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Ultimate Capacity Evaluations of Reinforced Concrete Slabs Using Yield Line Analysis
by G. E. Mertz Westinghouse Savannah River Company Savannah River Site Aiken, South Carolina 29808
A document prepared for FlFM DEPARTMENT OF ENERGY NATURAL PHENOMENA HAZARD MITIGATION SYMPOSIUM at Denver from 11/13/95 - 11/17/95.
DOE Contract No. DE-AC09-89SR18035
This paper was prepared in connection with work done under the above contract number with the U. S. Department of Energy. By acceptance of this paper, the publisher and/or recipient acknowledges the U. S. Government’s right to retain a nonexclusive, royalty-free license in and to any copyright covering this paper, along with the right to reproduce and to authorize others to reproduce all or part of the copyrighted paper.
. I
DISCLA.IMER This report was prepared-'as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, o r represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, o r otherwise does not necessarily constitute o r imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United- States - Government or any agency thereof.
This report has been reproduced directly from the best available copy.
.z
Available to DOE and DOE contractors from the Office of Scientific and Technical Information, P. 0. Box 62. Oak Ridge, T N 37831: prices available from (615) 576-8401.
Available to the public from the National Technical Information Service. U. S. Department of Commerce, 5285 Port Royal Rd.. Springfield, VA 22161
ULTIMATE CAPACITY EVALUATIONS OF REINFORCED CONCRETE SLASS USING YIELD LINE ANALYSES
Greg E. Mertz Structural Mechanics Section
Westinghouse Savannah River Company Aiken,SC 29808
ABSTRACT
Yield line theory offers a simplified nonlinear analytical method that can determine the ultimate bending capacity of flat reinforced concrete plates subject to distributed and concentrated loads. Alternately, yield line theory, combined with hinge rotation limits can determine the energy absorption capacity of plates subject to impulsive and impact loads. This method is especially useful in evaluating existing structures that can not be qualified using conservative simplifying analytical assumptions. Typical components analyzed by yield line theory are basemats, floor and m f slabs subject to vertical loads along with walls subject to out of plane wall loads.
One practical limitation of yield line theory is that it is comput&onally difficult to evaluate some mechanisms. This problem is aggravated by the complex geometry and reinforcing layouts commonly found in practice. A yield line evaluation methodology is proposed to solve computationally tedious yield line mechanisms. This methodology is implemented in a small, PC based computer program, that allows the engineer to quickly evaluate multiple yield line mechanisms.
INTRODUCTION
At the Savannah River Site, structures, systems and components are being assessed to determine if their design basis is adequate as required by DOE Order 5480.28, Natural Phenomena Hazards Mitigation. The magnitude of these NPH loads often exceeds the original design basis for existing facilities. Elastic d y s i s of these facilities may indicate unsatisfactory stxuctllzal performance because the elastic analyses typically indicate the load level at first yield instead of the ultimate load carryiflg capacity of the structure. In lightly reinforced concrete slabs, the redistribution of bending moments often provides an ultimate load capacity that is significantly greater than the load capacity at fmt yield.
The ultiknate load capacity can be determined by either nonlinear analysis methods or testing. Nonlinear finite element analysis of reinforced concrete structures is an active research area and there are several commercially available finite element codes which have nonlinear concrete analysis capability. However, these codes are typically research oriented and analyses using these codes can be very expensive.
Yield line theory is a relatively simple analysis method which is accepted by ACI to calculate the ultimate bending capacity of flat reinforced concrete slabs. It is based on the observed failure mechanisms in reinforced concrete slabs which suggest that all of the yielding in a slab can be lumped into discrete plastic hinges or yield lines. Thus, a slab is idealized as a series of rigid bodies which are connected together by yield lines. At the ultimate load, the total plastic strain energy in the yield lines equated to the external work done by the moving the loads to the displaced shape of the postulated yield line mechanism. Yield line theory is an upper bound energy method, and the
quality of the solution depends on the postulated yield line mechanism.
A yield line analysis typically consists of three steps: (1) postulating a yield mechanism; (2) evaluating the capacity of the yield mechanism; and (3) iterating on steps 1 and 2 until the engineer judges that a reasonable approximation of the slabs minimum capacity has been reached.
The first step requires that the engineer visualize the geometry of a failed slab and identify the possible locations of yield l ies . The second step is relatively straight forward but computation?dly tedious. A PC based yield Line Evaluator (YLE) computer program was developed to facilitate the evaluation of multiple yield line mechanisms. For complex geometries or loading conditions, where the geometry of the yield l i e mechanism is not intuitively obvious, multiple yield line mechanisms are postulated as the engineer searches for the mechanism corresponding to the minimum ultimate capacity. Thus, the t h i i analysis step requires judgment to determine when the mechanism is sufficiently refied to provide the desired accuracy.
The ultimate load capacity of typical reinforced concrete slabs is usually governed by bending, and yield line theory calculates the ultimate beGding capacity. Yield line theory does not evaluate the transverse shear capacity of a slab and additional calculations must be made to preclude a shear failure.
YIELD LINE EVALUATION METHODOLOGY
Two possible yield line mechanisms for edge supported (fmed) square plates are shown in Figure 1. The region between yield lines is a rigid plate. The rigid plates are bounded by three or more yield lines, resulting in many possible rigid plate geometries. and the analytical treatment
of each possible plate geometry is very cumbersome. To make the problem more tractable, each rigid plate is divided into triangular elements and, each edge of the triangular element represents a possible yield line. Complex rigid plates are assembled by combining triangular plates and setting the rotation of interior yield
Nodes at the corners of the triangular elements are used to define the geometry of the yield l i e mechanism, as shown in Figure 1. Each plate is defied by three nodes, specified in counter clockwise order. The plates lie in an XY plane with 2 projecting upward. The displacement of the yield line mechanism is specified by the 2 coordinate.
lies equal to zero.
The sign convention for plate rotations and bending moments is tension on the top of the slab is positive and tension on the bottom of the slab is negative.
PLASTIC STRAM ENeRGY
The design bending capacity for a unit strip of slab, +Mn, is calculated in accordanee with the ACI code. The strain energy dissipated by a differential length of yield l i e , i, is
where +Mn(x,y) is the bending moment capacity of the yield l i e at the coordinates x and y,
8i is the rotation of the,yield line i, and dl is a differential length of a yield line.
Since the ultimate capacity may vary with location, the
Plates with yield line mechanisms Triangular plate elements
-I .- Discritized model
Figure 1 Typical Mge Supported PIate Yield Line Mechanism and Discretized Model
total strain energy dissipated by yield line, i, is obtained by numerically integrating Equation 1 over the length of the yield line
The total strain energy is determined by s h i n g up the strain energy of each of the yield lines
n SE= SEi
i=l (3)
where n is the total number of yield lines in the structure.
