Calculating Points of Contraflexure in Continuous Beam

Continuous Beam

When dealing with continuous beams or one-way slabs, one should use approximate moments and shears provided by the local code regulations.

From: Concrete Portable Handbook , 2012

Behavior of Statically Indeterminate Structures at Elevated Temperatures

Zhenhai Guo , Xudong Shi , in Experiment and Calculation of Reinforced Concrete at Elevated Temperatures, 2011

10.1.1 Specimen Design and Testing Content

A continuous beam of two spans is the simplest statically indeterminate structure containing only one indeterminacy, but it reflects the basic characteristic behavior of a statically indeterminate structure. The mechanical conditions are simple and clear; the deformations, redistribution of internal force, and failure process are observed and measured easily during testing. The experimental devices needed are readily available and the experimental costs are low. Therefore, it is a suitable device for testing a statically indeterminate structure at elevated temperatures.

The continuous beam specimen (Fig. 10-1) is designed with two equal spans of 1.2 m each. The total length and section are 2.5 m and 100 mm × 180 mm, respectively, which fit within the existing furnace (see Fig. 8-3) for testing a structural member with three surfaces exposed to high temperature. The upper and lower reinforcements on the section of the specimen are symmetrical and its construction is shown in Fig. 10-1. Both ends of the specimen are supported on rollers, which permit longitudinal (horizontal) displacement, and one concentrated load acts symmetrically on each span.

FIGURE 10-1. Construction and loading pattern of a continuous beam specimen^[1-12]: (a) specimen; (b) loading pattern.

Six specimens of a continuous beam were produced to investigate the load position (the distance (βl) between the concentrated load and the end support), the initial loading level (the value of constant load), and the heating conditions (heating simultaneously on both spans or heating on one span only while the other span is kept at room temperature). The numbers and experimental parameters of the specimens are listed in Table 10-1.

TABLE 10-1. Numbers and Experimental Parameters for Continuous Beam Specimens

Number of specimen	Load position (β)	Heating spans	Value of constant load (kN)
TCB1-1	$\frac{1}{3}$	2	10
TCB1-2		2	20
TCB1-3		1	20
TCB2-1	$\frac{2}{3}$	2	25
TCB2-2		2	35
TCB2-3		1	10

A single-bay and single-story frame was selected as the statically indeterminate structure. Although it is the simplest form of frame structure, it includes both basic structural members, i.e., beam and column, and has three degrees of indeterminacy. Therefore, it still reflects the main characteristics of the frame structure.

The construction of the frame specimen is shown in Fig. 10-2. Two concentrated loads act symmetrically on the third point of the beam. The span and height of the frame and the sizes of beam and column sections fit well with the existing experimental furnace (see Fig. 8-3). The bottoms of the two columns are connected with two stiff base beams.

FIGURE 10-2. Construction and loading pattern of a frame specimen.^[1-12]

Five frame specimens were designed and their numbers and experimental parameters are listed in Table 10-2. One specimen (TFC-1), used for comparison, was tested at room temperature, and the value of the ultimate load obtained is P _u (kN). Four other specimens were tested at elevated temperatures and the factors investigated are the initial load level (P ₀/P _u ) and the ratio between the linear elastic rigidities of the beam and column:

TABLE 10-2. Numbers and Experimental Parameters for Frame Specimens

Number of specimen	Span of beam L (mm)	Height of column H (mm)	Depth of beam section h b (mm)	Depth of column section h c (mm)	Ratio between linear elastic rigidities of beam and column (i b /i c )	Load level P 0/P u
TFC-1	1700	1425	150	200	0.354	1.0
TFC-2	1700	1425	150	200	0.354	0.49
TFC-3	1700	1425	150	200	0.354	0.30
TFC-4	1650	1425	150	150	0.877	0.29
TFC-5	1650	1400	200	150	2.011	0.31

The width of the beam and column sections is b _b = b _c = 100 mm. The beam span L and the column height H are varied because of the different depths of their sections, and are adjusted to fit the size of the experimental furnace.

