisson ratio for steel are always close to, MPa and.3, respectively. Concrete is less predictable, but under short-duration compressive stress not greater than u /3 u /, its behavior is reasonably linear such as the commonly used.% 3 4
(Fig. b in which typical values for u are: 3 MPa u 5 MPa). An elastic modulus of E =, MPa and Poisson ratio of =.5 are typical for concrete. In using concrete for analysis, the ACI code specifies using the gross cross area properties to perform analyses to determine the force distributions in frame structures, i.e., ignore the reinforcing steel and tension cracking in calculating the force distributions. 5 Work and Energy The principle of conservation of energy is fundamentally important in structural analysis. This principle, expressed as energy or work balance, is applicable to both rigid and deformable structures. Rigid structures only require multiplying the external forces by the respective displacements. Deformable structures also require the summation of the internal stresses acting through the 6 respective deformations. Internal work is called strain energy and must be accounted for in the energy balance. The work dw of a force F acting through a change in displacement d in the direction of F is dw Fd () Over, the total work is W Fd () imiting attention to gradually applied forces, i.e., ignoring inertial forces caused by dynamic loads, and linear elastic response leads to W Fdk d k F F F k (3) 7 8
Expanding to a vector of forces and displacements leads to W F { } (4) The special case shown in the right figure: u W F x Fxu v where U= strain energy for the element. Equation (5) is a homogeneous, quadratic polynomial in terms of the local coordinate element displacements {u} or global coordinate element displacement {v}. Expanding (4) for a single element ({F} = [k] {u} or {F} = [K] {v}): W u [k]{u} v [K]{v} U (5) 9 Principle of Virtual Displacements to constructing stiffness equations. In prior chapters we established The principle of virtual the relationships of framework displacements can be stated as analysis directly utilizing the basic If a deformable structure is in conditions of equilibrium and equilibrium and remains in displacement continuity. Henceforth, we will use energy principles, equilibrium while it is subject to a virtual distortion, the external specifically the principle of virtual virtual work done by the external displacements since it permits forces acting on the structure is mathematical manipulations that equal to the internal virtual work are not possible with direct done by the stress resultants. procedures. We restrict our attention to virtual displacements Recall: virtual imaginary, not real, or in essence but not in fact since this principle is applicable 3
The principle of virtual displacements is expressed mathematically as W ext = W int (6) F F W ext W where W ext = F = external virtual work (shaded blue area in the figure) and W int = internal virtual work. 3 Equation (6) is based on the conservation of energy principle, i.e. the work done by the external forces going through a virtual displacement equals the work done by the internal forces due to the same virtual displacement. The external virtual work can be generalized to a system of forces as s ext i i (7) i W qdx ( )P 4 The internal virtual work (W int ) is a function of the structure type. Since this course focuses on frame members, only axial and bending deformations will be considered. Axial Deformation Consider the axial force system shown in Fig.. The differential internal virtual work (dw int ) is d( u) dw int Fdx x (8a) dx where u = virtual axial displacement and F x = real axial force. Recalling from your mechanics of materials class that axial strain e x = du/dx and the axial force F x = x A (axial stress times area), (8a) can be rewritten as dw int ex x Adx (8b) Fig. : Axial Deformation 5 Integrating (8b) over the length of 6 the element and substituting 4
Hooke s law ( x = Ee x ) leads to W e e dx int x x d( u) du dx (9) dx dx For the beam bending (flexure) case (Fig. 3), the internal virtual work is Wint z Mz dx z EIz dx d ( v) d v () EI dx dx dx where v = virtual transverse displacement; z = d(v)/dx = virtual rotation; M z = real moment about the z-axis; z = d v/dx = curvature strain about the z-axis; and M z = EI k z. Fig. 3: Bending Deformation 7 8 NOTE: A difficulty in applying the principle of virtual displacements is that functions must be assumed or developed for the real and virtual displacement functions in (9) and (). Development of these expressions will follow finite element mechanics, which is covered in a later section. 9 Analytical Solutions Using Principle of Virtual Displacements Consider the simple axial force structure shown in Fig. 4. The real x, u F x, u Fig. 4: Axial Deformation Structure displacement u: u = x/ u The real strain is e x = du/dx = u / Imposing a virtual displacement 5
u results in an external virtual work of W ext = u F x In order to calculate the internal virtual work d( u) du Wint dx dx dx expressions for u and u over the length of the axial deformation structure must be assumed. We will consistently assume the real displacement u: u = (x/) u We will consider various expressions for the virtual displacement to demonstrate the principle of virtual displacements. First, consider u = (x/) u The internal virtual work: u u int W dx u u Equating the external and internal virtual works gives u F x = u (/) u or u = F x / which is exact. Consider next: u = (x/) u The internal virtual work: u u int W xdx u u Which again gives the exact solution: u = F x / astly, consider: u = u sin(x/) 3 The internal virtual work: u x u Wint cos dx u u Which again gives the exact solution: u = F x / These three virtual displacement expressions all resulted in an exact solution since the real displacement solution was exact. If the chosen real displacements 4 6
correspond to stresses that identically satisfy the conditions of equilibrium, any form of admissible virtual displacement will suffice to produce the exact solution. Notice the adjective admissible in front of virtual displacement. Admissible means that the chosen function is physically continuous and satisfies all essential boundary conditions, i.e., is appropriately zero at all A = A (-x/) Consider next the nonprismatic axial deformation structure of Fig. 5. We will repeat the process considered for Fig. 4 with reference to the geometry of Fig. 5. Considering the first case: u = (x/) u 5 6 supports. F x x, u Fig. 5: Nonprismatic Axial Deformation Structure u x u int W A ( )dx E 3 W u u 4 int Equating the external and internal virtual works leads to 4Fx u 3 Considering the second virtual displacement expression: u = (x/) u leads to u x u int W x dx u u 3 7 Equating the external and internal virtual works leads to 3Fx u Considering the third virtual displacement expression: u = u sin(x/) leads to u x x u int W cos dx u u (.88) u u 8 7
Again, equating the external and internal virtual works leads to u.fx NOTE: None of the three solutions match. This is because neither the real or virtual displacements are exact. However, we produced three good approximate solutions. The exact solution for Fig. 5 is u.387fx The principle of virtual displacements has its greatest application in producing approximate solutions. The standard procedure is to adopt a virtual displacement of the same form as the real displacement. Adopting different forms for the real and virtual displacements can lead to unsymmetric stiffness matrices. 9 3 Special Transformations in Analysis Congruent Transformation A matrix triple product in which the pre-multiplying matrix is the transpose of the post-multiplying matrix, e.g. T T [C] [A] [B][A] or [D] [A][B][A] Significance of the transformation is that [C] and [D] will each be symmetric if [B] is symmetric, which is one of the reasons all our stiffness Contragradience Principal If one transformation is known, e.g., the local to global displacements, the force transformation will be transpose of the displacement transformation provided both sets of forces and displacements are conjugate and vice versa. Such a transformation is known as contragradient (or contragredient) under the stipulated conditions of conjugacy. Conjugate simply means that the force-displacement pair only produce work in the matrices were symmetric. 3 3 direction of the displacement. 8
For linear analysis, this is always the case when using orthogonal coordinate systems. A good example are the coordinate transformations for a truss member (7.) in which the transformation matrices are rectangular: {u a } = [T a ] {v a } T {F a} [T a] {Q a} cos sin [T a ] cos sin 33 9