Rotation
The yield line rotation is developed in this section for yield line 2-3 which is between plate elements A and B as shown in Figure 2. Define the unit vector721 towards joints 2 from joint 1 on plate A as
where xi ... z2 are the coordinates of nodes 1 and 2, 4-t i, j, and % are the unit vectors in the global X, Y
The unit vector 731 towards joint 3 from joint 1 On plate A is similarly defined. The normal vector to plate A, %=A, is the cross product of vectors 7 2 1 and 731 or
and 2 directions
similarly, the n o m vector to plate ~,+b, is the cross
product of vectors~34 and 724. The angle of rotation, 8, for the yield l i e is given by
m e n notie 4 lies below the p h e of element A, the top of the slab is in tension and the sign of 8 is positive, as shown in Figure 2. When node 4 lies above the plane of element A, the bottom of the slab is in tension and the sign of 8 is negative.
At fied boundaries and at lines of symmetry where only one plate element is modeled, the n o d vector to plate B is replaced by the vector, 7". Assume that yield line 2-3 in Figure 2 is a line of symmetry on the edge of Element A.
Let& be the vector defining the direction of the yield lie, thenTv is given by
'
(7)
The angle of rotation, 8, for the yield line is given by substituting7" f o r T a in Equation 6.
-Moment Canacity,
Consider the slab with reinforcing parallel to the global X and Y axes, shown in Figure 3. Let CpMnx be the unit width slab bending moment capacity for reinforcing steel parallel to the global X axis, and 4MnY be the unit width slab bending moment capacity for reinforcing steel parallel to the global Y axis.
Both $Mnx and $Mny are calculated in accordance to either ACI-318 or ACI-349, as applicable and vary with changes in reinforcement and slab thickness. Similar to the rotation angle sign convention, positive bending moments cause tension on the top of the slab, negative bending moments cause tension on the bottom of the slab.
The bending capacity on a yield line inclined a from the X axis, CpMn, varies along the length of the yield line as the components of the capacity, CpMnx and $MnY v y The bending capacity, CpMn, for a specific point, (xty), IS 111
CpMn(x,y) = CpMnAx,y) Sin2a + CpMny(xty1 Cos2a .
(8)
Y 2-Y 3 Tana =- X2-X3
(9)
The moment capacity acting at point (r,a) on a yield line with a radial reinforcing pattern, shown in Figure 4, is simiiar to the moment capacity previously developed. Comparing Figures 3 and 4 the angle between CpMnc and CpMn is (a-p), and the bending moment capacity on the yield line is
where b is the angle between the radial reinforcing bar and
CpMnAr) is the moment capacity due to radial steel
CpMnJr) is the moment capacity due to
the x axis,
at radius, r, and
circumferential steel at radius, r.
EXTERNAL WORK
The external work of a load moving the displaced shape of the yield line mechanism is discussed in this section. The external work is the sum of work due to distributed, Wdist, and concentrated, Wmnc , loads.
ne nc
i=l i=l Work = E Wdist i + Wconc i
Referring to Figure 2, the orientation angle, a, of the yield l i e between nodes 2 and 3 is
L
where ne is the total number of elements and nc is the number of concentrated loads.
Y f"
1 wa X
t " 0 & PlateA - N%e 1
Node 4 is below the plane of plate A, the top of the plate is in tension Node 4
Section a-a Figure 2 Yield Line Rotation
Y direction Rebar Bending MomentCpMuy y I -.
X dirtytion Rebar
Bending J
X
Cp Mnx
Positive Sign Convention
Figure 3 Bending Moment Capacity
Circumferential Reinforcing
Figure 4 Bending Moment Capacity for Radial Reinforcing
Distributed Load
The work of distributed load acting on a triangular plate and moving through an imposed displacement is
Wdist i = d X , Y ) s(X,Y) dA.* (12) Area
where W&t i is the work due to a distributed load acting
q(x,y) is the distributed load acting on a differential
S(x,y) is the imposed displacement of the plate at
on plate i,
element, and
the location of the differential element
Note that both q(x,y) and 6(x,y) may have a linear variation amss the plate.
3
1
Figure 5 Natural Coordinates For A Triangular Element
The triangular plates used to evaluate yield line mechanisms have many different possible geometries, complicating the evaluation of the above integral. However, a closed form solution is available in natural (triangular) coordinates 123. Consider the triangular element shown in Figure 5. Given point i on the triangle, the natural coordinates @,I, L2, L3) of that point are
where Area is the mil area of the triangular element,
1x3 Y3 4 XI, ... y3 are the x and y nodal coordinates of and AI, A2 A3 are the areas of the triangular
sub-elements in Figure 5 which are calculated in a manner similar to the are&
The vertical displacement of the plate eliment, in natural coordinates becomes
6&1&%L3) = 61 L1+ &2 L2 + 63 L3 (14)
where 61, &2. 63 are the nodal displacements.
Similarly, the distributed load on the triangular element acting in natural coordinates is
q(Ll&%L3) = 41 L1+ 42L2 + 93 L3 (15) - where qi,q2,q3 is the magnitude of the distributed load at nodes 1 to 3.
The work of a distributed load acting on a differential element, moving through a displacement is represented by
Integrating Equation 12 over the area of the triangular element yields [2]
Concentrated Load
The work due to a concentrated load acting on a triangular element is given by
wconc i = p(Xi.Yi) 661,L2.L3) (18)
where P(Xi,yi) is the magnitude of the concentrated load at the coordinates (Xi, yi).
EQUILIBRIUM CONDITION SE= W1+ f W2
Two different load paths, shown in Figure 6, may be used to reach the ultimate load capacity. For proportional loading, path 0-B in Figure 6, all of the loads are increased from zero to the ultimate load simultaneously. Typical analyses with a proportional load path involve the live load acting on a structure with an insignxcant dead load or a settlement problem, where all of the loads are acting on the structure as the supports are removed.
Contrarily, a non-proportional loading (path O-A-B in Figure 6) has one set of constant loads and a second set of loads that are varied until the ultimate capacity is reached. The maximum concentrated load acting on a slab with a constant uniform load is an example of an analysis with a non-proportional loading path.
At the ultimate load, the dissipated strain energy is equal to the external work. Since the loads acting on the structure may be more or less than the ultimate load a capacity factor, f, is introduced for structures with a proportional load path
SE = f Work
or solving for f,
SE f = - Work
Capacity factors greater than one indicate that the applied loading is less than the urtimate load capacity. For proportional load paths, the capacity factor is equal to the reciprocal of the demand capacity ratio.
1 Demand f - Capacity --
For non-proportional load paths, a portion of the load on the structure is held constant and the remainder of the load is factored to reach the ultimate capacity
t Ultimate Load Capacity
Non Proportional Load Path (O-A-B)
/
1 i . B
0 \ Proportional Load Path
(0-B)
0
t Time
Figure 6 Proportional and Non-ProportionaI Load Paths
or solving for f,
SE-W1 w2 f =
where W1 is the work due to loads which remain constant, and
W2 is the work due to loads which are varied to reach the ultimate load Capacity.