$i_b=\frac{E{b}_b{h}_b^3}{12L},i_c=\frac{E{b}_c{h}_c^3}{12H}$

(10.1) $\frac{i_b}{i_c}=\frac{b_b{h}_b^{3}H}{b_c{h}_c^{3}L}$

All the specimens were made in the laboratory using a steel mold. The continuous beams were cast in a normal position and the frames were cast horizontally. They were prepared for testing after casting, compacting, and curing of the concrete. The raw materials used for the concrete specimens were slag cement, river sand, and crushed limestone with maximum particle size of 15 mm. The mix of the concrete and its strength at room temperature while testing are shown in Table 10-3.

TABLE 10-3. Mix and Strength of Concrete

Number of specimen		Grade of cement	Mix in weight ratio				Cubic strength at room temperature f cu (MPa)
			Cement	Water	Sand	Crushed stone
Continuous beam	TCB (All)	325	1	0.48	1.54	2.99	29.50
Frame	TFC-1	325					29.94
	TFC-2
	TFC-3
	TFC-4	425					47.81
	TFC-5

The longitudinal reinforcement used in all the specimens was grade I (diameter is 10 mm), its yield strength at room temperature was f _y = 270 MPa, and the net thickness of the concrete cover was 10 mm. The stirrups were made of cold drawing steel wire (No. 8) after tempering.

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Optimization of typical structures

Leah W. Ratner , in Non-Linear Theory of Elasticity and Optimal Design, 2003

6 Continuous beam

A continuous beam, i.e. a beam that has more than two supports, is statically indeterminate. The reactions in the supports of a continuous beam cannot be obtained with the equations of static equilibrium only. For the calculation of the reactions in the supports each section is considered as an independent beam. The action of the adjacent part is substituted by a moment in the support between the sections. The equation for determining these moments is known as the theorem of three moments.

A beam that has one fixed end and n roller supports is n-fold statically indeterminate. If both ends of the beam are fixed, then the degree of indeterminacy is equal to the number of supports. The moment in a roller support for a beam with a console is equal to the moment from the load on the console. We calculate the necessary geometrical stiffness of the beam with the equation of elastic stability, R = $\mathit{R}=\sqrt{\mathit{M}/\mathit{E}{\mathit{C}}_{\mathit{s}}}\text{.}$ The stress in the beam should be less than the beam-specific and material limits, σ _max  < σ _cr  and σ _max  < σ _y.

Example:

The first equation of three moments includes an artificial section (M = 0) and section 1: here 2M ₀ L  + M ₁ L  =   0. This sum is equal to 0 because the section has no active load. The second equation of three moments relates to sections 1 and 2: M ₀ L ₁ + 2 M ₁(L ₁ +   L ₂)+   M ₂ L ₂ = −6(Ω ₁ a ₁/L₁+ Ω ₂ a ₂/L ₂), where Q_i, is the area of the diagram of moments for section i and a is the coordinate of center of gravity of the area.

Here, M ₂ = –PL ₃; L ₁ = L ₂ = 50 in; L ₃ = 25 in, ${\mathit{a}}^{\prime }=\frac{2}{3}\mathit{L}\text{,}$ ${\mathit{a}}^{″}=\frac{1}{3}\mathit{L}\text{.}$ After some substitutions the equations become $\frac{7}{2}{\mathit{M}}_1+2500=-6\left(\mathit{PL}*\mathit{L}/4\right)*\left(\mathit{a}\text{'}+\mathit{a}"\right)/\mathit{L}=\frac{1}{4}\mathit{PL}-2,500;{\mathit{M}}_1=5,000\mathrm{lb}-\mathrm{in}$ , M _max = 5,000   lb-in; the critical geometrical stiffness is R _cr = ${\mathit{R}}_{\mathrm{cr}}=\sqrt{\mathit{M}/\mathit{E}{\mathit{C}}_{\mathit{s}}}=0.0065{\mathrm{in}}^3;{\mathit{I}}_{\mathrm{cr}}0.325{\mathrm{in}}^4;{\mathit{d}}_{\mathrm{cr}}=1.6\mathrm{in};{\mathit{S}}_{\mathrm{cr}}=0.41{\mathrm{in}}^3;{\mathit{\sigma }}_{\mathrm{cr}}=12,145\mathrm{psi};{\mathit{d}}_{\mathrm{o}}=3.2\mathrm{in};{\mathit{\sigma }}_{\mathrm{cr}}<{\mathit{\sigma }}_{\mathrm{y}}\text{.}$