Individual loads are idenWied as part of the constant load or part of the varying load through the factor? r.
where r a t is the r factor for distributed loads, rmnc i is the r factor for concentrated load i, r=l for loads that are varied, and d, for loads which remain constant.
If a portion of a load remains constant then r can be between 0 and 1.
ROTATION LIMrrS
ACI 349 Appendix C specifies rotaEion limits, for impulsive and impact effects, as
re = 0.0065 E 10.07 radians
where d is the distance from the extreme compression fiber to the centroid of the tension steel, and
c is the distance from the extreme compression fiber to the neutral axis at ultimate strength.
The ACI 349 rotation limits are based on beam test performed by Mattock [SI, who reported the rotation corresponding to the ultimate load capacity. Exaxdance of these rotation limits will result in reduced bending capacity, with increased cracking and spalling. A displacement coqtrolled structure which does not rely on bending capacity to maintain structural stability can withstand rotations larger than the ACI 349 limits without collapse.
The ratio of the experimentally determined rotations to the calculated rotations is reported to have a mean of 1.47 and a standard deviation of 0.49 [7]. Assuming a log-normal distribution, the ACI 349 rotation limits have a failure frequency of approximately 15%. Recall that these rotation limits were developed for impulsive and impact loads which are the result of a severe accident or abnormal operating event Allowable rotations with a 15% failure
.
frequency are acceptable for low probability severe accident or abnormal operating events, and the ACI 349 code allowable rotation is also judged to be acceptable for NPH events. However, for normal operation, the code allowable strengths have target failure frequencies of 1% [lo]. Using the log-normal distribution to adjust the ACI 349 allowable rotation for a 1% failure frequency yields.
d (-2.33 x 0.324) re 1% =0.0065; 1.39 e
= 0.0643 5 0.07 radians (27) where -2.33 standard deviations from the mean
correspond to a 1% failure frequency, 0.324 is the log-normal standard deviation, and 1.39 is the median ratio of test to calculated
rotations.
I,,([ Capacity
I - - - - I - - - - - calculate
Hinge Rotations
, b, calculate Plastic
Strain Energy
calculate External
Work
Calculate
Factor capacity
t
Coordinates
Figure 7 Flow Chart for Yield Line Evaluator Computer Program
YLE - YIELD LINE EVALUATOR COMPUTER PROGRAM
The Yield Line Evaluator is a small computer program that calculates the rotation of a yield line (Equation 6), the internal strain energy (Equation 3), the external work (Equation 11) and the capacity factor (Equations 20 and 23) for a yield line mechanism. A flow chart for the Yield Line Evaluator is shown in Figure 7. The yield line mechaniism is described as a series of rigid triangular plates where the each edge of the triangular element represents a possible yield line. Nodes at the corners of the trianguh elements are used to define the geometry of the elements, as shown in Figure 1. The out of plane coordinate, 2, is used to specify the deformed shape of the yield line mechanism. Both the strain energy and external work are linearly proportional to the magnitude of the deformed shape. Thus, the ultimate capacity, Equations 22 and 25, is independent of the magnitude of the deformed shape.
At the engineers option, the YLE computer program can alter the geometry of the yield line mechanism by iterating the X and Y coordinates of selected nodes. This option allows the user to search for a minimum energy solution with a given topology.
EXAMPLE PROBLEMS
Two examples are presented in this section to demonstrate the practical application of yield line analysis to complex structures. The fmt example considers the load catrying capacity of an unsupported basemat, and demonstrates a process of evaluating multiple yield line mechanisms in a search for the minimum load capacity.
The second example considers an impact load on a slab. This example demonstrates the application of non-proportional loading and the use of rotation limits to determine the energy absorbimg capacity of the slab.
BMEMAT SETTLEMENT
A structural assessment of cylindrical reinforced concrete waste storage vaults was recently carried out to determine the damage susceptibility from post-seismic differential settlement [SI. The basemat failure mechanism, shown in Figure 8, was one of several vault failure mechanisms evaluated.
A portion of the basemat is assumed to be unsupported in the basemat failure mechanism. Successive yield line analyses of the circular basemat are performed to determine the maximum width of the settled region that the basemat can span. The following yield line analysis is presented for one trial width of settled region.
The circular basemat is 95 feet in diameter and 3.6 feet thick The center portion of the basemat is thickened to 6.3 feet to support a central column. Reinforcing in the bulk of the basemat is oriented radially and circumferentially, with
c
Elevation
Widthof I Settled Region +-+I
Figure 8 Basemat Failure Mechanism
reinforcing ratios varying from about 0.1% to 1%, depending on the location, orientation, and which face of the slab is in tension. Bending moment capacities, based on the ACI 349 code [3] are used in the following mechanisms. Symmetxy is used in this analysis and only one half of the basemat is analyzed, as shown in Figure 8
The yield line method is an upper bound solution and the quality of the solution is dependent on the assumed yield line mechanism. Since the specific geometry of the yield line mechanism is not obvious, different geometries are analyzed and the specific geometry with the lowest capacity is adopted.
These mechanisms begin with a simple triangular mechanism and evolve into more complicated mechanisms. Convergence is observed by comparing the change in solutions as each mechanism is refined.
Mechanism lA, shown in Figure 9, is a simple triangular mechanism with an unsupported region of approximately U3 of the vault radius. Node 200 is assumed to displace vertically, forming yield lines between each of the nodes. The initial coordinates of nodes 101 and 200 is arbitrary because the coordinates of these nodes are varied, and the yield line capacity for each geometry is determined by the Yield Line Evaluator computer program. (In Figures 9 through 16, nodes with varying coordinates are denoted by filled circles, 0 ) Note that node 101 is varied along the circular arc, while node 200 is constrained to move along the horizontal line of symmetry, The uniforpl load capacity of Mechanism 1A is 12.7 ksf.
Mechanism IB, shown in Figure 10, is similar to Mechanism 1A except that node 101 is varied along the edge of the supported region. The uniform load capacity of Mechanism 1B is 13 ksf.
50 T
0 10 20 30 40 50 Figure 9 Basemat Mechanism lA
50 T
40
30
20
10
0 0 10 20 30 40 50
Figure 10 Basemat Mechanism lB
Mechanism-2, shown in Figure 11, represents the combination of Mechanisms 1A and 1B along with the addition of an internal node (201). Both nodes 200 and 201- are assumed to displace vertically. The coordinates of the solid nodes (.) are varied until a minimum capacity is found. Node 301 is varied along the circular arc, node 101 varied along the edge of the settled region, node 200 is constrained to move along the X axis, and node 201 is free to move both in the X and Y directions. The uniform load capacity of Mechanism 2 is 7.8 ksf.