The actual geometrical stiffness of the sections of a continuous beam and the distributed elastic forces in these sections can be obtained by measuring the deflections of the individual sections. However, the analysis of deformation and geometrical stiffness cannot be performed within the linear theory of elasticity. Such analysis can be done only within the system of the equations of the non-linear theory of elasticity.

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Design Examples

DERIC JOHN OEHLERS , RUDOLF SERACINO , in Design of FRP and Steel Plated RC Structures, 2004

(a) Beam strength

The internal continuous beam is shown in Figs 7.17(a) and (d). We will assume that it is encastre and subjected to uniformly distributed loads. The beam has a hogging capacity of 339 kNm and a sagging capacity of 173 kNm, so that the distribution of moment at ultimate flexural failure is given by the strength curve in Fig. 7.17(b) where the static moment is 512 kNm. For all intents and purposes, the elastic distribution in Fig. 7.17(b) can be considered to be identical to the strength distribution as the moment redistribution is only 0.6%. Hence, the unplated beam has been designed without moment redistribution.

Figure 7.17. Beam structure

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Moment distribution methods

Alan Williams Ph.D., S.E., C.ENG. , in Structural Analysis, 2009

Supplementary problems

Use the moment distribution technique to solve the following problems.

S7.1 The continuous beam shown in Figure S7.1 has a second moment of area for member 34, which is 3×that for members 12 and 23. Determine the reactions at supports 2, 3, and 4.

Figure S7.1.

S7.2 Figure S7.2 shows a propped rigid frame with the relative EI/L values shown ringed alongside the members. Determine the reactions at supports 1 and 4.

Figure S7.2.

S7.3 The continuous beam shown in Figure S7.3 has a second moment of area of 376   in⁴ and a modulus of elasticity of 29,000   kips/in². Determine the bending moments produced in the beam by a settlement of support 1 by 1   in, support 2 by 2   in, and support 3 by 1   in.

Figure S7.3.

S7.4 Determine the moments produced in the members of the rigidly jointed frame shown in Figure S7.4 by the indicated load of 10   kips. The second moment of area of all members is 20   in⁴. The cross-sectional area of all members is 2   in². The modulus of elasticity is constant for all members.

Figure S7.4.

S7.5 Figure S7.5 indicates a sway frame, with pinned supports, with the relative EI/L values shown ringed alongside the members. Determine the bending moments at joints 2 and 3 caused by the lateral load.

Figure S7.5.

S7.6 Determine the forces produced at the joints of the rigidly jointed frame shown in Figure S7.6 by the indicated loads. The EI/L value is constant for all members.

Figure S7.6.

S7.7 Determine the bending moments produced at the joints of the rigidly jointed frame shown in Figure S7.7 by the indicated loads. The relative EI/L values are shown ringed alongside the members.

Figure S7.7.

S7.8 The dimensions and loading on a symmetrical, single-bay, two-story frame are shown in Figure S7.8. The relative second moment of area values are shown ringed alongside the members. Determine the bending moments, shears, and axial forces in the members.

Figure S7.8.

S7.9 Determine the bending moments produced at the joints of the Vierendeel girder shown in Figure S7.9 by the indicated loads. The relative EI/L values are shown ringed alongside the members.

Figure S7.9.

S7.10 Determine the bending moments produced at the joints of the rigidly jointed frame shown in Figure S7.10 by the indicated loads. The EI/L value is constant for all members.

Figure S7.10.

S7.11 Determine the bending moments produced in the left-hand columns of the rigidly jointed frame shown in Figure S7.11 by the indicated loads. The relative EI/L values are shown ringed alongside the members.

Figure S7.11.

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Railway Plane and Profile Design

Sirong Yi , in Principles of Railway Location and Design, 2018

3.4.1.2 Profile Design

The culverts and ballasted deck bridges can be set on a grade with any profile.

The profiles of continuous beam bridges, steel beam bridges, and large span bridges shall be designed in accordance with the technical requirements of bridge design.

Open deck bridges should set on the near-level grade. If the bridges are set on a grade, it is hard to determine the line location due to the rail creeping. The track gauge is difficult to keep, causing difficulties in line maintenance and affecting the running safety. If the bridges must be set on a grade, the grade should not exceed 4‰ to avoid increasing the rail creeping when the trains run downgrade and brake on the bridge. The open deck bridge with the span exceeding 40   m or the length exceeding 100   m is set on the grade greater than 4‰, providing the sufficient technical and economic bases are ensured.

The vertical curve shall not be set on an open deck bridge to avoid paving and maintenance difficulties due to the adjustment of rail top elevation. Thus the distance from the grade change point to two ends of the open deck bridge shall not be less than the vertical curve tangent during the profile design, as shown in Fig. 3.43.

Fig. 3.43. Distance from grade change point to an open deck bridge.

For the designed formation level, it shall not be less than the minimum height at the culverts as required by hydrological and construction conditions. It shall not be less than the minimum height at the bridges as required by hydrological conditions and the clearance height under bridge. To ensure necessary navigation clearance under large bridges on riverways in flat areas, the bridge approach elevations on both ends can be reduced. In addition, the convex profile can be set on the bridge.

When a PDL crosses other railways and highways (roads), the profile design elevation shall meet its clearance height; the clearance height under the overpass crossing PDL shall not be less than 7.25   m. When a PDL passes underneath the existing overpass, the lower clearance height can be adopted through the technical and economic comparison.

In addition to meeting the hydrologic conditions and bridge structure, the profile design for bridges that cross river shall conform to the navigation clearance requirements.

For elevated lines for crossing roads or zone with poor geological conditions, the bridges mainly adopt the simply supported beams with span not greater than 32   m or continuous box beam with equal spans generally not greater than 15   m.

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Intermediate Crack (IC) Debonding

DERIC JOHN OEHLERS , RUDOLF SERACINO , in Design of FRP and Steel Plated RC Structures, 2004

2.3.2.1 IC interface crack propagation

The two span continuous beam in Fig. 2.18 is plated over its hogging region. The plates are terminated well past the points of contraflexure and close to the applied loads so that it can safely be said that the plates are anchored in an uncracked region.

Figure 2.18. Two span continuous beam plated in hogging region

IC debonding of a steel tension face plate is shown in Fig. 2.19. These beams were specifically designed so that IC debonding would precede PE debonding as well as precede CDC debonding and, hence, the absence of any critical diagonal cracks in the beam. The point of contraflexure can be estimated to lie approximately midway along the flexurally uncracked region E-D and so it can be seen that the plate was terminated well into the compression face and, hence, anchored in an uncracked region. As designed for, PE debonding which propagates from the plate ends inwards, did not occur so that the only mechanism of debonding which occurred in this beam was IC debonding.

Figure 2.19. IC debonding failure of a steel tension face plated beam

The sequence of cracking is illustrated in Figs. 2.20 and 2.21 for a CFRP plate of the same length and width as the steel plate in Figs. 2.18 and 2.19. A flexural crack first formed over the support at a load that is marked 9 in Fig. 2.20(a) and from this intermediate crack, interface cracks propagated in both directions toward the plate ends. Further flexural cracks then occurred as the load was increased from 24 kN to 35 kN in Fig. 2.20(b) and from each emanated an IC interface crack which propagated towards the nearest plate end, that is away from the internal support. Hence in general, except for the flexural crack at the position of maximum moment where the IC interface crack propagated in both directions, the IC interface crack propagates in one direction towards a region of lower moment. At the load of 35 kN, the IC interface cracks are still concentrated close to the position of maximum moment. A further increase in load from 35 kN to 45 kN in Fig. 2.21 caused IC debonding as the IC interface crack propagated more rapidly to the point of contraflexure, and then at 45 kN it continued very rapidly towards the plate end in the anchorage zone although the plate end still remained attached.