50
40
30
20
10
0
T
0 10 20 30 40 50 Figure 11 Basemat Mechanism 2
Approximately 90,OOO different yield line geometries were evaluated for Mechanism 2 as the coordinates of nodes 101,200,201 and 301 were varied by approximately 1 foot for each successive geometry. The final geometry is shown in ,Figure 12 with the nodal coordinates which yielded an
ultimate capacity within 5% of the minimum capacity. For this problem, the ultimate capacity is not very sensitive to the exact location of the yield lines.
c
corresponding to a capacity within 5%
factor geometry ---I--
- of the minimum
I
-
-
1 I
Mechanism 3, shown in Figure 13, builds on Mechanism 2 by adding and additional node (302) along the circular arc. The coordinates of the solid nodes (.) are varied with the same constraints as Mechanism 2, until the miniium capacity is found. The additional node, 302, is constrained to vary along the circular arc. The capacity of Mechanism 3 is less than the capacity of Mechanism 2, indicating that the refrnements to the yield line mechaniSm are proceeding in the right direction. The uniform load capacity of Mechanism 3 is 7.1 ksf.
50 T
0 10 20 30 40 50 Figure 13 Basemat Mechanism 3
Mechanism 4, shown in Figure 14, builds on Mechanism 3 by adding an additional node (303) along the circular arc. The coordinates of the solid nodes (.) are varied with the same constraints as Mechanism 3, until a minimum capacity factor is found. The additional node, 303, is constrained to vary along the circular arc. The minimum capacity of Mechanism 4 is 6.4 ksf which is 10% less than the capacity of Mechanism 3.
50 T
0 I 1 - 1 1 1
0 10 20 30 40 50 Figure 14 Basemat Mechanism 4
Mechanism 5, shown in Figure 15, builds on Mechanism 4 by adding nodes on the ck.ular arc between nodes 301, 302 and 303 in Mechanism 4. The capacity of Mechanism 5 is 6.1 ksf which represents a 5% decrease in capacity from Mechanism 4. Since the change in capacity between Mechanisms 4 and 5 is small, the refrnement of the yield line mechanism along the slabs curved edge is judged to be sufficient
I
40 -- 30 -- 20 -- lo -- 0 I I I
1 - 1 1
0 10 20 30 40 50 Figure 15 Bibemat Mechanism 5
Classical solutions of square slabs (11 show that adding fan shaped yield lines to a comer reduce the capacity by approximately 10%. Mechanism 6, shown in Figure 16, is constructed by adding fan shaped yield lines near node 101 in Mechanism 5.
The coordinates of the solid nodes (.) are varied until a minimum capacity factor is found. Node 101 is varied along the edge of the settled region. Node 306 is varied along a line betwFn nodes 305 and 101 in Mechanism 5. The uniform load capacity of Mechanism 6 is 5.9 ksf which is less than a 5% decrease in capacity from Mechanism 5.
The capacity factor for the preceding six yield line mechanisms is summarized in Figure 17. This figure shows that the yield line Mechanism6 is sufficiently refined to give a reasonable load capacity, and that additional refinement is not necessary.
50 T
40
30
20
10
0 0 10 20 30 40 50
Figure 16 Basemat Mechanism 6
l4 T
1 2 3 4 5 6 Mechanism
Figure 17 Ultimate Load Capacity for Yield L i e Mechanisms 1 through 6
Mechanism 7, shown in Figure 18, is a highly refined version of Mechanism 6 ana was developed solely to demonstrate the marginal change in capacity achieved with further mechanism refinement. The capacity of Mechanism 7 is 5.8 ksf which is less than 3% than the capacity of Mechanism 6 and is sufficiently accurate for the settlement analysis.
IMPACT
50 T
40
30
20
10
0 0 10 20 30 40 50 Figure 18 Yield Line Mechanism 7
A review of non-safety class equipment mounted over the top slab of a safety class waste tank i nd icae the potential for 2-over-1 interaction should the non-safety class equipment fail due to NPH loads. Thus, the tank top is evaluated to determine its susceptibility to damage from impact loads. The impact evaluation considered global collapse of the tank top, shown in Figure 19, in addition to the local effects of scabbing, spalling, and penetration. The following discussion is limited to global bending failure.
Impact Load
Figure 19 Tank Top Slab with an Impact Load And A GIobal Collapse Mechanism
A simplified impact criteria which equates the kinetic or potential energy of the missile to the plastic strain energy capacity of the slab is used to perform the impact analysis.
During n o d operation, a uniform dead and Live load, w, acts on the tank top. During an accident an additional concentrated impact load, P, is applied to the tank top. Equation 23 is used to evaluate the capacity factor for this non-proportional loading with the uniform load- contributing to W1, -0, and the concentrated load contributing to W2, r=l. The ultimate static concentrated load capacity is Pstatic = P x f, where f is the capacity factor given by EQuation 23, and P is the concentrated load used in the analysis..
An iterative process of evaluating multiple yield line mechanisms was performed to search for the minimum capacity solution. A circular pie shaped yield line mechanism (similar to the mechanism shown in Figure 19) with 16 slices was found to give a reasonable estimate of the collapse load.
ACI 349 Appendix C rotation limits, re, are used to determine the allowable out-of-plane deformation of the slab. The Yield Line Evaluator computer program calculates the hinge rotation, 8 , based on the assumed displaced geometry of the yield line mechanism. The energy available to absorb an impact, &pact, is
The impact energy is then equated to either the kinetic or potential energy of the missile.
CON I - 7 3 / I I d 6 -"Ly WSRC-MS-95-0398
Ultimate Capacity Evaluations of Reinforced Concrete Slabs Using Yield Line Analysis
by G. E. Mertz Westinghouse Savannah River Company Savannah River Site Aiken, South Carolina 29808
A document prepared for FlFM DEPARTMENTOF ENERGY NATURAL PHENOMENA HAZARD MITIGATION SYMPOSIUM at Denver from 11/13/95 - 11/17/95.
DOE Contract No. DE-AC09-89SR18035
This paper was prepared in connection with work done under the above contract number with the U. S. Department of Energy. By acceptance of this paper, the publisher and/or recipient acknowledges the U. S. Government's right to retain a nonexclusive, royalty-free license in and to any copyright covering this paper, along with the right to reproduce and to authorize others to reproduce all or part of the copyrighted paper.
----- 81mtrWnON OF THIS DOCUMENT IS UNLIMITED, ASTE
DISCLAIMER This report was prepared-'as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, o r assumes any legal liability o r responsibiiity for the accuracy, completeness, o r usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, o r otherwise does not necessarily constitute o r imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United- States - Government or any agency thereof.
This report has been reproduced directly from the best available 'copy.