Figure 2.20. IC interface cracking

Figure 2.21. IC debonding crack propagation to uncracked anchorage zone

Figure 2.19 illustrates the difference between IC debonding in a beam and IC debonding in a pull-push test as in Fig. 2.11(b). In Fig. 2.19, the force in the plate is transmitted from the concrete in the uncracked zone E-D which is equivalent to the pull-push specimen in Fig. 2.11(b), plus the force in the concrete between what is sometimes termed the concrete teeth D-C, C-B and B-A. Hence in this example, there are 4 zones in the shear span, bordered by the intermediate cracks, through which shear forces can transmit the axial forces in the plate. These four zones pull the plate but may not necessarily achieve their maximum force at the same time which adds to the complexity of the problem. These concrete teeth are equivalent to the shear connectors in composite steel and concrete beams and their individual σp. _ic /δ _ic behaviours (as described for pull-push tests in Section 2.3.1.3 and in Fig. 2.17) control the axial force in the plate. It is an irony of this mechanism that intermediate cracking may actually help to increase the force in the plate above that achieved from pull-push tests. It must be emphasised that this is a simplistic description of the shear force mechanism which it is felt is much more complex as it is also affected by the curvature in the beam.

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Flexural Strength and Ductility

DERIC JOHN OEHLERS , RUDOLF SERACINO , in Design of FRP and Steel Plated RC Structures, 2004

3.5.2.1 Beam specifications

The details of the continuous beam are given in Fig. 3.41. The hogging flexural capacity is 339 kNm. The variation of the moment distribution at failure, that is the strength, is very close to the elastic distribution so that the beam has been designed without moment redistribution. The concrete, reinforcing bar, and FRP plate material properties are the same as those described in Section 3.5.1. The hogging region of the beam has k_u = 0.29 and V_c = 134 kN and the sagging region k_u = 0.14 and V_c = 104 kN.

Figure 3.41. Beam structure

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The Displacement Method

Branko S. Bedenik PhD, DIC, Dipl.Ing. , Colin B. Besant PhD, DIC, BSc(Eng), FEng. , in Analysis of Engineering Structures, 1999

9.6 The moment distribution method (Cross's method)

The method of moment distribution is a numerical application of the displacement method in which the desired quantities are determined by a method of successive approximation that is suitable for longhand calculations.

The method is applicable to structures that satisfy the following:

Plane problem

Shear deformations can be neglected

Deformations are small

Equation {F}= [K]·{u} is written in a form for plane problems where rotations of joints 1 and 2 are only considered:

(9.26) $\left\{\begin{array}{l}{M}_{Z1}\\ M_{Z2}\end{array}\right\}=\left[\begin{array}{ll}{K}_{6,6}& K_{6,12}\\ K_{12,6}& K_{12,12}\end{array}\right]\dot{}\left\{\begin{array}{l}{\varphi }_{Z1}\\ {\varphi }_{Z2}\end{array}\right\}$

(9.27) $\left\{\begin{array}{l}{M}_{Z1}\\ M_{Z2}\end{array}\right\}=\left[\begin{array}{ll}\frac{4E{I}_Z}{L}& \frac{2E{I}_Z}{L}\\ \frac{2E{I}_Z}{L}& \frac{4E{I}_Z}{L}\end{array}\right]\dot{}\left\{\begin{array}{l}{\varphi }_{Z1}\\ {\varphi }_{Z2}\end{array}\right\}$

Loading by a counter-clockwise positive moments causes bending moments in a beam in accordance with the convention in the figure below:

Due to the above convention the equation is rewritten as:

(9.28) $\left\{\begin{array}{l}{M}_{jk}\\ M_{kj}\end{array}\right\}=\frac{2E{I}_Z}{L}\dot{}\left[\begin{array}{ll}2& 1\\ 1& 2\end{array}\right]\dot{}\left\{\begin{array}{l}{\varphi }_j\\ {\varphi }_k\end{array}\right\}$

From the condition that φ _k = 0 both end moments are

(9.29) $M_{jk}=\frac{4EI}{L}\dot{}{\varphi }_j$

(9.30) $M_{kj}=\frac{2EI}{L}\dot{}{\varphi }_j=\frac{M_{jk}}{2}$

from which we can see that for prismatic members a carry-over factor is 0.50.

Consider now a case when joint B from Fig. 9.21 is loaded by a moment M:

Figure 9.21. Distribution of moments

The sum of all moments at the rotation of the joint must be zero, hence:

(9.31) $M=M_{BA}+M_{BC}=\left(\frac{4\dot{}{(EI)}_1}{L_1}+\frac{4\dot{}{(EI)}_2}{L_2}\right)\dot{}{\varphi }_B$

(9.32) $\begin{array}{l}{\varphi }_B=\frac{M}{\left(\frac{4\dot{}{(EI)}_1}{L_1}+\frac{4\dot{}{(EI)}_2}{L_2}\right)}\\ M_{AB}=\frac{2\dot{}{(EI)}_1}{L_1}\dot{}{\varphi }_B=\frac{M_{BA}}{2}\\ M_{CB}=\frac{2\dot{}{(EI)}_2}{L_2}\dot{}{\varphi }_B=\frac{M_{BC}}{2}\\ M_{BA}=\frac{4\dot{}{(EI)}_1}{L_1}\dot{}{\varphi }_B=\frac{4\dot{}{(EI)}_1}{L_1}\dot{}\frac{M}{\frac{4\dot{}{(EI)}_1}{L_1}+\frac{4\dot{}{(EI)}_2}{L_2}}\end{array}$

In the majority of structures the modulus of elasticity is the same for all members therefore E cancels out:

(9.33) $M_{BA}=M\dot{}\frac{\frac{I_1}{L_1}}{\frac{I_1}{L_1}+\frac{I_2}{L_2}}=\frac{k_1}{\Sigma k}\dot{}M=r_{BA}\dot{}M$

The coefficient r is the distribution factor.

If joint C would be pinned, the equations would become:

$\begin{array}{l}M=M_{BA}+M_{BC}=\frac{4\dot{}{(EI)}_1}{L_1}\dot{}{\varphi }_B+\frac{3\dot{}{(EI)}_2}{L_2}\dot{}{\varphi }_B\\ M_{AB}=\frac{2\dot{}{(EI)}_1}{L_1}\dot{}{\varphi }_B=\frac{M_{BA}}{2}\\ M_{BC}=\frac{3\dot{}{(EI)}_2}{L_2}\dot{}{\varphi }_B\\ M_{CB}=0\end{array}$

and the moment at B on the element adjacent to A is:

(9.34) $M_{BA}=M\dot{}\frac{\frac{4\dot{}{(EI)}_1}{L_1}}{\frac{4\dot{}{(EI)}_1}{L_1}+\frac{3\dot{}{(EI)}_2}{L_2}}=M\dot{}\frac{\frac{I_1}{L_1}}{\frac{I_1}{L_1}+\frac{3}{4}\dot{}\frac{I_2}{L_2}}$

(9.35) $M_{BC}=M\dot{}\frac{\frac{3}{4}\dot{}\frac{I_2}{L_2}}{\frac{I_1}{L_1}+\frac{3}{4}\dot{}\frac{I_2}{L_2}}$

The stiffness coefficient for a one-sided pinned element is 0.75 of the value for a fixed element and the carry-over factor is zero.

The moment distribution procedure begins with the moments due to loads on a geometrically determinate structure; that is, all joints are prevented from movement by fixed-end moments.

The structure is next gradually released into its final deformed shape by allowing one joint at the time to rotate. Each time the joint is released the unbalanced moment on the joint is distributed to adjacent elements (whose opposite ends are fixed at this stage) in accordance to the distribution factors at that joint.

A fraction of these moments is carried over to the far end of the elements in accordance with the carry-over factors.