.' Available to DOE and DOE contractors from the Office of Scientific and Technical Infoxmation. P. 0. Box 62. Oak Ridge, TN 37831: prices available from (615) 576-8401.
Available to the public from the National Technical Information Service, U. S. Department of Commerce, 5285 Port Royal Rd., Springfield. VA 22161
ULTIMATE CAPACITY EVALUATIONS OF REINFORCED CONCRETE SLABS USING YIELDLINE ANALYSES
Greg E. Mertz Structural Mechanics Section
Westinghouse Savannah River Company A&en,SC 29808
ABSTRACT
Yield line theory offers a simplified nonlinear analytical method that can determine the ultimate bending capacity of fIat reinforced concrete plates subject to distributed and concentrated loads. Alternately, yield line theory, combined with hinge rotation limits can determine the energy absorption capacity of plates subject to impulsive and impact loads. This method is especially useful in evaluating existing structures that can not be qualified using conservative simplifying analytical assumptions. Typical components analyzed by yield line theory are basemats, floor and roof slabs subject to vertical loads along with walls subject to out of plane wall loads.
One practical limitation of yield line theory is that it is computationally difficult to evaluate some mechanisms. This problem is aggravated by the complex geometry and reinforcing layouts commonly found in practice. A yield line evaluation methodology is proposed to solve computationally tedious yield line mechanisms. This methodology is implemented in a small, PC based computer program, that allows the engineer to quickly evaluate multiple yield line mechanisms.
INTRODUCTION
At the Savannah River Site, structures, systems and components are being assessed to determine if their design basis is adequate as required by DOE Order 5480.28, Natural Phenomena Hazards Mitigation. The magnitude of these NPH loads often exceeds the original design basis for existing facilities. Elastic andysis of these facilities may indicate unsatisfactory structural performance because the elastic analyses typically indicate the load level at first yield instead of the ultimate load w g capacity of the structure. In lightly reinforced concrete slabs, the redistribution of bending moments often provides an ultimate load capacity that is significantly greater than the load capacity at fmt yield.
The ultimate load capacity can be determined by either nonlinear analysis methods or testing. Nonlinear finite element analysis of reinforced concrete structures is an active research area and there are several commercially available finite element codes which have nonlinear concrete analysis capability. However, these codes are typically research oriented and analyses using these codes can be very expensive.
Yield line theory is a relatively simple analysis method which is accepted by ACI to calculate the ultimate bending capacity of flat reinforced concrete slabs. It is based on the observed failure mechanisms in reinforced concrete slabs which suggest that all of the yielding in a slab can be lumped into discrete plastic hinges or yield lines. Thus, a slab is idealized as a series of rigid bodies which are connected together by yield lines. At the ultimate load, the total plastic strain energy in the yield lines equated to the external work done by the moving the loads to the displaml shape of the postulated yield line mechanism. Yield line theory is an upper bound energy method, and the
quality of the solution depends on the postulated yield line mechanism.
A yield line analysis typically consists of three steps: (1) postulating a yield mechanism; (2) evaluating the capacity of the yield mechanism; and (3) iterating on steps 1 and 2 until the engineer judges that a reasonable approximation of the slabs minium capacity has been reached.
The first step requires that the engineer visualize the geometry of a failed slab and identify the possible locations of yield lines. The second step is relatively straight forward but computationally tedious. A PC based Yield Line Evaluator (YLE) computer program was developed to facilitate the evaluation of multiple yield line mechanisms. For complex geometries or loading conditions, where the geometry of the yield l i e mechanism is not intuitively obvious, multiple yield line mechanisms are postulated as the engineer searches for the mechanism corresponding to the minimum ultimate capacity. Thus, the third analysis step requires judgment to determine when the mechanism is sufficiently refined to provide the desired accuracy.
The ultimate load capacity of typical reinforced concrete slabs is usually governed by bending, and yield line theory calculates the ultimate be;ding capacity. Yield line theory does not evaluate the transverse shear capacity of a slab and additional calculations must be made to preclude a shear failure.
YIELD U N E EVALUATION METHODOWGY
Two possible yield line mechanisms for edge supported (fixed) square plates are shown in Figure 1. The region between yield lines is a rigid plate. The rigid plates are bounded by three or more yield lines, resulting in many possible rigid plate geometries. and the analytical treatment
of each possible plate geometry is very cumbersome. To make the problem more tractable, each rigid plate is divided into triangular elements and, each edge of the triangular element represents a possible yield line. Complex rigid plates are assembled by combining triangular plates and setting the rotation of interior yield
Nodes at the corners of the triangular elements are used to define the geomepy of the yield l i e mechanism, as shown in Figure 1. Each plate is defined by three nodes, specifted in counter clockwise order. The plates lie in an XY plane with 2 projecting upward. The displacement of the yield line mechanism is specified by the 2, coordinate.
l ies equal to zero.
The sign convention for plate rotations and bending moments is tension on the top of the slab is positive and tension on the bottom of the slab is negative.
PLASTIC STRAIN ENERGY
The design bending capacity for a unit strip of slab, +Mn, is calculated in accordance with the ACI code. The strain energy dissipated by a differential length of yield line, i, is
where +Mn(x,y) is the bending moment capacity of the yield l i e at the coordinates x and y,
8i is the rotation of.the*yield line i, and dl is a differential length of a yield line.
Since the ultimate capacity may vary with location, the
I Fixed Edges (typ)
Plates with yield line mechanisms Triangular late elements
Discritized model Figure 1 Typical Edge Supported Plate Yield Line
Mechanism and Discretized Model
total strain energy dissipated by yield line, i, is obtained by numerically integrating Equation 1 over the length of the yield line
The total strain energy is determined by s&ing up the strain energy of each of the yield lines
(3)
where n is the total number of yield lines in the structure.
Rotation
The yield line rotation is developed in this section for yield line 2-3 which is between plate elements A and B as shown in Figure 2. Define the unit vector321 towards joints 2 from joint 1 on plate A as
where XI . . . z2 are the coordinates of nodes 1 and 2, 4-
i, j, and % are the unit vectors in the global X, Y and Z directions
The unit vector 731 towards joint 3 from joint 1 On plate A is similarly defined. The normal vector to plate A, % n ~ , is the cross product of vectors 7 2 1 and 531 or
similarly, the n o d vector to plate ~,-b, is the cross product of vectors734 andTj24. The angle of rotation, 8, for the yield l i e is given by
m e n node 4 lies MOW the plane of element A, the top of the slab is in tension and the sign of 8 is positive, as shown in Figure 2. When node 4 lies above the plane of element A, !he bottom of the slab is in tension and the sign of 8 is negative.
At fixed boundaries and at lines of symmetry where only one plate element is modeled, the normal vector to plate B is replaced by the vector, TV. Assume that yield line 2-3 in Figure 2 is a line of symmetry on the edge of Element A.