As the joints are successively released the residual unbalanced moments become smaller and smaller and finally converges to the correct solution.

The successful application of the procedure depends on an efficient tabular scheme as shown in the following example.

Example 9.8

Analyse the continuous beam of the Fig. 9.22 by the moment distribution method (E = const., I₁ = 3I₂ )

1.

Fixed-end moments:

$M_{BC}=q\dot{}\frac{6^2}{8}=4.5\dot{}q$

Free-span moment:

$M_{BC}^0=\frac{q\dot{}{L}^2}{8}=4.5\dot{}q$

2.

Carry-over factors:

$\begin{array}{lll}fromBtoA\hfill & \to \hfill & 0.5\hfill \\ fromBtoC\hfill & \to \hfill & 0\hfill \end{array}$

3.

Stiffness:

$\begin{array}{l}{k}_{AB}=\frac{4\dot{}E{I}_{AB}}{L_{AB}}=\frac{4\dot{}3}{5}=2.4\\ k_{BC}=\frac{3\dot{}E{I}_{BC}}{L_{BC}}=\frac{3\dot{}1}{6}=0.5\end{array}$

Total stiffness:

$k=2.4+0.5=2.9$

4.

Distribution coefficients:

$\begin{array}{l}{r}_{AB}=\frac{2.4}{2.9}=0.83\\ r_{BC}=\frac{0.5}{2.9}=0.17\end{array}$

Figure 9.22. Continuous beam

Iteration scheme:

All reactions and internal forces can be obtained by statics from these end moments and the moment and shear forces diagrams are drawn as shown in figure below.

Figure 9.23. Free bodies and internal force diagrams

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Behavior of Flexural Members at Elevated Temperatures

Zhenhai Guo , Xudong Shi , in Experiment and Calculation of Reinforced Concrete at Elevated Temperatures, 2011

8.2.1 Testing Method and Specimens

When a reinforced concrete continuous beam or frame beam is being considered, the positive bending moment occurs in the middle part of the span and the negative bending moment occurs near the support. If three surfaces of the beam are exposed to a high temperature during a fire accident, the tension zone of the section within the middle part of the span and the compression zone of the section near the support are exposed to high temperature. Even if the section and the reinforcement along the beam are the same, the thermal behavior of these two sections differ considerably.

Beams with the tension and the compression zones exposed to high temperatures have different experimental and loading methods (Figs. 8-1 and 8-2). The specimens for both conditions have to be designed and manufactured separately, and the sizes and construction are shown in Fig. 8-5. The strengths of concrete and longitudinal reinforcements of the specimen at room temperature are listed in Table 8-1.

FIGURE 8-5. Specimens for measuring the behavior of beams at elevated temperatures: (a) tension zone exposed to a high temperature ^[8-4] ; (b) compression zone exposed to a high temperature. ^[8-3]

TABLE 8-1. Strength of Specimens at Room Temperature

Specimen	Concrete f cu (MPa)	Longitudinal reinforcement
		Diameter (mm)	f y (MPa)
Tension zone exposed to a high temperature	29.45	10	270
Compression zone exposed to a high temperature	39.20	12	310

The mechanical behavior of a beam specimen at different temperatures is measured under the path of loading under constant temperature, and the method and procedure are as follows: the specimen is manufactured and cured for 28 days, kept in the laboratory and tested after aging for 60 days, the specimen is installed, and the longitudinal, transverse, and vertical positions of the experimental furnace are adjusted to place the experimental part of the specimen in the middle of the furnace. Various transducers are set up and connected to the measuring instrument, the experimental furnace is electrified and the specimen is heated; the supports of the specimen permit free expansion deformation. When the temperature in the chamber reaches the predetermined temperature (20–950 °C) and is held for 10 min, the specimen is loaded continuously until failure, i.e., loss of bearing capacity. In the meantime, the data on the temperature in the chamber, and the temperature, load, and deformation of the specimen are measured and recorded.

The thermocouples are fixed into the chamber of the furnace and the interior of the specimen, and can measure the temperatures and their variation at the corresponding places during the heating and loading process. Using these data, the vertical and transverse temperature distributions on the middle section of the specimen at different times can be obtained (Fig. 8-6).