Let& be the vector defining the direction of the yield line, then?,, is given by
Y 2-Y 3 Tana =- X2-X3
$9)
(7)
The angle of rotation, 8, for the yield l i e is given by substitutingTv for?& in Quation 6.
-Moment CaDacity
Consider the slab with reinforcing parallel to the global X and Y axes, shown in Figure 3. Let $Mnx be the unit width slab bending moment capacity for seeinforcing steel parallel to the global X axis, and $Mny be the unit width slab bending moment capacity for reinforcing steel parallel to the global Y axis.
Both $Mnx and $Mny am calculated in accordance to either ACI-318 or ACI-349, as applicable and vary with changes in reinforcement and slab thickness. Similar to the rotation angle sign convention, positive bending moments cause tension on the top of the slab, negative bending moments cause tension on the bottom of the slab.
The bending capacity on a yield line inclined a from the X axis, $Mn, varies along the length of the yield line as the components of the capacity, $Mnx and $Mny v y The bending capacity, $Mn, for a specific point, (x,y), IS [l]
$M~(x,Y) = $M~x(x,Y) S h 2 ~ -I- $Mnyky) C0s2ct .
(8)
Referring to Figure 2, the orientation angle, a, of the yield l i e between nodes 2 and 3 is
t' f" I S A
1 I 3- X
/->+e late B i Node 4 is below the plane of plate A, the top of the plate is in tension Node
Section a-a t
Figure 2 Yield Line Rotation
The moment capacity acting at point (r,a) on a yield line with a radial reinforcing pattern, shown in Figure 4, is simiiar to the moment capacity previously developed. Comparing Figures 3 and 4 the angle between $Mnc and $Mn is (a-p), and the bending moment capacity on the yield line is
where b is the angle between the radial reinforcing bar and
$MnAr) is rhe moment capacity due to radial steel
$MnJr) is the moment capacity due to
the x axis,
at radius, r, and
circumferential steel at radius, r.
EXTERNAL WORK
The external work of a load moving the displaced shape of the yield l i e mechanism is discussed in this section. The external work is the sum of work due to distributed, W&t, and concentrated, Wconc, loads.
where ne is the total number of elements and nc is the number of concentrated loads. Y direction Rebar
y Bending Moment$Muy I -- X direction Rebar
Bending J L
.,
4 Mnx
Positive Sign Convention
Figure 3 Bending Moment Capacity,
Circumferential Reinforcing
Area=xp
Figure 4 Bending Moment Capacity for Radial Reinforcing
X l Y1 1
y2 1
Distributed Load
The work of distributed load acting on a triangular plate and moving through an imposed displacement is
where Wdit i is the work due to a distributed load acting
q(x,y) is the distributed load acting on a differential
6(x,y) is the imposed displacement of the plate at
on plate i,
element, and
the location of the differential element
Note that both q(x,y) and 6(x,y) may have a linear variation across the plate.
3
1
Figure 5 Natural Coordinates For A Triangular Element
The triangular plates used to evaluate yield line mechanisms have many different possible geometries, complicating the evaluation of the above integral. However, a closed form solution is available in natural (triangular) coordinates [2]. Consider the triangular element shown in Figure 5. Given point i on the triangle, the natural coordinates (L1, L2, L3) of that point are
A3 L2=- A2 I L3 =-& Area
where Area is the totd area of the triangular element,
1x3 ~3 4 XI, ... y3 are the x and y nodal coordinates of and AI, A2 A3 are the areas of the triangular
sub-elements in Figure 5 which are calculated in a manner similar to the an%
The vertical displacement of the plate element, in natural coordinates becomes
where 61,&, 63 are the nodal displacements.
Similarly, the distributed load on the triangular element acting in natural coordinates is
where 41, Q, 43 is the magnitude of the distributed load at nodes 1 to 3.
The work of a distributed load acting on a differential element, moving through a displamment is represented by
Integrating Equation 12 over the area of the triangular element yields [2]
61(24l + 42 + 43)
+62(91 + a 2 +93)
+63(41+42+293)
Wdist i =y
Concentrated Load
The work due to a concentrated load acting on a triangular element is given by
where P(Xi,yi) is the magnitude of the concentrated load at the coordinates (Xi, yi).
EQUILIBRIUM CONDITION
Two different load paths, shown in Figure 6, may be used to reach the ultimate load capacity. For proportional loading, path 0-B in Figure 6, all of the loads are inaeased from zero to the ultimate load simultaneously. Typical analyses with a proportional load path involve the live load acting on a structure with an insignificant dead load or a settlement problem, where all of the loads are acting on the structure as the supports are removed.
Contrarily, a non-proportional loading (path 0-A-B in Figure 6) has one set of constant loads and a second set of loads that are varied until the ultimate capacity is reached. The maximum concentrated load acting on a slab with a constant uniform load is an example of an analysis with a non-proportional loading path.
At the ultimate load, the dissipated strain energy is equal to the external work Since the loads acting on the structure may be more or less than the ultimate load a capacity factor, f. is introduced for structures with a proportional load path
SE = f Work
or solving for f,
SE f = - Work
Capacity factors greater than one indi th
t 19)
the applied loading is less than the uftimate load capacity. For proportional load paths, the capacity factor is equal to the reciprocal of the demand capacity ratio.
-- 1 Demand f - Capacity
For non-proportional load paths, a portion of the load on the structure is held constant and the remainder of the load is factored to mch the ultimate capacity
t Ultimate Load Capacity
\
0 \ Proportional Load Path 0
Figure 6 Proportional and Non-Proportional Load Paths
SE = W1+ f W2
or solving for f,
SE-W1 w2 f =
(22)
where W1 is the work due to loads which remain constant,
W2 is the work due to loads which are varied to and
reach the ultimate load Capacity.
Individual loads are identified as part of the constant load or part of the varying load through the factor, r.
ne nc
i=l i=l ne nc
i=l i=l
W1= C tl-rdist) Wdist i +C tl-rconc i) Wconc i
W2= Z rdist wdist i +C rconc i Wconc i
tW
(25)
where r a t is the r factor for distributed loads, rmnc i is the r factor for concentrated load i, 1=1 for loads that arevaried, and ~
1=0, for loads which remain constant.
If a portion of a load remains constant then r can be between 0 and 1.
ROTATION LIMITS
ACI 349 Appendix C specifies rotation limits, for impulsive and impact effects, as
(26) d re = 0.0065 ; 5 0.07 radians
where d is the distance from the extreme compression fiber to the centroid of the tension steel, and
c is the distance from the extreme compression fiber to the neutral axis at ultimate strength.
The ACI 349 rotation limits are based on beam test performed by Mattock [9], who reported the rotation corresponding to the ultimate load capacity. Exaxdance of these rotation limits will result in reduced bending capacity, with increased cracking and spalling. A displacement coqtrolled structure which does not rely on bending capacity to maintain structural stability can withstand rotations larger than the ACI 349 limits without collapse.
The ratio of the experimentally determined rotations to the calculated rotations is reported to have a mean of 1.47 and a standard deviation of 0.49 [7]. Assuming a log-normal distribution, the ACI 349 rotation limits have a failure frequency of approximately 15%. Recall that these rotation limits were developed for impulsive and impact loads which are the result of a severe accident or abnormal operating event Allowable rotations with a 15% failure
frequency are acceptable for low probability severe accident or abnormal operating events, and the ACI 349 code allowable rotation is also judged to be acceptable for NPH events. However, for normal operation, the code allowable stmigths have target failure frequencies of 1% [lo]. Using the log-normal distribution to adjust the ACI 349 allowable rotation for a 1% failure frequency yields.
d (-2.33 x 0.324) re 1% =0.0065; 1.39 e
(27) d = 0.043 S 0.07 radians
where -2.33 standard deviations from the mean correspond to a 1% failure frequency,
0.324 is the log-normal standard deviation, and 1.39 is the median ratio of test to calculated
rotations.
Plastic
wculate External
Work
calculate Capacity Factor v
Coordinates
- - h------ Results
- t_ I5 1 I to
Figure 7 Flow Chart for Yield Line Evaluator Computer Program
Y U - YIELD LINE EVALUATOR COMPUTER PROGRAM
The Yield Line Evaluator is a small computer program that calculates the rotation of a yield line (Equation 6), the internal strain energy (Equation 3), the external work (Equation 11) and the capacity factor (Equations 20 and 23) for a yield line mechanism. A flow chart for the Yield Line Evaluator is shown in Figure 7. The yield line mechaniim is described as a series of rigid triangular plates where the each edge of the triangular element represents a possible yield line. Nodes at the comers of the triangular elements are used to define the geometry of the elements, as shown in Figure 1. The out of plane coordinate, 2, is used to specify the deformed shape of the yield line mechanism. Both the strain energy and external work are linearly proportional to the magnitude of the deformed shape. Thus, the ultimate capacity, Equations 22 and 25, is independent of the magnitude of the deformed shape.
At the engineers option, the YLE computer program can alter the geometry of the yield line mechanism by iterating the X and Y coordinates of selected nodes. This option allows the user to search for a minimum energy solution with a given topology.
EXAMPLE PROBLEMS
Two examples are presented in this section to demonstrate the practical application of yield line analysis to complex structures. The fKst example considers the load carrying capacity of an unsupported basemat, and demonstrates a process of evaluating multiple yield line mechanisms in a search for the minimum load capacity.
The second example considers an impact load on a slab. This example demonstrates the application of non-proportional loading and the use of rotation limits to determine the energy absorbiig capacity of the slab.
BASEMAT SETlZEMENT
A structural assessment of cylindrical reinforced concrete waste storage vaults was recently carried out to determine the damage susceptibility from post-seismic differential settlement [SI. The basemat failure mechanism, shown in Figure 8, was one of several vault failure mechanisms evaluated.
A portion of the basemat is assumed to be unsupported in the basemat failure mechanism. Successive yield line analyses of the circular basemat are performed to determine the maximum width of the settled region that the basemat can span. The following yield line analysis is presented for one trial width of settled region.
The circular basemat is 95 feet in diameter and 3.6 feet thick The center portion of the basemat is thickened to 6.3 feet to support a central column. Reinforcing in the bulk of the basemat is oriented radially and circumferentially, with
I Sector
Width of I Elevation Settled Region +4
Figure 8 Basemat Failure Mechanism
reinforcing ratios varying from about 0.1% to 1%, depending on the location, orientation, and which face of the slab is in tension. Bending moment capacities, based on the ACI 349 code [3] are used in the following mechanisms. Symmetry is used in this analysis and only one half of the basemat is analyzed, as shown in Figure 8
The yield line method is an upper bound solution and the quality of the solution is dependent on the assumed yield line mechanism. Since the specifc geometry of the yield line mechanism is not obvious, different geometries are analyzed and the specific geometry with the lowest capacity is adopted.
These mechanisms begin with a simple triangular mechanism and evolve into more complicated mechanisms. Convergence is observed by comparing the change in solutions as each mechanism is refined.
Mechanism lA, shown in Figure 9, is a simple triangular mechanism with an unsupported region of approximately U3 of the vault radius. Node 200 is assumed to displace vertically, forming yield lines between each of the nodes. The initial coordinates of nodes 101 and 200 is arbitrary because the coordinates of these nodes are varied, and the yield line capacity for each geometry is determined by the Yield Line Evaluator computer program. (In Figures 9 through 16, nodes with varying coordinates are denoted by filled circles, *) Note that node 101 is varied along the circular arc, while node 200 is constrained to move along the horizontal line of symmetry, The uniforp load capacity of Mechanism 1A is 12.7 ksf.
Mechanism lB , shown in Figure 10, is similar to Mechanism 1A except that node 101 is varied along the edge of the supported region. The uniform load capacity of Mechanism 1B is 13 ksf.
,
0 10 20 30 40 50 Figure 9 Basemat Mechanism lA
50 T
40
30
20
10
0 0 10 20 30 40 50
Figure 10 Basemat Mechanism lB
Mechanism -2, shown in Figure 11, represents the combination of Mechanisms 1A and 1B along with the addition of an internal node (201). Both nodes 200 and 201- are assumed to disphce vertidy. The coordinates of the solid nodes (.) are varied until a minimum capacity is found. Node 301 is varied along the circular arc, node 101 varied along the edge of the settled region, node 200 is constrained to move along the X axis, and node 201 is free to move both in the X and Y directions. The uniform load capacity of Mechanism 2 is 7.8 ksf.
50 T
40
30
20
10
0 0 10 20 30 40 50
Figure 11 Basemat Mechanism 2
Approximately 90,OOO different yield line geometries were evaluated for Mechanism 2 as the coordinates of nodes 101,200,201 and 301 were varied by approximately 1 foot for each successive geometry. The final geometry is shown in,Figure 12 with the nodal coordinates which yielded an
ultimate capacity within 5% of the minimum capacity. For this problem, the ultimate capacity is not very sensitive to the exact location of the yield lines.
t
factor geometry corresponding to a capacity withii 5% -* ----
- of the minimum
- -
-
I I
Mechanism 3, shown in Figure 13, builds on Mechanism 2 by adding and additional node (302) along the circular arc. The coordinates of the solid nodes (.) are varied with the same constraints as Mechanism 2, until the minimum capacity is found. The additional node, 302, is constrained to vary along the circular arc. The capacity of Mechanism 3 is less than the capacity of Mechanism 2, indicating that the refinements to the yield line mechanism are proceeding in the right direction. The uniform load capacity of Mechanism 3 is 7.1 ksf.