FIGURE 8-6. Temperature distribution on a section of the specimen. ^[8-5]

The maximum temperature occurs on the bottom and both side surfaces for a beam specimen with three surfaces exposed to high temperature, but it is slightly lower than that of the chamber, and the temperature on the side surface decreases gradually from bottom to top. Therefore, the temperature on the section is not uniformly distributed along both vertical and transverse directions. The temperature gradient within the range 20–30 mm of the outer layer of the section is high, but the temperature variation in the interior of the section is small. During the initial stages of heating, the temperature is not very high (e.g., <400 °C), but the temperature gradient in the outer layer is much greater, although it decreases gradually later. The top surface of the specimen does not come in direct contact with the high temperature and has the lowest temperature. However, as the heating time continues and the temperature in the interior of the specimen increases, the temperature on the top surface also elevates gradually, because the heat being transferred to the top surface exceeds that transferred to the air. When the temperature (T) in the chamber of the experimental furnace is <600 °C, the temperature on the top surface is <80 °C. When the experimental temperature reaches 950 °C, the temperature on the top surface may reach 200–300 °C.

Under the same heating conditions, beams with tension and the compression zones exposed to high temperature have the same temperature field on their sections, but the mechanical behavior differs considerably. The existing experimental investigations ^[8-6–8-10] were conducted for the beams with compression zone exposed to high temperatures.

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Time of Flight Mass Spectrometers

K.G. Standing , W. Ens , in Encyclopedia of Spectroscopy and Spectrometry, 1999

TOF measurements with a continuous beam

As remarked above, a continuous beam must be formed into pulses before it is introduced into the TOF spectrometer. This requirement is an extra complication that is not present if the beam is intrinsically pulsed. However, there are several cases in which mass analysis of ions produced in a continuous beam can benefit considerably from the features of TOF instruments. For example, electrospray ionization has been the most successful technique for producing ions from intact noncovalent complexes, but these ions are often formed with high mass-to-charge ratios, beyond the range of quadrupole mass filters; TOF imposes no limit on the mass-to-charge ratio ( m/z) range. A second example involves coupling of separation techniques such as high-performance liquid chromatography with mass spectrometry. The sensitivity and fast time response of TOF instruments are well suited for such an application, but most separation techniques produce a continuous output. The same can be said of coupling TOF instruments with other types of mass spectrometer in order to perform MS/MS measurements as discussed below. For these reasons, there is clearly a need for an effective method for coupling continuous sources to TOF instruments.

A continuous beam can be injected into a TOF spectrometer in the longitudinal geometry of Figure 1, but only with very low efficiency. A more practical arrangement is illustrated in Figure 3; here a continuous beam of ions enters the TOF instrument perpendicular to its axis with low velocity and is injected into the flight path by the electrical pulses indicated in the figure. This technique takes advantage of the relative insensitivity of TOF measurements to spatial spreads in a plane perpendicular to the TOF axis. Such 'orthogonal injection' geometries were first introduced in the early 1960s, but acquired particular relevance when used with an electrospray source and an ion mirror by Dodonov. Limits on the injection efficiency and the resolution are set by the spatial and velocity spreads of the injected beam as described above. The properties of these instruments are therefore considerably improved if they are preceded by an ion guide running at relatively high pressure (up to ∼ 10 Pa) to provide collisional translational cooling of the ions before they enter the TOF spectrometer. In this way, a beam is produced with a small energy spread, limited by thermal velocities, allowing effective spatial focusing as described above.

Figure 3. A schematic diagram of an orthogonal-injection TOF instrument with an electrospray ionization source. Collisional cooling is used in a quadrupole ion guide to produce a beam with a small energy spread, and a small cross section. The pressure in the ion guide is typically tens of millitorr; the main TOF chamber is under high vacuum, typically 10⁻⁷ torr. Ions are pulsed into the spectrometer at a repetition rate of several kilohertz; packets of ions for one mass are shown at several positions along the ion path.

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