50 r
0 10 20 30 40 50 Figure 13 Basemat Mechanism 3
Mechanism 4, shown in Figure 14, builds on Mechanism 3 by adding an additional node (303) along the circular arc. The coordinates of the solid nodes (.) are varied with the same constraints as Mechanism 3, until a minimum capacity factor is found. The additional node, 303, is constrained to vary along the circular arc. The minimum capacity of Mechanism 4 is 6.4 ksf which is 10% less than the capacity of Mechanism 3.
0 10 20 30 40 50 Figure 14 Basemat Mechanism 4
Mechanism 5, shown in Figure 15, builds on Mechanism 4 by adding nodes on the circ.uIar arc between nodes 301, 302 and 303 in Mechanism 4. The capacity of Mechanism 5 is 6.1 ksf which represents a 5% decrease in capacity from Mechanism 4. Since the change in capacity between Mechanisns 4 and 5 is small, the refiement of the yield line mechanism along the slabs curved edge is judged to be sufficient.
50 T
40
30
20
10
0 . 0 10 20 30 40 50 Figure 15 Bsisemat Mechanism 5
Classical solutions of square slabs [l] show that adding fan shaped yield lines to a comer reduce the capacity by approximately 10%. Mechanism 6, shown in Figure 16, is constructed by adding fan shaped yield lines near node 101 in Mechanism 5.
The coordinates of the solid nodes (.) are varied until a minimum capacity factor is found. Node 101 is varied along the edge of the settled region. Node 306 is varied along a line between nodes 305 and 101 in Mechanism 5. The uniform load capacity of Mechanism 6 is 5.9 ksf which is less than a 5% decrease in capacity from Mecbanii 5.
The capacity factor for the preceding six yield line mechanisms is summarized in Figure 17. This figure shows that the yield line Mechanism6 is sufficiently refined to give a reasonable load capacity, and that additional refinement is not necessary.
So T
30 -- 101
20 -- lo -- 0 - I I I
0 10 20 30 40 50 Figure 16 Basemat Mechanism 6
l4 7
1 2 3 4 5 6 Mechanism
Figure 17 Ultimate Load Capacity for Yield Line Mechanisms 1 through 6
Mechanism 7, shown in Figure 18, is a highly refined,, version of Mechanism 6 ana was developed solely to demonstrate the marginal change in capacity achieved with further mechanism refinement. The capacity of Mechanism 7 is 5.8 ksf which is less than 3% than the capacity of Mechanism 6 and is sufficiently accurate for the settlement analysis.
IMPACT
50 T
40
30
20
10
0 4
A review of non-safety class equipment mounted over the top slab of a safety class waste tank indica@ the potential for 2-over-1 interaction should the non-safety class equipment fail due to NPH loads. Thus, the tank top is evaluated to determine its susceptibility to damage from impact loads. The impact evaluation considered global collapse of the tank top, shown in Figure 19, in addition to the local effects of scabbing, spalling, and pen-on. The following discussion is limited to global bending failure.
Impact Load
’ TankTop Slab
Global Collapse
Figure 19 Tank Top Slab with an Impact Load And A GIobaI ColIapse Mechanism
A simplified impact criteria which equates the kinetic or potential energy of the missile to the plastic stxain energy capacity of the slab is used to perform the impact analysis.
During normal operation, a uniform dead and live load, w, acts on the tank top. During an accident an additional concentrated impact load, P, is applied to the tank top. Equation 23 is used to evaluate the capacity factor for this non-proportional loading with the uniform loa& contributing to W1, 1=0, and the concentrated load contributing to W 2 , s l . The ultimate static concentrated load capacity is Pstatic = P x f, where f is the capacity factor given by Equation 23, and P is the concentrated load used in the analysis..
An iterative process of evaluating multiple yield line mechanisms was performed to search for the minimum capacity solution. A circular pie shaped yield line mechanism (similar to the mechanism shown in Figure 19) with 16 slices was found to give a reasonable estimate of the collapse load.
ACI 349 Appendix C rotation lipits, re, are used to determine the allowable out-of-plane deformation of the slab. The Yield Line Evaluator computer program calculates the hinge rotation, 8, based on the assumed displaced geometry of the yield line mechanism. The energy available to absorb an impact, &pact is
The impact energy is then equated to either the kinetic or potential energy of the missile.
0 10 20 30 40 50 Figure 18 Yield Line Mechanism 7
&pact = Pmissle H 3 M i l e = I Pstatic (29)
where Pmiisle is the weight of the missile, H is the drop height, measured from the deformed
V is the impact velocity, and g is the acceleration due to gravity.
geometry
Allowable equipment weight versus drop height charts for global collapse are developed by plotting Equation 29, and a typical chart is shown in Figure 20. These charts, combined with similar charts for scabbing, spalling and penetration, represent the design envelope for the placement of equipment above waste tanks.
1 Equipment Weight , Pstatic
Figure 20 Allowable Equipment Weight YS. Drop Height to Preclude Global Collapse
CONCLUSION
Yield line theory is capable of determining the ultimate bending capacity of complex slabs, and when combined with rotation limits, yield line theory can also be used to evaluate slabs for impact loads.
The implementation of the analysis methodology outlined in this paper via the PC based Yield Line Evaluator computer program makes yield line theory a practical, cost effective engineering analysis tool.
ACKNOWLEDGMENTS
The information contained in this paper was developed during the course of work done under Contract No. DE- AC09-89SR18035 with the U.S. Department of Energy. Calculation of the impact yield line mechanisms by S. M. Kahn of the Savannah River Site is gratefully acknowledged.
REFERENCES
1) Wang, C. K., and Salmon, C. G. Reinforced Concrete Design, Second Edition, Intex Education Publishers, 1973.
2) Gallagher, R. H., Finite Element Analysis Fundamentals, Prentice Hall, 1975.
3) MacGregor, J. G., Reinforced Concrete Mechanics and Design, Second Edition, Prentice Hall, 1992.
4) Scott, R. F., Foundation Analysis, Prentice Hall, 1981.
5) Cracking, Deflection and Ultimate Load of Concrete Slab Systems, American Concrete Institute Publication SP-30,1971.
6) ACI 318-89, Building Code Requirements for Reinforced Concrete, American Concrete Institute, 1989.
7) ACI 349-85, Code Requirements for Nuclear Safety Related Concrete Structures, American Concrete Institute, 1986.
8) Mertz, G. E., Houston, T. W. and Flanders, H. E., Overview of the Savannah River Site High Level Waste Storage Tank Structural Qualification Program, to be presented at the 1995 DOE NPH Conference, Denver CO, Nov. 13-14,1995.
9) A. H. Mattock, Rotational Capacities of Hinging Regions in Reinforced Concrete Beams, ACI SP-12, 1965.
10) G. Winter, Safety and Serviceability Provisions in the ACI Building Code, ACI SP 59-3,1979.
